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+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+1
+Knockoffs-SPR: Clean Sample Selection in
+Learning with Noisy Labels
+Yikai Wang, Yanwei Fu, and Xinwei Sun.
+Abstract—A noisy training set usually leads to the degradation of the generalization and robustness of neural networks. In this paper,
+we propose a novel theoretically guaranteed clean sample selection framework for learning with noisy labels. Specifically, we first
+present a Scalable Penalized Regression (SPR) method, to model the linear relation between network features and one-hot labels. In
+SPR, the clean data are identified by the zero mean-shift parameters solved in the regression model. We theoretically show that SPR
+can recover clean data under some conditions. Under general scenarios, the conditions may be no longer satisfied; and some noisy
+data are falsely selected as clean data. To solve this problem, we propose a data-adaptive method for Scalable Penalized Regression
+with Knockoff filters (Knockoffs-SPR), which is provable to control the False-Selection-Rate (FSR) in the selected clean data. To
+improve the efficiency, we further present a split algorithm that divides the whole training set into small pieces that can be solved in
+parallel to make the framework scalable to large datasets. While Knockoffs-SPR can be regarded as a sample selection module for a
+standard supervised training pipeline, we further combine it with a semi-supervised algorithm to exploit the support of noisy data as
+unlabeled data. Experimental results on several benchmark datasets and real-world noisy datasets show the effectiveness of our
+framework and validate the theoretical results of Knockoffs-SPR. Our code and pre-trained models will be released.
+Index Terms—Learning with Noisy Labels, Knockoffs Method, Type-Two Error Control.
+!
+1
+INTRODUCTION
+D
+EEP learning has achieved remarkable success on
+many supervised learning tasks trained by millions
+of labeled training data. The performance of deep models
+heavily relies on the quality of label annotation since neural
+networks are susceptible to noisy labels and even can easily
+memorize randomly labeled annotations [1]. Such noisy
+labels can lead to the degradation of the generalization
+and robustness of such models. Critically, it is expensive
+and difficult to obtain precise labels in many real-world
+scenarios, thus exposing a realistic challenge for supervised
+deep models to learn with noisy data.
+There are many previous efforts in tackling this challenge by
+making the models robust to noisy data, such as modifying
+the network architectures [2]–[5] or loss functions [6]–[9].
+This paper addresses the challenge by directly selecting
+clean samples. Inspired by the dynamic sample selection
+methods [9]–[16], we construct a “virtuous” cycle between
+sample selection and network training: the selected clean
+samples can improve the network training; and on the
+other hand, the improved network has a more powerful
+ability in picking up clean data. As this cycle evolves, the
+performance can be improved. To well establish this cycle, a
+key question remains: how to effectively differentiate clean data
+from noisy ones?
+Preliminary. Typical principles in existing works [9]–[16]
+to differentiate clean data from noisy data include large
+loss [11], inconsistent prediction [17], and irregular feature
+•
+Yikai Wang and Yanwei Fu contribute equally.
+•
+Xinwei Sun is the corresponding author.
+•
+Yikai Wang, Yanwei Fu and Xinwei Sun are with the School of
+Data Science, Fudan University. E-mail: {yikaiwang19, yanweifu,
+sunxinwei}@fudan.edu.cn
+representation [18]. The former two principles identify
+irregular behaviors in the label space, while the last one
+analyzes the instance representations of the same class in
+the feature space. In this paper, we propose unifying the
+label and feature space by making the linear relationship,
+yi = x⊤
+i β + ε,
+(1)
+between feature-label pair (xi ∈ Rp: feature vector; yi ∈ Rc:
+one-hot label vector) of data i. We also have the fixed
+(unknown) coefficient matrix β ∈ Rp×c, and random noise
+ε ∈ Rc. Essentially, the linear relationship here is an ideal
+approximation, as the networks are trained to minimize
+the divergence between a (soft-max) linear projection of
+the feature and a one-hot label vector. For a well-trained
+network, the output prediction of clean data is expected
+to be as similar to a one-hot vector as possible, while the
+entropy of the output of noisy data should be large. Thus if
+the underlying linear relation is well-approximated without
+soft-max operation, the corresponding data is likely to be
+clean. In contrast, the feature-label pair of noisy data may
+not be approximated well by the linear model.
+The simplest way to measure the goodness of the linear
+model in fitting the feature-label pair is to check the
+prediction error, or residual, ri = yi − x⊤
+i ˆβ, where ˆβ is the
+estimate of β. The larger ∥r∥ indicates a larger fitting error
+and thus more possibility for instance i to be outlier/noisy
+data. Many methods have been proposed to test whether ri
+is non-zero. Particularly, we highlight the classical statistical
+leave-one-out approach [19] that computes the studentized
+residual as,
+ti =
+yi − x⊤
+i ˆβ−i
+ˆσ−i
+�
+1 + x⊤
+i
+�X⊤
+−iX−i
+�−1 xi
+�1/2 ,
+(2)
+arXiv:2301.00545v1  [cs.LG]  2 Jan 2023
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+2
+Images
+Features
+Noisy Labels
+Stage 1:
+Network Learning
+Classifier
+Stage 2:
+Sample Selection
+β ∈ Rd×c
+Noisy Labels
+Y ∈ Rn×c
+X ∈ Rn×d
+Features
+Noisy Data Indicator
+=
++
+γ ∈ Rn×c
+Yi (1, 0, 0)
+(0, 1, 0) ˜Yi
+Permute
+Compare
+Selected Clean Data
+Fig. 1. Knockoffs-SPR runs a cycle between network learning and sample selection, where clean data are selected via the comparison of the
+mean-shift parameters between its original label and permuted label.
+where ˆσ is the scale estimate and the subscript −i indicates
+estimates based on the n − 1 observations, leaving out
+the i-th data for testing. Equivalently, the linear regression
+model can be re-formulated into explicitly representing the
+residual,
+Y = Xβ + γ + ε,
+εi,j ∼ N(0, σ2),
+(3)
+by introducing a mean-shift parameter γ as in [20] with the
+feature X ∈ Rn×p, and label Y ∈ Rn×c paired and stacked
+by rows. For each row of γ ∈ Rn×c, γi represents the predict
+residual of the corresponding data. This formulation has
+been widely studied in different research topics, including
+economics [21]–[24], robust regression [20], [25], statistical
+ranking [26], face recognition [27], semi-supervised few-shot
+learning [28], [29], and Bayesian preference learning [30],
+to name a few. This formulation is differently focused on
+the specific research tasks. For example, for the robust
+regression problem [20], [25], the target is to get a robust
+estimate ˆβ against the influence of γ. Here for solving the
+problem of learning with noisy labels, we are interested
+in recovering zeros elements of γ, since these elements
+correspond to clean data.
+SPR [31]. To this end, from the statistical perspective,
+our conference report [31] starts from Eq. (3) to build up
+a sample selection framework, dubbed Scalable Penalized
+Regression (SPR). With a sparse penalty P(γ; λ) on γ, the
+SPR obtains a regularization solution path of γ(λ) by
+evolving λ from ∞ to 0. Then it identifies those samples
+that are earlier (or at larger λ) selected to be non-zeros
+as noisy data and those later selected as clean data, with
+a manually specified ratio of selected data. Under the
+irrepresentable condition [4], [33], the SPR enjoys model
+selection consistency in the sense that it can recover the set
+of noisy data. By feeding only clean data into next-round
+training, the trained network is less corrupted by the noisy
+data and hence performs well empirically.
+Knockoffs-SPR. However, the irrepresentable condition
+demands the prior of the ground-truth noisy set, which
+is not accessible in practice. When this condition fails,
+the trained network with SPR may be still corrupted by
+a large proportion of noisy data, leading to performance
+degradation as empirically verified in our experiments. To
+amend this problem, we provide a data-adaptive sample
+selection algorithm, in order to well control the expected
+rate of noisy data in the selected data under the desired level
+q, e.g., q = 0.05. As the goal is to identify clean data for the
+next-round training, we term this rate as the False-Selection-
+Rate (FSR). The FSR is the expected rate of the type-II error
+in sparse regression, as non-zero elements correspond to
+the noisy data. Our method to achieve the FSR control is
+inspired by the ideas of Knockoffs in Statistics, which is
+a recently developed framework for variable selection [1],
+[2], [34], [35]. The Knockoffs framework aims at selecting
+non-null variables and controlling the False-Discovery-Rate
+(FDR), by taking as negative controls knockoff features ˜
+X,
+which are constructed as a fake copy for the original features
+X. Here, the FDR corresponds to the expectation of the
+type-I error rate in sparse regression. Therefore, the vanilla
+Knockoffs cannot be directly applied to our SPR framework,
+since FSR is the expected rate of the type-II error and there
+is no theoretical guarantee in Knockoffs for this control. To
+achieve the FSR control, we propose Knockoffs-SPR, which
+turns to construct the knockoff labels ˜Y via permutation for
+the original label Y , and incorporates it into a data-partition
+strategy for FSR control.
+Formally, we repurpose the knockoffs in Statistics in our
+SPR method; and propose a novel data-adaptive sample
+selection algorithm, dubbed Knockoffs-SPR. It extends SPR
+in controlling the ratio of noisy data among the selected
+clean data. With this new property, Knockoffs-SPR ensures
+that the clean pattern is dominant in the data and hence
+leads to better network training. Specifically, we partition
+the whole noisy training set into two random subsets and
+apply the Knockoffs-SPR to two subsets separately. For each
+time, we use one subset to estimate the intercept β and the
+other to select the clean data by comparing between the
+solution paths of γ(λ) and ˜γ(λ) that respectively obtained
+via regression on noisy labels and the permuted labels. With
+such a decoupled structure between β and γ, we prove
+that the FSR can be controlled by any prescribed level.
+Compared with the original theory of SPR, our new theory
+enables us to effectively select clean data under general
+conditions. Besides, Knockoffs-SPR also enjoys a superior
+performance over the original SPR.
+Together with network training, the whole framework is
+illustrated in Fig. 1 in which the sample selection and
+the network learning are well incorporated into each
+other. Specifically, we run the network learning process
+and sample selection process iteratively and repeat this
+cycle until convergence. To incorporate Knockoffs-SPR into
+the end-to-end training pipeline of deep architecture, the
+simplest way is to directly solve Knockoffs-SPR for each
+
+featureA
+B
+C
+DJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+3
+training mini-batch or training epoch to select clean data.
+Solving Knockoffs-SPR for each mini-batch is efficient but
+suffers from the identifiability issue. The sample size in a
+mini-batch may be too small to distinguish clean patterns
+from noisy ones among all classes, especially for large
+datasets with small batch size. Solving Knockoffs-SPR for
+the whole training set is powerful but suffers from the
+complexity issue, leading to an unacceptable computation
+cost. To resolve these two problems, we strike a balance
+between complexity and identifiability by proposing a
+splitting strategy that divides the whole data into small
+pieces such that each piece is class-balanced with the proper
+sample size. In this regard, the sample size of each piece is
+small enough to be solved efficiently and large enough to
+distinguish clean patterns from noisy ones. Then Knockoffs-
+SPR runs on each piece in parallel, making it scalable to
+large datasets.
+As the removed noisy data still contain useful information
+for network training, we adopt the semi-supervised training
+pipeline with CutMix [38] where the noisy data are utilized
+as unlabeled data. We conduct extensive experiments to
+validate the effectiveness of our framework on several
+benchmark datasets and real-world noisy datasets. The
+results show the efficacy of our Knockoffs-SPR algorithm.
+Contributions. Our contributions are as follows:
+• Ideologically, we propose to control the False-Selection-
+Rate in selecting clean data, under general scenarios.
+• Methodologically, we propose Knockoffs-SPR, a data-
+adaptive method to control the FSR.
+• Theoretically, we prove that the Knockoffs-SPR can
+control the FSR under any desired level.
+• Algorithmically, we propose a splitting algorithm for
+better sample selection with balanced identifiability and
+complexity in large datasets.
+• Experimentally, we demonstrate the effectiveness and
+efficiency of our method on several benchmark datasets
+and real-world noisy datasets.
+Extensions. Our conference version of this work, SPR, was
+published in [31]. Compared with SPR [31], we have the
+following extensions.
+• We identify the limitation of the SPR and consider the FSR
+control in selecting clean data.
+• We propose a new framework: Knockoffs-SPR which is
+effective in selecting clean data under general scenarios,
+theoretically and empirically.
+• We apply our method on Clothing1M and achieve better
+results than compared baselines.
+Logistics. The rest of this paper is organized as follows:
+• In Section 9, we introduce our SPR algorithm with its
+noisy set recovery theory.
+• In Section 3, the Knockoffs-SPR algorithm is introduced
+with its FSR control theorem.
+• In Section 4, several training strategies are proposed to
+well incorporate the Knockoffs-SPR with the network
+training.
+• In Section 5, connections are made between our proposed
+works and several previous works.
+• In Section 6, we conduct experiments on several synthetic
+and real-world noisy datasets.
+• Section 7 concludes this paper.
+2
+CLEAN SAMPLE SELECTION
+2.1
+Problem Setup
+We are given a dataset of image-label pairs {(imgi, yi)}n
+i=1,
+where the noisy label yi is corrupted from the ground-
+truth label y∗
+i . The ground-truth label y∗
+i and the corruption
+process are unknown. Our target is to learn a recognition
+model f(·) such that it can recognize the true category y∗
+i
+from the image imgi, i.e., f(imgi) = y∗
+i , after training on the
+noisy label yi.
+In this paper, we adopt deep neural networks as the
+recognition model and divide the f(·) into fc(g(·)) where
+g(·) is the deep model for feature extraction and fc(·) is
+the final fully-connected layer for classification. For each
+input image imgi, the feature extractor g(·) is used to encode
+the feature xi := g(imgi). Then the fully-connected layer is
+used to output the score vector ˆyi = fc(xi) which indicates
+the chance it belongs to each class and the prediction is
+provided with ˆyi = argmax(ˆyi).
+As the training data contain many noisy labels, simply
+training from all the data leads to severe degradation
+of generalization and robustness. Intuitively, if we could
+identify the clean labels from the noisy training set, and
+train the network with the clean data, we can reduce the
+influence of noisy labels and achieve better performance and
+robustness of the model. To achieve this, we thus propose a
+sample selection algorithm to identify the clean data in the
+noisy training set with theoretical guarantees.
+Notation. In this paper, we will use a to represent scalar, a
+to represent a vector, and A to represent a matrix. We will
+annotate a∗ to denote the ground-truth value of a. We use
+∥ · ∥F to denote the Frobenius norm.
+2.2
+Clean Sample Selection via Penalized Regression
+Motivated
+by
+the
+leave-one-out
+approach
+for
+outlier
+detection, we introduce an explicit noisy data indicator γi
+for each data and assume a linear relation between extracted
+feature xi and one-hot label yi with noisy data indicator as,
+yi = x⊤
+i β + γi + εi,
+(4)
+where yi ∈ Rc is one-hot vector; and xi ∈ Rp, β ∈
+Rp×c, γi ∈ Rc, εi ∈ Rc. The noisy data indicator γi can be
+regarded as the correction of the linear prediction. For clean
+data, yi ∼ N(x⊤
+i β∗, σ2Ic) with γ∗
+i = 0, and for noisy data
+y∗
+i = yi −γ∗
+i ∼ N(x⊤
+i β∗, σ2). We denote C := {i : γ∗
+i = 0}
+as the ground-truth clean set.
+To select clean data for training, we propose Scalable
+Penalized Regression (SPR), designed as the following sparse
+learning paradigm,
+argmin
+β,γ
+1
+2 ∥Y − Xβ − γ∥2
+F + P(γ; λ),
+(5)
+where we have the matrix formulation X ∈ Rn×p, and
+Y
+∈
+Rn×c of {xi, yi}n
+i=1; and P(·; λ) is a row-wise
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+4
+Fig. 2. Solution Path of SPR. Red lines indicate noisy data while blue
+lines indicate clean data. As λ decreases, the γi gradually solved with
+non-zero values.
+sparse penalty with coefficient parameter λ. So we have
+P(γ; λ) = �n
+j=1 P(γi; λ), e.g., group-lasso sparsity with
+P(γ; λ) = λ �
+i ∥γi∥2.
+To estimate C, we only need to solve γ with no need to
+estimate β. Thus to simplify the optimization, we substitute
+the Ordinary Least Squares (OLS) estimate for β with γ
+fixed into Eq. (5). To ensure that ˆβ is identifiable, we apply
+PCA on X to make p ≪ n so that the X has full-column
+rank. Denote
+˜
+X = I − X
+�X⊤X
+�† X⊤, ˜Y
+=
+˜
+XY , the
+Eq. (5) is transformed into
+argmin
+γ
+1
+2
+��� ˜Y − ˜
+Xγ
+���
+2
+F + P(γ; λ),
+(6)
+which is a standard sparse linear regression for γ. Note that
+in practice we can hardly choose a proper λ that works well
+in all scenarios. Furthermore, from the equivalence between
+the penalized regression problem and Huber’s M-estimate,
+the solution of γ is returned with soft-thresholding. Thus
+it is not worth finding the precise solution of a single γ.
+Instead, we use a block-wise descent algorithm [39] to solve
+γ with a list of λs and generate the solution path. As
+λ changes from ∞ to 0, the influence of sparse penalty
+decreases, and γi are gradually solved with non-zero values,
+in other words, selected by the model, as visualized in
+Fig. 2. Since earlier selected instance is more possible to be
+noisy, we rank all samples in the descendent order of their
+selecting time defined as:
+Zi = sup {λ : γi (λ) ̸= 0} .
+(7)
+A large Zi means that the γi is earlier selected. Then the top
+samples are identified as noisy data and the other samples
+are selected as clean data. In practice, we select 50% of the
+data as clean data.
+2.3
+The Theory of Noisy Set Recovery in SPR
+The SPR enjoys theoretical guarantees that the noisy data
+set can be fully recovered with high probability, under
+the irrepresentable condition [33]. Formally, consider the
+vectorized version of Eq. (6):
+argmin
+⃗γ
+1
+2
+���⃗y − ˚
+X⃗γ
+���
+2
+2 + λ ∥⃗γ∥1 ,
+(8)
+where ⃗y, ⃗γ is vectorized from Y , γ in Eq. (6); ˚
+X = Ic ⊗ ˜
+X
+with ⊗ denoting the Kronecker product operator. Denote
+S := supp(⃗γ∗), which is the noisy set Cc. We further denote
+˚
+XS (resp. ˚
+XSc) as the column vectors of ˚
+X whose indexes
+are in S (resp. Sc) and µ ˚
+X = maxi∈Sc ∥ ˚
+X∥2
+2. Then we have
+Theorem 1 (Noisy set recovery). Assume that:
+C1, Restricted eigenvalue: λmin( ˚
+X⊤
+S ˚
+XS) = Cmin > 0;
+C2, Irrepresentability: there exists a η ∈ (0, 1], such that
+∥ ˚
+X⊤
+Sc ˚
+XS( ˚
+X⊤
+S ˚
+XS)−1∥∞ ≤ 1 − η;
+C3, Large error:
+⃗γ∗
+min := mini∈S |⃗γ∗
+i | > h(λ, η, ˚
+X, ⃗γ∗);
+where ∥A∥∞
+:=
+maxi
+�
+j |Ai,j|, and h(λ, η, ˚
+X, ⃗γ∗)
+=
+λη/�Cminµ ˚
+X + λ∥( ˚
+X⊤
+S ˚
+XS)−1sign(⃗γ∗
+S)∥∞.
+Let λ ≥
+2σ√µ ˚
+X
+η
+√log cn. Then with probability greater than
+1 − 2(cn)−1, model Eq. (8) has a unique solution ˆ⃗γ such that: 1)
+If C1 and C2 hold, ˆ
+Cc ⊆ Cc;2) If C1, C2 and C3 hold, ˆ
+Cc = Cc.
+We present the proof in the appendix, following the
+treatment in [4], [40]. In this theorem, C1 is necessary to get
+a unique solution, and in our case is mostly satisfied with
+the natural assumption that the clean data is the majority
+in the training data. If C2 holds, the estimated noisy data
+is the subset of truly noisy data. This condition is the key
+to ensuring the success of SPR, which requires divergence
+between clean and noisy data such that we cannot represent
+clean data with noisy data. If C3 further holds, the estimated
+noisy data is exactly all the truly noisy data. C3 requires the
+error measured by γi is large enough to be identified from
+random noise. If the conditions fail, SPR will fail in a non-
+vanishing probability, not deterministic.
+3
+CONTROLLED CLEAN SAMPLE SELECTION
+In the last section, we stop the solution path at λ such
+that 50% samples are selected as clean data. If this happens
+to be the rate of clean data, Thm. 1 shows that our SPR
+can identify the clean data C under the irrepresentable
+condition. However, the irrepresentable condition and the
+information of the ground-truth clean set C are practically
+unknown, making this theory hard to be used in the real
+life. Particularly, with |Cc| unknown, the algorithm can stop
+at an improper time such that the noisy rate of the selected
+clean data ˆC can be still high, making the next-round trained
+model corrupted a lot by noisy patterns.
+To resolve the problem of false selection in SPR , we in this
+section propose a data-adaptive early stopping method for
+the solution path, that targets controlling the expected noisy
+rate of the selected data dubbed as False-Selection-Rate (FSR)
+under the desired level q (0 < q < 1):
+FSR = E
+�
+�#
+�
+j : j ̸∈ H0 ∩ ˆC
+�
+#
+�
+j : j ∈ ˆC
+�
+∨ 1
+�
+� ,
+(9)
+where ˆC = {j : ˆγj = 0} is the recovered clean set of
+γ, and H0 : γ∗
+i = 0 denotes the null hypothesis, i.e., the
+sample i belonging to the clean dataset. Therefore, the FSR
+in Eq. (9) targets controlling the false rate among selected
+null hypotheses, which is also called the expected rate of
+the type-II error in hypothesis testing.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+5
+3.1
+Knockoffs-SPR
+To achieve the FSR control, we propose the Knockoffs-
+SPR for clean sample selection. Our method is inspired
+by knockoff methods [1], [2], [34], [35], [41] with the
+different focus that we target selecting clean labels via
+permutation instead of constructing knockoff features to
+select explanatory variables. Specifically, under model (4)
+we permute the label for each data and construct the
+permutation ˜y. Then model (4) can be solved for y and
+˜y to obtain the solution paths γ(λ) and ˜γ(λ), respectively.
+We will show that this construction can pick up clean data
+from noisy ones, by comparing the selecting time (Eq. (7))
+between γ(λ) and ˜γ(λ) for each data. On the basis of this
+construction, we propose to partition the whole dataset
+into two disjoint parts, with one part for estimating β
+and the other for learning γ(λ) and ˜γ(λ). We will show
+that the independent structure with such a data partition
+enables us to construct the comparison statistics whose sign
+patterns among alternative hypotheses (noisy data) are the
+independent Bernoulli processes, which is crucial for FSR
+control.
+Formally speaking, we split the whole data D into D1 :=
+(X1, Y1) and D2
+:=
+(X2, Y2) with ni
+:=
+|Di|, and
+implement Knockoffs-SPR on both D1 and D2. In the
+following, we only introduce the procedure on D2, as the
+procedure for D1 shares the same spirit. Roughly speaking,
+the procedure is composed of three steps: i) estimate β on
+D1; ii) estimate ˜γ(λ)) on D2; and iii) construct the comparison
+statistics and selection filters. We leave detailed discussions for
+each step in Sec. 3.2.
+Step i): Estimating β on D1. Our target is to provide
+an estimate of β that is independent of D2. The simplest
+strategy is to use the standard OLS estimator to obtain
+ˆβ1. However, this estimator may not be accurate since it
+is corrupted by noisy samples. For this consideration, we
+first run SPR on D1 to get clean data and then solve β via
+OLS on the estimated clean data.
+Step ii): Estimating (γ(λ), ˜γ(λ)) on D2. After obtaining the
+solution ˆβ1 on D1 , we learn the γ(λ) on D2:
+1
+2
+���Y2 − X2 ˆβ1 − γ2
+���
+2
+F + P(γ2; λ).
+(10)
+For each one-hot encoded vector y2,j, we randomly permute
+the position of 1 and obtain another one-hot vector ˜y2,j ̸=
+y2,j. For clean data j, the ˜y2,j turns to be a noisy label;
+while for noisy data, the ˜y2,j is switched to another noisy
+label with probability c−2
+c−1 or clean label with probability
+1
+c−1. After obtaining the permuted matrix as ˜Y2, we learn
+the solution paths (γ2(λ), ˜γ2(λ)) using the same algorithm
+as SPR via:
+�
+�
+�
+�
+�
+1
+2
+���Y2 − X2 ˜β1 − γ2
+���
+2
+F + �
+j P(γ2,j; λ),
+1
+2
+��� ˜Y2 − X2 ˜β1 − ˜γ2
+���
+2
+F + �
+j P(˜γ2,j; λ).
+(11)
+Step iii): Comparison statistics and selection filters.
+After obtaining the solution path (γ2(λ), ˜γ2(λ)), we define
+sample significance scores with respect to y2,i and ˜y2,i of
+each i respectively, as the selection time: Zi := sup{λ :
+Algorithm 1 Knockoffs-SPR
+Input: subsets D1 and D2.
+Output: clean set of D2.
+1: Use D1 to fit an linear regression model and get β(D1);
+2: Generate permuted label of each sample i in D2;
+3: Solve Eq. (26) for D2 and generate {Wi} by Eq. (12);
+4: Initialize q = 0.02 and T = 0;
+5: while q < 0.5 and T = 0 do
+6:
+Compute T by Eq. (13);
+7:
+q = q + 0.02;
+8: end while
+9: if T is 0 then
+10:
+Construct clean set via half of the samples with largest
+Wi in Eq. (14) with T = ∞;
+11: else
+12:
+Construct clean set via samples in Eq. (14);
+13: end if
+14: return clean set.
+∥γ2,i(λ)∥2 ̸= 0} and ˜Zi := sup{λ : ∥˜γ2,i(λ)∥2 ̸= 0}. With
+Zi, ˜Zi, we define the Wi as:
+Wi := Zi · sign(Zi − ˜Zi).
+(12)
+Based on these statistics, we define a data-dependent
+threshold T as
+T = max
+�
+t > 0 : 1 + # {j : 0 < Wj ≤ t}
+# {j : −t ≤ Wj < 0} ∨ 1 ≤ q
+�
+,
+(13)
+or T = 0 if this set is empty, where q is the pre-defined upper
+bound. Our algorithm will select the clean subset identified
+by
+C2 := {j : −T ≤ Wj < 0}.
+(14)
+Empirically, T may be equal to 0 if the threshold q is
+sufficiently small. In this regard, no clean data are selected,
+which is meaningless. Therefore, we start with a small q and
+iteratively increase q and calculate T, until an attainable T
+such that T > 0 to bound the FSR as small as possible. In
+practice, when the FSR cannot be bounded by q = 50%,
+we will end the selection and simply select half of the most
+possible clean examples via {Wj}. The whole procedure of
+Knockoffs-SPR is shown in Algorithm 1.
+3.2
+Statistical Analysis about Knockoffs-SPR
+In this part, we present the motivations and intuitions of
+each step in Knockoffs-SPR.
+Data Partition. Knockoffs-SPR partitions the dataset D
+into two subset D1 and D2. This step decomposes the
+dependency of the estimate of β and γ in that we use D1/D2
+to estimate β/γ, respectively. Then ˆβ(D1) is independent of
+ˆγ(D2) if D1 and D2 are disjoint. The independent estimation
+of β and γ makes it provable for FSR control on D2.
+Permutation. As we discussed in step ii, when the original
+label is clean, its permuted label will be a noisy label. On
+the other hand, if the original label is noisy, its permuted
+label changes to clean with probability
+1
+c−1 and noisy with
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+6
+probability
+c−2
+c−1, where c denotes the number of classes.
+Note that γ of noisy data is often selected earlier than that
+of clean data in the solution path. This implies larger Z
+values for noisy data than those for clean data. As a result,
+according to the definition of W, a clean sample will ideally
+have a small negative of W := Z · sign(Z − ˜Z), where Z
+and ˜Z respectively correspond to the clean label and noisy
+label. In contrast for a noisy sample, the W tends to have
+a large magnitude and has approximately equal probability
+to be positive or negative. Such a different behavior of W
+between clean and noisy data can help us to identify clean
+samples from noisy ones.
+Asymmetric comparison statistics W. The classical way to
+define comparison statistics is in a symmetric manner, i.e.,
+Wi := Zi ∨ ˜Zi · sign(Zi − ˜Zi). In this way, a clean sample
+with a noisy permuted label tends to have a large |Wi|, as
+we expect the noisy label to have a large ˜Zi. However, this is
+against our target as we only require clean samples to have
+a small magnitude. For this purpose, we design asymmetric
+comparison statistics that only consider the magnitude of
+the original labels.
+To see the asymmetric behavior of W
+for noisy and
+clean data, we consider the Karush–Kuhn–Tucker (KKT)
+conditions of Eq. (26) with respect to (γ2,i, ˜γ2,i)
+γ2,i + ∂P(γ2,i; λ)
+∂γ2,i
+= x⊤
+2,i(β∗ − ˆβ1) + γ∗
+2,i + ε(2),i,
+(15a)
+˜γ2,i + ∂P(˜γ2,i; λ)
+∂˜γ2,i
+= x⊤
+2,i(β∗ − ˆβ1) + ˜γ∗
+2,i + ˜ε(2),i,
+(15b)
+where ε(2),i ∼i.i.d ˜ε(2),i, |γ∗
+2,i| = |˜γ∗
+2,i| if both y2,i and
+˜y2,i are noisy, and P(γ2,i; λ) := λ|γ2,i| as an example. By
+conditioning on ˆβ1 and denoting ai := x⊤
+2,i(β∗ − ˆβ1), we
+have that
+P(Wi > 0) = P(|ai+γ∗
+2,i+ε2,i| > |ai+ ˜γ∗
+2,i+ ˜ε(2),i|). (16)
+Then it can be seen that if i is clean, we have γ∗
+2,i = 0.
+Then Zi tends to be small and besides, it is probable to have
+Zi < ˜Zi if ˆβ1 can estimate β∗ well. As a result, Wi tends
+to be a small negative. On the other hand, if i is noisy, then
+Zi tends to be large for γi to account for the noisy pattern,
+and besides, it has equal probability between Zi < ˜Zi and
+Zi ≥ ˜Zi when ˜y2,i is switched to another noisy label, with
+probability
+c−2
+c−1. So Wi tends to have a large value and
+besides,
+P(Wi > 0) = P(Wi > 0|˜y2,i is noisy)P(˜y2,i is noisy)
++ P(Wi > 0|˜y2,i is clean)P(˜y2,i is clean) = 1
+2 · c − 2
+c − 1
++ P(Wi > 0|˜y2,i is clean) ·
+1
+c − 1,
+(17)
+which falls in the interval of
+�
+c−2
+c−1 · 1
+2,
+c
+c−1 · 1
+2
+�
+. That is to say,
+P(Wi > 0) ≈ 1
+2. In this regard, the clean data corresponds
+to small negatives of W in the ideal case, which can help
+to discriminate noisy data with large W with almost equal
+probability to be positive or negative.
+Remark. For noisy y2,i, we have P(Wi > 0|˜y2,i is noisy) =
+1/2 by assuming |γ∗
+2,i| = |˜γ∗
+2,i|. However, it may not hold
+in practice when y2,i corresponds to the noisy pattern that
+has been learned by the model. In this regard, it may
+have |γ∗
+2,i| < |˜γ∗
+2,i| for a randomly permuted label ˜y2,i.
+To resolve this problem, we instead set the permutation
+label as the most confident candidate of the model, please
+refer to Sec. 4.1 for details. Besides, if ˆβ1 can accurately
+estimate β∗, according to KKT conditions in Eq. (15), we
+have P(Wi > 0) < 1/2. That is Wi tends to be negative for
+the clean data, which is beneficial for clean sample selection.
+Data-adaptive
+threshold.
+The
+proposed
+data-adaptive
+threshold T
+is directly designed to control the FSR.
+Specifically, the FSR defined in Eq. (9) is equivalent to
+FSR(t) = E
+�# {j : γj ̸= 0 and − t ≤ Wj < 0}
+# {j : −t ≤ Wj < 0} ∨ 1
+�
+,
+(18)
+where the denominator denotes the number of selected
+clean data according to Eq. (14) and the nominator denotes
+the number of falsely selected noisy data. This form of
+Eq. (18) can be further decomposed into,
+E
+� # {γj ̸= 0, −t ≤ Wj < 0}
+1 + # {γj ̸= 0, 0 < Wj ≤ t} · 1 + # {γj ̸= 0, 0 < Wj ≤ t}
+# {−t ≤ Wj < 0} ∨ 1
+�
+≤ E
+� # {γj ̸= 0, −t ≤ Wj < 0}
+1 + # {γj ̸= 0, 0 < Wj ≤ t}
+1 + # {0 < Wj ≤ t}
+# {−t ≤ Wj < 0} ∨ 1
+�
+≤ E
+� # {γj ̸= 0, −t ≤ Wj < 0}
+1 + # {γj ̸= 0, 0 < Wj ≤ t}q
+�
+,
+(19)
+where the last inequality comes from the definition of
+T in Eq. (13). To control the FSR, it suffices to bound
+E
+�
+#{γj̸=0, −t≤Wj<0}
+1+#{γj̸=0, 0<Wj≤t}
+�
+. Roughly speaking, this term means
+the number of negative W to the number of positive
+W, among noisy data. Since W
+for noisy data has
+approximately equal probability to be positive/negative
+as mentioned earlier, intuitively we have this term ≈
+1
+2.
+Formally, we construct a martingale process of 1(Wi > 0)
+among noisy data, which is independent of the magnitude
+|W| due to data partition. We leave these details in the
+appendix.
+3.3
+FSR Control of Knockoffs-SPR
+Our target is to show that FSR ≤ q with our data-adaptive
+threshold T in Eq. (13). Our main result is as follows:
+Theorem 2 (FSR control). For c-class classification task, and for
+all 0 < q ≤ 1, the solution of Knockoffs-SPR holds
+FSR(T) ≤ q
+(20)
+with the threshold T for two subsets defined respectively as
+Ti = max
+�
+t ∈ W : 1 + # {j : 0 < Wj ≤ t}
+# {j : −t ≤ Wj < 0} ∨ 1 ≤ c − 2
+2c q
+�
+.
+We present the proof in the appendix. The coefficient
+1/2 comes from the subset-partition strategy that we run
+Knockoffs-SPR on two D1 and D2, and the term
+c−2
+c
+comes from the upper-bound of the first part in Eq. (19).
+This theorem tells us that FSR can be controlled by the
+given threshold q using the procedure of Knockoffs-SPR.
+Compared to SPR, this procedure is more practical and
+useful in real-world experiments and we demonstrate its
+utility in Sec. 6.3 for more details.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+7
+Algorithm 2 Knockoffs-SPR on full training set
+Input: Noisy feature-label pairs {(xi, yi)}n
+i=1, group class
+size N sample size m, (Optional) clean set.
+Output: clean set.
+1: if Number of classes > N then
+2:
+Compute class prototypes using Eq. (22);
+3:
+Divide classes into groups using Eq. (21);
+4: else
+5:
+Use all classes as a single group;
+6: end if
+7: Construct pieces with uniformly sampled m examples
+for each class (total=N × m);
+8: for each piece do
+9:
+Randomly partition the piece into two sub-pieces A and
+B (each contains Nm/2 examples);
+10:
+Run Algorithm 1 (B, A) on A to get clean-set-A;
+11:
+Run Algorithm 1 (A, B) on B to get clean-set-B;
+12:
+Concat clean-set-A and clean-set-B to get clean-set-piece;
+13: end for
+14: Concat clean-set-pieces to get clean set;
+15: return clean set.
+4
+LEARNING WITH KNOCKOFFS-SPR
+In this section, we introduce how to incorporate Knockoffs-
+SPR into the training of neural networks. We first introduce
+several implementation details of Knockoffs-SPR, then we
+introduce a splitting algorithm that makes Knockoffs-SPR
+scalable to large-scale datasets. Finally, we discuss some
+training strategies to better utilize the selected clean data.
+4.1
+Knockoffs-SPR in Practice
+We introduce several strategies to improve FSR control and
+the power of selecting clean samples, which are inspired by
+different behaviors of W between noisy and clean samples.
+Ideally, for a clean sample i, Wi is expected to be a small
+negative; if i is noisy data, Wi tends to be large and is
+approximately 50% to be positive or negative, as shown
+in Eq. (17). To achieve these properties for better clean
+sample selection, the following strategies are proposed, in
+the procedure of feature extractor, data-preprocessing, label
+permutation strategy, estimating β on D1, and clean data
+identification in Eq. (13), (14).
+Feature Extractor. A good feature extractor is essential for
+clean sample selection algorithms. In our experiments, we
+adopt the self-supervised training method SimSiam [42] to
+pre-train the feature extractor, to make X well encode the
+information of the training data in the early stages.
+Data Preprocessing. We implement PCA on the features
+extracted by neural network for dimension reduction. This
+can make X of full rank, which ensures the identifiability of
+ˆβ in SPR. Besides, such a low dimensionality can make the
+model estimate β more accurately. According to the KKT
+conditions Eq. (15), we have that Wi of clean data i tends
+to be negative with small magnitudes. In this regard, the
+model can have better power of clean sample selection, i.e.,
+selecting more clean samples while controlling FSR.
+Label
+Permutation
+Strategy.
+Instead
+of
+the
+random
+permutation strategy, our Knckoff-SPR permutes the label
+as the most-confident candidate provided by the model at
+each training stage, for FSR consideration especially when
+the noise rate is high or some noisy pattern is dominant
+in the data. Specifically, if the pattern of some noisy label
+y2,i is learned by the model, then γ∗
+2,i may have a smaller
+magnitude than that of ˜γ∗
+2,i for a randomly permuted
+label ˜y2,i that may not be learned by the model, violating
+P(Wi > 0|˜y2,i) = 1/2 and hence P(Wi > 0) ≈ 1/2 in
+practice. In contrast, the most confident permutation can
+alleviate this problem, because the most confident label ˜y2,i
+can naturally have a small magnitude of ˜γ∗
+2,i.
+Estimating β on D1. We implement SPR as the first step
+to learn β on D1. Compared to vanilla OLS, the SPR
+can remove some noisy patterns from data, and hence
+can achieve an accurate estimate of β. Similar to the data
+processing step, such an accurate estimation can improve the
+power of selecting clean samples.
+Clean data identification in Eq. (13), (14). We calculate T
+among W for each class, and identify the clean subset for
+each class, to improve the power of clean data for each
+class. In practice, since some classes may be easier to learn
+than others, the Wi for i in these classes have smaller
+magnitudes. Therefore, data from these classes will take the
+main proportion if we calculate T and identify C2 among all
+classes. With this design, the clean data are more balanced,
+which facilitates the training in the next epochs.
+4.2
+Scalable to Large Dataset
+The computation cost of the sample selection algorithm
+increases with the growth of the training sample, making
+it not scalable to large datasets. To resolve this problem,
+we propose to split the total training set into many pieces,
+each of which contains a small portion of training categories
+with a small number of training data. With the splitting
+strategy, we can run the Knockoffs-SPR on several pieces
+in parallel and significantly reduce the running time. For
+the splitting strategy, we notice that the key to identifying
+clean data is leveraging different behavior in terms of the
+magnitude and the sign of W. Such a difference can be
+alleviated if the patterns from clean classes are similar to the
+noisy ones, which may lead to unsatisfactory recall/power
+of identifying the clean set. This motivates us to group
+similar categories together, to facilitate the discrimination
+of clean data from noisy ones.
+Formally speaking, we define the similarity between the
+class i and j as
+s(i, j) = p⊤
+i pj,
+(21)
+where p represents the class prototype. To obtain pi for the
+class i, we take the clean features xi of each class extracted
+by the network along the training iteration, and average
+them to get the class prototype pc after the current training
+epoch ends, as
+pc =
+�n
+i=1 xi1(yi = c, i ∈ C)
+�n
+i=1 1(yi = c, i ∈ C) ,
+(22)
+Then the most similar classes are grouped together. In
+the initialization step when the clean set has not been
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+8
+Algorithm 3 Training with Knockoffs-SPR.
+Input: Noisy dataset {(imgi, xi, yi)}n
+i=1, p.
+Output: Trained network.
+Initialization :
+1: Model: A self-supervised pre-trained backbone with
+a random initialized fully-connected layer, an EMA
+model;
+2: Initial clean set: Run Algorithm 2 with self-supervised
+pre-trained feature and noisy labels;
+Training Process:
+3: for ep = 0 to max epochs do
+4:
+for each mini-batch do
+5:
+Sample r from U(0, 1);
+6:
+if r > p then
+7:
+Train the network using Eq. (25);
+8:
+else
+9:
+Train the network using Eq. (24);
+10:
+end if
+11:
+Update features x visited in current mini-batch;
+12:
+Update EMA model;
+13:
+end for
+14:
+Run Algorithm 2 on {(xi, yi)}n
+i=1 to get clean set;
+15: end for
+16: return Trained network.
+estimated yet, we simply use all the data to calculate the
+class prototypes. In our experiments, each group is designed
+to have 10 classes.
+For the instances in each group, we split the training data
+of each class in a balanced way such that each piece contains
+the same number of instances for each class. The number
+is determined to ensure that the clean pattern remains the
+majority in the piece, such that optimization can be done
+easily. In practice, we select 75 training data from each
+class to construct the piece. When the class proportion
+is imbalanced in the original dataset, we adopt the over-
+sampling strategy to sample the instance of each class with
+less training data multiple times to ensure that each training
+instance is selected once in some piece. The pipeline of our
+splitting algorithm is described in Algorithm 2.
+4.3
+Network Learning with Knockoffs-SPR
+When training with Knockoffs-SPR, we can further exploit
+the support of noisy data by incorporating Knockoffs-
+SPR with semi-supervised algorithms. In this paper, we
+interpolate part of images between clean data and noisy
+data as in CutMix [38],
+˜
+img = M ⊙ imgclean + (1 − M) ⊙ imgnoisy
+(23a)
+˜y = λyclean + (1 − λ)ynoisy
+(23b)
+where M ∈ {0, 1}W ×H is a binary mask, ⊙ is element-
+wise multiplication, λ ∼ Beta(0.5, 0.5) is the interpolation
+coefficient, and the clean and noisy data are identified
+by Knockoffs-SPR. Then we train the network using the
+interpolated data using
+L
+�
+˜
+img, ˜y
+�
+= LCE
+�
+˜
+img, ˜y
+�
+.
+(24)
+Empirically, we could switch between this semi-supervised
+training with standard supervised training on estimated
+clean data.
+L (imgi, yi) = 1i/∈O · LCE (imgi, yi) ,
+(25)
+where 1i/∈O is the indicator function, which means that
+only the cross-entropy loss of estimated clean data is
+used to calculate the loss. We further store a model with
+EMA-updated weights. Our full algorithm is illustrated in
+Algorithm 3. Neural networks trained with this pipeline
+enjoy powerful recognition capacity in several synthetic and
+real-world noisy datasets.
+5
+RELATED WORK
+Here we make the connections between our Knockoffs-SPR
+and previous research efforts.
+5.1
+Learning with Noisy Labels
+The target of Learning with Noisy Labels (LNL) is to
+train a more robust model from the noisy dataset. We can
+roughly categorize LNL algorithms into two groups: robust
+algorithm and noise detection. A robust algorithm does not
+focus on specific noisy data but designs specific modules
+to ensure that networks can be well-trained even from the
+noisy datasets. Methods following this direction includes
+constructing robust network [2]–[5], robust loss function [6]–
+[9] robust regularization [43]–[46] against noisy labels.
+The noise detection method aims to identify the noisy
+data and design specific strategies to deal with the noisy
+data, including down-weighting the importance in the loss
+function for the network training [47], re-labeling them to
+get correct labels [48], or regarding them as unlabeled data
+in the semi-supervised manner [49], etc.
+For the noise detection algorithm, noisy data are identified
+by some irregular patterns, including large error [14],
+gradient directions [50], disagreement within multiple
+networks [15], inconsistency along the training path [17] and
+some spatial properties in the training data [18], [51]–[53].
+Some algorithms [50], [54] rely on the existence of an extra
+clean set to detect noisy data.
+After detecting the clean data, the simplest strategy is to
+train the network using the clean data only or re-weight
+the data [55] to eliminate the noise. Some algorithms [49],
+[56] regard the detected noisy data as unlabeled data to
+fully exploit the distribution support of the training set in
+the semi-supervised learning manner. There are also some
+studies of designing label-correction module [2], [48], [54],
+[57]–[59] to further pseudo-labeling the noisy data to train
+the network. Few of these approaches are designed from the
+statistical perspective with non-asymptotic guarantees, in
+terms of clean sample selection. In contrast, our Knockoffs-
+SPR can theoretically control the false-selected rate in
+selecting clean samples under general scenarios.
+5.2
+Mean-Shit Parameter
+Mean-shift
+parameters
+or
+incidental
+parameters
+[21]
+originally tackled to solve the robust estimation problem
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+9
+via penalized estimation [60] . With a different focus on
+specific parameters, this formulation address wide attention
+in different research topics, including economics [21]–[24],
+robust regression [20], [25], statistical ranking [26], face
+recognition [27], semi-supervised few-shot learning [28],
+[29], and Bayesian preference learning [30], to name a
+few. Previous work usually uses this formulation to solve
+robust linear models, while in this paper we adopt this to
+select clean data and help the training of neural networks.
+Furthermore, we design an FSR control module and a
+scalable sample selection algorithm based on mean-shift
+parameters with theoretical guarantees.
+5.3
+Knockoffs
+Knockoffs was first proposed in [34] as a data-adaptive
+method to control FDR of variable selection in the sparse
+regression problem. This method was then extended to
+high-dimension regression [1], [61], multi-task regression
+[35], outlier detection [41] and structural sparsity [2]. The
+core of Knockoffs is to construct a fake copy of X as
+negative controls of original features, in order to select true
+positive features with FDR control. Our Knockoffs-SPR is
+inspired by but different from the classical knockoffs in the
+following aspects: i) the Knockoff is to control the FDR,
+i.e., the expected rate of the type-I error while our goal is
+to control the expected rate of the type-II error, a.k.a, FSR,
+in the noisy data scenario; ii) instead of constructing copy
+for X, we construct the copy ˜Y via permutation. Equipped
+with a calibrated data-partitioning strategy, our method can
+control the FER under any desired level.
+6
+EXPERIMENTS
+Datasets. We validate the effectiveness of Knockoffs-
+SPR on synthetic noisy datasets CIFAR-10 and CIFAR-
+100 [62], and real-world noisy datasets WebVision [63]
+and Clothing1M [2]. We consider two types of noisy
+labels for CIFAR: (i) Symmetric noise: Every class is
+corrupted uniformly with all other labels; (ii) Asymmetric
+noise: Labels are corrupted by similar (in pattern) classes.
+WebVision has 2.4 million images collected from the
+internet with the same category list as ImageNet ILSVRC12.
+Clothing1M has 1 million images collected from the internet
+and labeled by the surrounding texts. Thus, the WebVision
+and Clothing1M datasets can be regarded as real-world
+challenges.
+Backbones. For CIFAR, we use ResNet-18 [64] as our
+backbone. For WebVision we use Inception-ResNet [65] to
+extract features to follow previous works. For Clothing1M
+we use ResNet-50 as backbones. For CIFAR and WebVision,
+we respectively self-supervised pretrain for 100 epochs and
+350 epochs using SimSiam [42]. For Clothing1M, we use
+ImageNet pre-trained weights to follow previous works.
+Hyperparameter setting. We use SGD to train all the
+networks with a momentum of 0.9 and a cosine learning
+rate decay strategy. The initial learning rate is set as 0.01.
+The weight decay is set as 1e-4 for Clothing1M, and 5e-
+4 for other datasets. We use a batch size of 128 for all
+experiments. We use random crop and random horizontal
+flip as augmentation strategies. The network is trained
+for 180 epochs for CIFAR, 300 epochs for WebVision, and
+5 epochs for Clothing1M. Network training strategy is
+selected with p = 0.5 (line 6 in Alg. 3) for Clothing1M, while
+for other datasets we only use CutMix training. For features
+used in Knockoffs-SPR, we reduce the dimension of X to
+the number of classes. For Clothing1M, this is 14 and for
+other datasets the reduced dimension is 10 (each piece of
+CIFAR-100 and WebVision contains 10 classes). We also run
+SPR with our new network training algorithm (Alg. 3).
+6.1
+Evaluation on Synthetic Label Noise
+Competitors. We use cross-entropy loss (Standard) as the
+baseline algorithm for two datasets. We compare Knockoffs-
+SPR with algorithms that include Forgetting [66] with
+train the network using dropout strategy, Bootstrap [67]
+which trains with bootstrapping, Forward Correction [55]
+which corrects the loss function to get a robust model,
+Decoupling
+[68]
+which
+uses
+a
+meta-update
+strategy
+to
+decouple
+the
+update
+time
+and
+update
+method,
+MentorNet [12] which uses a teacher network to help train
+the network, Co-teaching [11] which uses two networks to
+teach each other, Co-teaching+ [15] which further uses an
+update by disagreement strategy to improve Co-teaching,
+IterNLD [51] which uses an iterative update strategy,
+RoG [52] which uses generated classifiers, PENCIL [59]
+which uses a probabilistic noise correction strategy, GCE [7]
+and SL [8] which are extensions of the standard cross-
+entropy loss function, and TopoFilter [18] which uses feature
+representation to detect noisy data. For each dataset, all the
+experiments are run with the same backbone to make a
+fair comparison. We randomly run all the experiments five
+times and calculate the average and standard deviation of
+the accuracy of the last epoch. The results of competitors are
+reported in [18].
+As in Table 1, Knockoffs-SPR enjoys a higher performance
+compared with other competitors on CIFAR, validating the
+effectiveness of Knockoffs-SPR on different noise scenarios.
+SPR enjoys better performance on higher symmetric noise
+rate of CIFAR-100. This may contributes to the manual
+selection threshold of 50% of the data. Then SPR will
+select more data than Knockoffs-SPR, for example in Sym.
+80% noise scenario SPR will select 24816 clean data while
+Knockoffs-SPR will select 18185 clean data. This leads to
+a better recovery of clean data (recall of 94.22% while
+Knockoffs-SPR is 81.20%) and thus a better recognition
+capacity.
+6.2
+Evaluation on Real-World Noisy Datasets
+In this part, we compare Knockoffs-SPR with other methods
+on real-world noisy datasets: WebVision and Clothing1M.
+We follow previous work to train and test on the first 50
+classes of WebVision. We also evaluate models trained on
+WebVision to ILSVRC12 to test the cross-dataset accuracy.
+Competitors.
+For
+WebVision,
+we
+compare
+with
+CE
+that
+trains
+with
+cross-entropy
+loss
+(CE),
+as
+well
+as
+Decoupling [68], D2L [69], MentorNet [12], Co-teaching [11],
+Iterative-CV [13], and DivideMix [49]. For clothing1M, we
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+10
+TABLE 1
+Test accuracies(%) on several benchmark datasets with different settings.
+Dataset
+Method
+Sym. Noise Rate
+Asy. Noise Rate
+0.2
+0.4
+0.6
+0.8
+0.2
+0.3
+0.4
+CIFAR-10
+Standard
+85.7 ± 0.5
+81.8 ± 0.6
+73.7 ± 1.1
+42.0 ± 2.8
+88.0 ± 0.3
+86.4 ± 0.4
+84.9 ± 0.7
+Forgetting
+86.0 ± 0.8
+82.1 ± 0.7
+75.5 ± 0.7
+41.3 ± 3.3
+89.5 ± 0.2
+88.2 ± 0.1
+85.0 ± 1.0
+Bootstrap
+86.4 ± 0.6
+82.5 ± 0.1
+75.2 ± 0.8
+42.1 ± 3.3
+88.8 ± 0.5
+87.5 ± 0.5
+85.1 ± 0.3
+Forward
+85.7 ± 0.4
+81.0 ± 0.4
+73.3 ± 1.1
+31.6 ± 4.0
+88.5 ± 0.4
+87.3 ± 0.2
+85.3 ± 0.6
+Decoupling
+87.4 ± 0.3
+83.3 ± 0.4
+73.8 ± 1.0
+36.0 ± 3.2
+89.3 ± 0.3
+88.1 ± 0.4
+85.1 ± 1.0
+MentorNet
+88.1 ± 0.3
+81.4 ± 0.5
+70.4 ± 1.1
+31.3 ± 2.9
+86.3 ± 0.4
+84.8 ± 0.3
+78.7 ± 0.4
+Co-teaching
+89.2 ± 0.3
+86.4 ± 0.4
+79.0 ± 0.2
+22.9 ± 3.5
+90.0 ± 0.2
+88.2 ± 0.1
+78.4 ± 0.7
+Co-teaching+
+89.8 ± 0.2
+86.1 ± 0.2
+74.0 ± 0.2
+17.9 ± 1.1
+89.4 ± 0.2
+87.1 ± 0.5
+71.3 ± 0.8
+IterNLD
+87.9 ± 0.4
+83.7 ± 0.4
+74.1 ± 0.5
+38.0 ± 1.9
+89.3 ± 0.3
+88.8 ± 0.5
+85.0 ± 0.4
+RoG
+89.2 ± 0.3
+83.5 ± 0.4
+77.9 ± 0.6
+29.1 ± 1.8
+89.6 ± 0.4
+88.4 ± 0.5
+86.2 ± 0.6
+PENCIL
+88.2 ± 0.2
+86.6 ± 0.3
+74.3 ± 0.6
+45.3 ± 1.4
+90.2 ± 0.2
+88.3 ± 0.2
+84.5 ± 0.5
+GCE
+88.7 ± 0.3
+84.7 ± 0.4
+76.1 ± 0.3
+41.7 ± 1.0
+88.1 ± 0.3
+86.0 ± 0.4
+81.4 ± 0.6
+SL
+89.2 ± 0.5
+85.3 ± 0.7
+78.0 ± 0.3
+44.4 ± 1.1
+88.7 ± 0.3
+86.3 ± 0.1
+81.4 ± 0.7
+TopoFilter
+90.2 ± 0.2
+87.2 ± 0.4
+80.5 ± 0.4
+45.7 ± 1.0
+90.5 ± 0.2
+89.7 ± 0.3
+87.9 ± 0.2
+SPR
+92.0 ± 0.1
+94.6 ± 0.2
+91.6 ± 0.2
+80.5 ± 0.6
+89.0 ± 0.8
+90.3 ± 0.8
+91.0 ± 0.6
+Knockoffs-SPR
+95.4 ± 0.1
+94.5 ± 0.1
+93.3 ± 0.1
+84.6 ± 0.8
+95.1 ± 0.1
+94.5 ± 0.2
+93.6 ± 0.2
+CIFAR-100
+Standard
+56.5 ± 0.7
+50.4 ± 0.8
+38.7 ± 1.0
+18.4 ± 0.5
+57.3 ± 0.7
+52.2 ± 0.4
+42.3 ± 0.7
+Forgetting
+56.5 ± 0.7
+50.6 ± 0.9
+38.7 ± 1.0
+18.4 ± 0.4
+57.5 ± 1.1
+52.4 ± 0.8
+42.4 ± 0.8
+Bootstrap
+56.2 ± 0.5
+50.8 ± 0.6
+37.7 ± 0.8
+19.0 ± 0.6
+57.1 ± 0.9
+53.0 ± 0.9
+43.0 ± 1.0
+Forward
+56.4 ± 0.4
+49.7 ± 1.3
+38.0 ± 1.5
+12.8 ± 1.3
+56.8 ± 1.0
+52.7 ± 0.5
+42.0 ± 1.0
+Decoupling
+57.8 ± 0.4
+49.9 ± 1.0
+37.8 ± 0.7
+17.0 ± 0.7
+60.2 ± 0.9
+54.9 ± 0.1
+47.2 ± 0.9
+MentorNet
+62.9 ± 1.2
+52.8 ± 0.7
+36.0 ± 1.5
+15.1 ± 0.9
+62.3 ± 1.3
+55.3 ± 0.5
+44.4 ± 1.6
+Co-teaching
+64.8 ± 0.2
+60.3 ± 0.4
+46.8 ± 0.7
+13.3 ± 2.8
+63.6 ± 0.4
+58.3 ± 1.1
+48.9 ± 0.8
+Co-teaching+
+64.2 ± 0.4
+53.1 ± 0.2
+25.3 ± 0.5
+10.1 ± 1.2
+60.9 ± 0.3
+56.8 ± 0.5
+48.6 ± 0.4
+IterNLD
+57.9 ± 0.4
+51.2 ± 0.4
+38.1 ± 0.9
+15.5 ± 0.8
+58.1 ± 0.4
+53.0 ± 0.3
+43.5 ± 0.8
+RoG
+63.1 ± 0.3
+58.2 ± 0.5
+47.4 ± 0.8
+20.0 ± 0.9
+67.1 ± 0.6
+65.6 ± 0.4
+58.8 ± 0.1
+PENCIL
+64.9 ± 0.3
+61.3 ± 0.4
+46.6 ± 0.7
+17.3 ± 0.8
+67.5 ± 0.5
+66.0 ± 0.4
+61.9 ± 0.4
+GCE
+63.6 ± 0.6
+59.8 ± 0.5
+46.5 ± 1.3
+17.0 ± 1.1
+64.8 ± 0.9
+61.4 ± 1.1
+50.4 ± 0.9
+SL
+62.1 ± 0.4
+55.6 ± 0.6
+42.7 ± 0.8
+19.5 ± 0.7
+59.2 ± 0.6
+55.1 ± 0.7
+44.8 ± 0.1
+TopoFilter
+65.6 ± 0.3
+62.0 ± 0.6
+47.7 ± 0.5
+20.7 ± 1.2
+68.0 ± 0.3
+66.7 ± 0.6
+62.4 ± 0.2
+SPR
+72.5 ± 0.2
+75.0 ± 0.1
+70.9 ± 0.3
+38.1 ± 0.8
+71.9 ± 0.2
+72.4 ± 0.3
+70.9 ± 0.5
+Knockoffs-SPR
+77.5 ± 0.2
+74.3 ± 0.2
+67.8 ± 0.4
+30.5 ± 1.0
+77.3 ± 0.4
+76.3 ± 0.3
+73.9 ± 0.6
+TABLE 2
+Test accuracies(%) on WebVision and ILSVRC12 (trained on
+WebVision).
+Method
+WebVision
+ILSVRC12
+top1
+top5
+top1
+top5
+F-correction
+61.12
+82.68
+57.36
+82.36
+Decoupling
+62.54
+84.74
+58.26
+82.26
+D2L
+62.68
+84.00
+57.80
+81.36
+MentorNet
+63.00
+81.40
+57.80
+79.92
+Co-teaching
+63.58
+85.20
+61.48
+84.70
+Iterative-CV
+65.24
+85.34
+61.60
+84.98
+DivideMix
+77.32
+91.64
+75.20
+90.84
+SPR
+77.08
+91.40
+72.32
+90.92
+Knockoffs-SPR
+78.20
+92.36
+74.72
+92.88
+compare with F-correction [55], M-correction [56], Joint-
+Optim [48], Meta-Cleaner [70], Meta-Learning [71], P-
+correction [59], TopoFilter [18] and DivideMix [49].
+The results of real-world datasets are shown in Table 3
+and Table 2, where the results of competitors are reported
+in [49]. Our algorithm Knockoffs-SPR enjoys superior
+performance to almost all the competitors, showing the
+ability of handling real-world challenges. Compared with
+SPR, Knockoffs-SPR also achieves better performance,
+indicating the beneficial of FSR control in real-world
+problems of learning with noisy labels.
+TABLE 3
+Test accuracies(%) on Clothing1M.
+Method
+Accuracy
+Cross-Entropy
+69.21
+F-correction
+69.84
+M-correction
+71.00
+Joint-Optim
+72.16
+Meta-Cleaner
+72.50
+Meta-Learning
+73.47
+P-correction
+73.49
+TopoFiler
+74.10
+DivideMix
+74.76
+SPR
+71.16
+Knockoffs-SPR
+75.25
+6.3
+Evaluation of Sample Selection Quality
+To test whether Knockoffs-SPR leads to better sample
+selection quality, we test the following statistics on CIFAR-
+10 with different noise scenarios, including Sym. 40%, Sym.
+80%, and Asy. 40%. (1) FSR: the ratio of falsely selected
+noisy data in the estimated clean data, which is the target
+that Knockoffs-SPR aims to control; (2) Recall: the ratio of
+selected ground-truth clean data in the full ground-truth
+clean data, which indicates the power of sample selection
+algorithms; (3) F1-score: the harmonic mean of precision (1-
+FSR) and recall, which measures the balanced performance
+of FSR control and power. We plot the corresponding
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+11
+0
+25
+50
+75
+100
+125
+150
+175
+0
+2
+4
+6
+8
+10
+12
+14
+16
+FSR
+Symmetric-40%
+0
+25
+50
+75
+100
+125
+150
+175
+0
+10
+20
+30
+40
+50
+60
+70
+80
+Symmetric-80%
+0
+25
+50
+75
+100
+125
+150
+175
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+Asymmetric-40%
+Knockoff-SPR
+Estimated q
+SPR
+TopoFilter
+0
+25
+50
+75
+100
+125
+150
+175
+60
+65
+70
+75
+80
+85
+90
+95
+100
+Recall
+0
+25
+50
+75
+100
+125
+150
+175
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+0
+25
+50
+75
+100
+125
+150
+175
+55
+60
+65
+70
+75
+80
+85
+90
+95
+100
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+Training Epochs
+70
+75
+80
+85
+90
+95
+100
+F score
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+Training Epochs
+10
+20
+30
+40
+50
+60
+70
+80
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+Training Epochs
+65
+70
+75
+80
+85
+90
+95
+100
+Fig. 3. Performance(%) comparison on sample selection along the
+training path on CIFAR 10 with different noise scenarios. In the FSR,
+we also visualize the estimated FSR (q) by Knockoffs-SPR, which is the
+threshold we use to select clean data.
+statistics of each algorithm along the training epochs in
+Fig. 3. We further visualize the estimated FSR, q, of
+Knockoffs-SPR to compare with the ground-truth FSR. As
+we use the splitting algorithm, where each piece contains
+10 classes with each class containing a subset of data, we
+estimate FSR for each piece and report their average and
+standard deviation.
+FSR control in practice. (1) When the noise rate is not
+high, for example in Sym. 40% and Asy. 40% scenarios, the
+ground-truth FSR is well upper-bounded by the estimated
+FSR (with no larger than a single standard deviation). When
+the noise rate is high, for example in Sym. 80% noise
+scenario, the FSR cannot get controlled in the early stage.
+However, as the training goes on, FSR can be well-bounded
+by Knockoffs-SPR.
+(2) When the training set is not very noisy, for example in
+Sym. 40% scenario, the true FSR is far below the estimated
+q. This gap can be explained by a good estimation of
+β due to the small noisy rate. When ˆβ1 can accurately
+estimate β∗, the ˜γ∗
+2,i dominate in Eq. (15). Therefore, the
+P(Wi > 0|˜y2,i is clean) > 1
+2, making P(Wi > 0) > 1/2 >
+c−2
+2(c−1). Since the true FSR bound is inversely proportional to
+P(Wi > 0) (FSR ∝ maxi∈Cc 1/P(Wi > 0) − 1), it is smaller
+than the theoretical bound q.
+Sample selection quality comparison. We compare the
+sample selection quality of Knockoffs-SPR with SPR and
+TopoFilter [18]. (1) Knockoffs-SPR enjoys the (almost) best
+FSR control capacity in all noise scenarios, especially in the
+high noise rate setting. Other algorithms can suffer from
+failure in controlling the FSR (for example in Sym. 80%
+scenario). (2) The power of Knockoffs-SPR is comparable to
+the best algorithms in Sym. 40% and Asy. 40% scenarios. For
+the Sym. 80% case, Knockoffs-SPR sacrifices some power for
+FSR control. (3) Compared together, Knockoffs-SPR enjoys
+the best F1 score on sample selection quality, which well-
+establishes its superiority in selecting clean data with FSR
+control.
+TABLE 4
+Ablation(%) of Knockoffs-SPR on CIFAR-10.
+SPR
+∗-random
+∗-multi
+∗-noPCA
+Knockoffs-SPR
+Sym. 40%
+Acc.
+94.0
+92.0
+94.4
+81.7
+94.7
+FSR
+0.82
+23.04
+1.31
+11.51
+1.27
+q
+-
+4.31±0.73
+2.00±0.00
+14.18±7.62
+5.59±1.11
+Sym. 80%
+Acc.
+78.0
+84.6
+83.0
+10.0
+84.3
+FSR
+60.47
+49.76
+25.77
+78.06
+26.72
+q
+-
+9.47±4.39
+2.22±0.62
+25.95±11.88
+19.52±12.77
+Asy. 40%
+Acc.
+89.5
+84.4
+93.4
+93.7
+93.5
+FSR
+2.19
+16.94
+2.97
+7.62
+2.84
+q
+-
+4.15±2.59
+2.00±0.00
+5.22±2.85
+4.45±2.68
+6.4
+Further Analysis
+Influence
+of
+Knockoffs-SPR
+strategies.
+We
+compare
+Knockoffs-SPR
+with
+several
+variants,
+including:
+SPR
+(The original SPR algorithm), ∗-random (Knockoffs-SPR
+with
+randomly
+permuted
+labels),
+∗-multi
+(Knockoffs-
+SPR without class-specific selection), ∗-noPCA (Knockoffs-
+SPR without using PCA to pre-process the features).
+Experiments are conducted on CIFAR-10 with different
+noise scenarios, as in Table 4. We observe the following
+results:
+(1) As also shown in Fig. 3, the SPR can control the FSR in
+Sym. 40% and Asy. 40% but fails in Sym. 80%. This may
+be due to that when the noisy pattern is not significant,
+the collinearity is weak between noisy samples and clean
+ones, as shown by the distribution of irrepresentable value
+{∥(X⊤
+S XS)−1X⊤
+S Xj∥1}j∈Sc
+in Fig. 1 in the appendix.
+In this regard, most of the earlier (resp. later) selected
+samples in the solution path tend to be noisy (resp. clean)
+samples. When there is strong multi-collinearity and the
+irrepresentable condition is violated seriously, our proposed
+Knockoff procedure can help to control the FSR. The higher
+accuracy of Knockoffs-SPR over SPR can be explained by
+consistent improvements in terms of the F1 score of sample
+selection capacity, as shown in Fig. 3.
+(2) Compared with the random permutation strategy,
+Knockoffs-SPR with most-confident permutation enjoys
+much better FSR control and works much better in Sym.
+40% and Asy. 40% noise scenarios. In Sym. 80% noise
+scenario, the accuracy is comparable, but the most-confident
+permutation still enjoys much better FSR control. This
+result empirically demonstrates the advantage of the most-
+confident permutation over the random permutation.
+(3) Running Knockoffs-SPR on each class separately is
+beneficial for the FSR control capacity and recognition
+capacity. When the noise rate is high, running Knockoffs-
+SPR on multiple classes cannot control the FSR properly by
+the estimated q.
+(4) Using PCA on features as the pre-processing is beneficial
+for FSR control in all cases and will increase the recognition
+capacity in some cases, especially when the noise rate is
+high.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+12
+TABLE 5
+Ablation of the splitting algorithm in computation efficiency (for one
+epoch) on CIFAR-10.
+Model
+Training Time
+Knockoffs-SPR w/o split algorithm
+about 6h
+Knockoffs-SPR w/ split algorithm
+66s
+TABLE 6
+Ablation(%) of training strategies on CIFAR-10.
+Method
+Sym. 40%
+Sym. 80%
+Asy. 40%
+Knockoffs-SPR - Self
+92.5
+24.3
+92.2
+Knockoffs-SPR - Semi
+91.3
+54.0
+88.5
+Knockoffs-SPR - EMA
+94.5
+83.8
+93.2
+Knockoffs-SPR
+94.7
+84.3
+93.5
+Influence of Scalable. In our framework, we propose a split
+algorithm to divide the whole training set into small pieces
+to run Knockoffs-SPR in parallel. In this part, we compare
+the running time between using the split algorithm and not
+using it. Results are shown in Tab. 5. We can see that the
+splitting algorithm can significantly reduce the computation
+time. This is important in large-scale applications.
+Influence of network training strategies. To better train
+the network, we adopt the self-supervised pre-trained
+backbone and the semi-supervised learning framework with
+an EMA update model. In this part, we test the influence of
+these strategies on CIFAR-10 with different noise scenarios.
+Concretely, we compare the full framework with Knockoffs-
+SPR - Self which uses a randomly initialized backbone,
+Knockoffs-SPR - Semi which uses supervised training, and
+Knockoffs-SPR - EMA which does not use the EMA update
+model. Results are summarized in table. 6. We can find
+that: (1) The self-supervised pre-training is important for
+high noise rate scenarios, while for other settings, it is
+not so essential; (2) Semi-supervised training consistently
+improves the recognition capacity, indicating the utility of
+leveraging the support of noisy data; (3) The EMA model
+will slightly improve the recognition capacity.
+airplane
+frog
+automobile
+truck
+bird
+deer
+cat
+dog
+deer
+cat
+dog
+cat
+frog
+deer
+horse
+dog
+ship
+airplane
+truck
+ship
+Fig. 4. Qualitative results of falsely selected examples by Knockoffs-
+SPR. The black words are the labeled classes while the real classes
+are denoted by red words.
+Qualitative visualization. We randomly visualize some
+falsely selected examples of CIFAR-10 in Fig. 4. Most of
+these cases have some patterns that confuse the noisy
+label and the true label, thus making Knockoffs-SPR falsely
+identify them as clean samples.
+7
+CONCLUSION
+This
+paper
+proposes
+a
+statistical
+sample
+selection
+framework – Scalable Penalized Regression with Knockoff
+Filters (Knockoffs-SPR) to identify noisy data with a
+controlled false selection rate. Specifically, we propose an
+equivalent leave-one-out t-test approach as a penalized
+linear model, in which non-zero mean-shift parameters can
+be induced as an indicator for noisy data. We propose a
+delicate Knockoff-SPR algorithm to identify clean samples
+in a way that the false selection rate is controlled by
+the user-provided upper bound. Such an upper bound is
+proved theoretically and works well in empirical results.
+Experiments on several synthetic and real-world datasets
+demonstrate the effectiveness of Knockoff-SPR.
+REFERENCES
+[1]
+C. Zhang, S. Bengio, M. Hardt, B. Recht, and O. Vinyals,
+“Understanding
+deep
+learning
+requires
+rethinking
+generalization,” in ICLR, 2017. (document)
+[2]
+T. Xiao, T. Xia, Y. Yang, C. Huang, and X. Wang, “Learning from
+massive noisy labeled data for image classification,” in CVPR,
+2015. (document), 5.1, 6
+[3]
+J. Goldberger and E. Ben-Reuven, “Training deep neural-networks
+using a noise adaptation layer,” in ICLR, 2017. (document), 5.1
+[4]
+X.
+Chen
+and
+A.
+Gupta,
+“Webly
+supervised
+learning
+of
+convolutional networks,” in ICCV, 2015. (document), 5.1
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+Yikai
+Wang
+is
+a
+PhD
+candidate
+at
+the
+School
+of
+Data
+Science,
+Fudan
+University,
+under the supervision of Prof. Yanwei Fu. He
+received a Bachelor’s degree in mathematics
+from the School of Mathematical Sciences,
+Fudan University, in 2019. He published 1
+IEEE TPAMI paper and 2 CVPR papers. His
+current research interests include theoretically
+guaranteed machine learning algorithms and
+applications to computer vision.
+Yanwei Fu received his PhD degree from the
+Queen Mary University of London, in 2014. He
+worked as a post-doctoral researcher at Disney
+Research, Pittsburgh, PA, from 2015 to 2016.
+He is currently a tenure-track professor at Fudan
+University. He was appointed as the Professor
+of Special Appointment (Eastern Scholar) at
+Shanghai Institutions of Higher Learning. He
+published
+more
+than
+80
+journal/conference
+papers including IEEE TPAMI, TMM, ECCV,
+and CVPR. His research interests are one-
+shot/meta-learning, learning-based 3D reconstruction, and image and
+video understanding in general.
+Xinwei Sun is currently an assistant professor
+at the School of Data Science, Fudan University.
+He
+received
+his
+Ph.D.
+in
+the
+school
+of
+mathematical sciences, at Peking University in
+2018. His research interests mainly focus on
+high-dimensional statistics and causal inference,
+with their applications in machine learning and
+medical imaging.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+15
+Supplementary Material
+Yikai Wang, Yanwei Fu, and Xinwei Sun.
+In this supplementary material, we formally present the
+proof of FSR control theorem of knockoff-SPR in Sec. 8.
+For consistency, we also provide the proof of the noisy
+set recovery theorem of SPR in Sec. 9. Some additional
+experimental results are provided in Sec. 10.
+8
+FSR CONTROL THEOREM OF KNOCKOFF-SPR
+Recall that we are solving the problem of
+�
+�
+�
+�
+�
+1
+2
+���Y2 − X2 ˜β1 − γ2
+���
+2
+F + �
+j P(γ2,j; λ),
+1
+2
+��� ˜Y2 − X2 ˜β1 − ˜γ2
+���
+2
+F + �
+j P(˜γ2,j; λ).
+(26)
+We introduce the following lemma from [1] and [2].
+Lemma 3. Suppose that B1, . . . , Bn are indenpendent variables,
+with Bi ∼ Bernoulli(ρi) for each i, where mini ρi ≥ ρ > 0. Let
+J be a stopping time in reverse time with respect to the filtration
+{Fj}, where
+Fj = σ ({B1 + · · · + Bj, Bj+1, . . . , Bn}) .
+Then
+E
+�
+1 + J
+1 + B1 + · · · + BJ
+�
+≤ ρ−1.
+Proof. We first follow [1] to prove the case when {Bi} are
+i.i.d. variables with Bi ∼ Bernoulli(ρ), where ρ > 0. Then
+we follow [2] to generalize the conclusion to non-identical
+case.
+Define the stochastic process
+Mj := 1 + j
+1 + Sj
+with
+Sj := B1 + · · · + Bj
+(27)
+We show that {Mj} is a super-martingale with respect to
+the reverse filtration {Fj}. It is trivial that {Mj} is {Fj}-
+adapted and {Fj} is reverse filtration, that is a decreasing
+sequence
+Fj ⊂ Fj−1 · · · ⊂ {Bi}n
+i=1
+(28)
+with each Fj be a sub-σ-algebras of σ({Bi}n
+i=1). Further,
+we have E [|Mj|] ≤ 1 + j ≤ 1 + n < ∞ with fixed n. Now
+we bound the conditional expectation E[Mj | Fj+1]. Note
+that since {Bj}i+1
+j=1 are i.i.d. variable and thus exchangeable
+when conditioned on Fj+1, then we have
+P(Bj+1 = 1 | Fj+1) = Sj+1
+j + 1
+(29)
+When Sj+1 = 0, it is natural that Sj = 0 thus Mj = 1 + j <
+1 + (j + 1) = Mj+1. When Sj+ > 0, we have
+E[Mj | Fj+1] =
+1 + j
+1 + Sj+1 − 1 · P(Bj+1 = 1 | Fj+1)
++
+1 + j
+1 + Sj+1
+· P(Bj+1 = 0 | Fj+1)
+=1 + j
+Sj+1
+· Sj+1
+j + 1 +
+1 + j
+1 + Sj+1
+· j + 1 − Sj+1
+j + 1
+=1 + (j + 1)
+1 + Sj+1
+=Mj+1.
+(30)
+Hence we have E[Mj | Fj+1] ≤ Mj+1, which finishes the
+proof for the super-martingale {Mj}. Then by the Doob’s
+optional sampling theorem [3], we have
+E[Mj] ≤ E[Mn].
+(31)
+Finally, we have
+E[Mn] = E[ 1 + n
+1 + Sn
+]
+= (1 + n)
+n
+�
+m=0
+1
+1 + m ·
+n!
+m!(n − m)!ρm(1 − ρ)n−m
+= ρ−1(1 − (1 − ρ)n+1)
+≤ ρ−1.
+(32)
+Now it suffices to show that the conclusion also holds for
+non-identical Bernoulli variables. Following [2], for each
+Bi ∼ Bernoulli(ρi), we construct the following disjoint
+Borel sets {Ai
+j}4
+j=1 such that R = ∪4
+j=1Aj with
+P(Ai
+1) = 1 − ρi;
+P(Ai
+2) = ρ1 − ρi
+1 − ρ ;
+P(Ai
+3) = ρρi − ρ
+1 − ρ ;
+P(Ai
+4) = ρi − ρ.
+(33)
+Define Ui = Ai
+1∪Ai
+2, Vi = Ai
+2∪Ai
+3, Gi = Ai
+2∪Ai
+3∪Ai
+4. Based
+on the specific construction we can set Gi = Bi. Further
+define Qi = 1{ξi ∈ Vi} and a random set A = {i : ξi ∈ Ui}.
+Then we have
+Qi · 1{i ∈ A} + 1{i /∈ A}
+= 1{{ξi ∈ Vi ∩ Ui} ∪ {ξi ∈ U C
+i }}
+= 1{{ξi ∈ Ai
+2} ∪ {ξi ∈ Ai
+3 ∪ Ai
+4}}
+= 1{{ξi ∈ Ai
+2 ∪ Ai
+3 ∪ Ai
+4}} = Bi.
+(34)
+Hence
+1 + j
+1 + Sj
+= 1 + |i ≤ j : i ∈ A| + |i ≤ j : i /∈ A|
+1 + �
+i≤j,i∈A Qi + |i ≤ j : i /∈ A|
+≤ 1 + |i ≤ j : i ∈ A|
+1 + �
+i≤j,i∈A Qi
+.
+(35)
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+16
+The inequality holds because a+c
+b+c ≤ a
+b for 0 < b ≤ a, c ≥ 0.
+Note that by definition
+P(Qi = 1 | i ∈ A) = P(ξi ∈ Vi | ξi ∈ Ui)
+=
+P(Ai
+2)
+P(Ai
+1 ∪ Ai
+2)
+= ρ = P(Qi = 1),
+P(Qi = 1 | i ̸∈ A) = P(ξi ∈ Vi | ξi /∈ Ui)
+=
+P(Ai
+3)
+P(Ai
+3 ∪ Ai
+4)
+= ρ = P(Qi = 1).
+(36)
+indicating that Qi and A are independent.
+For any fixed A, define ˜Qi = Qi · 1{i ∈ A} and the reverse
+filtration ˜Fj = σ({�j
+k=1 ˜Qk, ˜Qj+1, . . . , ˜Qn, A}). Then when
+conditioned on A, the established result suggests that
+E
+�
+1 + |i ≤ j : i ∈ A|
+1 + �
+i≤j,i∈A Qi
+����A
+�
+≤ ρ−1.
+(37)
+Take expectation over A finishes the proof.
+8.1
+Proof of Theorem 2
+Proof. We first control the FSR rate of the second subset.
+Specifically, we have
+FSR(T) ≤ E
+� # {j : γj ̸= 0 and − T ≤ Wj < 0}
+1 + # {j : γj ̸= 0 and 0 < Wj ≤ T}
+· 1 + # {j : 0 < Wj ≤ T}
+# {j : −T ≤ Wj < 0} ∨ 1
+�
+≤ q · E
+� # {j : γj ̸= 0 and − T ≤ Wj < 0}
+1 + # {j : γj ̸= 0 and 0 < Wj ≤ T}
+�
+.
+(38)
+The second inequality holds by the definition of T. Now it
+suffices to show that
+E
+� # {j : γj ̸= 0 and − T ≤ Wj < 0}
+1 + # {j : γj ̸= 0 and 0 < Wj ≤ T}
+�
+≤
+c
+c − 2.
+(39)
+For γj ̸= 0, we have a probability of
+1
+c−1 to get a clean
+˜γj, leading to Zj > Zj+n with non-zero probability, and
+a probability of c−2
+c−1 to get a noisy ˜γj, where we have no
+information and hence assume a equal probability of Zj >
+Zj+n and Zj < Zj+n. Then we have
+P(Wi > 0) =P(Wi > 0|˜γ∗
+j ̸= 0)P(˜γ∗
+j ̸= 0)
++ P(Wi > 0|˜γ∗
+j = 0)P(˜γ∗
+j = 0)
+≥1
+2 × c − 2
+c − 1 + 0 =
+c − 2
+2(c − 1)
+(40)
+Hence the random variable Bj := 1{Wj>0} ∼ Bernoulli(ρj)
+for γj ̸= 0 with ρj ≥ (c − 2)/(2(c − 1)).
+Now we consider all the Wj of non-null variables, and
+assumes |W1| ≤ · · · ≤ |Wn| with the abuse of subscripts.
+We have
+γj ̸= 0 and − T ≤ Wj < 0
+⇐⇒
+j ≤ J and Bj = 0
+and
+γj ̸= 0 and 0 < Wj ≤ T
+⇐⇒
+j ≤ J and Bj = 1
+Hence
+# {j : γj ̸= 0 and − T ≤ Wj < 0}
+1 + # {j : γj ̸= 0 and 0 < Wj ≤ T}
+= (1 − B1) + · · · + (1 − BJ)
+1 + B1 + · · · + BJ
+=
+1 + J
+1 + B1 + · · · + BJ
+− 1.
+(41)
+If we can use Lemma 3, then
+E
+� # {j : γj ̸= 0 and − T ≤ Wj < 0}
+1 + # {j : γj ̸= 0 and 0 < Wj ≤ T}
+�
+≤ ρ−1−1 ≤
+c
+c − 2
+(42)
+Then we finally get
+FSR(T) ≤ q
+c
+c − 2.
+(43)
+as long as c > 2. Now it suffices to show that our
+random variables {Bj} are mutually independent. This is
+straightforward as we set P(α2; λ) as a sparse penalty for
+each row α2,j in Eq. (26), respectively. Then problem of
+Eq. (26) now is a combination of independent sub-problems
+for each row α2,j, and the solution only depends on
+(x2,j, y2,j, β(λ; D1)). Then with fixed β(λ; D1), the mutual
+independence naturally exist.
+Finally, after we control the FSR rate for the second subset,
+we can get the estimate of β(λ; D2) based on the identified
+clean data in the second subset, and return to run knockoff-
+SPR on the first subset in a similar pipeline. Then we have
+for the whole dataset:
+FSR = E
+�|S1 ∩ C1| + |S2 ∩ C2|
+|C1| + |C2|
+�
+≤ E
+�|S1 ∩ C1|
+|C1|
+�
++ E
+�|S2 ∩ C2|
+|C2|
+�
+≤ 2
+c
+c − 2q.
+(44)
+To control the FSR with q, the threshold of T should be
+defined as c−2
+2c q, which leads to Theorem 2.
+9
+NOISY SET RECOVERY THEOREM OF SPR
+Recall that we are solving the problem of
+min
+⃗γ
+���⃗y − ˚
+X⃗γ
+���
+2
+2 + λ ∥⃗γ∥1 .
+(45)
+Proposition 4. Assume that ˚
+X⊤ ˚
+X is invertible. If
+����λ ˚
+X⊤
+Sc ˚
+XS
+�
+˚
+X⊤
+S ˚
+XS
+�−1
+ˆvS + ˚
+X⊤
+Sc (I − IS) ( ˚
+Xε)
+����
+∞
+< λ
+(46)
+holds for all ˆvS ∈ [−1, 1]S, where IS = ˚
+XS
+�
+˚
+X⊤
+S ˚
+XS
+�−1 ˚
+X⊤
+S ,
+then the estimator ˆ⃗γ of Eq. (45) satisfies that
+ˆS = supp
+�ˆ⃗γ
+�
+⊆ supp (⃗γ∗) = S.
+Moreover, if the sign consistency
+sign
+�ˆ⃗γS
+�
+= sign (⃗γ∗
+S)
+(47)
+holds, Then ˆ⃗γ is the unique solution of (45) with the same sign as
+ˆ⃗γ∗.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+17
+Proof. Note that Eq. (45) is convex that has global minima.
+Denote Eq. (45) as L, the solution of ∂L/∂⃗γ = 0 is the
+unique minimizer. Hence we have
+∂L
+∂⃗γ = − ˚
+X⊤ �
+⃗y − ˚
+X⃗γ
+�
++ λv = 0
+(48)
+where v = ∂ ∥⃗γ∥1 /∂⃗γ. Note that ∥⃗γ∥1 is non-differentiable,
+so we instead compute its sub-gradient. Further note that
+vi = ∂ ∥⃗γ∥1 /∂⃗γi = ∂ |⃗γi| /∂γi. Hence vi = sign (⃗γi) if ⃗γi ̸=
+0 and vi ∈ [−1, 1] if ⃗γi = 0. To distinguish between the two
+cases, we assume vi ∈ (−1, 1) if ⃗γi = 0. Hence there exists
+ˆv ∈ Rn×1 such that
+− ˚
+X⊤ �
+⃗y − ˚
+X ˆ⃗γ
+�
++ λˆv = 0,
+(49)
+ˆvi = sign
+�ˆ⃗γi
+�
+if i ∈ ˆS and ˆvi ∈ (−1, 1) if i ∈ ˆSc.
+To obtain ˆS ⊆ S, we should have ˆ⃗γi = 0 for i ∈ Sc, that is,
+∀i ∈ Sc, |ˆvi| < 1, i.e.,
+��� ˚
+X⊤
+Sc
+�
+⃗y − ˚
+XS ˆ⃗γS
+����
+∞ < λ,
+(50)
+For i ∈ S, we have
+− ˚
+X⊤
+S
+�
+⃗y − ˚
+XS ˆ⃗γS
+�
++ λˆvS = 0.
+(51)
+If ˚
+X⊤ ˚
+X is invertible then
+ˆ⃗γS =
+�
+˚
+X⊤
+S ˚
+XS
+�−1 �
+˚
+X⊤
+S ⃗y − λˆvS
+�
+(52)
+Recall that we have
+⃗y = ˚
+XS⃗γ∗
+S + ˚
+X⃗ε
+(53)
+Hence
+ˆ⃗γS = ⃗γ∗
+S+δS,
+δS :=
+�
+˚
+X⊤
+S ˚
+XS
+�−1 �
+˚
+X⊤
+S ˚
+X⃗ε − λˆvS
+�
+. (54)
+Plugging (54) and (53) into (50) we have
+���� ˚
+X⊤
+Sc ˚
+X⃗ε − ˚
+X⊤
+Sc ˚
+XS
+�
+˚
+X⊤
+S ˚
+XS
+�−1 �
+˚
+X⊤
+S ˚
+X⃗ε − λˆvS
+�����
+∞
+< λ,
+(55)
+or equivalently
+����λ ˚
+X⊤
+Sc ˚
+XS
+�
+˚
+X⊤
+S ˚
+XS
+�−1
+ˆvS + ˚
+X⊤
+Sc (I − IS) ˚
+X⃗ε
+����
+∞
+< λ,
+(56)
+where IS
+=
+˚
+XS
+�
+˚
+X⊤
+S ˚
+XS
+�−1 ˚
+X⊤
+S . To ensure the sign
+consistency, replacing ˆvS
+= sign (⃗γ∗
+S) in the inequality
+above leads to the final result.
+Lemma 5. Assume that ⃗ε is indenpendent sub-Gaussian with
+zero mean and bounded variance Var (⃗εi) ≤ σ2.
+Then with probability at least
+1 − 2cn exp
+�
+�
+�−
+λ2η2
+2σ2 maxi∈Sc
+��� ˚
+Xi
+���
+2
+2
+�
+�
+�
+(57)
+there holds
+��� ˚
+X⊤
+Sc (I − IS)
+�
+˚
+X⃗ε
+����
+∞ ≤ λη
+(58)
+and����
+�
+˚
+X⊤
+S ˚
+XS
+�−1 ˚
+X⊤
+S ˚
+X⃗ε
+����
+∞
+≤
+λη
+√Cmin maxi∈Sc
+��� ˚
+Xi
+���
+2
+. (59)
+Proof. Let zc = ˚
+X⊤
+Sc (I − IS)
+�
+˚
+X⃗ε
+�
+, for each i ∈ Sc the
+variance can be bounded by
+Var (zc
+i ) ≤ σ2 ˚
+X⊤
+i (I − IS)2 ˚
+Xi ≤ σ2 max
+i∈Sc
+��� ˚
+Xi
+���
+2
+2 .
+Hoeffding inequality implies that
+P
+���� ˚
+X⊤
+Sc (I − IS)
+�
+˚
+X⃗ε
+����
+∞ ≥ t
+�
+≤ 2 |Sc| exp
+�
+�
+�−
+t2
+2σ2 maxi∈Sc
+��� ˚
+Xi
+���
+2
+2
+�
+�
+� ,
+Setting t = λη leads to the result.
+Now let z =
+�
+˚
+X⊤
+S ˚
+XS
+�−1 ˚
+X⊤
+S ˚
+X⃗ε, we have
+Var (z) =
+�
+˚
+X⊤
+S ˚
+XS
+�−1 ˚
+X⊤
+S ˚
+XVar (⃗ε) ˚
+X⊤ ˚
+XS
+�
+˚
+X⊤
+S ˚
+XS
+�−1
+≤ σ2 �
+˚
+X⊤
+S ˚
+XS
+�−1
+≤
+σ2
+Cmin
+I.
+Then
+P
+�����
+�
+˚
+X⊤
+S ˚
+XS
+�−1 ˚
+X⊤
+S ˚
+X⃗ε
+����
+∞
+≥ t
+�
+≤ 2 |S| exp
+�
+−t2Cmin
+2σ2
+�
+.
+Choose
+t =
+λη
+√Cmin maxi∈Sc
+��� ˚
+Xi
+���
+2
+,
+(60)
+then there holds
+P
+�
+�
+�∥
+�
+˚
+X⊤
+S ˚
+XS
+�−1 ˚
+X⊤
+S ˚
+X⃗ε∥∞ ≥
+λη
+√Cmin maxi∈Sc
+��� ˚
+Xi
+���
+2
+�
+�
+�
+≤ 2 |S| exp
+�
+�
+�−
+λ2η2
+2σ2 maxi∈Sc
+��� ˚
+Xi
+���
+2
+2
+�
+�
+� .
+9.1
+Proof of Theorem 1
+Proof. The proof essentially follows the treatment in [4].
+The results follow by applying Lemma 5 to Proposition 4.
+Inequality (46) holds if condition C2 and the first bound (58)
+hold, which proves the first part of the theorem. The
+sign consistency (47) holds if condition C3 and the second
+bound (59) hold, which gives the second part of the theorem.
+It suffices to show that ˆS ⊆ S implies ˆCc ⊆ Cc. Consider
+one instance i, there are three possible cases for γ∗
+i ∈ R1×c:
+(1) γ∗
+i,j ̸= 0, ∀j ∈ [c]; (2) γ∗
+i,j = 0, ∀j ∈ [c]; (3) ∃j, k ∈
+[c] , s.t. γ∗
+i,j = 0, γ∗
+i,k ̸= 0. If instance i follows case (1) or
+case (3), then i ∈ Cc. If it follows case (2), then i ∈ C, and
+the indexes of all elements of γi are in Sc. Since we have
+ˆS ⊆ S, all elements of γi is in ˆSc, hence i ∈ ˆC. Then we
+have ˆCc ⊆ Cc.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+18
+10
+MORE EXPERIMENTAL RESULTS
+Histogram of the median value of IRR condition of
+SPR. We visualize the median value of the irrepresentable
+(IRR) value, i.e., {∥(X⊤
+S XS)−1X⊤
+S Xj∥1}j of SPR final epoch
+on CIFAR10 with various noisy scenarios in Fig. 5. As
+SPR is running on each piece split from the training set,
+we calculate matrix ˚
+X⊤
+Sc ˚
+XS( ˚
+X⊤
+S ˚
+XS)−1 in irrepresentable
+condition (C2 in Theorem 1) for each piece at the final
+epoch. Then the L1 norm of each row of the matrix is the
+IRR value of corresponding clean data. The median value
+of IRR values in a single piece is used to construct the
+histogram. For the noise scenario of Asy. 40% and Sym. 40%,
+the median IRR value is small, indicating weak collinearity
+between clean data and noisy data. In these cases, SPR
+has more chance to distinguish noisy data from clean data
+and thus leads to a good FSR control capacity. For the
+noise scenario of Sym. 80%, the median IRR values are
+much larger, indicating a strong multi-collinearity. Thus SPR
+can hardly distinguish between clean data and noisy data,
+leading to a high FSR rate.
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+Symmetric-40%
+4
+6
+8
+10
+12
+14
+16
+0
+5
+10
+15
+20
+25
+Symmetric-80%
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+0
+10
+20
+30
+40
+50
+60
+Asymmetric-40%
+Fig. 5. Histogram of the median value of the IRR value of SPR on
+CIFAR10 with various noisy scenarios.
+REFERENCES
+[1] Rina Foygel Barber and Emmanuel J. Cand‘es. A knockoff filter
+for high-dimensional selective inference. The Annals of Statistics,
+47(5):2504 – 2537, 20 (document), 3.1, 5.3, 8, 8
+[2] Yang Cao, Xinwei Sun, and Yuan Yao. Controlling the false
+discovery rate in transformational sparsity: Split knockoffs. In
+arXiv, 2021. (document), 3.1, 5.3, 8, 8, 8
+[3] Joseph L Doob. Stochastic processes. Wiley New York, 195 8
+[4] M.
+J.
+Wainwright,
+“Sharp
+thresholds
+for
+high-dimensional
+and noisy sparsity recovery using l1 -constrained quadratic
+programming (lasso),” IEEE transactions on information theory,
+2009. (document), 2.3, 9.1
+
diff --git a/-tAyT4oBgHgl3EQfqfjJ/content/tmp_files/load_file.txt b/-tAyT4oBgHgl3EQfqfjJ/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..ed3edcec2ba65330f954084d9c4e26293783f822
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+++ b/-tAyT4oBgHgl3EQfqfjJ/content/tmp_files/load_file.txt
@@ -0,0 +1,2141 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf,len=2140
+page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 8, AUGUST 2015  1  Knockoffs-SPR: Clean Sample Selection in  Learning with Noisy Labels  Yikai Wang, Yanwei Fu, and Xinwei Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Abstract—A noisy training set usually leads to the degradation of the generalization and robustness of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In this paper,  we propose a novel theoretically guaranteed clean sample selection framework for learning with noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Specifically, we first  present a Scalable Penalized Regression (SPR) method, to model the linear relation between network features and one-hot labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In  SPR, the clean data are identified by the zero mean-shift parameters solved in the regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We theoretically show that SPR  can recover clean data under some conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Under general scenarios, the conditions may be no longer satisfied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' and some noisy  data are falsely selected as clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' To solve this problem, we propose a data-adaptive method for Scalable Penalized Regression  with Knockoff filters (Knockoffs-SPR), which is provable to control the False-Selection-Rate (FSR) in the selected clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' To  improve the efficiency, we further present a split algorithm that divides the whole training set into small pieces that can be solved in  parallel to make the framework scalable to large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' While Knockoffs-SPR can be regarded as a sample selection module for a  standard supervised training pipeline, we further combine it with a semi-supervised algorithm to exploit the support of noisy data as  unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Experimental results on several benchmark datasets and real-world noisy datasets show the effectiveness of our  framework and validate the theoretical results of Knockoffs-SPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Our code and pre-trained models will be released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Index Terms—Learning with Noisy Labels, Knockoffs Method, Type-Two Error Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  1  INTRODUCTION  D  EEP learning has achieved remarkable success on  many supervised learning tasks trained by millions  of labeled training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The performance of deep models  heavily relies on the quality of label annotation since neural  networks are susceptible to noisy labels and even can easily  memorize randomly labeled annotations [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Such noisy  labels can lead to the degradation of the generalization  and robustness of such models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Critically, it is expensive  and difficult to obtain precise labels in many real-world  scenarios, thus exposing a realistic challenge for supervised  deep models to learn with noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  There are many previous efforts in tackling this challenge by  making the models robust to noisy data, such as modifying  the network architectures [2]–[5] or loss functions [6]–[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  This paper addresses the challenge by directly selecting  clean samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Inspired by the dynamic sample selection  methods [9]–[16], we construct a “virtuous” cycle between  sample selection and network training: the selected clean  samples can improve the network training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' and on the  other hand, the improved network has a more powerful  ability in picking up clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' As this cycle evolves, the  performance can be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' To well establish this cycle, a  key question remains: how to effectively differentiate clean data  from noisy ones?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Preliminary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Typical principles in existing works [9]–[16]  to differentiate clean data from noisy data include large  loss [11], inconsistent prediction [17], and irregular feature Yikai Wang and Yanwei Fu contribute equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Xinwei Sun is the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Yikai Wang, Yanwei Fu and Xinwei Sun are with the School of  Data Science, Fudan University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' E-mail: {yikaiwang19, yanweifu,  sunxinwei}@fudan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='cn  representation [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The former two principles identify  irregular behaviors in the label space, while the last one  analyzes the instance representations of the same class in  the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In this paper, we propose unifying the  label and feature space by making the linear relationship,  yi = x⊤  i β + ε,  (1)  between feature-label pair (xi ∈ Rp: feature vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' yi ∈ Rc:  one-hot label vector) of data i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We also have the fixed  (unknown) coefficient matrix β ∈ Rp×c, and random noise  ε ∈ Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Essentially, the linear relationship here is an ideal  approximation, as the networks are trained to minimize  the divergence between a (soft-max) linear projection of  the feature and a one-hot label vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For a well-trained  network, the output prediction of clean data is expected  to be as similar to a one-hot vector as possible, while the  entropy of the output of noisy data should be large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Thus if  the underlying linear relation is well-approximated without  soft-max operation, the corresponding data is likely to be  clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In contrast, the feature-label pair of noisy data may  not be approximated well by the linear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  The simplest way to measure the goodness of the linear  model in fitting the feature-label pair is to check the  prediction error, or residual, ri = yi − x⊤  i ˆβ, where ˆβ is the  estimate of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The larger ∥r∥ indicates a larger fitting error  and thus more possibility for instance i to be outlier/noisy  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Many methods have been proposed to test whether ri  is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Particularly, we highlight the classical statistical  leave-one-out approach [19] that computes the studentized  residual as,  ti =  yi − x⊤  i ˆβ−i  ˆσ−i  �  1 + x⊤  i  �X⊤  −iX−i  �−1 xi  �1/2 ,  (2)  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='00545v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='LG]  2 Jan 2023  JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 8, AUGUST 2015  2  Images  Features  Noisy Labels  Stage 1:  Network Learning  Classifier  Stage 2:  Sample Selection  β ∈ Rd×c  Noisy Labels  Y ∈ Rn×c  X ∈ Rn×d  Features  Noisy Data Indicator  =  +  γ ∈ Rn×c  Yi (1, 0, 0)  (0, 1, 0) ˜Yi  Permute  Compare  Selected Clean Data  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Knockoffs-SPR runs a cycle between network learning and sample selection, where clean data are selected via the comparison of the  mean-shift parameters between its original label and permuted label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  where ˆσ is the scale estimate and the subscript −i indicates  estimates based on the n − 1 observations, leaving out  the i-th data for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Equivalently, the linear regression  model can be re-formulated into explicitly representing the  residual,  Y = Xβ + γ + ε,  εi,j ∼ N(0, σ2),  (3)  by introducing a mean-shift parameter γ as in [20] with the  feature X ∈ Rn×p, and label Y ∈ Rn×c paired and stacked  by rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For each row of γ ∈ Rn×c, γi represents the predict  residual of the corresponding data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' This formulation has  been widely studied in different research topics, including  economics [21]–[24], robust regression [20], [25], statistical  ranking [26], face recognition [27], semi-supervised few-shot  learning [28], [29], and Bayesian preference learning [30],  to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' This formulation is differently focused on  the specific research tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For example, for the robust  regression problem [20], [25], the target is to get a robust  estimate ˆβ against the influence of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Here for solving the  problem of learning with noisy labels, we are interested  in recovering zeros elements of γ, since these elements  correspond to clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  SPR [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' To this end, from the statistical perspective,  our conference report [31] starts from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (3) to build up  a sample selection framework, dubbed Scalable Penalized  Regression (SPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' With a sparse penalty P(γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' λ) on γ, the  SPR obtains a regularization solution path of γ(λ) by  evolving λ from ∞ to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Then it identifies those samples  that are earlier (or at larger λ) selected to be non-zeros  as noisy data and those later selected as clean data, with  a manually specified ratio of selected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Under the  irrepresentable condition [4], [33], the SPR enjoys model  selection consistency in the sense that it can recover the set  of noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' By feeding only clean data into next-round  training, the trained network is less corrupted by the noisy  data and hence performs well empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Knockoffs-SPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' However, the irrepresentable condition  demands the prior of the ground-truth noisy set, which  is not accessible in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' When this condition fails,  the trained network with SPR may be still corrupted by  a large proportion of noisy data, leading to performance  degradation as empirically verified in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' To  amend this problem, we provide a data-adaptive sample  selection algorithm, in order to well control the expected  rate of noisy data in the selected data under the desired level  q, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=', q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' As the goal is to identify clean data for the  next-round training, we term this rate as the False-Selection-  Rate (FSR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The FSR is the expected rate of the type-II error  in sparse regression, as non-zero elements correspond to  the noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Our method to achieve the FSR control is  inspired by the ideas of Knockoffs in Statistics, which is  a recently developed framework for variable selection [1],  [2], [34], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The Knockoffs framework aims at selecting  non-null variables and controlling the False-Discovery-Rate  (FDR), by taking as negative controls knockoff features ˜  X,  which are constructed as a fake copy for the original features  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Here, the FDR corresponds to the expectation of the  type-I error rate in sparse regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Therefore, the vanilla  Knockoffs cannot be directly applied to our SPR framework,  since FSR is the expected rate of the type-II error and there  is no theoretical guarantee in Knockoffs for this control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' To  achieve the FSR control, we propose Knockoffs-SPR, which  turns to construct the knockoff labels ˜Y via permutation for  the original label Y , and incorporates it into a data-partition  strategy for FSR control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Formally, we repurpose the knockoffs in Statistics in our  SPR method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' and propose a novel data-adaptive sample  selection algorithm, dubbed Knockoffs-SPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' It extends SPR  in controlling the ratio of noisy data among the selected  clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' With this new property, Knockoffs-SPR ensures  that the clean pattern is dominant in the data and hence  leads to better network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Specifically, we partition  the whole noisy training set into two random subsets and  apply the Knockoffs-SPR to two subsets separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For each  time, we use one subset to estimate the intercept β and the  other to select the clean data by comparing between the  solution paths of γ(λ) and ˜γ(λ) that respectively obtained  via regression on noisy labels and the permuted labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' With  such a decoupled structure between β and γ, we prove  that the FSR can be controlled by any prescribed level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Compared with the original theory of SPR, our new theory  enables us to effectively select clean data under general  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Besides, Knockoffs-SPR also enjoys a superior  performance over the original SPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Together with network training, the whole framework is  illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 1 in which the sample selection and  the network learning are well incorporated into each  other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Specifically, we run the network learning process  and sample selection process iteratively and repeat this  cycle until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' To incorporate Knockoffs-SPR into  the end-to-end training pipeline of deep architecture, the  simplest way is to directly solve Knockoffs-SPR for each  featureA  B  C  DJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 8, AUGUST 2015  3  training mini-batch or training epoch to select clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Solving Knockoffs-SPR for each mini-batch is efficient but  suffers from the identifiability issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The sample size in a  mini-batch may be too small to distinguish clean patterns  from noisy ones among all classes, especially for large  datasets with small batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Solving Knockoffs-SPR for  the whole training set is powerful but suffers from the  complexity issue, leading to an unacceptable computation  cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' To resolve these two problems, we strike a balance  between complexity and identifiability by proposing a  splitting strategy that divides the whole data into small  pieces such that each piece is class-balanced with the proper  sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In this regard, the sample size of each piece is  small enough to be solved efficiently and large enough to  distinguish clean patterns from noisy ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Then Knockoffs-  SPR runs on each piece in parallel, making it scalable to  large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  As the removed noisy data still contain useful information  for network training, we adopt the semi-supervised training  pipeline with CutMix [38] where the noisy data are utilized  as unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We conduct extensive experiments to  validate the effectiveness of our framework on several  benchmark datasets and real-world noisy datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The  results show the efficacy of our Knockoffs-SPR algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Our contributions are as follows:  Ideologically, we propose to control the False-Selection-  Rate in selecting clean data, under general scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Methodologically, we propose Knockoffs-SPR, a data-  adaptive method to control the FSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Theoretically, we prove that the Knockoffs-SPR can  control the FSR under any desired level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Algorithmically, we propose a splitting algorithm for  better sample selection with balanced identifiability and  complexity in large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Experimentally, we demonstrate the effectiveness and  efficiency of our method on several benchmark datasets  and real-world noisy datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Our conference version of this work, SPR, was  published in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Compared with SPR [31], we have the  following extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  We identify the limitation of the SPR and consider the FSR  control in selecting clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  We propose a new framework: Knockoffs-SPR which is  effective in selecting clean data under general scenarios,  theoretically and empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  We apply our method on Clothing1M and achieve better  results than compared baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Logistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The rest of this paper is organized as follows:  In Section 9, we introduce our SPR algorithm with its  noisy set recovery theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  In Section 3, the Knockoffs-SPR algorithm is introduced  with its FSR control theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  In Section 4, several training strategies are proposed to  well incorporate the Knockoffs-SPR with the network  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  In Section 5, connections are made between our proposed  works and several previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  In Section 6, we conduct experiments on several synthetic  and real-world noisy datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Section 7 concludes this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  2  CLEAN SAMPLE SELECTION  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1  Problem Setup  We are given a dataset of image-label pairs {(imgi, yi)}n  i=1,  where the noisy label yi is corrupted from the ground-  truth label y∗  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The ground-truth label y∗  i and the corruption  process are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Our target is to learn a recognition  model f(·) such that it can recognize the true category y∗  i  from the image imgi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=', f(imgi) = y∗  i , after training on the  noisy label yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  In this paper, we adopt deep neural networks as the  recognition model and divide the f(·) into fc(g(·)) where  g(·) is the deep model for feature extraction and fc(·) is  the final fully-connected layer for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For each  input image imgi, the feature extractor g(·) is used to encode  the feature xi := g(imgi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Then the fully-connected layer is  used to output the score vector ˆyi = fc(xi) which indicates  the chance it belongs to each class and the prediction is  provided with ˆyi = argmax(ˆyi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  As the training data contain many noisy labels, simply  training from all the data leads to severe degradation  of generalization and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Intuitively, if we could  identify the clean labels from the noisy training set, and  train the network with the clean data, we can reduce the  influence of noisy labels and achieve better performance and  robustness of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' To achieve this, we thus propose a  sample selection algorithm to identify the clean data in the  noisy training set with theoretical guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In this paper, we will use a to represent scalar, a  to represent a vector, and A to represent a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We will  annotate a∗ to denote the ground-truth value of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We use  ∥ · ∥F to denote the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2  Clean Sample Selection via Penalized Regression  Motivated  by  the  leave-one-out  approach  for  outlier  detection, we introduce an explicit noisy data indicator γi  for each data and assume a linear relation between extracted  feature xi and one-hot label yi with noisy data indicator as,  yi = x⊤  i β + γi + εi,  (4)  where yi ∈ Rc is one-hot vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' and xi ∈ Rp, β ∈  Rp×c, γi ∈ Rc, εi ∈ Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The noisy data indicator γi can be  regarded as the correction of the linear prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For clean  data, yi ∼ N(x⊤  i β∗, σ2Ic) with γ∗  i = 0, and for noisy data  y∗  i = yi −γ∗  i ∼ N(x⊤  i β∗, σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We denote C := {i : γ∗  i = 0}  as the ground-truth clean set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  To select clean data for training, we propose Scalable  Penalized Regression (SPR), designed as the following sparse  learning paradigm,  argmin  β,γ  1  2 ∥Y − Xβ − γ∥2  F + P(γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' λ),  (5)  where we have the matrix formulation X ∈ Rn×p, and  Y  ∈  Rn×c of {xi, yi}n  i=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' and P(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' λ) is a row-wise  JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 8, AUGUST 2015  4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Solution Path of SPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Red lines indicate noisy data while blue  lines indicate clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' As λ decreases, the γi gradually solved with  non-zero values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  sparse penalty with coefficient parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' So we have  P(γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' λ) = �n  j=1 P(γi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' λ), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=', group-lasso sparsity with  P(γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' λ) = λ �  i ∥γi∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  To estimate C, we only need to solve γ with no need to  estimate β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Thus to simplify the optimization, we substitute  the Ordinary Least Squares (OLS) estimate for β with γ  fixed into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' To ensure that ˆβ is identifiable, we apply  PCA on X to make p ≪ n so that the X has full-column  rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Denote  ˜  X = I − X  �X⊤X  �† X⊤, ˜Y  =  ˜  XY , the  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (5) is transformed into  argmin  γ  1  2  ��� ˜Y − ˜  Xγ  ���  2  F + P(γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' λ),  (6)  which is a standard sparse linear regression for γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Note that  in practice we can hardly choose a proper λ that works well  in all scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Furthermore, from the equivalence between  the penalized regression problem and Huber’s M-estimate,  the solution of γ is returned with soft-thresholding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Thus  it is not worth finding the precise solution of a single γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Instead, we use a block-wise descent algorithm [39] to solve  γ with a list of λs and generate the solution path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' As  λ changes from ∞ to 0, the influence of sparse penalty  decreases, and γi are gradually solved with non-zero values,  in other words, selected by the model, as visualized in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Since earlier selected instance is more possible to be  noisy, we rank all samples in the descendent order of their  selecting time defined as:  Zi = sup {λ : γi (λ) ̸= 0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (7)  A large Zi means that the γi is earlier selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Then the top  samples are identified as noisy data and the other samples  are selected as clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In practice, we select 50% of the  data as clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3  The Theory of Noisy Set Recovery in SPR  The SPR enjoys theoretical guarantees that the noisy data  set can be fully recovered with high probability, under  the irrepresentable condition [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Formally, consider the  vectorized version of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (6):  argmin  ⃗γ  1  2  ���⃗y − ˚  X⃗γ  ���  2  2 + λ ∥⃗γ∥1 ,  (8)  where ⃗y, ⃗γ is vectorized from Y , γ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' ˚  X = Ic ⊗ ˜  X  with ⊗ denoting the Kronecker product operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Denote  S := supp(⃗γ∗), which is the noisy set Cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We further denote  ˚  XS (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' ˚  XSc) as the column vectors of ˚  X whose indexes  are in S (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Sc) and µ ˚  X = maxi∈Sc ∥ ˚  X∥2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Then we have  Theorem 1 (Noisy set recovery).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Assume that:  C1, Restricted eigenvalue: λmin( ˚  X⊤  S ˚  XS) = Cmin > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  C2, Irrepresentability: there exists a η ∈ (0, 1], such that  ∥ ˚  X⊤  Sc ˚  XS( ˚  X⊤  S ˚  XS)−1∥∞ ≤ 1 − η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  C3, Large error:  ⃗γ∗  min := mini∈S |⃗γ∗  i | > h(λ, η, ˚  X, ⃗γ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  where ∥A∥∞  :=  maxi  �  j |Ai,j|, and h(λ, η, ˚  X, ⃗γ∗)  =  λη/�Cminµ ˚  X + λ∥( ˚  X⊤  S ˚  XS)−1sign(⃗γ∗  S)∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Let λ ≥  2σ√µ ˚  X  η  √log cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Then with probability greater than  1 − 2(cn)−1, model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (8) has a unique solution ˆ⃗γ such that: 1)  If C1 and C2 hold, ˆ  Cc ⊆ Cc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2) If C1, C2 and C3 hold, ˆ  Cc = Cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  We present the proof in the appendix, following the  treatment in [4], [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In this theorem, C1 is necessary to get  a unique solution, and in our case is mostly satisfied with  the natural assumption that the clean data is the majority  in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' If C2 holds, the estimated noisy data  is the subset of truly noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' This condition is the key  to ensuring the success of SPR, which requires divergence  between clean and noisy data such that we cannot represent  clean data with noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' If C3 further holds, the estimated  noisy data is exactly all the truly noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' C3 requires the  error measured by γi is large enough to be identified from  random noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' If the conditions fail, SPR will fail in a non-  vanishing probability, not deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  3  CONTROLLED CLEAN SAMPLE SELECTION  In the last section, we stop the solution path at λ such  that 50% samples are selected as clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' If this happens  to be the rate of clean data, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 1 shows that our SPR  can identify the clean data C under the irrepresentable  condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' However, the irrepresentable condition and the  information of the ground-truth clean set C are practically  unknown, making this theory hard to be used in the real  life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Particularly, with |Cc| unknown, the algorithm can stop  at an improper time such that the noisy rate of the selected  clean data ˆC can be still high, making the next-round trained  model corrupted a lot by noisy patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  To resolve the problem of false selection in SPR , we in this  section propose a data-adaptive early stopping method for  the solution path, that targets controlling the expected noisy  rate of the selected data dubbed as False-Selection-Rate (FSR)  under the desired level q (0 < q < 1):  FSR = E  �  �#  �  j : j ̸∈ H0 ∩ ˆC  �  #  �  j : j ∈ ˆC  �  ∨ 1  �  � ,  (9)  where ˆC = {j : ˆγj = 0} is the recovered clean set of  γ, and H0 : γ∗  i = 0 denotes the null hypothesis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=', the  sample i belonging to the clean dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Therefore, the FSR  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (9) targets controlling the false rate among selected  null hypotheses, which is also called the expected rate of  the type-II error in hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 8, AUGUST 2015  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1  Knockoffs-SPR  To achieve the FSR control, we propose the Knockoffs-  SPR for clean sample selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Our method is inspired  by knockoff methods [1], [2], [34], [35], [41] with the  different focus that we target selecting clean labels via  permutation instead of constructing knockoff features to  select explanatory variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Specifically, under model (4)  we permute the label for each data and construct the  permutation ˜y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Then model (4) can be solved for y and  ˜y to obtain the solution paths γ(λ) and ˜γ(λ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  We will show that this construction can pick up clean data  from noisy ones, by comparing the selecting time (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (7))  between γ(λ) and ˜γ(λ) for each data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' On the basis of this  construction, we propose to partition the whole dataset  into two disjoint parts, with one part for estimating β  and the other for learning γ(λ) and ˜γ(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We will show  that the independent structure with such a data partition  enables us to construct the comparison statistics whose sign  patterns among alternative hypotheses (noisy data) are the  independent Bernoulli processes, which is crucial for FSR  control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Formally speaking, we split the whole data D into D1 :=  (X1, Y1) and D2  :=  (X2, Y2) with ni  :=  |Di|, and  implement Knockoffs-SPR on both D1 and D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In the  following, we only introduce the procedure on D2, as the  procedure for D1 shares the same spirit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Roughly speaking,  the procedure is composed of three steps: i) estimate β on  D1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' ii) estimate ˜γ(λ)) on D2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' and iii) construct the comparison  statistics and selection filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We leave detailed discussions for  each step in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Step i): Estimating β on D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Our target is to provide  an estimate of β that is independent of D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The simplest  strategy is to use the standard OLS estimator to obtain  ˆβ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' However, this estimator may not be accurate since it  is corrupted by noisy samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For this consideration, we  first run SPR on D1 to get clean data and then solve β via  OLS on the estimated clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Step ii): Estimating (γ(λ), ˜γ(λ)) on D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' After obtaining the  solution ˆβ1 on D1 , we learn the γ(λ) on D2:  1  2  ���Y2 − X2 ˆβ1 − γ2  ���  2  F + P(γ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (10)  For each one-hot encoded vector y2,j, we randomly permute  the position of 1 and obtain another one-hot vector ˜y2,j ̸=  y2,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For clean data j, the ˜y2,j turns to be a noisy label;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  while for noisy data, the ˜y2,j is switched to another noisy  label with probability c−2  c−1 or clean label with probability  1  c−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' After obtaining the permuted matrix as ˜Y2, we learn  the solution paths (γ2(λ), ˜γ2(λ)) using the same algorithm  as SPR via:  �  �  �  �  �  1  2  ���Y2 − X2 ˜β1 − γ2  ���  2  F + �  j P(γ2,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' λ),  1  2  ��� ˜Y2 − X2 ˜β1 − ˜γ2  ���  2  F + �  j P(˜γ2,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (11)  Step iii): Comparison statistics and selection filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  After obtaining the solution path (γ2(λ), ˜γ2(λ)), we define  sample significance scores with respect to y2,i and ˜y2,i of  each i respectively, as the selection time: Zi := sup{λ :  Algorithm 1 Knockoffs-SPR  Input: subsets D1 and D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Output: clean set of D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  1: Use D1 to fit an linear regression model and get β(D1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  2: Generate permuted label of each sample i in D2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  3: Solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (26) for D2 and generate {Wi} by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  4: Initialize q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='02 and T = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  5: while q < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5 and T = 0 do  6:  Compute T by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (13);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  7:  q = q + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='02;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  8: end while  9: if T is 0 then  10:  Construct clean set via half of the samples with largest  Wi in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (14) with T = ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  11: else  12:  Construct clean set via samples in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (14);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  13: end if  14: return clean set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  ∥γ2,i(λ)∥2 ̸= 0} and ˜Zi := sup{λ : ∥˜γ2,i(λ)∥2 ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' With  Zi, ˜Zi, we define the Wi as:  Wi := Zi · sign(Zi − ˜Zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (12)  Based on these statistics, we define a data-dependent  threshold T as  T = max  �  t > 0 : 1 + # {j : 0 < Wj ≤ t}  # {j : −t ≤ Wj < 0} ∨ 1 ≤ q  �  ,  (13)  or T = 0 if this set is empty, where q is the pre-defined upper  bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Our algorithm will select the clean subset identified  by  C2 := {j : −T ≤ Wj < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (14)  Empirically, T may be equal to 0 if the threshold q is  sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In this regard, no clean data are selected,  which is meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Therefore, we start with a small q and  iteratively increase q and calculate T, until an attainable T  such that T > 0 to bound the FSR as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In  practice, when the FSR cannot be bounded by q = 50%,  we will end the selection and simply select half of the most  possible clean examples via {Wj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The whole procedure of  Knockoffs-SPR is shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2  Statistical Analysis about Knockoffs-SPR  In this part, we present the motivations and intuitions of  each step in Knockoffs-SPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Data Partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Knockoffs-SPR partitions the dataset D  into two subset D1 and D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' This step decomposes the  dependency of the estimate of β and γ in that we use D1/D2  to estimate β/γ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Then ˆβ(D1) is independent of  ˆγ(D2) if D1 and D2 are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The independent estimation  of β and γ makes it provable for FSR control on D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' As we discussed in step ii, when the original  label is clean, its permuted label will be a noisy label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' On  the other hand, if the original label is noisy, its permuted  label changes to clean with probability  1  c−1 and noisy with  JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 8, AUGUST 2015  6  probability  c−2  c−1, where c denotes the number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Note that γ of noisy data is often selected earlier than that  of clean data in the solution path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' This implies larger Z  values for noisy data than those for clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' As a result,  according to the definition of W, a clean sample will ideally  have a small negative of W := Z · sign(Z − ˜Z), where Z  and ˜Z respectively correspond to the clean label and noisy  label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In contrast for a noisy sample, the W tends to have  a large magnitude and has approximately equal probability  to be positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Such a different behavior of W  between clean and noisy data can help us to identify clean  samples from noisy ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Asymmetric comparison statistics W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The classical way to  define comparison statistics is in a symmetric manner, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=',  Wi := Zi ∨ ˜Zi · sign(Zi − ˜Zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In this way, a clean sample  with a noisy permuted label tends to have a large |Wi|, as  we expect the noisy label to have a large ˜Zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' However, this is  against our target as we only require clean samples to have  a small magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For this purpose, we design asymmetric  comparison statistics that only consider the magnitude of  the original labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  To see the asymmetric behavior of W  for noisy and  clean data, we consider the Karush–Kuhn–Tucker (KKT)  conditions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (26) with respect to (γ2,i, ˜γ2,i)  γ2,i + ∂P(γ2,i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' λ)  ∂γ2,i  = x⊤  2,i(β∗ − ˆβ1) + γ∗  2,i + ε(2),i,  (15a)  ˜γ2,i + ∂P(˜γ2,i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' λ)  ∂˜γ2,i  = x⊤  2,i(β∗ − ˆβ1) + ˜γ∗  2,i + ˜ε(2),i,  (15b)  where ε(2),i ∼i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='d ˜ε(2),i, |γ∗  2,i| = |˜γ∗  2,i| if both y2,i and  ˜y2,i are noisy, and P(γ2,i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' λ) := λ|γ2,i| as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' By  conditioning on ˆβ1 and denoting ai := x⊤  2,i(β∗ − ˆβ1), we  have that  P(Wi > 0) = P(|ai+γ∗  2,i+ε2,i| > |ai+ ˜γ∗  2,i+ ˜ε(2),i|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (16)  Then it can be seen that if i is clean, we have γ∗  2,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Then Zi tends to be small and besides, it is probable to have  Zi < ˜Zi if ˆβ1 can estimate β∗ well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' As a result, Wi tends  to be a small negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' On the other hand, if i is noisy, then  Zi tends to be large for γi to account for the noisy pattern,  and besides, it has equal probability between Zi < ˜Zi and  Zi ≥ ˜Zi when ˜y2,i is switched to another noisy label, with  probability  c−2  c−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' So Wi tends to have a large value and  besides,  P(Wi > 0) = P(Wi > 0|˜y2,i is noisy)P(˜y2,i is noisy)  + P(Wi > 0|˜y2,i is clean)P(˜y2,i is clean) = 1  2 · c − 2  c − 1  + P(Wi > 0|˜y2,i is clean) ·  1  c − 1,  (17)  which falls in the interval of  �  c−2  c−1 · 1  2,  c  c−1 · 1  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' That is to say,  P(Wi > 0) ≈ 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In this regard, the clean data corresponds  to small negatives of W in the ideal case, which can help  to discriminate noisy data with large W with almost equal  probability to be positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For noisy y2,i, we have P(Wi > 0|˜y2,i is noisy) =  1/2 by assuming |γ∗  2,i| = |˜γ∗  2,i|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' However, it may not hold  in practice when y2,i corresponds to the noisy pattern that  has been learned by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In this regard, it may  have |γ∗  2,i| < |˜γ∗  2,i| for a randomly permuted label ˜y2,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  To resolve this problem, we instead set the permutation  label as the most confident candidate of the model, please  refer to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Besides, if ˆβ1 can accurately  estimate β∗, according to KKT conditions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (15), we  have P(Wi > 0) < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' That is Wi tends to be negative for  the clean data, which is beneficial for clean sample selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Data-adaptive  threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  The  proposed  data-adaptive  threshold T  is directly designed to control the FSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Specifically, the FSR defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (9) is equivalent to  FSR(t) = E  �# {j : γj ̸= 0 and − t ≤ Wj < 0}  # {j : −t ≤ Wj < 0} ∨ 1  �  ,  (18)  where the denominator denotes the number of selected  clean data according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (14) and the nominator denotes  the number of falsely selected noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' This form of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (18) can be further decomposed into,  E  � # {γj ̸= 0, −t ≤ Wj < 0}  1 + # {γj ̸= 0, 0 < Wj ≤ t} · 1 + # {γj ̸= 0, 0 < Wj ≤ t}  # {−t ≤ Wj < 0} ∨ 1  �  ≤ E  � # {γj ̸= 0, −t ≤ Wj < 0}  1 + # {γj ̸= 0, 0 < Wj ≤ t}  1 + # {0 < Wj ≤ t}  # {−t ≤ Wj < 0} ∨ 1  �  ≤ E  � # {γj ̸= 0, −t ≤ Wj < 0}  1 + # {γj ̸= 0, 0 < Wj ≤ t}q  �  ,  (19)  where the last inequality comes from the definition of  T in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' To control the FSR, it suffices to bound  E  �  #{γj̸=0, −t≤Wj<0}  1+#{γj̸=0, 0<Wj≤t}  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Roughly speaking, this term means  the number of negative W to the number of positive  W, among noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Since W  for noisy data has  approximately equal probability to be positive/negative  as mentioned earlier, intuitively we have this term ≈  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Formally, we construct a martingale process of 1(Wi > 0)  among noisy data, which is independent of the magnitude  |W| due to data partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We leave these details in the  appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3  FSR Control of Knockoffs-SPR  Our target is to show that FSR ≤ q with our data-adaptive  threshold T in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Our main result is as follows:  Theorem 2 (FSR control).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For c-class classification task, and for  all 0 < q ≤ 1, the solution of Knockoffs-SPR holds  FSR(T) ≤ q  (20)  with the threshold T for two subsets defined respectively as  Ti = max  �  t ∈ W : 1 + # {j : 0 < Wj ≤ t}  # {j : −t ≤ Wj < 0} ∨ 1 ≤ c − 2  2c q  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  We present the proof in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The coefficient  1/2 comes from the subset-partition strategy that we run  Knockoffs-SPR on two D1 and D2, and the term  c−2  c  comes from the upper-bound of the first part in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  This theorem tells us that FSR can be controlled by the  given threshold q using the procedure of Knockoffs-SPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Compared to SPR, this procedure is more practical and  useful in real-world experiments and we demonstrate its  utility in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 8, AUGUST 2015  7  Algorithm 2 Knockoffs-SPR on full training set  Input: Noisy feature-label pairs {(xi, yi)}n  i=1, group class  size N sample size m, (Optional) clean set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Output: clean set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  1: if Number of classes > N then  2:  Compute class prototypes using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (22);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  3:  Divide classes into groups using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (21);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  4: else  5:  Use all classes as a single group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  6: end if  7: Construct pieces with uniformly sampled m examples  for each class (total=N × m);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  8: for each piece do  9:  Randomly partition the piece into two sub-pieces A and  B (each contains Nm/2 examples);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  10:  Run Algorithm 1 (B, A) on A to get clean-set-A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  11:  Run Algorithm 1 (A, B) on B to get clean-set-B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  12:  Concat clean-set-A and clean-set-B to get clean-set-piece;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  13: end for  14: Concat clean-set-pieces to get clean set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  15: return clean set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  4  LEARNING WITH KNOCKOFFS-SPR  In this section, we introduce how to incorporate Knockoffs-  SPR into the training of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We first introduce  several implementation details of Knockoffs-SPR, then we  introduce a splitting algorithm that makes Knockoffs-SPR  scalable to large-scale datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Finally, we discuss some  training strategies to better utilize the selected clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1  Knockoffs-SPR in Practice  We introduce several strategies to improve FSR control and  the power of selecting clean samples, which are inspired by  different behaviors of W between noisy and clean samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Ideally, for a clean sample i, Wi is expected to be a small  negative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' if i is noisy data, Wi tends to be large and is  approximately 50% to be positive or negative, as shown  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' To achieve these properties for better clean  sample selection, the following strategies are proposed, in  the procedure of feature extractor, data-preprocessing, label  permutation strategy, estimating β on D1, and clean data  identification in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (13), (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Feature Extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' A good feature extractor is essential for  clean sample selection algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In our experiments, we  adopt the self-supervised training method SimSiam [42] to  pre-train the feature extractor, to make X well encode the  information of the training data in the early stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Data Preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We implement PCA on the features  extracted by neural network for dimension reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' This  can make X of full rank, which ensures the identifiability of  ˆβ in SPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Besides, such a low dimensionality can make the  model estimate β more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' According to the KKT  conditions Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (15), we have that Wi of clean data i tends  to be negative with small magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In this regard, the  model can have better power of clean sample selection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=',  selecting more clean samples while controlling FSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Label  Permutation  Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Instead  of  the  random  permutation strategy, our Knckoff-SPR permutes the label  as the most-confident candidate provided by the model at  each training stage, for FSR consideration especially when  the noise rate is high or some noisy pattern is dominant  in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Specifically, if the pattern of some noisy label  y2,i is learned by the model, then γ∗  2,i may have a smaller  magnitude than that of ˜γ∗  2,i for a randomly permuted  label ˜y2,i that may not be learned by the model, violating  P(Wi > 0|˜y2,i) = 1/2 and hence P(Wi > 0) ≈ 1/2 in  practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In contrast, the most confident permutation can  alleviate this problem, because the most confident label ˜y2,i  can naturally have a small magnitude of ˜γ∗  2,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Estimating β on D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We implement SPR as the first step  to learn β on D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Compared to vanilla OLS, the SPR  can remove some noisy patterns from data, and hence  can achieve an accurate estimate of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Similar to the data  processing step, such an accurate estimation can improve the  power of selecting clean samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Clean data identification in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (13), (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We calculate T  among W for each class, and identify the clean subset for  each class, to improve the power of clean data for each  class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In practice, since some classes may be easier to learn  than others, the Wi for i in these classes have smaller  magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Therefore, data from these classes will take the  main proportion if we calculate T and identify C2 among all  classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' With this design, the clean data are more balanced,  which facilitates the training in the next epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2  Scalable to Large Dataset  The computation cost of the sample selection algorithm  increases with the growth of the training sample, making  it not scalable to large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' To resolve this problem,  we propose to split the total training set into many pieces,  each of which contains a small portion of training categories  with a small number of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' With the splitting  strategy, we can run the Knockoffs-SPR on several pieces  in parallel and significantly reduce the running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For  the splitting strategy, we notice that the key to identifying  clean data is leveraging different behavior in terms of the  magnitude and the sign of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Such a difference can be  alleviated if the patterns from clean classes are similar to the  noisy ones, which may lead to unsatisfactory recall/power  of identifying the clean set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' This motivates us to group  similar categories together, to facilitate the discrimination  of clean data from noisy ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Formally speaking, we define the similarity between the  class i and j as  s(i, j) = p⊤  i pj,  (21)  where p represents the class prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' To obtain pi for the  class i, we take the clean features xi of each class extracted  by the network along the training iteration, and average  them to get the class prototype pc after the current training  epoch ends, as  pc =  �n  i=1 xi1(yi = c, i ∈ C)  �n  i=1 1(yi = c, i ∈ C) ,  (22)  Then the most similar classes are grouped together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In  the initialization step when the clean set has not been  JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 8, AUGUST 2015  8  Algorithm 3 Training with Knockoffs-SPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Input: Noisy dataset {(imgi, xi, yi)}n  i=1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Output: Trained network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Initialization :  1: Model: A self-supervised pre-trained backbone with  a random initialized fully-connected layer, an EMA  model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  2: Initial clean set: Run Algorithm 2 with self-supervised  pre-trained feature and noisy labels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Training Process:  3: for ep = 0 to max epochs do  4:  for each mini-batch do  5:  Sample r from U(0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  6:  if r > p then  7:  Train the network using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (25);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  8:  else  9:  Train the network using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (24);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  10:  end if  11:  Update features x visited in current mini-batch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  12:  Update EMA model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  13:  end for  14:  Run Algorithm 2 on {(xi, yi)}n  i=1 to get clean set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  15: end for  16: return Trained network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  estimated yet, we simply use all the data to calculate the  class prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In our experiments, each group is designed  to have 10 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  For the instances in each group, we split the training data  of each class in a balanced way such that each piece contains  the same number of instances for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The number  is determined to ensure that the clean pattern remains the  majority in the piece, such that optimization can be done  easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In practice, we select 75 training data from each  class to construct the piece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' When the class proportion  is imbalanced in the original dataset, we adopt the over-  sampling strategy to sample the instance of each class with  less training data multiple times to ensure that each training  instance is selected once in some piece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The pipeline of our  splitting algorithm is described in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3  Network Learning with Knockoffs-SPR  When training with Knockoffs-SPR, we can further exploit  the support of noisy data by incorporating Knockoffs-  SPR with semi-supervised algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In this paper, we  interpolate part of images between clean data and noisy  data as in CutMix [38],  ˜  img = M ⊙ imgclean + (1 − M) ⊙ imgnoisy  (23a)  ˜y = λyclean + (1 − λ)ynoisy  (23b)  where M ∈ {0, 1}W ×H is a binary mask, ⊙ is element-  wise multiplication, λ ∼ Beta(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5) is the interpolation  coefficient, and the clean and noisy data are identified  by Knockoffs-SPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Then we train the network using the  interpolated data using  L  �  ˜  img, ˜y  �  = LCE  �  ˜  img, ˜y  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (24)  Empirically, we could switch between this semi-supervised  training with standard supervised training on estimated  clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  L (imgi, yi) = 1i/∈O · LCE (imgi, yi) ,  (25)  where 1i/∈O is the indicator function, which means that  only the cross-entropy loss of estimated clean data is  used to calculate the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We further store a model with  EMA-updated weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Our full algorithm is illustrated in  Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Neural networks trained with this pipeline  enjoy powerful recognition capacity in several synthetic and  real-world noisy datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  5  RELATED WORK  Here we make the connections between our Knockoffs-SPR  and previous research efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1  Learning with Noisy Labels  The target of Learning with Noisy Labels (LNL) is to  train a more robust model from the noisy dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We can  roughly categorize LNL algorithms into two groups: robust  algorithm and noise detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' A robust algorithm does not  focus on specific noisy data but designs specific modules  to ensure that networks can be well-trained even from the  noisy datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Methods following this direction includes  constructing robust network [2]–[5], robust loss function [6]–  [9] robust regularization [43]–[46] against noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  The noise detection method aims to identify the noisy  data and design specific strategies to deal with the noisy  data, including down-weighting the importance in the loss  function for the network training [47], re-labeling them to  get correct labels [48], or regarding them as unlabeled data  in the semi-supervised manner [49], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  For the noise detection algorithm, noisy data are identified  by some irregular patterns, including large error [14],  gradient directions [50], disagreement within multiple  networks [15], inconsistency along the training path [17] and  some spatial properties in the training data [18], [51]–[53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Some algorithms [50], [54] rely on the existence of an extra  clean set to detect noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  After detecting the clean data, the simplest strategy is to  train the network using the clean data only or re-weight  the data [55] to eliminate the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Some algorithms [49],  [56] regard the detected noisy data as unlabeled data to  fully exploit the distribution support of the training set in  the semi-supervised learning manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' There are also some  studies of designing label-correction module [2], [48], [54],  [57]–[59] to further pseudo-labeling the noisy data to train  the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Few of these approaches are designed from the  statistical perspective with non-asymptotic guarantees, in  terms of clean sample selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In contrast, our Knockoffs-  SPR can theoretically control the false-selected rate in  selecting clean samples under general scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2  Mean-Shit Parameter  Mean-shift  parameters  or  incidental  parameters  [21]  originally tackled to solve the robust estimation problem  JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 8, AUGUST 2015  9  via penalized estimation [60] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' With a different focus on  specific parameters, this formulation address wide attention  in different research topics, including economics [21]–[24],  robust regression [20], [25], statistical ranking [26], face  recognition [27], semi-supervised few-shot learning [28],  [29], and Bayesian preference learning [30], to name a  few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Previous work usually uses this formulation to solve  robust linear models, while in this paper we adopt this to  select clean data and help the training of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Furthermore, we design an FSR control module and a  scalable sample selection algorithm based on mean-shift  parameters with theoretical guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3  Knockoffs  Knockoffs was first proposed in [34] as a data-adaptive  method to control FDR of variable selection in the sparse  regression problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' This method was then extended to  high-dimension regression [1], [61], multi-task regression  [35], outlier detection [41] and structural sparsity [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The  core of Knockoffs is to construct a fake copy of X as  negative controls of original features, in order to select true  positive features with FDR control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Our Knockoffs-SPR is  inspired by but different from the classical knockoffs in the  following aspects: i) the Knockoff is to control the FDR,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=', the expected rate of the type-I error while our goal is  to control the expected rate of the type-II error, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='a, FSR,  in the noisy data scenario;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' ii) instead of constructing copy  for X, we construct the copy ˜Y via permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Equipped  with a calibrated data-partitioning strategy, our method can  control the FER under any desired level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  6  EXPERIMENTS  Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We validate the effectiveness of Knockoffs-  SPR on synthetic noisy datasets CIFAR-10 and CIFAR-  100 [62], and real-world noisy datasets WebVision [63]  and Clothing1M [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We consider two types of noisy  labels for CIFAR: (i) Symmetric noise: Every class is  corrupted uniformly with all other labels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (ii) Asymmetric  noise: Labels are corrupted by similar (in pattern) classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  WebVision has 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4 million images collected from the  internet with the same category list as ImageNet ILSVRC12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Clothing1M has 1 million images collected from the internet  and labeled by the surrounding texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Thus, the WebVision  and Clothing1M datasets can be regarded as real-world  challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For CIFAR, we use ResNet-18 [64] as our  backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For WebVision we use Inception-ResNet [65] to  extract features to follow previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For Clothing1M  we use ResNet-50 as backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For CIFAR and WebVision,  we respectively self-supervised pretrain for 100 epochs and  350 epochs using SimSiam [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For Clothing1M, we use  ImageNet pre-trained weights to follow previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Hyperparameter setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We use SGD to train all the  networks with a momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='9 and a cosine learning  rate decay strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The initial learning rate is set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  The weight decay is set as 1e-4 for Clothing1M, and 5e-  4 for other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We use a batch size of 128 for all  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We use random crop and random horizontal  flip as augmentation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The network is trained  for 180 epochs for CIFAR, 300 epochs for WebVision, and  5 epochs for Clothing1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Network training strategy is  selected with p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5 (line 6 in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 3) for Clothing1M, while  for other datasets we only use CutMix training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For features  used in Knockoffs-SPR, we reduce the dimension of X to  the number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For Clothing1M, this is 14 and for  other datasets the reduced dimension is 10 (each piece of  CIFAR-100 and WebVision contains 10 classes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We also run  SPR with our new network training algorithm (Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1  Evaluation on Synthetic Label Noise  Competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We use cross-entropy loss (Standard) as the  baseline algorithm for two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We compare Knockoffs-  SPR with algorithms that include Forgetting [66] with  train the network using dropout strategy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Bootstrap [67]  which trains with bootstrapping,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Forward Correction [55]  which corrects the loss function to get a robust model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Decoupling  [68]  which  uses  a  meta-update  strategy  to  decouple  the  update  time  and  update  method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  MentorNet [12] which uses a teacher network to help train  the network,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Co-teaching [11] which uses two networks to  teach each other,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Co-teaching+ [15] which further uses an  update by disagreement strategy to improve Co-teaching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  IterNLD [51] which uses an iterative update strategy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  RoG [52] which uses generated classifiers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' PENCIL [59]  which uses a probabilistic noise correction strategy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' GCE [7]  and SL [8] which are extensions of the standard cross-  entropy loss function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' and TopoFilter [18] which uses feature  representation to detect noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For each dataset, all the  experiments are run with the same backbone to make a  fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We randomly run all the experiments five  times and calculate the average and standard deviation of  the accuracy of the last epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The results of competitors are  reported in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  As in Table 1, Knockoffs-SPR enjoys a higher performance  compared with other competitors on CIFAR, validating the  effectiveness of Knockoffs-SPR on different noise scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  SPR enjoys better performance on higher symmetric noise  rate of CIFAR-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' This may contributes to the manual  selection threshold of 50% of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Then SPR will  select more data than Knockoffs-SPR, for example in Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  80% noise scenario SPR will select 24816 clean data while  Knockoffs-SPR will select 18185 clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' This leads to  a better recovery of clean data (recall of 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='22% while  Knockoffs-SPR is 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='20%) and thus a better recognition  capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2  Evaluation on Real-World Noisy Datasets  In this part, we compare Knockoffs-SPR with other methods  on real-world noisy datasets: WebVision and Clothing1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  We follow previous work to train and test on the first 50  classes of WebVision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We also evaluate models trained on  WebVision to ILSVRC12 to test the cross-dataset accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  For  WebVision,  we  compare  with  CE  that  trains  with  cross-entropy  loss  (CE),  as  well  as  Decoupling [68], D2L [69], MentorNet [12], Co-teaching [11],  Iterative-CV [13], and DivideMix [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For clothing1M, we  JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 8, AUGUST 2015  10  TABLE 1  Test accuracies(%) on several benchmark datasets with different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Dataset  Method  Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Noise Rate  Asy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Noise Rate  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4  CIFAR-10  Standard  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='6  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='8  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7  Forgetting  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='8  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0  Bootstrap  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='6  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='8  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3  Forward  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='6 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='0  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='2  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='1  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='9  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='5  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='2  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='6  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='8  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='4  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='3  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='4  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='0  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='7  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='1  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7  TopoFilter  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2  SPR  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='6  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='8  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='8  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='6  Knockoffs-SPR  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='8  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2  CIFAR-100  Standard  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='8  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7  Forgetting  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='9  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='8  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='8  Bootstrap  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='6  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='8  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='6  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='9  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='9  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0  Forward  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0  Decoupling  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='9  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='9  MentorNet  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='9  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='8  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='2  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='3  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='1  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='1  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='8  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='7  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='6  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='2  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='8  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5  Knockoffs-SPR  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='6  TABLE 2  Test accuracies(%) on WebVision and ILSVRC12 (trained on  WebVision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Method  WebVision  ILSVRC12  top1  top5  top1  top5  F-correction  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='12  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='68  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='36  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='36  Decoupling  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='54  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='74  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='26  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='26  D2L  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='68  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='00  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='80  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='36  MentorNet  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='00  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='40  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='80  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='92  Co-teaching  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='58  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='20  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='48  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='70  Iterative-CV  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='24  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='34  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='60  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='98  DivideMix  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='32  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='64  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='20  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='84  SPR  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='08  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='40  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='32  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='92  Knockoffs-SPR  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='20  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='36  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='72  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='88  compare with F-correction [55], M-correction [56], Joint-  Optim [48], Meta-Cleaner [70], Meta-Learning [71], P-  correction [59], TopoFilter [18] and DivideMix [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  The results of real-world datasets are shown in Table 3  and Table 2, where the results of competitors are reported  in [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Our algorithm Knockoffs-SPR enjoys superior  performance to almost all the competitors, showing the  ability of handling real-world challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Compared with  SPR, Knockoffs-SPR also achieves better performance,  indicating the beneficial of FSR control in real-world  problems of learning with noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  TABLE 3  Test accuracies(%) on Clothing1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Method  Accuracy  Cross-Entropy  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='21  F-correction  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='84  M-correction  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='00  Joint-Optim  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='16  Meta-Cleaner  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='50  Meta-Learning  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='47  P-correction  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='49  TopoFiler  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='10  DivideMix  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='76  SPR  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='16  Knockoffs-SPR  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='25  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3  Evaluation of Sample Selection Quality  To test whether Knockoffs-SPR leads to better sample  selection quality, we test the following statistics on CIFAR-  10 with different noise scenarios, including Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 40%, Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  80%, and Asy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (1) FSR: the ratio of falsely selected  noisy data in the estimated clean data, which is the target  that Knockoffs-SPR aims to control;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (2) Recall: the ratio of  selected ground-truth clean data in the full ground-truth  clean data, which indicates the power of sample selection  algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (3) F1-score: the harmonic mean of precision (1-  FSR) and recall, which measures the balanced performance  of FSR control and power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We plot the corresponding  JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' AUGUST 2015  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='FSR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='Symmetric-40%  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='120  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='140  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='160  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='180  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='Training Epochs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='65  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='70  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='75  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='85  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='90  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='95  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Performance(%) comparison on sample selection along the  training path on CIFAR 10 with different noise scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In the FSR,  we also visualize the estimated FSR (q) by Knockoffs-SPR, which is the  threshold we use to select clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  statistics of each algorithm along the training epochs in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We further visualize the estimated FSR, q, of  Knockoffs-SPR to compare with the ground-truth FSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' As  we use the splitting algorithm, where each piece contains  10 classes with each class containing a subset of data, we  estimate FSR for each piece and report their average and  standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  FSR control in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (1) When the noise rate is not  high, for example in Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 40% and Asy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 40% scenarios, the  ground-truth FSR is well upper-bounded by the estimated  FSR (with no larger than a single standard deviation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' When  the noise rate is high, for example in Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 80% noise  scenario, the FSR cannot get controlled in the early stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  However, as the training goes on, FSR can be well-bounded  by Knockoffs-SPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (2) When the training set is not very noisy, for example in  Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 40% scenario, the true FSR is far below the estimated  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' This gap can be explained by a good estimation of  β due to the small noisy rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' When ˆβ1 can accurately  estimate β∗, the ˜γ∗  2,i dominate in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Therefore, the  P(Wi > 0|˜y2,i is clean) > 1  2, making P(Wi > 0) > 1/2 >  c−2  2(c−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Since the true FSR bound is inversely proportional to  P(Wi > 0) (FSR ∝ maxi∈Cc 1/P(Wi > 0) − 1), it is smaller  than the theoretical bound q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Sample selection quality comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We compare the  sample selection quality of Knockoffs-SPR with SPR and  TopoFilter [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (1) Knockoffs-SPR enjoys the (almost) best  FSR control capacity in all noise scenarios, especially in the  high noise rate setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Other algorithms can suffer from  failure in controlling the FSR (for example in Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 80%  scenario).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (2) The power of Knockoffs-SPR is comparable to  the best algorithms in Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 40% and Asy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 40% scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For  the Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 80% case, Knockoffs-SPR sacrifices some power for  FSR control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (3) Compared together, Knockoffs-SPR enjoys  the best F1 score on sample selection quality, which well-  establishes its superiority in selecting clean data with FSR  control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  TABLE 4  Ablation(%) of Knockoffs-SPR on CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  SPR  ∗-random  ∗-multi  ∗-noPCA  Knockoffs-SPR  Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 40%  Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7  FSR  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='82  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='31  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='51  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='27  q 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='31±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='73  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='00±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='00  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='18±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='62  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='59±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='11  Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 80%  Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='6  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3  FSR  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='47  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='76  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='77  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='06  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='72  q 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='47±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='39  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='22±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='62  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='95±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='88  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='52±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='77  Asy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 40%  Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5  FSR  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='19  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='94  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='97  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='62  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='84  q 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='15±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='59  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='00±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='00  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='22±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='85  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='45±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='68  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4  Further Analysis  Influence  of  Knockoffs-SPR  strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  We  compare  Knockoffs-SPR  with  several  variants,  including:  SPR  (The original SPR algorithm), ∗-random (Knockoffs-SPR  with  randomly  permuted  labels),  ∗-multi  (Knockoffs-  SPR without class-specific selection), ∗-noPCA (Knockoffs-  SPR without using PCA to pre-process the features).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Experiments are conducted on CIFAR-10 with different  noise scenarios, as in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We observe the following  results:  (1) As also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 3, the SPR can control the FSR in  Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 40% and Asy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 40% but fails in Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 80%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' This may  be due to that when the noisy pattern is not significant,  the collinearity is weak between noisy samples and clean  ones, as shown by the distribution of irrepresentable value  {∥(X⊤  S XS)−1X⊤  S Xj∥1}j∈Sc  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 1 in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  In this regard, most of the earlier (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' later) selected  samples in the solution path tend to be noisy (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' clean)  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' When there is strong multi-collinearity and the  irrepresentable condition is violated seriously, our proposed  Knockoff procedure can help to control the FSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The higher  accuracy of Knockoffs-SPR over SPR can be explained by  consistent improvements in terms of the F1 score of sample  selection capacity, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (2) Compared with the random permutation strategy,  Knockoffs-SPR with most-confident permutation enjoys  much better FSR control and works much better in Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  40% and Asy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 40% noise scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 80% noise  scenario, the accuracy is comparable, but the most-confident  permutation still enjoys much better FSR control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' This  result empirically demonstrates the advantage of the most-  confident permutation over the random permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (3) Running Knockoffs-SPR on each class separately is  beneficial for the FSR control capacity and recognition  capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' When the noise rate is high, running Knockoffs-  SPR on multiple classes cannot control the FSR properly by  the estimated q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (4) Using PCA on features as the pre-processing is beneficial  for FSR control in all cases and will increase the recognition  capacity in some cases, especially when the noise rate is  high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 8, AUGUST 2015  12  TABLE 5  Ablation of the splitting algorithm in computation efficiency (for one  epoch) on CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Model  Training Time  Knockoffs-SPR w/o split algorithm  about 6h  Knockoffs-SPR w/ split algorithm  66s  TABLE 6  Ablation(%) of training strategies on CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Method  Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 40%  Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 80%  Asy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 40%  Knockoffs-SPR - Self  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2  Knockoffs-SPR - Semi  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5  Knockoffs-SPR - EMA  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='8  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2  Knockoffs-SPR  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='7  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5  Influence of Scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In our framework, we propose a split  algorithm to divide the whole training set into small pieces  to run Knockoffs-SPR in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In this part, we compare  the running time between using the split algorithm and not  using it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Results are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We can see that the  splitting algorithm can significantly reduce the computation  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' This is important in large-scale applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Influence of network training strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' To better train  the network, we adopt the self-supervised pre-trained  backbone and the semi-supervised learning framework with  an EMA update model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In this part, we test the influence of  these strategies on CIFAR-10 with different noise scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Concretely, we compare the full framework with Knockoffs-  SPR - Self which uses a randomly initialized backbone,  Knockoffs-SPR - Semi which uses supervised training, and  Knockoffs-SPR - EMA which does not use the EMA update  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Results are summarized in table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We can find  that: (1) The self-supervised pre-training is important for  high noise rate scenarios, while for other settings, it is  not so essential;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (2) Semi-supervised training consistently  improves the recognition capacity, indicating the utility of  leveraging the support of noisy data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (3) The EMA model  will slightly improve the recognition capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  airplane  frog  automobile  truck  bird  deer  cat  dog  deer  cat  dog  cat  frog  deer  horse  dog  ship  airplane  truck  ship  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Qualitative results of falsely selected examples by Knockoffs-  SPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The black words are the labeled classes while the real classes  are denoted by red words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Qualitative visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We randomly visualize some  falsely selected examples of CIFAR-10 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Most of  these cases have some patterns that confuse the noisy  label and the true label, thus making Knockoffs-SPR falsely  identify them as clean samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  7  CONCLUSION  This  paper  proposes  a  statistical  sample  selection  framework – Scalable Penalized Regression with Knockoff  Filters (Knockoffs-SPR) to identify noisy data with a  controlled false selection rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Specifically, we propose an  equivalent leave-one-out t-test approach as a penalized  linear model, in which non-zero mean-shift parameters can  be induced as an indicator for noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We propose a  delicate Knockoff-SPR algorithm to identify clean samples  in a way that the false selection rate is controlled by  the user-provided upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Such an upper bound is  proved theoretically and works well in empirical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Experiments on several synthetic and real-world datasets  demonstrate the effectiveness of Knockoff-SPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content=' Kankanhalli, “Learning to learn  from noisy labeled data,” in Proceedings of the IEEE/CVF Conference  on Computer Vision and Pattern Recognition, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 5051–5059.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2  Yikai  Wang  is  a  PhD  candidate  at  the  School  of  Data  Science,  Fudan  University,  under the supervision of Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Yanwei Fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' He  received a Bachelor’s degree in mathematics  from the School of Mathematical Sciences,  Fudan University, in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' He published 1  IEEE TPAMI paper and 2 CVPR papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' His  current research interests include theoretically  guaranteed machine learning algorithms and  applications to computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Yanwei Fu received his PhD degree from the  Queen Mary University of London, in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' He  worked as a post-doctoral researcher at Disney  Research, Pittsburgh, PA, from 2015 to 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  He is currently a tenure-track professor at Fudan  University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' He was appointed as the Professor  of Special Appointment (Eastern Scholar) at  Shanghai Institutions of Higher Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' He  published  more  than  80  journal/conference  papers including IEEE TPAMI, TMM, ECCV,  and CVPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' His research interests are one-  shot/meta-learning, learning-based 3D reconstruction, and image and  video understanding in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Xinwei Sun is currently an assistant professor  at the School of Data Science, Fudan University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  He  received  his  Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  in  the  school  of  mathematical sciences, at Peking University in  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' His research interests mainly focus on  high-dimensional statistics and causal inference,  with their applications in machine learning and  medical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 8, AUGUST 2015  15  Supplementary Material  Yikai Wang, Yanwei Fu, and Xinwei Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  In this supplementary material, we formally present the  proof of FSR control theorem of knockoff-SPR in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  For consistency, we also provide the proof of the noisy  set recovery theorem of SPR in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Some additional  experimental results are provided in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  8  FSR CONTROL THEOREM OF KNOCKOFF-SPR  Recall that we are solving the problem of  �  �  �  �  �  1  2  ���Y2 − X2 ˜β1 − γ2  ���  2  F + �  j P(γ2,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' λ),  1  2  ��� ˜Y2 − X2 ˜β1 − ˜γ2  ���  2  F + �  j P(˜γ2,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (26)  We introduce the following lemma from [1] and [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Suppose that B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' , Bn are indenpendent variables,  with Bi ∼ Bernoulli(ρi) for each i, where mini ρi ≥ ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Let  J be a stopping time in reverse time with respect to the filtration  {Fj}, where  Fj = σ ({B1 + · · · + Bj, Bj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' , Bn}) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Then  E  �  1 + J  1 + B1 + · · · + BJ  �  ≤ ρ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We first follow [1] to prove the case when {Bi} are  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' variables with Bi ∼ Bernoulli(ρ), where ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Then  we follow [2] to generalize the conclusion to non-identical  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Define the stochastic process  Mj := 1 + j  1 + Sj  with  Sj := B1 + · · · + Bj  (27)  We show that {Mj} is a super-martingale with respect to  the reverse filtration {Fj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' It is trivial that {Mj} is {Fj}-  adapted and {Fj} is reverse filtration, that is a decreasing  sequence  Fj ⊂ Fj−1 · · · ⊂ {Bi}n  i=1  (28)  with each Fj be a sub-σ-algebras of σ({Bi}n  i=1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Further,  we have E [|Mj|] ≤ 1 + j ≤ 1 + n < ∞ with fixed n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Now  we bound the conditional expectation E[Mj | Fj+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Note  that since {Bj}i+1  j=1 are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' variable and thus exchangeable  when conditioned on Fj+1, then we have  P(Bj+1 = 1 | Fj+1) = Sj+1  j + 1  (29)  When Sj+1 = 0, it is natural that Sj = 0 thus Mj = 1 + j <  1 + (j + 1) = Mj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' When Sj+ > 0, we have  E[Mj | Fj+1] =  1 + j  1 + Sj+1 − 1 · P(Bj+1 = 1 | Fj+1)  +  1 + j  1 + Sj+1  P(Bj+1 = 0 | Fj+1)  =1 + j  Sj+1  Sj+1  j + 1 +  1 + j  1 + Sj+1  j + 1 − Sj+1  j + 1  =1 + (j + 1)  1 + Sj+1  =Mj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (30)  Hence we have E[Mj | Fj+1] ≤ Mj+1, which finishes the  proof for the super-martingale {Mj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Then by the Doob’s  optional sampling theorem [3], we have  E[Mj] ≤ E[Mn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (31)  Finally, we have  E[Mn] = E[ 1 + n  1 + Sn  ]  = (1 + n)  n  �  m=0  1  1 + m ·  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (n − m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='ρm(1 − ρ)n−m  = ρ−1(1 − (1 − ρ)n+1)  ≤ ρ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (32)  Now it suffices to show that the conclusion also holds for  non-identical Bernoulli variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Following [2], for each  Bi ∼ Bernoulli(ρi), we construct the following disjoint  Borel sets {Ai  j}4  j=1 such that R = ∪4  j=1Aj with  P(Ai  1) = 1 − ρi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  P(Ai  2) = ρ1 − ρi  1 − ρ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  P(Ai  3) = ρρi − ρ  1 − ρ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  P(Ai  4) = ρi − ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (33)  Define Ui = Ai  1∪Ai  2, Vi = Ai  2∪Ai  3, Gi = Ai  2∪Ai  3∪Ai  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Based  on the specific construction we can set Gi = Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Further  define Qi = 1{ξi ∈ Vi} and a random set A = {i : ξi ∈ Ui}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Then we have  Qi · 1{i ∈ A} + 1{i /∈ A}  = 1{{ξi ∈ Vi ∩ Ui} ∪ {ξi ∈ U C  i }}  = 1{{ξi ∈ Ai  2} ∪ {ξi ∈ Ai  3 ∪ Ai  4}}  = 1{{ξi ∈ Ai  2 ∪ Ai  3 ∪ Ai  4}} = Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (34)  Hence  1 + j  1 + Sj  = 1 + |i ≤ j : i ∈ A| + |i ≤ j : i /∈ A|  1 + �  i≤j,i∈A Qi + |i ≤ j : i /∈ A|  ≤ 1 + |i ≤ j : i ∈ A|  1 + �  i≤j,i∈A Qi  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (35)  JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 8, AUGUST 2015  16  The inequality holds because a+c  b+c ≤ a  b for 0 < b ≤ a, c ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Note that by definition  P(Qi = 1 | i ∈ A) = P(ξi ∈ Vi | ξi ∈ Ui)  =  P(Ai  2)  P(Ai  1 ∪ Ai  2)  = ρ = P(Qi = 1),  P(Qi = 1 | i ̸∈ A) = P(ξi ∈ Vi | ξi /∈ Ui)  =  P(Ai  3)  P(Ai  3 ∪ Ai  4)  = ρ = P(Qi = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (36)  indicating that Qi and A are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  For any fixed A, define ˜Qi = Qi · 1{i ∈ A} and the reverse  filtration ˜Fj = σ({�j  k=1 ˜Qk, ˜Qj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' , ˜Qn, A}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Then when  conditioned on A, the established result suggests that  E  �  1 + |i ≤ j : i ∈ A|  1 + �  i≤j,i∈A Qi  ����A  �  ≤ ρ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (37)  Take expectation over A finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1  Proof of Theorem 2  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We first control the FSR rate of the second subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Specifically, we have  FSR(T) ≤ E  � # {j : γj ̸= 0 and − T ≤ Wj < 0}  1 + # {j : γj ̸= 0 and 0 < Wj ≤ T}  1 + # {j : 0 < Wj ≤ T}  # {j : −T ≤ Wj < 0} ∨ 1  �  ≤ q · E  � # {j : γj ̸= 0 and − T ≤ Wj < 0}  1 + # {j : γj ̸= 0 and 0 < Wj ≤ T}  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (38)  The second inequality holds by the definition of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Now it  suffices to show that  E  � # {j : γj ̸= 0 and − T ≤ Wj < 0}  1 + # {j : γj ̸= 0 and 0 < Wj ≤ T}  �  ≤  c  c − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (39)  For γj ̸= 0, we have a probability of  1  c−1 to get a clean  ˜γj, leading to Zj > Zj+n with non-zero probability, and  a probability of c−2  c−1 to get a noisy ˜γj, where we have no  information and hence assume a equal probability of Zj >  Zj+n and Zj < Zj+n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Then we have  P(Wi > 0) =P(Wi > 0|˜γ∗  j ̸= 0)P(˜γ∗  j ̸= 0)  + P(Wi > 0|˜γ∗  j = 0)P(˜γ∗  j = 0)  ≥1  2 × c − 2  c − 1 + 0 =  c − 2  2(c − 1)  (40)  Hence the random variable Bj := 1{Wj>0} ∼ Bernoulli(ρj)  for γj ̸= 0 with ρj ≥ (c − 2)/(2(c − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Now we consider all the Wj of non-null variables, and  assumes |W1| ≤ · · · ≤ |Wn| with the abuse of subscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  We have  γj ̸= 0 and − T ≤ Wj < 0  ⇐⇒  j ≤ J and Bj = 0  and  γj ̸= 0 and 0 < Wj ≤ T  ⇐⇒  j ≤ J and Bj = 1  Hence  # {j : γj ̸= 0 and − T ≤ Wj < 0}  1 + # {j : γj ̸= 0 and 0 < Wj ≤ T}  = (1 − B1) + · · · + (1 − BJ)  1 + B1 + · · · + BJ  =  1 + J  1 + B1 + · · · + BJ  − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (41)  If we can use Lemma 3, then  E  � # {j : γj ̸= 0 and − T ≤ Wj < 0}  1 + # {j : γj ̸= 0 and 0 < Wj ≤ T}  �  ≤ ρ−1−1 ≤  c  c − 2  (42)  Then we finally get  FSR(T) ≤ q  c  c − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (43)  as long as c > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Now it suffices to show that our  random variables {Bj} are mutually independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' This is  straightforward as we set P(α2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' λ) as a sparse penalty for  each row α2,j in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (26), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Then problem of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (26) now is a combination of independent sub-problems  for each row α2,j, and the solution only depends on  (x2,j, y2,j, β(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' D1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Then with fixed β(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' D1), the mutual  independence naturally exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Finally, after we control the FSR rate for the second subset,  we can get the estimate of β(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' D2) based on the identified  clean data in the second subset, and return to run knockoff-  SPR on the first subset in a similar pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Then we have  for the whole dataset:  FSR = E  �|S1 ∩ C1| + |S2 ∩ C2|  |C1| + |C2|  �  ≤ E  �|S1 ∩ C1|  |C1|  �  + E  �|S2 ∩ C2|  |C2|  �  ≤ 2  c  c − 2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (44)  To control the FSR with q, the threshold of T should be  defined as c−2  2c q, which leads to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  9  NOISY SET RECOVERY THEOREM OF SPR  Recall that we are solving the problem of  min  ⃗γ  ���⃗y − ˚  X⃗γ  ���  2  2 + λ ∥⃗γ∥1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (45)  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Assume that ˚  X⊤ ˚  X is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' If  ����λ ˚  X⊤  Sc ˚  XS  �  ˚  X⊤  S ˚  XS  �−1  ˆvS + ˚  X⊤  Sc (I − IS) ( ˚  Xε)  ����  ∞  < λ  (46)  holds for all ˆvS ∈ [−1, 1]S, where IS = ˚  XS  �  ˚  X⊤  S ˚  XS  �−1 ˚  X⊤  S ,  then the estimator ˆ⃗γ of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (45) satisfies that  ˆS = supp  �ˆ⃗γ  �  ⊆ supp (⃗γ∗) = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Moreover, if the sign consistency  sign  �ˆ⃗γS  �  = sign (⃗γ∗  S)  (47)  holds, Then ˆ⃗γ is the unique solution of (45) with the same sign as  ˆ⃗γ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 8, AUGUST 2015  17  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (45) is convex that has global minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Denote Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (45) as L, the solution of ∂L/∂⃗γ = 0 is the  unique minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Hence we have  ∂L  ∂⃗γ = − ˚  X⊤ �  ⃗y − ˚  X⃗γ  �  + λv = 0  (48)  where v = ∂ ∥⃗γ∥1 /∂⃗γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Note that ∥⃗γ∥1 is non-differentiable,  so we instead compute its sub-gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Further note that  vi = ∂ ∥⃗γ∥1 /∂⃗γi = ∂ |⃗γi| /∂γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Hence vi = sign (⃗γi) if ⃗γi ̸=  0 and vi ∈ [−1, 1] if ⃗γi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' To distinguish between the two  cases, we assume vi ∈ (−1, 1) if ⃗γi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Hence there exists  ˆv ∈ Rn×1 such that  − ˚  X⊤ �  ⃗y − ˚  X ˆ⃗γ  �  + λˆv = 0,  (49)  ˆvi = sign  �ˆ⃗γi  �  if i ∈ ˆS and ˆvi ∈ (−1, 1) if i ∈ ˆSc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  To obtain ˆS ⊆ S, we should have ˆ⃗γi = 0 for i ∈ Sc, that is,  ∀i ∈ Sc, |ˆvi| < 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=',  ��� ˚  X⊤  Sc  �  ⃗y − ˚  XS ˆ⃗γS  ����  ∞ < λ,  (50)  For i ∈ S, we have  − ˚  X⊤  S  �  ⃗y − ˚  XS ˆ⃗γS  �  + λˆvS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  (51)  If ˚  X⊤ ˚  X is invertible then  ˆ⃗γS =  �  ˚  X⊤  S ˚  XS  �−1 �  ˚  X⊤  S ⃗y − λˆvS  �  (52)  Recall that we have  ⃗y = ˚  XS⃗γ∗  S + ˚  X⃗ε  (53)  Hence  ˆ⃗γS = ⃗γ∗  S+δS,  δS :=  �  ˚  X⊤  S ˚  XS  �−1 �  ˚  X⊤  S ˚  X⃗ε − λˆvS  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (54)  Plugging (54) and (53) into (50) we have  ���� ˚  X⊤  Sc ˚  X⃗ε − ˚  X⊤  Sc ˚  XS  �  ˚  X⊤  S ˚  XS  �−1 �  ˚  X⊤  S ˚  X⃗ε − λˆvS  �����  ∞  < λ,  (55)  or equivalently  ����λ ˚  X⊤  Sc ˚  XS  �  ˚  X⊤  S ˚  XS  �−1  ˆvS + ˚  X⊤  Sc (I − IS) ˚  X⃗ε  ����  ∞  < λ,  (56)  where IS  =  ˚  XS  �  ˚  X⊤  S ˚  XS  �−1 ˚  X⊤  S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' To ensure the sign  consistency, replacing ˆvS  = sign (⃗γ∗  S) in the inequality  above leads to the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Assume that ⃗ε is indenpendent sub-Gaussian with  zero mean and bounded variance Var (⃗εi) ≤ σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Then with probability at least  1 − 2cn exp  �  �  �−  λ2η2  2σ2 maxi∈Sc  ��� ˚  Xi  ���  2  2  �  �  �  (57)  there holds  ��� ˚  X⊤  Sc (I − IS)  �  ˚  X⃗ε  ����  ∞ ≤ λη  (58)  and����  �  ˚  X⊤  S ˚  XS  �−1 ˚  X⊤  S ˚  X⃗ε  ����  ∞  ≤  λη  √Cmin maxi∈Sc  ��� ˚  Xi  ���  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (59)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Let zc = ˚  X⊤  Sc (I − IS)  �  ˚  X⃗ε  �  , for each i ∈ Sc the  variance can be bounded by  Var (zc  i ) ≤ σ2 ˚  X⊤  i (I − IS)2 ˚  Xi ≤ σ2 max  i∈Sc  ��� ˚  Xi  ���  2  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Hoeffding inequality implies that  P  ���� ˚  X⊤  Sc (I − IS)  �  ˚  X⃗ε  ����  ∞ ≥ t  �  ≤ 2 |Sc| exp  �  �  �−  t2  2σ2 maxi∈Sc  ��� ˚  Xi  ���  2  2  �  �  � ,  Setting t = λη leads to the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Now let z =  �  ˚  X⊤  S ˚  XS  �−1 ˚  X⊤  S ˚  X⃗ε, we have  Var (z) =  �  ˚  X⊤  S ˚  XS  �−1 ˚  X⊤  S ˚  XVar (⃗ε) ˚  X⊤ ˚  XS  �  ˚  X⊤  S ˚  XS  �−1  ≤ σ2 �  ˚  X⊤  S ˚  XS  �−1  ≤  σ2  Cmin  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Then  P  �����  �  ˚  X⊤  S ˚  XS  �−1 ˚  X⊤  S ˚  X⃗ε  ����  ∞  ≥ t  �  ≤ 2 |S| exp  �  −t2Cmin  2σ2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Choose  t =  λη  √Cmin maxi∈Sc  ��� ˚  Xi  ���  2  ,  (60)  then there holds  P  �  �  �∥  �  ˚  X⊤  S ˚  XS  �−1 ˚  X⊤  S ˚  X⃗ε∥∞ ≥  λη  √Cmin maxi∈Sc  ��� ˚  Xi  ���  2  �  �  �  ≤ 2 |S| exp  �  �  �−  λ2η2  2σ2 maxi∈Sc  ��� ˚  Xi  ���  2  2  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1  Proof of Theorem 1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The proof essentially follows the treatment in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  The results follow by applying Lemma 5 to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Inequality (46) holds if condition C2 and the first bound (58)  hold, which proves the first part of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The  sign consistency (47) holds if condition C3 and the second  bound (59) hold, which gives the second part of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  It suffices to show that ˆS ⊆ S implies ˆCc ⊆ Cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Consider  one instance i, there are three possible cases for γ∗  i ∈ R1×c:  (1) γ∗  i,j ̸= 0, ∀j ∈ [c];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (2) γ∗  i,j = 0, ∀j ∈ [c];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (3) ∃j, k ∈  [c] , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' γ∗  i,j = 0, γ∗  i,k ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' If instance i follows case (1) or  case (3), then i ∈ Cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' If it follows case (2), then i ∈ C, and  the indexes of all elements of γi are in Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Since we have  ˆS ⊆ S, all elements of γi is in ˆSc, hence i ∈ ˆC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Then we  have ˆCc ⊆ Cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 8, AUGUST 2015  18  10  MORE EXPERIMENTAL RESULTS  Histogram of the median value of IRR condition of  SPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' We visualize the median value of the irrepresentable  (IRR) value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=', {∥(X⊤  S XS)−1X⊤  S Xj∥1}j of SPR final epoch  on CIFAR10 with various noisy scenarios in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' As  SPR is running on each piece split from the training set,  we calculate matrix ˚  X⊤  Sc ˚  XS( ˚  X⊤  S ˚  XS)−1 in irrepresentable  condition (C2 in Theorem 1) for each piece at the final  epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Then the L1 norm of each row of the matrix is the  IRR value of corresponding clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The median value  of IRR values in a single piece is used to construct the  histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For the noise scenario of Asy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 40% and Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 40%,  the median IRR value is small, indicating weak collinearity  between clean data and noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In these cases, SPR  has more chance to distinguish noisy data from clean data  and thus leads to a good FSR control capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' For the  noise scenario of Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 80%, the median IRR values are  much larger, indicating a strong multi-collinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Thus SPR  can hardly distinguish between clean data and noisy data,  leading to a high FSR rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='5  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0  Symmetric-40%  4  6  8  10  12  14  16  0  5  10  15  20  25  Symmetric-80%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='4  0  10  20  30  40  50  60  Asymmetric-40%  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Histogram of the median value of the IRR value of SPR on  CIFAR10 with various noisy scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  REFERENCES  [1] Rina Foygel Barber and Emmanuel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Cand‘es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' A knockoff filter  for high-dimensional selective inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' The Annals of Statistics,  47(5):2504 – 2537, 20 (document), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3, 8, 8  [2] Yang Cao, Xinwei Sun, and Yuan Yao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Controlling the false  discovery rate in transformational sparsity: Split knockoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' In  arXiv, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (document), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3, 8, 8, 8  [3] Joseph L Doob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' Wiley New York, 195 8  [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='  Wainwright,  “Sharp  thresholds  for  high-dimensional  and noisy sparsity recovery using l1 -constrained quadratic  programming (lasso),” IEEE transactions on information theory,  2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content=' (document), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='3, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
+page_content='1' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfqfjJ/content/2301.00545v1.pdf'}
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+Generic transversality of radially symmetric
+stationary solutions stable at infinity for
+parabolic gradient systems
+Emmanuel Risler
+January 9, 2023
+This paper is devoted to the generic transversality of radially symmetric
+stationary solutions of nonlinear parabolic systems of the form
+∂tw(x, t) = −∇V
+�w((x, t))
+� + ∆xw(x, t) ,
+where the space variable x is multidimensional and unbounded. It is proved
+that, generically with respect to the potential V , radially symmetric stationary
+solutions that are stable at infinity (in other words, that approach a minimum
+point of V at infinity in space) are transverse; as a consequence, the set of
+such solutions is discrete. This result can be viewed as the extension to
+higher space dimensions of the generic elementarity of symmetric standing
+pulses, proved in a companion paper. It justifies the generic character of the
+discreteness hypothesis concerning this set of stationary solutions, made in
+another companion paper devoted to the global behaviour of (time dependent)
+radially symmetric solutions stable at infinity for such systems.
+2020 Mathematics Subject Classification: 35K57, 37C20, 37C29.
+Key words and phrases: parabolic gradient systems, radially symmetric stationary solutions, generic
+transversality, Morse–Smale theorem.
+1
+arXiv:2301.02605v1  [math.AP]  6 Jan 2023
+
+Contents
+1
+Introduction
+3
+1.1
+An insight into the main result . . . . . . . . . . . . . . . . . . . . . . . .
+3
+1.2
+Radially symmetric stationary solutions stable at infinity
+. . . . . . . . .
+3
+1.3
+Differential systems governing radially symmetric stationary solutions
+. .
+4
+1.4
+Transversality of radially symmetric stationary solutions stable at infinity
+7
+1.5
+The space of potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+1.6
+Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+1.7
+Key differences with the generic transversality of standing pulses in space
+dimension one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+2
+Preliminary properties
+10
+2.1
+Proof of Lemma 1.4
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+2.2
+Transversality of homogeneous radially symmetric stationary solutions
+stable at infinity
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+11
+2.3
+Additional properties close to the origin . . . . . . . . . . . . . . . . . . .
+13
+2.4
+Additional properties close to infinity . . . . . . . . . . . . . . . . . . . . .
+14
+3
+Tools for genericity
+15
+4
+Generic transversality among potentials that are quadratic past a given radius 17
+4.1
+Notation and statement . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+17
+4.2
+Reduction to a local statement
+. . . . . . . . . . . . . . . . . . . . . . . .
+17
+4.3
+Proof of the local statement (Proposition 4.2) . . . . . . . . . . . . . . . .
+18
+4.3.1
+Setting
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+18
+4.3.2
+Equivalent characterizations of transversality . . . . . . . . . . . .
+19
+4.3.3
+Checking hypothesis 1 of Theorem 4.2 of [1] . . . . . . . . . . . . .
+20
+4.3.4
+Checking hypothesis 2 of Theorem 4.2 of [1] . . . . . . . . . . . . .
+21
+4.3.5
+Conclusion
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+23
+5
+Proof of the main results
+24
+2
+
+1 Introduction
+1.1 An insight into the main result
+The purpose of this paper is to prove the generic transversality of radially symmetric
+stationary solutions stable at infinity for gradient systems of the form
+(1.1)
+∂tw(x, t) = −∇V
+�w((x, t))
+� + ∆xw(x, t) ,
+where time variable t is real, space variable x lies in the spatial domain Rdsp with dsp an
+integer not smaller than 2, the state function (x, t) �→ w(x, t) takes its values in Rdst with
+dst a positive integer, and the nonlinearity is the gradient of a scalar potential function
+V : Rdst → R, which is assumed to be regular (of class at least C2). An insight into the
+main result of this paper (Theorem 1 on page 8) is provided by the following corollary.
+Corollary 1.1. For a generic potential V , the following conclusions hold:
+1. every radially symmetric stationary solution stable at infinity of system (1.1) is
+robust with respect to small perturbations of V ;
+2. the set of all such solutions is discrete.
+The discreteness stated in conclusion 2 of this corollary is a required assumption for
+the main result of [4], which describes the global behaviour of radially symmetric (time
+dependent) solutions stable at infinity for the parabolic system (1.1). Corollary 1.1
+provides a rigorous proof that this assumption holds generically with respect to V .
+This paper can be viewed as a supplement of the article [1], which is devoted to the
+generic transversality of bistable travelling fronts and standing pulses stable at infinity
+for parabolic systems of the form (1.1) in (unbounded) space dimension one, and which
+provides a rigorous proof of the genericity of similar assumptions made in [2, 3, 5]. The
+ideas, the nature of the results, and the scheme of the proof are the same.
+1.2 Radially symmetric stationary solutions stable at infinity
+A function u : [0, +∞) → Rdst, r �→ u(r) defines a radially symmetric stationary solution
+of the parabolic system (1.1) if and only if it satisfies, on (0, +∞), the (non-autonomous)
+differential system
+(1.2)
+¨u(r) = −dsp − 1
+r
+˙u(r) + ∇V
+�u(r)
+� ,
+where ˙u and ¨u stand for the first and second derivatives of r �→ u(r), together with the
+limit
+(1.3)
+˙u(r) → 0
+as
+r → 0+ .
+Observe that, in this case, u(·) is actually the restriction to [0, +∞) of an even function in
+C3(R, Rd
+st) which is a solution (on R) of the differential system (1.2) (the limit (1.3) ensures
+3
+
+that equality (1.2) still makes sense and holds at r equals 0). In other words, provided
+that condition (1.3) holds, it is equivalent to assume that system (1.2) holds on (0, +∞)
+or on [0, +∞). By abuse of language, the terminology radially symmetric stationary
+solution of system (1.1) will refer, all along the paper, to functions u : [0, +∞) → Rdst
+satisfying these conditions (1.2) and (1.3) (even if, formally, it is rather the function
+Rd
+sp → Rd
+st, x �→ u
+�|x|
+� that fits with this terminology).
+Let us denote by Σmin(V ) the set of nondegenerate (local or global) minimum points
+of V ; with symbols,
+Σmin(V ) =
+�u ∈ Rdst : ∇V (u) = 0 and D2V (u) > 0
+� .
+Throughout all the paper, the words minimum point will be used to denote a local or
+global minimum point of a (potential) function.
+Definition 1.2. A (global) solution (0, +∞) → Rdst, r �→ u(r), of the differential system
+(1.2) (in particular a radially symmetric stationary solution of system (1.1)) is said to be
+stable at infinity if u(r) approaches a point of Σmin(V ) as r goes to +∞. If this point of
+Σmin(V ) is denoted by u∞, then the solution is said to be stable close to u∞ at infinity.
+Notation. For every u∞ in Σmin(V ), let SV, u∞ denote the set of the radially symmetric
+stationary solutions of system (1.1) that are stable close to u∞ at infinity. With symbols,
+SV, u∞ =
+�u : [0, +∞) → Rdst : u satisfies (1.2) and (1.3) and u(r) −−−−→
+r→+∞ u∞
+� .
+Let
+S0
+V, u∞ =
+�
+u(0) : u ∈ SV, u∞
+�
+,
+and let
+(1.4)
+SV =
+�
+u∞∈Σmin(V )
+SV, u∞
+and
+S0
+V =
+�
+u∞∈Σmin(V )
+S0
+V, u∞ .
+The following statement is an equivalent (simpler) formulation of conclusion 2 of
+Corollary 1.1.
+Corollary 1.3. For a generic potential V , the subset S0
+V of Rdst is discrete.
+1.3 Differential systems governing radially symmetric stationary solutions
+The second-order differential system (1.2) is equivalent to the (non-autonomous) 2dst-
+dimensional first order differential differential system
+(1.5)
+�
+�
+�
+�
+�
+˙u = v
+˙v = −dsp − 1
+r
+v + ∇V (u) .
+Introducing the auxiliary variables τ and c defined as
+(1.6)
+τ = log(r)
+and
+c = 1
+r ,
+4
+
+the previous 2dst-dimensional differential system (1.5) is equivalent to each of the following
+two 2dst + 1-dimensional autonomous differential systems:
+(1.7)
+�
+�
+�
+�
+�
+�
+�
+uτ = rv
+vτ = −(dsp − 1)v + r∇V (u)
+rτ = r ,
+and
+(1.8)
+�
+�
+�
+�
+�
+�
+�
+ur = v
+vr = −(dsp − 1)cv + ∇V (u)
+cr = −c2 .
+Remark. Integrating the third equations of systems (1.7) and (1.8) yields
+r = r0eτ−τ0
+and
+1
+c − 1
+c0
+= r − r0 ,
+and the parameters τ0 and c0 (which determine in each case the origin of “time”) do
+not matter in principle, since those systems are autonomous. However, if the “initial
+conditions” r0 and c0 are positive (which is true for the solutions that describe radially
+symmetric stationary solutions of system (1.1)), it is natural to choose, in each case, the
+origins of time according to equalities (1.6), that is :
+τ0 = ln(r0)
+and
+c0 = 1
+r0
+.
+Properties close to origin.
+System (1.7) is relevant to provide an insight into the limit
+system (1.5) as r goes to 0. The subspace R2dst × {0} (r equal to 0) is invariant by the
+flow of this system, and the system reduces on this invariant subspace to
+(1.9)
+�
+uτ = 0
+vτ = −(dsp − 1)v ,
+see figure 1.1. For every u0 in Rdst, the point (u0, 0Rdst, 0) is an equilibrium of sys-
+tem (1.7); let us denote by W u, 0
+V
+(u0) the (one-dimensional) unstable manifold of this
+equilibrium, for this system, let
+(1.10)
+W u, 0, +
+V
+(u0) = W u, 0
+V
+(u0) ∩
+�R2dst × (0, +∞)
+� ,
+and let
+W u, 0, +
+V
+=
+�
+u0∈Rdst
+W u, 0, +
+V
+(u0) .
+The subspace
+(1.11)
+Ssym = Rdst × {0Rdst} × {0}
+of R2dst+1 can be seen as the higher space dimension analogue of the symmetry (reversibil-
+ity) subspace Rdst × {0Rdst} of R2dst (which is relevant for symmetric standing pulses in
+space dimension 1, see [1] and subsection 1.7 below); the set W u, 0, +
+V
+can be seen as the
+unstable manifold of this subspace Ssym.
+5
+
+Figure 1.1: Dynamics of the (equivalent) differential systems (1.7) (for r nonnegative
+finite) and (1.8) (for c = 1/r nonnegative finite) in Rdst × Rdst × [0, +∞] (this domain is
+three-dimensional if dst is equal to 1, as on the figure). For the limit differential system
+(1.9) in the subspace r = 0 (in green), the trajectories are vertical and the solutions
+converge towards the horizontal u-axis, defined as Ssym in (1.11), and which is the higher
+space dimensional analogue of the symmetry subspace for symmetric standing pulses
+in space dimension 1. The point u∞ is a local minimum point of V , so that the point
+(u∞, 0Rdst) is a hyperbolic equilibrium for the limit differential system (1.12) in the
+subspace c = 0 ⇐⇒ r = +∞ (in blue). Systems (1.7) and (1.8) are autonomous, but the
+quantity r (the quantity c) goes monotonously from 0 to +∞ (from +∞ to 0) for all the
+solutions in the subspace r > 0 ⇐⇒ c > 0, so that those solutions can be parametrized
+with r (with c) as time. The unstable manifold W u, 0, +
+V
+(u0) is one-dimensional and is a
+transverse intersection between the unstable set W u, 0, +
+V
+of the subspace {r = 0, v = 0Rdst}
+and the centre stable manifold W cs, ∞, +
+V
+(u∞) of the equilibrium (u∞, 0Rdst, c = 0). To
+prove the generic transversality of this intersection is the main goal of the paper. The
+dotted red curve is the projection onto the (u, r)-subspace of this intersection. The part of
+W cs, ∞, +
+V
+(u∞) which is displayed on the figure can also be seen as the local centre stable
+manifold W cs, ∞, +
+loc, V, ε1, c1(u∞) defined in (2.10) (with u∞ equal to the point u∞,1 introduced
+there).
+6
+
+wo
+u
+sym
+om
+L
+0
+(8m)
+loc, V, E1, C1
+C1
+L0=
+u
+8
+E1Properties close to infinity.
+System (1.8) is relevant to provide an insight into the limit
+system (1.5) as r goes to +∞. The subspace R2dst × {0} of R2dst+1 (c equal to 0, or
+in other words r equal to +∞) is invariant by the flow of this system, and the system
+reduces on this invariant subspace to
+(1.12)
+�
+ur = v
+vr = ∇V (u) .
+For every u∞ in Σmin(V ), the point (u∞, 0Rdst, 0) is an equilibrium of system (1.8); let
+us consider its global centre-stable manifold in R2dst × (0, +∞), defined as
+(1.13)
+W cs, ∞, +
+V
+(u∞) =
+�
+(u0, v0, c0) ∈ R2dst × (0, +∞) : the solution of system (1.8)
+with initial condition (u0, v0, c0) at “time” r0 = 1/c0 is
+defined up to +∞ and goes to (u∞, 0, 0) as r goes to +∞
+�
+.
+This set W cs, ∞, +
+V
+(u∞) is a dst + 1-dimensional submanifold of R2dst × (0, +∞) (see
+subsection 2.4).
+Radially symmetric stationary solutions.
+Let us consider the involution
+ι : R2dst × (0, +∞) → R2dst × (0, +∞) ,
+(u, v, r) �→ (u, v, 1/r) .
+The following lemma, proved in subsection 2.1, formalizes the correspondence between
+the radially symmetric stationary solutions stable at infinity for system (1.1) and the
+manifolds defined above.
+Lemma 1.4. Let u∞ be a point of Σmin(V ). A (global) solution [0, +∞) → Rdst, r �→ u(r)
+of system (1.2) belongs to SV, u∞ if and only if its trajectory (in R2dst × (0, +∞))
+(1.14)
+��u(r), ˙u(r), r
+� : r ∈ (0, +∞)
+�
+belongs to the intersection
+(1.15)
+W u, 0, +
+V
+∩ ι−1�W cs, ∞, +
+V
+(u∞)
+� .
+1.4 Transversality of radially symmetric stationary solutions stable at infinity
+Definition 1.5. Let u∞ be a point of Σmin(V ). A radially symmetric stationary solution
+stable close to u∞ at infinity for system (1.1) (in other words, a function u of SV, u∞) is
+said to be transverse if the intersection (1.15) is transverse, in R2dst × (0, +∞), along the
+trajectory (1.14).
+Remark. The natural analogue of radially symmetric stationary solutions stable at infinity
+when space dimension dsp is equal to 1 are symmetric standing pulses stable at infinity
+(see Definition 1.5 of [1]), and the natural analogue for such pulses of Definition 1.5 above
+7
+
+is their elementarity, not their transversality (see Definition 1.4 and Definition 1.6 of [1]).
+However, the transversality of a symmetric standing pulse (when the space dimension
+dsp equals 1) makes little sense in higher space dimension, because of the singularity at r
+equals 0 for the differential systems (1.2) and (1.5), or because of the related fact that
+the subspace {r = 0} is invariant for the differential system (1.7). For that reason, the
+adjective transverse (not elementary) is chosen to qualify the property considered in
+Definition 1.5 above.
+1.5 The space of potentials
+For the remaining of the paper, let us take and fix an integer k not smaller than 1. Let us
+consider the space Ck+1
+b
+(Rdst, R) of functions Rd → R of class Ck+1 which are bounded,
+as well as their derivatives of order not larger than k + 1, equipped with the norm
+∥W∥Ck+1
+b
+=
+max
+α multi-index, |α|≤k+1 ∥∂|α|
+uαW∥L∞(Rd,R) ,
+and let us embed the larger space Ck+1(Rdst, R) with the following topology: for V in
+this space, a basis of neighbourhoods of V is given by the sets V + O, where O is an
+open subset of Ck+1
+b
+(Rdst, R) embedded with the topology defined by ∥·∥Ck+1
+b
+(which can
+be viewed as an extended metric). For comments concerning the choice of this topology,
+see subsection 1.4 of [1].
+1.6 Main result
+The following generic transversality statement is the main result of this paper.
+Theorem 1 (generic transversality of radially symmetric stationary solutions stable at
+infinity). There exists a generic subset G of
+�
+Ck+1(Rdst, R), ∥·∥Ck+1
+b
+�
+such that, for every
+potential function V in G, every radially symmetric stationary solution stable at infinity
+of the parabolic system (1.1) is transverse.
+Theorem 1 can be viewed as the extension to higher space dimensions (for radially
+symmetric solutions) of conclusion 2 of Theorem 1.7 of [1] (which is concerned with
+elementary standing pulses stable at infinity in space dimension 1). A short comparison
+between these two results and their proofs is provided in the next subsection. For more
+comments and a short historical review on transversality results in similar contexts, see
+subsection 1.6 of the same reference.
+The core of the paper (section 4) is devoted to the proof of the conclusions of Theorem 1
+among potentials which are quadratic past a certain radius (defined in (3.2)), as stated
+in Proposition 4.1. The extension to general potentials of Ck+1
+b
+(Rdst, R) is carried out in
+section 5.
+Remark. As in [1] (see Theorem 1.8 of that reference), the same arguments could be
+called upon to prove that the following additional conclusions hold, generically with
+respect to the potential V :
+8
+
+1. for every minimum point of V , the smallest eigenvalue of D2V at this minimum
+point is simple;
+2. every radially symmetric stationary solution stable at infinity of the parabolic system
+(1.1) approaches its limit at infinity tangentially to the eigenspace corresponding to
+the smallest eigenvalue of D2V at this point.
+1.7 Key differences with the generic transversality of standing pulses in
+space dimension one
+Table 1.1 lists the key differences between the proof of the generic elementarity of
+symmetric standing pulses carried out in [1], and the proof of the generic transversality
+of radially symmetric stationary solutions carried out in the present paper (implicitly,
+the other steps/features of the proofs are similar or identical). The state dimension,
+which is simply denoted by d in [1], is here denoted by dst in both cases. Some of the
+notation/rigour is lightened.
+Symmetric standing pulse
+Radially symmetric
+stationary solution
+Critical point at infinity
+critical point e, E = (e, 0Rdst )
+minimum point u∞
+Symmetry subspace Ssym
+{(u, v) ∈ R2dst : v = 0},
+dimension dst
+{(u, v, r) ∈ R2dst+1 : (v, r) = (0, 0)},
+dimension dst
+Differential system
+governing the profiles
+autonomous, conservative,
+regular at Ssym
+non-autonomous, dissipative,
+singular at reversibility subspace
+Direction of the flow
+E → Ssym
+Ssym → u∞
+Invariant manifold at
+infinity
+W u(E), dimension dst − m(e)
+W cs, ∞, +(u∞), dimension dst + 1
+Invariant manifold at
+symmetry subspace
+none
+W u, 0, +, dimension dst + 1
+Transversality
+W u(E) ⋔ Ssym
+W cs, ∞, +(u∞) ⋔ W u, 0, +
+Transversality of spatially
+homogeneous solutions
+irrelevant
+Proposition 2.2
+Interval Ionce (values
+reached only once)
+“anywhere”
+close to Ssym
+M (departure set of Φ)
+parametrization of ∂W u
+loc, V (E)
+and time, dimension dst − m(e)
+Ssym and W cs, ∞, +
+loc
+(u∞) at r = N,
+dimension 2dst
+N (arrival set of Φ)
+R2dst
+R2dst × R2dst
+W (target manifold)
+Ssym
+diagonal of N
+dim(M) − codim(W)
+−m(e)
+0
+Condition to be fulfilled
+by perturbation W
+�
+DΦ(W)
+�� (0, ψ)�
+̸= 0
+�
+DΦu(W)
+�� (φ, ψ)�
+̸= 0
+Perturbation W, case 3
+precluded
+W(u0) ̸= 0
+Table 1.1: Formal comparison between the generic elementarity of symmetric standing
+pulses (space dimension 1) proved in [1], and the generic transversality of radially
+symmetric stationary solutions (higher space dimension dsp) proved in the present paper.
+Here are a few additional comments about these differences.
+9
+
+Concerning the critical point at infinity, u∞ is assumed (here) to be a minimum point,
+whereas (in [1]) the Morse index of e is any. Indeed, if the Morse index m(u∞) of u∞ was
+positive, then the dimension of the centre-stable manifold W cs, ∞, +
+V
+(u∞) would be equal
+to dst + m(u∞) + 1; as a consequence, proving the transversality of the intersection (1.15)
+in that case would require more stringent regularity assumptions on V (see hypothesis
+1 of Theorem 4.2 of [1]) while nothing particularly useful could be derived from this
+transversality. On the other hand, assuming that u∞ is a minimum point allows to view
+its local centre-stable manifold as a graph (u, c) �→ v (see Proposition 2.4), which is
+slightly simpler.
+Concerning the interval Ionce providing values u reached “only once” by the profile
+(Lemma 2.3), the proof of the present paper takes advantage of the dissipation to find a
+convenient interval close to the “departure point” u0, as was done in [1] for travelling
+fronts (whereas, for standing pulse, the interval is to be found “anywhere”, thanks to the
+conservative nature of the differential system governing the profiles, see conclusion 1 of
+Proposition 3.3 of [1]).
+Concerning the function Φ to which Sard–Smale theorem is applied in the present
+paper, both manifolds W u, 0, + and W cs, ∞, +(u∞) depend on the potential V . However,
+the transversality of an intersection between these two manifolds can be seen as the
+transversality of the image of Φ with the (fixed) diagonal of R2dst × R2dst, for a function
+Φ combining the parametrization of these two manifolds. This trick, which is the same
+as in [1] for travelling fronts, allows to apply Sard–Smale theorem to a function Φ with a
+fixed arrival space N containing a fixed target manifold W (in this case the diagonal of
+N). By contrast, for symmetric standing pulses in [1], since the subspace Ssym involved
+in the transverse intersection is fixed, the previous trick is unnecessary and the setting is
+simpler.
+Finally, a technical difference occurs in “case 3” of the proof that the degrees of freedom
+provided by perturbing the potential allow to reach enough directions in the arrival state
+of Φ (Lemma 4.6, which is the core of the proof). In [1], case 3 is shown to lead to a
+contradiction, not only for symmetric standing pulses, but also for asymmetric ones and
+for travelling fronts. Here, such a contradiction does not seem to occur (or at least is
+more difficult to prove), but this has no harmful consequence: a suitable perturbation of
+the potential can still be found in this case.
+2 Preliminary properties
+2.1 Proof of Lemma 1.4
+Let V denote a potential function in Ck+1(Rdst, R). Let (0, +∞) → Rdst, r �→ u(r) denote
+a (global) solution of system (1.2), assumed to be stable close to some point u∞ of
+Σmin(V ) at infinity (Definition 1.2). Lemma 1.4 follows from the next lemma.
+Lemma 2.1. The derivative ˙u(r) goes to 0 as r goes to +∞.
+10
+
+Proof. Let us consider the Hamiltonian function
+(2.1)
+HV : R2dst → R ,
+(u, v) �→ v2
+2 − V (u) ,
+and, for every r in (0, +∞), let
+h(r) = HV
+�u(r), ˙u(r)
+� .
+It follows from system (1.2) that, for every r in (0, +∞),
+(2.2)
+˙h(r) = −dsp − 1
+r
+˙u(r)2 ,
+thus the function h(·) decreases, and it follows from the expression (2.1) of the Hamiltonian
+that this function converges, as r goes to +∞, towards a finite limit h∞ which is not
+smaller than −V (u∞).
+Let us proceed by contradiction and assume that h∞ is larger than −V (u∞). Then, it
+follows again from the expression (2.1) of the Hamiltonian that the quantity ˙u(r)2 con-
+verges towards the positive quantity 2
+�h∞ + V (u∞)
+� as r goes to +∞. As a consequence,
+it follows from equality (2.2) that h(r) goes to −∞ as r goes to +∞, a contradiction.
+Lemma 2.1 is proved.
+2.2 Transversality of homogeneous radially symmetric stationary solutions
+stable at infinity
+Proposition 2.2. For every potential function V in Ck+1(Rdst, R) and for every nonde-
+generate minimum point u∞ of V , the constant function
+[0, +∞) → Rdst ,
+r �→ u∞ ,
+which defines an (homogeneous) radially symmetric stationary solution stable at infinity
+for system (1.1) , is transverse (in the sense of Definition 1.5).
+Proof. Let V denote a function in Ck+1(Rdst, R) and u∞ denote a nondegenerate minimum
+point of V . The function [0, +∞) → Rdst, r �→ u∞ is a (constant) solution of the
+differential system (1.5), and the linearization of this differential system around this
+solution reads
+(2.3)
+¨u = −dsp − 1
+r
+˙u + D2V (u∞) · u .
+Let (0, +∞) → Rdst, r �→ u(r) denote a nonzero solution of this differential system, and,
+for every r in (0, +∞), let
+v(r) = ˙u(r)
+and
+U(r) =
+�u(r), v(r)
+�
+and
+q(r) = u(r)2
+2
+.
+11
+
+Then (omitting the dependency on r),
+˙q = u · ˙u
+and
+¨q = ˙u2 + u · ¨u = ˙u2 − dsp − 1
+r
+˙q + D2V (u∞) · (u, u) ,
+so that
+d
+dr
+�rdsp−1 ˙q(r)
+� = rdsp−1
+�
+¨q + dsp − 1
+r
+˙q
+�
+= rdsp−1� ˙u2 + D2V (u∞) · (u, u)
+� .
+Since r �→ u(r) was assumed to be nonzero, it follows that the quantity rdsp−1 ˙q(r) is
+strictly increasing on (0, +∞). To prove the intended conclusion, let us proceed by
+contradiction and assume that, for every r in (0, +∞),
+�u(r), v(r), r
+� belongs:
+1. to the tangent space T(u∞,0Rdst ,r)W u, 0, +
+V
+(u∞),
+2. and to the tangent space T(u∞,0Rdst ,r)
+�
+ι−1�W cs, ∞, +
+V
+(u∞)
+��
+.
+As in (1.6), let us introduce the auxiliary variables τ (equal to log(r)) and c (equal to
+1/r). With this notation, system (2.3) is equivalent to
+(2.4)
+�
+�
+�
+�
+�
+�
+�
+uτ = rv
+vτ = −(dsp − 1)v + rD2V (u∞) · u
+rτ = r ,
+and to
+(2.5)
+�
+�
+�
+�
+�
+�
+�
+ur = v
+vr = −(dsp − 1)cv + D2V (u∞) · u
+cr = −c2 .
+Assumptions 1 and 2 above yield the following conclusions.
+1. In view of the limit of system (2.4) as r goes to 0+, it follows from assumption 1
+that there exists δu0 in Rdst such that
+�u(r), v(r)
+� goes to (δu0, 0Rdst) as r goes to
+0+;
+2. and in view of the limit of system (2.5) as c goes to 0+, it follows from assumption
+2 that
+�u(r), v(r)
+� goes to (0Rdst, 0Rdst), at an exponential rate, as r goes to +∞.
+It follows from these two conclusions that the quantity rdsp−1 ˙q(r) goes to 0 as r goes
+to 0+ and as r goes to +∞, a contradiction with the fact (observed above) that this
+quantity is strictly increasing with r. Proposition 2.2 is proved.
+12
+
+2.3 Additional properties close to the origin
+Let V denote a potential function in Ck+1(Rdst, R) and let u0 be a point in Rdst. Let
+us recall (see subsection 1.3) that the unstable manifold W u, 0
+V
+(u0) of the equilibrium
+(u0, 0Rdst, 0) for the autonomous differential system (1.7)) is one-dimensional.
+As a
+consequence there exists a unique solution r �→ u(r) of the differential system (1.2) such
+that the image of the map r �→
+�u(r), ˙u(r), r) lies in the intersection W u, 0, +
+V
+(u0) of this
+unstable manifold with the half-space where r is positive (this intersection was defined in
+(1.10)); or, in other words, such that
+�u(r), ˙u(r)
+� goes to (u0, 0) as r goes to 0+. This
+solution is defined on some (maximal) interval (0, rmax), where rmax is either a finite
+quantity or +∞. The following lemma provides properties of this solution that will be
+used in the sequel. To ease its statement, let us assume that rmax is equal to +∞ (only
+this case will turn out to be relevant), and let us consider the continuous extension of
+u(·) to the interval [0, +∞) (and let us still denote by u(·) this continuous extension).
+Lemma 2.3. If u(·) is not identically equal to u0 (in other words, if u0 is not a critical
+point of V ), then there exists a positive quantity ronce such that, denoting by Ionce the
+interval [0, ronce), the following conclusions hold:
+1. the function ˙u(·) does not vanish on Ionce,
+2. and, for every r∗ in Ionce and r in [0, +∞),
+u(r) = u(r∗) =⇒ r = r∗.
+Proof. The linearized system (1.7) at the equilibrium (u0, 0Rdst, 0) reads:
+d
+dτ
+�
+�
+�
+δu
+δv
+δr
+�
+�
+� =
+�
+�
+�
+0
+0
+0
+0
+−(dsp − 1)
+∇V (u0)
+0
+0
+1
+�
+�
+�
+�
+�
+�
+δu
+δv
+δr
+�
+�
+� ,
+thus the tangent space at (u0, 0Rdst, 0) to W u, 0
+V
+(u0) (the unstable eigenspace of the matrix
+of this system) is spanned by the vector
+�0, ∇V (u0)/dsp, 1
+�; it follows that
+(2.6)
+˙u(r) = r
+dsp
+∇V (u0)
+�1 + or→0+(r)
+� .
+Thus, if r0 is a sufficiently small positive quantity, then ˙u(·) does not vanish on (0, r0]
+(so that conclusion 1 of Lemma 2.3 holds provided that ronce is not larger than r0), and
+the map
+(2.7)
+[0, r0] → Rdst ,
+r �→ u(r)
+is a C1-diffeomorphism onto its image. For r in [0, +∞), let us denote
+�u(r), ˙u(r)
+� by
+U(r). According to the decrease (2.2) of the Hamiltonian, there exists a quantity ronce in
+(0, r0) such that, for every r∗ in [0, ronce),
+(2.8)
+HV
+�U(r0)
+� < −V
+�u(r∗)
+� .
+13
+
+Take r∗ in [0, ronce] and r in [0, +∞), and let us assume that u(r) equals u(r∗). If r was
+larger than r0 then it would follow from the expression (2.1) of the Hamiltonian, its
+decrease (2.2), and inequality (2.8) that
+−V
+�u(r)
+� ≤ HV
+�U(r)
+� ≤ HV
+�U(r0)
+� < −V
+�u(r∗)
+� ,
+a contradiction with the equality of u(r) and u(r∗). Thus r is not larger than r0, and
+it follows from the one-to-one property of the function (2.7) that r must be equal to
+r∗; conclusion 2 of Lemma 2.3 thus holds, and Lemma 2.3 is proved.
+2.4 Additional properties close to infinity
+Let V1 denote a potential function in Ck+1(Rdst, R) and u1,∞ denote a nondegenerate
+minimum point of V1. According to the implicit function theorem, there exists a (small)
+neighbourhood νrobust(V1, u1,∞) of Vquad-R and a Ck-function V �→ u∞(V ) defined on
+νrobust(V1, u1,∞) and with values in Rdst such that u∞(V1) equals u1,∞ and, for every
+V in νrobust(V1, u1,∞), u∞(V ) is a local minimum point of V . The following proposi-
+tion is nothing but the local centre-stable manifold theorem applied to the equilibrium
+�u∞(V ), 0Rdst, 0
+� of the (autonomous) differential system (1.8), for V close to V1. Addi-
+tional comments and references concerning local stable/centre/unstable manifolds are
+provided in subsection 2.2 of [1].
+Proposition 2.4 (local centre-stable manifold at infinity). There exist a neighbourhood
+ν of V1 in Ck+1(Rdst, R), included in νrobust(V1, u1,∞), such that, if ε1 and c1 denote
+sufficiently small positive quantities, then, for every V in ν, there exists a Ck-map
+(2.9)
+wcs, ∞
+loc, V : BRdst(u1,∞, ε1) × [0, c1] → Rdst ,
+(u, c) �→ wcs, ∞
+loc, V (u, c) ,
+such that, for every (u0, v0, c0) in BRdst(u1,∞, ε1) × Rdst × [0, c1], the following two
+statements are equivalent:
+1. v = wcs, ∞
+loc, V (u, c);
+2. the solution r �→
+�u(r), v(r), c(r)
+� of the differential system (1.8) with initial condi-
+tion (u0, v0, c0) at time r0 = 1/c0 is defined up to +∞, remains in BRdst(u1,∞, ε1)×
+Rdst × [0, c1] of all r larger than r0, and goes to
+�u∞(V ), 0Rdst, 0
+� as r goes to +∞.
+In particular, wcs, ∞
+loc, V
+�u∞(V ), 0
+� is equal to 0Rdst. In addition, the map
+BRdst(u1,∞, ε1) × [0, c1] × ν → Rdst ,
+(u, c, V ) �→ wcs, ∞
+loc, V (u, c)
+is of class Ck (with respect to u and c and V ), and, for every V in ν, the graph of the
+differential at
+�u∞(V ), 0) of the map (u, c) �→ wcs, ∞
+loc, V (u, c) is equal to the centre-stable
+subspace of the linearization at
+�u∞(V ), 0Rdst, 0
+� of the differential system (1.8).
+14
+
+Let us denote by W cs, ∞, +
+loc, V, ε1, c1
+�u∞(V )
+� the graph of the map (2.9) (restricted to positive
+values of c), see figure 1.1; with symbols,
+(2.10)
+W cs, ∞, +
+loc, V, ε1, c1
+�u∞(V )
+� =
+��u, wcs, ∞
+loc, V (u, c), c
+� : (u, c) ∈ BRdst(u1,∞, ε1) × (0, c1]
+�
+.
+This set defines a local centre-manifold (restricted to positive values of c) for the equilib-
+rium
+�u∞(V ), 0Rdst, 0
+� of the differential system (1.8). Its uniqueness (for positive values
+of c) is ensured by the dynamics of the centre component c, which, according to the
+third equation of system (1.8), decreases to 0 (see figure 1.1). The global centre-stable
+manifold W cs, ∞, +
+V
+�u∞(V )
+� already defined in (1.13) can be redefined as the points of
+R2dst ×(0, +∞) that eventually reach the local centre manifold W cs, ∞, +
+loc, V, ε1, c1
+�u∞(V )
+� when
+they are transported by the flow of the differential system (1.8).
+Remark. If the state dimension dst is equal to 1, then a calculation shows that
+wcs, ∞
+loc, V (u, c) = −
+�u − u∞(V )
+� ��
+V ′′�u∞(V )
+� + dsp − 1
+2
+c + . . .
+�
+,
+where “. . . ” stands for higher order terms in u − u∞(V ) and c. In particular the quantity
+∂c∂uwcs, ∞
+loc, V
+�u∞(V ), 0
+� is equal to the (negative) quantity −(dsp − 1)/2. The display of
+the local centre-stable manifold at infinity on figure 1.1 fits with the sign of this quantity.
+3 Tools for genericity
+Let
+(3.1)
+Vfull = Ck+1(Rdst, R) ,
+and, for a positive quantity R, let
+(3.2)
+Vquad-R =
+�
+V ∈ Vfull : for all u in Rd, |u| ≥ R =⇒ V (u) = u2
+2
+�
+.
+Let us recall the notation SV introduced in (1.4).
+Lemma 3.1. For every positive quantity R and for every potential V in Vquad-R, the
+following conclusions hold.
+1. The flow defined by the differential system (1.2) (governing radially symmetric
+stationary solutions of the parabolic system (1.1)) is global (that is, every solution
+is defined on (0, +∞)).
+2. For every u in SV , the following bound holds:
+(3.3)
+sup
+r∈(0,+∞)
+|u(r)| < R .
+15
+
+Proof. Let V be in Vquad-R. According to the definition (3.2) of Vquad-R, there exists a
+positive quantity K such that, for every u in Rdst,
+|∇V (u)| ≤ K + |u| .
+As a consequence, the following inequalities hold for the right-hand side of the first order
+differential system (1.5):
+����
+�
+v, −dsp − 1
+r
+v + ∇V (u)
+����� ≤ |v| + dsp − 1
+r
+|v| + K + |u| ≤ K +
+�
+2 + dsp − 1
+r
+�
+|(u, v)| ,
+and this bound prevents the solution from blowing up in finite time, which proves
+conclusion 1.
+Now, take a function u in SV . Let us still denote by u(·) the continuous extension of
+this solution to [0, +∞). For every r in [0, +∞), let
+q(r) = u(r)2
+2
+and
+Q(r) = rdsp−1 ˙q(r) .
+Then (omitting the dependency on r),
+˙q = u · ˙u
+and
+¨q = ˙u2 + u · ¨u = ˙u2 − dsp − 1
+r
+˙q + u · ∇V (u) ,
+so that
+˙Q = rdsp−1
+�
+¨q + dsp − 1
+r
+˙q
+�
+= rdsp−1� ˙u2 + u · ∇V (u)
+� .
+According to the definition (3.2) of Vquad-R, there exists a positive quantity δ (sufficiently
+small) so that, for every w in Rdst,
+(3.4)
+|w| ≥ R − δ =⇒ w · ∇V (w) ≥ w2
+2 .
+Let us proceed by contradiction and assume that supr∈(0,+∞) |u(r)| is not smaller than
+R. Since u(·) is stable at infinity and since the critical points of V belong to the open
+ball BRdst(0, R − δ), it follows that the set
+�r ∈ [0, +∞) : |u(r)| ≥ R
+�
+is nonempty; let rout denote the minimum of this set. For the same reason, the set
+�r ∈ (rout, +∞) : |u(r)| < R − δ
+�
+is also nonempty. Let rback denote the infimum of this last set. It follows from these
+definitions that rback is larger than rout and that, for every r in (rout, rback), according to
+inequality (3.4),
+(3.5)
+˙Q(r) ≥ rdsp−1
+�
+˙u2(r) + u2(r)
+2
+�
+> 0 .
+16
+
+If on the one hand rout equals 0 then |u(0)| is not smaller than R and, since Q(0) equals
+0, it follows from inequality (3.5) that Q(·) is positive on (0, rback), so that the same is
+true for ˙q(·). Thus q(·) is strictly increasing on [0, rback] and |u(rback)| must be larger
+than |u(rout)|, a contradiction with the definition of rback. If on the other hand rout is
+positive, then |u(rout)| is equal to R and ˙q(rout) is nonnegative so that the same is true
+for Q(rout), and it again follows from inequality (3.5) that Q(·) is positive on (0, rback),
+yielding the same contradiction. Conclusion 2 of Lemma 3.1 is proved.
+Notation. For every positive quantity R and every potential V in Vquad-R, let
+(3.6)
+SV : (0, +∞)2 × R2dst → R2dst ,
+�(rinit, r), (uinit, vinit)
+� �→ SV
+�(rinit, r), (uinit, vinit)
+�
+denote the (globally defined) flow of the (non-autonomous) differential system (1.5) for
+this potential V . In other words, for every rinit in (0, +∞) and (uinit, vinit) in R2dst, the
+function
+(0, +∞) → R2dst ,
+r �→ SV
+�(rinit, r1), (uinit, vinit)
+�
+is the solution of the differential system (1.5) for the initial condition (uinit, vinit) at r
+equals rinit. According to subsection 1.3, the flow SV may be extended to the larger set
+(0, +∞)2 × R2dst ∪ [0, +∞)2 × Rdst × {0Rdst} ;
+according to this extension, for every u0 in Rdst, the solution taking its values in the
+(one-dimensional) unstable manifold W u, 0, +
+V
+(u0) reads:
+(3.7)
+[0, +∞) → Rdst ,
+r �→ SV
+�(0, r), (u0, 0Rdst)
+� .
+4 Generic transversality among potentials that are quadratic
+past a given radius
+4.1 Notation and statement
+Let us recall the notation SV and SV, u∞ introduced in (1.4).
+Proposition 4.1. There exists a generic subset of Vquad-R such that, for every potential
+V in this subset, every radially symmetric stationary solution stable at infinity of the
+parabolic system (1.1) (in other words, every u in SV ) is transverse.
+4.2 Reduction to a local statement
+Let V1 denote a potential function in Vquad-R and u1,∞ denote a nondegenerate minimum
+point of V1. According to the implicit function theorem, there exists a (small) neighbour-
+hood νrobust(V1, u1,∞) of Vquad-R and a Ck-function u∞(·) defined on νrobust(V1, u1,∞) and
+with values in Rdst such that u∞(V1) equals u1,∞ and, for every V in νrobust(V1, u1,∞),
+u∞(V ) is a local minimum point of V . The following local generic transversality statement
+yields Proposition 4.1 (as shown below).
+17
+
+Proposition 4.2. There exists a neighbourhood νV1, u1,∞ of V1 in νrobust(V1, u1,∞) and
+a generic subset νV1, u1,∞, gen of νV1, u1,∞ such that, for every V in νV1, u1,∞, gen, every
+radially symmetric stationary solution stable close to u∞(V ) at infinity of the parabolic
+system (1.1) (in other words, every u in SV, u∞(V )) is transverse.
+Proof that Proposition 4.2 yields Proposition 4.1. Let us denote by Vquad-R-Morse the
+dense open subset of Vquad-R defined by the Morse property:
+(4.1)
+Vquad-R-Morse = {V ∈ Vquad-R : all critical points of V are nondegenerate} .
+Let V1 denote a potential function in Vquad-R-Morse. According to the Morse property
+its minimum points are isolated and since V1 is in Vquad-R they belong to the open ball
+BRd(0, R), so that those minimum points are in finite number. Assume that Proposi-
+tion 4.2 holds. With the notation of this proposition, let us consider the following two
+intersections, at each time over all minimum points u1,∞ of V1:
+(4.2)
+νV1 =
+�
+νV1, u1,∞
+and
+νV1, gen =
+�
+νV1, u1,∞, gen .
+Since those are finite intersections, νV1 is still a neighbourhood of V1 in Vquad-R and the
+set νV1, gen is still a generic subset of νV1. This shows that the set
+{V ∈ Vquad-R-Morse :
+every u in SV, u∞(V ) is transverse}
+is locally generic. Applying Lemma 4.3 of [1] as in Subsection 5.2 of this reference shows
+that this local genericity implies the global genericity stated in Proposition 4.1, which is
+therefore proved.
+4.3 Proof of the local statement (Proposition 4.2)
+4.3.1 Setting
+For the remaining part of this section, let us fix a potential function V1 in Vquad-R and a
+nondegenerate minimum point u1,∞ of V1. Let ν be a neighbourhood of V1 in Vquad-R,
+included in νrobust(V1, u1,∞), and let ε1 and c1 be positive quantities, with ν and ε1 and
+c1 small enough so that the conclusions of Proposition 2.4 hold. Let
+r1 = 1/c1
+and
+M = Rdst × BRdst(u1,∞, ε1)
+and
+Λ = ν ,
+and
+N = (R2dst)2
+and
+W = {(A, B) ∈ N : A = B}
+,
+thus W is the diagonal of N. Let N denote an integer not smaller than r1, and let us
+consider the functions
+Φu : Rdst × Λ → R2dst ,
+(u0, V ) �→ SV
+�(0, N), (u0, 0Rdst)
+� ,
+and
+Φcs : BRdst(u1,∞, ε1) × Λ → R2dst ,
+(uN, V ) �→
+�uN, wcs, ∞
+loc, V (uN, 1/N)
+� ,
+and the function
+(4.3)
+Φ : M × Λ → N ,
+(m, V ) = (u0, uN, V ) �→
+�Φu(u0, V ), Φcs(uN, V )
+� .
+18
+
+4.3.2 Equivalent characterizations of transversality
+Let us consider the set
+SΛ,u1,∞,N =
+�(V, u) : V ∈ Λ and u ∈ SV, u∞(V ) and u(N) ∈ BRdst(u1,∞, ε1)
+� .
+Proposition 4.3. The map
+(4.4)
+Φ−1(W) → SΛ,u1,∞,N ,
+(u0, u, V ) �→
+�
+V, r �→ SV
+�(0, r), (u0, 0Rdst
+��
+is well defined and one-to-one.
+Proof. The image by Φ of a point (u0, uN, V ) of M × Λ belongs to the diagonal W of
+N if and only if Φu(u0, V ) equals Φcs(uN, V ), and in this case the function u : r �→
+SV
+�(0, r), (u0, 0Rdst
+� belongs to SV, u∞(V ) and u(N) (which is equal to uN) belongs to
+BRdst(u1,∞, ε1), so that (V, u) belongs to SΛ,u1,∞,N. The map (4.4) above is thus well
+defined.
+Now, for every (V, u) in SΛ,u1,∞,N, if we denote by u0 the limit limr→0+ u(r) and by
+uN the vector u(N), then (u0, uN, V ) is the only possible antecedent of (V, u) by the map
+(4.4). In addition,
+SV
+�(0, N), (u0, 0Rdst)
+� =
+�uN, ˙u(N)
+� ,
+and since u(r) goes to u∞(V ) as r goes to +∞, the vector
+�u(N), ˙u(N), 1/N
+� must
+belong to the centre-stable manifold W cs, ∞, +
+V
+�u∞(V )
+� of u∞(V ), so that, according to
+the definition of wcs, ∞
+loc, V ,
+˙u(N) = wcs, ∞
+loc, V
+�u(N), 1/N
+� ,
+and this yields the equality between Φu(u0, V ) and Φcs(uN, V ). Thus Φ(V, u) belongs to
+W and (u0, uN, V ) belongs to Φ−1(W). Proposition 4.3 is proved.
+Proposition 4.4. For every potential function V in Λ, the following two statements are
+equivalent.
+1. The image of the function M → N, m �→ Φ(m, V ) is transverse to W.
+2. Every u in SV, u∞(V ) such that u(N) is in BRdst(u1,∞, ε1) is transverse.
+Remark. According to Proposition 2.2, for every V in Λ, the constant function r �→ u∞(V ),
+which belongs to SV , is already (a priori) known to be transverse, therefore only
+nonconstant solutions matter in statement 2 of this proposition.
+Proof. Let us consider (m2, V2) in M × Λ such that Φ(m2, V2) is in W, let (u2,0, u2,N)
+denote the components of m2, and let r �→ u2(r) and r �→ U2(r) denote the functions
+satisfying, for all r in [0, +∞),
+U2(r) =
+�u2(r), ˙u2(r)
+� = SV
+�(0, r), (u2,0, 0Rdst
+� .
+Let us consider the map
+∆Φ : M → R2dst ,
+(u0, uN) �→ Φu(u0, V2) − Φcs(uN, V2) ,
+19
+
+and let us write, only for this proof, DΦ and DΦu and DΦcs and D(∆Φ) for the
+differentials of Φ and Φu and Φcs and ∆Φ at (m2, V2) and with respect to all variables in
+M (but not with respect to V ). According to Definition 1.5, the transversality of u2 is
+defined as the transversality of the intersection W u, 0, +
+V2
+∩ ι−1�
+W cs, ∞, +
+V2
+�u∞(V2)
+��
+along
+the trajectory of U2. This transversality can be considered at a single point, no matter
+which, of the trajectory U2
+�(0, +∞)
+�, in particular at the point Φu(u2,0, V2) which is
+equal to Φcs�u2(N), V 2
+�, and is equivalent to the transversality of the dst-dimensional
+manifolds
+W u, 0, +
+V2
+∩
+�R2dst × {N}
+�
+and
+ι−1�
+W cs, ∞, +
+V2
+�u∞(V2)
+��
+∩
+�R2dst × {N}
+�
+in R2dst ×{N}. It is therefore equivalent to the surjectivity of the map D(∆Φ) (statement
+(B) in Lemma 4.5 below). On the other hand, the image of the function M → N,
+m �→ Φ(m, V2) is transverse at Φ(m, V2) to the diagonal W of N if and only if the image
+of DΦ contains a complementary space of this diagonal (statement (A) in Lemma 4.5
+below)). Thus Proposition 4.4 is a consequence of the next lemma.
+Lemma 4.5. The following two statements are equivalent.
+(A) The image of DΦ contains a complementary subspace of the diagonal W of N.
+(B) The map D(∆Φ) is surjective.
+Proof. If statement (A) holds, then, for every (α, β) in N, there exist γ in R2dst and δm
+in Tm2M such that
+(4.5)
+(γ, γ) + DΦ · δm = (α, β) ,
+so that
+(4.6)
+D(∆Φ) · δm = α − β ,
+and statement (B) holds. Conversely, if statement (B) holds, then, for every (α, β) in
+N, there exists δm in Tm2M such that (4.6) holds, and as a consequence, if (δu0, δuN)
+denote the components of δm, then α − DΦu(δu0) is equal to β − DΦcs(δuN), and if
+this vector is denoted by γ, then equality (4.5) holds, and this shows that statement (A)
+holds.
+As explained above, Proposition 4.4 follows from Lemma 4.5, and is therefore proved.
+4.3.3 Checking hypothesis 1 of Theorem 4.2 of [1]
+The function Φ is as regular as the flow SV , thus of class Ck. It follows from the definitions
+of M and N and W that
+dim(M) − codim(W) = (dst + dst) − 2dst = 0 ,
+so that hypothesis 1 of Theorem 4.2 of [1] is fulfilled.
+20
+
+4.3.4 Checking hypothesis 2 of Theorem 4.2 of [1]
+For every V in Vquad-R, let us recall the notation SV introduced in (3.6) and (3.7) for the
+flow of the differential system (1.5). Take (m2, V2) in the set Φ−1(W). Let (u2,0, u2,N)
+denote the components of m2, and, for every r in (0, +∞), let us write
+U2(r) =
+�u2(r), v2(r)
+� = SV2
+�(0, r), (u2,0, 0Rdst)
+� .
+Let us write
+DΦ
+and
+DΦu
+and
+DΦcs
+for the full differentials (with respect to arguments m in M and V in Λ) of the three
+functions Φ and Φu and Φcs respectively at the points
+�u2,0, u2,N, V2
+�,
+�u2,0, V2
+� and
+�u2,N, V2
+�. Checking hypothesis 2 of Theorem 4.2 of [1] amounts to prove that
+(4.7)
+im(DΦ) + W = N .
+If u2(·) is constant (that is, identically equal to u∞(V2)), then equality (4.7) follows from
+Proposition 2.2. Thus, let us assume that u2(·) is nonconstant. In this case, equality
+(4.7) is a consequence of the following lemma.
+Lemma 4.6. For every nonzero vector (φ2, ψ2) in R2dst, there exists a function W in
+Ck+1
+b
+(Rdst, R) such that
+supp(W) ⊂ BRd(0, R) ,
+(4.8)
+and
+�DΦu · (0, 0, W)
+�� (φ2, ψ2)
+� ̸= 0 ,
+(4.9)
+and
+DΦcs · (0, 0, W) = 0R2dst .
+(4.10)
+Proof that Lemma 4.6 yields equality (4.7). Inequality (4.9) shows that the orthogonal
+complement, in R2dst, of the directions that can be reached by DΦu·(0, 0, W) for potentials
+W satisfying (4.8) and (4.10) is reduced to 0R2dst; in other words, all directions of R2dst
+can be reached by that means. This shows that
+im(DΦ) ⊃ R2dst × {0R2dst} ,
+and since the subspace at the right-hand side of this inclusion is transverse to W in
+R4dst, this proves equality (4.7) (and shows that hypothesis 2 of Theorem 4.2 of [1] is
+fulfilled).
+Proof of Lemma 4.6. Let (φ2, ψ2) denote a nonzero vector in R2dst, let W be a function
+in Ck+1
+b
+(Rdst, R) satisfying the inclusion
+(4.11)
+supp(W) ⊂ BRd(0, R) \ BRdst(u1,∞, ε1) ,
+and observe that inclusion (4.8) and equality (4.10) follow from this inclusion (4.11). Let
+us consider the linearization of the differential system (1.2), for the potential V2, around
+the solution r �→ U2(r):
+(4.12)
+d
+dr
+�
+δu(r)
+δv(r)
+�
+=
+�
+0
+id
+D2V2
+�u2(r)
+�
+−dsp−1
+r
+� �
+δu(r)
+δv(r)
+�
+,
+21
+
+and let T(r, r′) denote the family of evolution operators obtained by integrating this
+linearized differential system between r and r′. It follows from the variation of constants
+formula that
+(4.13)
+DΦu · (0, 0, W) =
+� N
+−∞
+T(r, N)
+�
+0, ∇W
+�u2(r)
+��
+dr .
+For every r in (0, +∞), let T ∗(r, N) denote the adjoint operator of T(r, N), and let
+(4.14)
+�φ(r), ψ(r)
+� = T ∗(r, N) · (φ2, ψ2) .
+According to expression (4.13), inequality (4.9) reads
+� N
+−∞
+��
+0, ∇W
+�u2(r)
+�� ��� T ∗(r, N) · (φ2, ψ2)
+�
+dr ̸= 0 ,
+or equivalently
+(4.15)
+� N
+−∞
+∇W
+�u2(r)
+� · ψ(r) dr ̸= 0 .
+Due to the expression of the linearized differential system (4.12), (φ, ψ) is a solution of
+the adjoint linearized system
+(4.16)
+� ˙φ(r)
+˙ψ(r)
+�
+= −
+�
+0
+D2V2
+�u2(r)
+�
+id
+−dsp−1
+r
+� �
+φ(r)
+ψ(r)
+�
+.
+According to Lemma 2.3 (and since u2(·) was assumed to be nonconstant), there exists
+positive quantity ronce such that, if we denote by Ionce the interval (0, ronce], then ˙u2(·)
+does not vanish on Ionce, and, for all r∗ in Ionce and r in R,
+(4.17)
+u2(r) = u2(r∗) =⇒ r = r∗ .
+In addition, up to replacing ronce by a smaller positive quantity, it may be assumed that
+the following conclusions hold:
+u2(Ionce) ∩ BRdst(u1,∞, ε1) = ∅ .
+To complete the proof three cases have to be considered.
+Case 1.
+There exists r∗ in Ionce such that ψ(r∗) is not collinear to ˙u2(r∗).
+In this case, the construction of a potential function W satisfying inclusion (4.11) and
+inequality (4.9) (and thus the conclusions of Lemma 4.6) is the same as in the proof of
+Lemma 5.7 of [1].
+If case 1 does not occur, then ψ(r) is collinear to ˙u2(r), and since ˙u2(·) does not vanish
+on Ionce, there exists a C1-function α : Ionce → R such that, for every r in Ionce,
+(4.18)
+ψ(r) = α(r) ˙u2(r) .
+The next cases 2 and 3 differ according to whether the function α(·) is constant or not.
+22
+
+Case 2.
+For every r in Ionce, equality (4.18) holds for some nonconstant function α(·).
+In this case there exists r∗ in Ionce such that ˙α(r∗) is nonzero, and again the construction
+of a potential function W satisfying inclusion (4.11) and inequality (4.9) (and thus the
+conclusions of Lemma 4.6) is the same as in the proof of Lemma 5.7 of [1].
+Case 3.
+For every r in Ionce, ψ(r) = α ˙u2(r) for some real (constant) quantity α.
+In this case the quantity α cannot be 0 or else, due to (4.16) and (4.18), both φ(·)
+and ψ(·) would identically vanish on Ionce and thus on (0, +∞), a contradiction with the
+assumptions of Lemma 4.6. Thus, without loss of generality, we may assume that α is
+equal to 1. If supp(W) is included in a sufficiently small neighbourhood of u2,0, then
+W(·) vanishes on u2
+�[ronce, N]
+� and the integral on the left-hand side of inequality (4.15)
+reads
+� ronce
+0
+∇W
+�u2(r)
+� · ˙u2(r) dr = W
+�u2(ronce)
+� − W(u2,0) = −W(u2,0) ,
+so that inequality (4.15) holds as soon as W(u2,0) is nonzero. Lemma 4.6 is proved.
+Remark. By contrast with the proof of the generic elementarity of standing pulses in
+[1], case 3 above cannot be easily precluded. Indeed, let us assume that, for every r in
+Ionce, ψ(r) is equal to α ˙u2(r) for some nonzero (constant) quantity α. Without loss of
+generality, we may assume that α is equal to 1. Then, it follows from the second equation
+of (4.16) that, still for every r in Ionce (omitting the dependency on r),
+φ = dsp − 1
+r
+ψ − ˙ψ = dsp − 1
+r
+˙u2 − ¨u2 = 2(dsp − 1)
+r
+˙u2 − ∇V2(u2) ,
+and it follows from the first equation of (4.16) that
+−D2V2(u2) ˙u2 = ˙φ = −2(dsp − 1)
+r2
+˙u2 + 2(dsp − 1)
+r
+¨u2 − D2V2(u2) ˙u2 ,
+and thus, after simplification,
+¨u2 = 1
+r ˙u2 ,
+or equivalently
+˙u2 = r
+dsp
+∇V (u2) .
+As illustrated by equality (2.6), this last equality indeed holds if ∇V2 is constant on the
+set u2(Ionce). Case 3 can therefore not be a priori precluded, and if it may be argued
+that this case is “unlikely” (non generic), the direct argument provided above in this
+case is simpler. By contrast, in [1] for standing pulses in space dimension one (dsp equal
+to 1), this case could not occur because ψ was assumed to be nonzero on the symmetry
+subspace, defined here as {(v, r) = (0Rdst, 0)}, see (1.11).
+4.3.5 Conclusion
+As seen in sub-subsection 4.3.3, hypothesis 1 of Theorem 4.2 of [1] is fulfilled for the
+function Φ defined in (4.3), and since Lemma 4.6 yields equality (4.7), hypothesis 2 of this
+23
+
+theorem is also fulfilled. The conclusion of this theorem ensures that there exists a generic
+subset Λgen, N of Λ such that, for every V in Λgen, N, the image of the function M → N,
+m �→ Φ(m, V ) is transverse to the diagonal W of N. According to Proposition 4.4, it
+follows that every u in SV, u∞(V ) such that u(N) is in BRdst(u1,∞, ε1) is transverse. The
+set
+Λgen =
+�
+N∈N, N≥r0
+Λgen, N
+is still a generic subset of Λ. For every V in Λgen and every u in SV, u∞(V ), since u(r)
+goes to u∞(V ) as r goes to +∞, there exists N such that u(N) is in BRdst(u1,∞, ε1), and
+according to the previous statements u is transverse. In other words, the conclusions of
+Proposition 4.2 hold with:
+νV1, u1,∞ = ν = Λ
+and
+νV1, u1,∞, gen = Λgen .
+5 Proof of the main results
+Proposition 4.1 shows the genericity of the property considered in Theorem 1, but only
+inside the space Vquad-R of the potentials that are quadratic past some radius R. In this
+section, the arguments will be adapted to obtain the genericity of the same property
+in the space Vfull (that is Ck+1(Rdst, R)) of all potentials, endowed with the extended
+topology (see subsection 1.5). They are identical to those of section 9 of [1]. Let us recall
+the notation SV introduced in (1.4), and, for every positive quantity R, let us consider
+the set
+SV,R =
+�
+u ∈ SV :
+sup
+r∈[0,+∞)
+|u(r)| ≤ R
+�
+.
+Exactly as shown in subsection 9.1 of [1], Theorem 1 follows from the next proposition.
+Proposition 5.1. For every positive quantity R, there exists a generic subset Vfull-⋔-S-R
+of Vfull such that, for every potential V in this subset, every radially symmetric stationary
+solution stable at infinity in SV,R is transverse.
+Proof. Let R denote a positive quantity, let V1 denote a potential function in Vquad-(R+1),
+and let u1,∞ denote a nondegenerate minimum point of V1. Let us consider the neigh-
+bourhood νV1, u1,∞ of V1 in Vquad-(R+1) provided by Proposition 4.2 for these objects,
+together with the quantities ε1, c1, and r1 introduced in sub-subsection 4.3.1. Up to
+replacing νV1, u1,∞ by its interior, we may assume that it is open in Vquad-(R+1). As in
+sub-subsection 4.3.1, let us consider an integer N not smaller than r1, and the same
+function Φ : M × Λ → N as in (4.3).
+Here is the sole difference with the setting of sub-subsection 4.3.1: by contrast with the
+non-compact set M defining the departure set of Φ, let us consider the compact subset
+MN defined as:
+MN = BRdst(0Rdst, N) × BRdst(u1,∞, ε1) .
+Thus the integer N now serves two purposes: the “time” (radius) at which the intersection
+between unstable and centre-stable manifolds is considered, and the radius of the ball
+24
+
+containing the departure points of the unstable manifolds that are considered. These
+purposes are independent (two different integers instead of the single integer N may as
+well be introduced). Let us consider the set:
+OV1,u1,∞,N =
+�
+V ∈ νV1, u1,∞ : Φ(MN, V ) is transverse to W in N
+�
+.
+As shown in Proposition 4.4, this set OV1,u1,∞,N is made of the potential functions V in
+νV1, u1,∞ such that every u in SV, u∞(V ) such that u(N) is in BRdst(u1,∞, ε1) and u(0) is in
+BRdst(0Rdst, N), is transverse. This set contains the generic subset νV1, u1,∞, gen = Λgen of
+νV1, u1,∞ and is therefore generic (thus, in particular, dense) in νV1, u1,∞. By comparison
+with νV1, u1,∞, gen, the additional feature of this set OV1,u1,∞,N is that it is open: exactly
+as in the proof of Lemma 9.2 of [1], this openness follows from the intrinsic openness of a
+transversality property and the compactness of MN.
+Let us make the additional assumption that the potential V1 is a Morse function. Then,
+the set of minimum points of V1 is finite and depends smoothly on V in a neighbourhood
+νrobust(V1) of V1. Intersecting the sets νV1, u1,∞ and OV1,u1,∞,N above over all the minimum
+points u1,∞ of V1 provides an open neighbourhood νV1 of V1 and an open dense subset
+OV1,N of νV1 such that, for all V in νV1, every radially symmetric stationary solution
+stable close to a minimum point of V at infinity, and equal at origin to some point of
+BRdst(0Rdst, N), is transverse.
+Denoting by int(A) the interior of a set A and using the notation of subsection 4.4 of
+[1], let us introduce the sets
+˜νV1 = res−1
+R,∞ ◦ resR,(R+1)(νV1) ,
+and
+˜OV1,N = res−1
+R,∞ ◦ resR,(R+1)(OV1,N) ,
+and
+˜Oext
+V1,N = ˜OV1,N ⊔ int
+�Vfull \ ˜νV1
+� .
+It follows from these definitions that ˜Oext
+V1,N is a dense open subset of Vfull (for more
+details, see Lemma 9.3 of [1]).
+Since Vquad-(R+1) is a separable space, it is second-countable, and can be covered by a
+countable number of sets of the form νV1. With symbols, there exists a countable family
+(V1,i)i∈N of potentials of Vquad-(R+1)-Morse so that
+Vquad-(R+1)-Morse =
+�
+i∈N
+νV1,i .
+Let us consider the set
+Vfull-⋔-S-R = Vfull-Morse ∩
+�
+�
+�
+(i,N)∈N2
+˜Oext
+V1,i,N
+�
+� ,
+where Vfull-Morse is the set of potentials in Vfull which are Morse functions. This set is a
+countable intersection of dense open subsets of Vfull, and is therefore a generic subset of
+Vfull. And, for every potential V in this set Vfull-⋔-S-R, every radially symmetric stationary
+solution stable at infinity in SV,R is transverse (for more details, see Lemma 9.4 of [1]).
+Proposition 5.1 is proved.
+25
+
+As already mentioned at the beginning of this section, Theorem 1 follows from Proposi-
+tion 5.1. Finally, Corollary 1.1 follows from Theorem 1 (for more details, see subsection 9.4
+of [1]).
+Acknowledgements
+This paper owes a lot to numerous fruitful discussions with Romain
+Joly, about both its content and the content of the companion paper [1] written in
+collaboration with him.
+References
+[1]
+R. Joly and E. Risler. “Generic transversality of travelling fronts, standing fronts,
+and standing pulses for parabolic gradient systems”. In: arXiv (2023), pp. 1–69.
+arXiv: 2301.02095 (cit. on pp. 3, 5, 7–10, 14, 18, 20–26).
+[2]
+E. Risler. “Global behaviour of bistable solutions for gradient systems in one
+unbounded spatial dimension”. In: arXiv (2022), pp. 1–91. arXiv: 1604.02002
+(cit. on p. 3).
+[3]
+E. Risler. “Global behaviour of bistable solutions for hyperbolic gradient systems
+in one unbounded spatial dimension”. In: arXiv (2022), pp. 1–75. arXiv: 1703.01221
+(cit. on p. 3).
+[4]
+E. Risler. “Global behaviour of radially symmetric solutions stable at infinity for
+gradient systems”. In: arXiv (2022), pp. 1–52. arXiv: 1703.02134 (cit. on p. 3).
+[5]
+E. Risler. “Global relaxation of bistable solutions for gradient systems in one
+unbounded spatial dimension”. In: arXiv (2022), pp. 1–69. arXiv: 1604.00804
+(cit. on p. 3).
+Emmanuel Risler
+Université de Lyon, INSA de Lyon, CNRS UMR 5208, Institut Camille Jordan,
+F-69621 Villeurbanne, France.
+emmanuel.risler@insa-lyon.fr
+26
+
diff --git a/19E0T4oBgHgl3EQfugE5/content/tmp_files/load_file.txt b/19E0T4oBgHgl3EQfugE5/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf,len=1198
+page_content='Generic transversality of radially symmetric  stationary solutions stable at infinity for  parabolic gradient systems  Emmanuel Risler  January 9, 2023  This paper is devoted to the generic transversality of radially symmetric  stationary solutions of nonlinear parabolic systems of the form  ∂tw(x, t) = −∇V  �w((x, t))  � + ∆xw(x, t) ,  where the space variable x is multidimensional and unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' It is proved  that, generically with respect to the potential V , radially symmetric stationary  solutions that are stable at infinity (in other words, that approach a minimum  point of V at infinity in space) are transverse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' as a consequence, the set of  such solutions is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' This result can be viewed as the extension to  higher space dimensions of the generic elementarity of symmetric standing  pulses, proved in a companion paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' It justifies the generic character of the  discreteness hypothesis concerning this set of stationary solutions, made in  another companion paper devoted to the global behaviour of (time dependent)  radially symmetric solutions stable at infinity for such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  2020 Mathematics Subject Classification: 35K57, 37C20, 37C29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Key words and phrases: parabolic gradient systems, radially symmetric stationary solutions, generic  transversality, Morse–Smale theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='02605v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='AP]  6 Jan 2023  Contents  1  Introduction  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1  An insight into the main result .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
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+page_content='2  Radially symmetric stationary solutions stable at infinity  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  17  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3  Proof of the local statement (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  18  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1  Setting  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  18  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2  Equivalent characterizations of transversality .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  19  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3  Checking hypothesis 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2 of [1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  20  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4  Checking hypothesis 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2 of [1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  21  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5  Conclusion  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  23  5  Proof of the main results  24  2  1 Introduction  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1 An insight into the main result  The purpose of this paper is to prove the generic transversality of radially symmetric  stationary solutions stable at infinity for gradient systems of the form  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1)  ∂tw(x, t) = −∇V  �w((x, t))  � + ∆xw(x, t) ,  where time variable t is real, space variable x lies in the spatial domain Rdsp with dsp an  integer not smaller than 2, the state function (x, t) �→ w(x, t) takes its values in Rdst with  dst a positive integer, and the nonlinearity is the gradient of a scalar potential function  V : Rdst → R, which is assumed to be regular (of class at least C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' An insight into the  main result of this paper (Theorem 1 on page 8) is provided by the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' For a generic potential V , the following conclusions hold:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' every radially symmetric stationary solution stable at infinity of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1) is  robust with respect to small perturbations of V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' the set of all such solutions is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  The discreteness stated in conclusion 2 of this corollary is a required assumption for  the main result of [4], which describes the global behaviour of radially symmetric (time  dependent) solutions stable at infinity for the parabolic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1  provides a rigorous proof that this assumption holds generically with respect to V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  This paper can be viewed as a supplement of the article [1], which is devoted to the  generic transversality of bistable travelling fronts and standing pulses stable at infinity  for parabolic systems of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1) in (unbounded) space dimension one, and which  provides a rigorous proof of the genericity of similar assumptions made in [2, 3, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The  ideas, the nature of the results, and the scheme of the proof are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2 Radially symmetric stationary solutions stable at infinity  A function u : [0, +∞) → Rdst, r �→ u(r) defines a radially symmetric stationary solution  of the parabolic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1) if and only if it satisfies, on (0, +∞), the (non-autonomous)  differential system  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2)  ¨u(r) = −dsp − 1  r  ˙u(r) + ∇V  �u(r)  � ,  where ˙u and ¨u stand for the first and second derivatives of r �→ u(r), together with the  limit  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3)  ˙u(r) → 0  as  r → 0+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Observe that, in this case, u(·) is actually the restriction to [0, +∞) of an even function in  C3(R, Rd  st) which is a solution (on R) of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2) (the limit (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3) ensures  3  that equality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2) still makes sense and holds at r equals 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' In other words, provided  that condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3) holds, it is equivalent to assume that system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2) holds on (0, +∞)  or on [0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' By abuse of language, the terminology radially symmetric stationary  solution of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1) will refer, all along the paper, to functions u : [0, +∞) → Rdst  satisfying these conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3) (even if, formally, it is rather the function  Rd  sp → Rd  st, x �→ u  �|x|  � that fits with this terminology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Let us denote by Σmin(V ) the set of nondegenerate (local or global) minimum points  of V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' with symbols,  Σmin(V ) =  �u ∈ Rdst : ∇V (u) = 0 and D2V (u) > 0  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Throughout all the paper, the words minimum point will be used to denote a local or  global minimum point of a (potential) function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' A (global) solution (0, +∞) → Rdst, r �→ u(r), of the differential system  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2) (in particular a radially symmetric stationary solution of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1)) is said to be  stable at infinity if u(r) approaches a point of Σmin(V ) as r goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' If this point of  Σmin(V ) is denoted by u∞, then the solution is said to be stable close to u∞ at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' For every u∞ in Σmin(V ), let SV, u∞ denote the set of the radially symmetric  stationary solutions of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1) that are stable close to u∞ at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' With symbols,  SV, u∞ =  �u : [0, +∞) → Rdst : u satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3) and u(r) −−−−→  r→+∞ u∞  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Let  S0  V, u∞ =  �  u(0) : u ∈ SV, u∞  �  ,  and let  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4)  SV =  �  u∞∈Σmin(V )  SV, u∞  and  S0  V =  �  u∞∈Σmin(V )  S0  V, u∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  The following statement is an equivalent (simpler) formulation of conclusion 2 of  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' For a generic potential V , the subset S0  V of Rdst is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3 Differential systems governing radially symmetric stationary solutions  The second-order differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2) is equivalent to the (non-autonomous) 2dst-  dimensional first order differential differential system  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5)  �  �  �  �  �  ˙u = v  ˙v = −dsp − 1  r  v + ∇V (u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Introducing the auxiliary variables τ and c defined as  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='6)  τ = log(r)  and  c = 1  r ,  4  the previous 2dst-dimensional differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5) is equivalent to each of the following  two 2dst + 1-dimensional autonomous differential systems:  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7)  �  �  �  �  �  �  �  uτ = rv  vτ = −(dsp − 1)v + r∇V (u)  rτ = r ,  and  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='8)  �  �  �  �  �  �  �  ur = v  vr = −(dsp − 1)cv + ∇V (u)  cr = −c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Integrating the third equations of systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='8) yields  r = r0eτ−τ0  and  1  c − 1  c0  = r − r0 ,  and the parameters τ0 and c0 (which determine in each case the origin of “time”) do  not matter in principle, since those systems are autonomous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' However, if the “initial  conditions” r0 and c0 are positive (which is true for the solutions that describe radially  symmetric stationary solutions of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1)), it is natural to choose, in each case, the  origins of time according to equalities (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='6), that is :  τ0 = ln(r0)  and  c0 = 1  r0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Properties close to origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7) is relevant to provide an insight into the limit  system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5) as r goes to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The subspace R2dst × {0} (r equal to 0) is invariant by the  flow of this system, and the system reduces on this invariant subspace to  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='9)  �  uτ = 0  vτ = −(dsp − 1)v ,  see figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' For every u0 in Rdst, the point (u0, 0Rdst, 0) is an equilibrium of sys-  tem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' let us denote by W u, 0  V  (u0) the (one-dimensional) unstable manifold of this  equilibrium, for this system, let  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='10)  W u, 0, +  V  (u0) = W u, 0  V  (u0) ∩  �R2dst × (0, +∞)  � ,  and let  W u, 0, +  V  =  �  u0∈Rdst  W u, 0, +  V  (u0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  The subspace  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='11)  Ssym = Rdst × {0Rdst} × {0}  of R2dst+1 can be seen as the higher space dimension analogue of the symmetry (reversibil-  ity) subspace Rdst × {0Rdst} of R2dst (which is relevant for symmetric standing pulses in  space dimension 1, see [1] and subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7 below);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' the set W u, 0, +  V  can be seen as the  unstable manifold of this subspace Ssym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  5  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1: Dynamics of the (equivalent) differential systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7) (for r nonnegative  finite) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='8) (for c = 1/r nonnegative finite) in Rdst × Rdst × [0, +∞] (this domain is  three-dimensional if dst is equal to 1, as on the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' For the limit differential system  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='9) in the subspace r = 0 (in green), the trajectories are vertical and the solutions  converge towards the horizontal u-axis, defined as Ssym in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='11), and which is the higher  space dimensional analogue of the symmetry subspace for symmetric standing pulses  in space dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The point u∞ is a local minimum point of V , so that the point  (u∞, 0Rdst) is a hyperbolic equilibrium for the limit differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='12) in the  subspace c = 0 ⇐⇒ r = +∞ (in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='8) are autonomous, but the  quantity r (the quantity c) goes monotonously from 0 to +∞ (from +∞ to 0) for all the  solutions in the subspace r > 0 ⇐⇒ c > 0, so that those solutions can be parametrized  with r (with c) as time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The unstable manifold W u, 0, +  V  (u0) is one-dimensional and is a  transverse intersection between the unstable set W u, 0, +  V  of the subspace {r = 0, v = 0Rdst}  and the centre stable manifold W cs, ∞, +  V  (u∞) of the equilibrium (u∞, 0Rdst, c = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' To  prove the generic transversality of this intersection is the main goal of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The  dotted red curve is the projection onto the (u, r)-subspace of this intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The part of  W cs, ∞, +  V  (u∞) which is displayed on the figure can also be seen as the local centre stable  manifold W cs, ∞, +  loc, V, ε1, c1(u∞) defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='10) (with u∞ equal to the point u∞,1 introduced  there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  6  wo  u  sym  om  L  0  (8m)  loc, V, E1, C1  C1  L0=  u  8  E1Properties close to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='8) is relevant to provide an insight into the limit  system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5) as r goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The subspace R2dst × {0} of R2dst+1 (c equal to 0, or  in other words r equal to +∞) is invariant by the flow of this system, and the system  reduces on this invariant subspace to  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='12)  �  ur = v  vr = ∇V (u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  For every u∞ in Σmin(V ), the point (u∞, 0Rdst, 0) is an equilibrium of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' let  us consider its global centre-stable manifold in R2dst × (0, +∞), defined as  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='13)  W cs, ∞, +  V  (u∞) =  �  (u0, v0, c0) ∈ R2dst × (0, +∞) : the solution of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='8)  with initial condition (u0, v0, c0) at “time” r0 = 1/c0 is  defined up to +∞ and goes to (u∞, 0, 0) as r goes to +∞  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  This set W cs, ∞, +  V  (u∞) is a dst + 1-dimensional submanifold of R2dst × (0, +∞) (see  subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Radially symmetric stationary solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Let us consider the involution  ι : R2dst × (0, +∞) → R2dst × (0, +∞) ,  (u, v, r) �→ (u, v, 1/r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  The following lemma, proved in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1, formalizes the correspondence between  the radially symmetric stationary solutions stable at infinity for system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1) and the  manifolds defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Let u∞ be a point of Σmin(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' A (global) solution [0, +∞) → Rdst, r �→ u(r)  of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2) belongs to SV, u∞ if and only if its trajectory (in R2dst × (0, +∞))  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='14)  ��u(r), ˙u(r), r  � : r ∈ (0, +∞)  �  belongs to the intersection  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='15)  W u, 0, +  V  ∩ ι−1�W cs, ∞, +  V  (u∞)  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4 Transversality of radially symmetric stationary solutions stable at infinity  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Let u∞ be a point of Σmin(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' A radially symmetric stationary solution  stable close to u∞ at infinity for system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1) (in other words, a function u of SV, u∞) is  said to be transverse if the intersection (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='15) is transverse, in R2dst × (0, +∞), along the  trajectory (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The natural analogue of radially symmetric stationary solutions stable at infinity  when space dimension dsp is equal to 1 are symmetric standing pulses stable at infinity  (see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5 of [1]), and the natural analogue for such pulses of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5 above  7  is their elementarity, not their transversality (see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4 and Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='6 of [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  However, the transversality of a symmetric standing pulse (when the space dimension  dsp equals 1) makes little sense in higher space dimension, because of the singularity at r  equals 0 for the differential systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5), or because of the related fact that  the subspace {r = 0} is invariant for the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' For that reason, the  adjective transverse (not elementary) is chosen to qualify the property considered in  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5 The space of potentials  For the remaining of the paper, let us take and fix an integer k not smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Let us  consider the space Ck+1  b  (Rdst,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' R) of functions Rd → R of class Ck+1 which are bounded,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  as well as their derivatives of order not larger than k + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' equipped with the norm  ∥W∥Ck+1  b  =  max  α multi-index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' |α|≤k+1 ∥∂|α|  uαW∥L∞(Rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='R) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  and let us embed the larger space Ck+1(Rdst,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' R) with the following topology: for V in  this space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' a basis of neighbourhoods of V is given by the sets V + O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' where O is an  open subset of Ck+1  b  (Rdst,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' R) embedded with the topology defined by ∥·∥Ck+1  b  (which can  be viewed as an extended metric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' For comments concerning the choice of this topology,  see subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4 of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='6 Main result  The following generic transversality statement is the main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Theorem 1 (generic transversality of radially symmetric stationary solutions stable at  infinity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' There exists a generic subset G of  �  Ck+1(Rdst, R), ∥·∥Ck+1  b  �  such that, for every  potential function V in G, every radially symmetric stationary solution stable at infinity  of the parabolic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1) is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Theorem 1 can be viewed as the extension to higher space dimensions (for radially  symmetric solutions) of conclusion 2 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7 of [1] (which is concerned with  elementary standing pulses stable at infinity in space dimension 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' A short comparison  between these two results and their proofs is provided in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' For more  comments and a short historical review on transversality results in similar contexts, see  subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='6 of the same reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  The core of the paper (section 4) is devoted to the proof of the conclusions of Theorem 1  among potentials which are quadratic past a certain radius (defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2)), as stated  in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The extension to general potentials of Ck+1  b  (Rdst, R) is carried out in  section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' As in [1] (see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='8 of that reference), the same arguments could be  called upon to prove that the following additional conclusions hold, generically with  respect to the potential V :  8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' for every minimum point of V , the smallest eigenvalue of D2V at this minimum  point is simple;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' every radially symmetric stationary solution stable at infinity of the parabolic system  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1) approaches its limit at infinity tangentially to the eigenspace corresponding to  the smallest eigenvalue of D2V at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7 Key differences with the generic transversality of standing pulses in  space dimension one  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1 lists the key differences between the proof of the generic elementarity of  symmetric standing pulses carried out in [1], and the proof of the generic transversality  of radially symmetric stationary solutions carried out in the present paper (implicitly,  the other steps/features of the proofs are similar or identical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The state dimension,  which is simply denoted by d in [1], is here denoted by dst in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Some of the  notation/rigour is lightened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Symmetric standing pulse  Radially symmetric  stationary solution  Critical point at infinity  critical point e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' E = (e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' 0Rdst )  minimum point u∞  Symmetry subspace Ssym  {(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' v) ∈ R2dst : v = 0},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  dimension dst  {(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' r) ∈ R2dst+1 : (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' r) = (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' 0)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  dimension dst  Differential system  governing the profiles  autonomous,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' conservative,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  regular at Ssym  non-autonomous,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' dissipative,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  singular at reversibility subspace  Direction of the flow  E → Ssym  Ssym → u∞  Invariant manifold at  infinity  W u(E),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' dimension dst − m(e)  W cs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' +(u∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' dimension dst + 1  Invariant manifold at  symmetry subspace  none  W u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' dimension dst + 1  Transversality  W u(E) ⋔ Ssym  W cs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' +(u∞) ⋔ W u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' +  Transversality of spatially  homogeneous solutions  irrelevant  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2  Interval Ionce (values  reached only once)  “anywhere”  close to Ssym  M (departure set of Φ)  parametrization of ∂W u  loc, V (E)  and time, dimension dst − m(e)  Ssym and W cs, ∞, +  loc  (u∞) at r = N,  dimension 2dst  N (arrival set of Φ)  R2dst  R2dst × R2dst  W (target manifold)  Ssym  diagonal of N  dim(M) − codim(W)  −m(e)  0  Condition to be fulfilled  by perturbation W  �  DΦ(W)  �� (0, ψ)�  ̸= 0  �  DΦu(W)  �� (φ, ψ)�  ̸= 0  Perturbation W, case 3  precluded  W(u0) ̸= 0  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1: Formal comparison between the generic elementarity of symmetric standing  pulses (space dimension 1) proved in [1], and the generic transversality of radially  symmetric stationary solutions (higher space dimension dsp) proved in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Here are a few additional comments about these differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  9  Concerning the critical point at infinity, u∞ is assumed (here) to be a minimum point,  whereas (in [1]) the Morse index of e is any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Indeed, if the Morse index m(u∞) of u∞ was  positive, then the dimension of the centre-stable manifold W cs, ∞, +  V  (u∞) would be equal  to dst + m(u∞) + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' as a consequence, proving the transversality of the intersection (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='15)  in that case would require more stringent regularity assumptions on V (see hypothesis  1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2 of [1]) while nothing particularly useful could be derived from this  transversality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' On the other hand, assuming that u∞ is a minimum point allows to view  its local centre-stable manifold as a graph (u, c) �→ v (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4), which is  slightly simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Concerning the interval Ionce providing values u reached “only once” by the profile  (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3), the proof of the present paper takes advantage of the dissipation to find a  convenient interval close to the “departure point” u0, as was done in [1] for travelling  fronts (whereas, for standing pulse, the interval is to be found “anywhere”, thanks to the  conservative nature of the differential system governing the profiles, see conclusion 1 of  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3 of [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Concerning the function Φ to which Sard–Smale theorem is applied in the present  paper, both manifolds W u, 0, + and W cs, ∞, +(u∞) depend on the potential V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' However,  the transversality of an intersection between these two manifolds can be seen as the  transversality of the image of Φ with the (fixed) diagonal of R2dst × R2dst, for a function  Φ combining the parametrization of these two manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' This trick, which is the same  as in [1] for travelling fronts, allows to apply Sard–Smale theorem to a function Φ with a  fixed arrival space N containing a fixed target manifold W (in this case the diagonal of  N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' By contrast, for symmetric standing pulses in [1], since the subspace Ssym involved  in the transverse intersection is fixed, the previous trick is unnecessary and the setting is  simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Finally, a technical difference occurs in “case 3” of the proof that the degrees of freedom  provided by perturbing the potential allow to reach enough directions in the arrival state  of Φ (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='6, which is the core of the proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' In [1], case 3 is shown to lead to a  contradiction, not only for symmetric standing pulses, but also for asymmetric ones and  for travelling fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Here, such a contradiction does not seem to occur (or at least is  more difficult to prove), but this has no harmful consequence: a suitable perturbation of  the potential can still be found in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  2 Preliminary properties  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1 Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4  Let V denote a potential function in Ck+1(Rdst, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Let (0, +∞) → Rdst, r �→ u(r) denote  a (global) solution of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2), assumed to be stable close to some point u∞ of  Σmin(V ) at infinity (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4 follows from the next lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The derivative ˙u(r) goes to 0 as r goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  10  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Let us consider the Hamiltonian function  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1)  HV : R2dst → R ,  (u, v) �→ v2  2 − V (u) ,  and, for every r in (0, +∞), let  h(r) = HV  �u(r), ˙u(r)  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  It follows from system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2) that, for every r in (0, +∞),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2)  ˙h(r) = −dsp − 1  r  ˙u(r)2 ,  thus the function h(·) decreases, and it follows from the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1) of the Hamiltonian  that this function converges, as r goes to +∞, towards a finite limit h∞ which is not  smaller than −V (u∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Let us proceed by contradiction and assume that h∞ is larger than −V (u∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Then, it  follows again from the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1) of the Hamiltonian that the quantity ˙u(r)2 con-  verges towards the positive quantity 2  �h∞ + V (u∞)  � as r goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' As a consequence,  it follows from equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2) that h(r) goes to −∞ as r goes to +∞, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2 Transversality of homogeneous radially symmetric stationary solutions  stable at infinity  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' For every potential function V in Ck+1(Rdst, R) and for every nonde-  generate minimum point u∞ of V , the constant function  [0, +∞) → Rdst ,  r �→ u∞ ,  which defines an (homogeneous) radially symmetric stationary solution stable at infinity  for system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1) , is transverse (in the sense of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Let V denote a function in Ck+1(Rdst, R) and u∞ denote a nondegenerate minimum  point of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The function [0, +∞) → Rdst, r �→ u∞ is a (constant) solution of the  differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5), and the linearization of this differential system around this  solution reads  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3)  ¨u = −dsp − 1  r  ˙u + D2V (u∞) · u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Let (0, +∞) → Rdst, r �→ u(r) denote a nonzero solution of this differential system, and,  for every r in (0, +∞), let  v(r) = ˙u(r)  and  U(r) =  �u(r), v(r)  �  and  q(r) = u(r)2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  11  Then (omitting the dependency on r),  ˙q = u · ˙u  and  ¨q = ˙u2 + u · ¨u = ˙u2 − dsp − 1  r  ˙q + D2V (u∞) · (u, u) ,  so that  d  dr  �rdsp−1 ˙q(r)  � = rdsp−1  �  ¨q + dsp − 1  r  ˙q  �  = rdsp−1� ˙u2 + D2V (u∞) · (u, u)  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Since r �→ u(r) was assumed to be nonzero, it follows that the quantity rdsp−1 ˙q(r) is  strictly increasing on (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' To prove the intended conclusion, let us proceed by  contradiction and assume that, for every r in (0, +∞),  �u(r), v(r), r  � belongs:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' to the tangent space T(u∞,0Rdst ,r)W u, 0, +  V  (u∞),  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' and to the tangent space T(u∞,0Rdst ,r)  �  ι−1�W cs, ∞, +  V  (u∞)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  As in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='6), let us introduce the auxiliary variables τ (equal to log(r)) and c (equal to  1/r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' With this notation, system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3) is equivalent to  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4)  �  �  �  �  �  �  �  uτ = rv  vτ = −(dsp − 1)v + rD2V (u∞) · u  rτ = r ,  and to  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5)  �  �  �  �  �  �  �  ur = v  vr = −(dsp − 1)cv + D2V (u∞) · u  cr = −c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Assumptions 1 and 2 above yield the following conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' In view of the limit of system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4) as r goes to 0+, it follows from assumption 1  that there exists δu0 in Rdst such that  �u(r), v(r)  � goes to (δu0, 0Rdst) as r goes to  0+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' and in view of the limit of system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5) as c goes to 0+, it follows from assumption  2 that  �u(r), v(r)  � goes to (0Rdst, 0Rdst), at an exponential rate, as r goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  It follows from these two conclusions that the quantity rdsp−1 ˙q(r) goes to 0 as r goes  to 0+ and as r goes to +∞, a contradiction with the fact (observed above) that this  quantity is strictly increasing with r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  12  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3 Additional properties close to the origin  Let V denote a potential function in Ck+1(Rdst, R) and let u0 be a point in Rdst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Let  us recall (see subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3) that the unstable manifold W u, 0  V  (u0) of the equilibrium  (u0, 0Rdst, 0) for the autonomous differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7)) is one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  As a  consequence there exists a unique solution r �→ u(r) of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2) such  that the image of the map r �→  �u(r), ˙u(r), r) lies in the intersection W u, 0, +  V  (u0) of this  unstable manifold with the half-space where r is positive (this intersection was defined in  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='10));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' or, in other words, such that  �u(r), ˙u(r)  � goes to (u0, 0) as r goes to 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' This  solution is defined on some (maximal) interval (0, rmax), where rmax is either a finite  quantity or +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The following lemma provides properties of this solution that will be  used in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' To ease its statement, let us assume that rmax is equal to +∞ (only  this case will turn out to be relevant), and let us consider the continuous extension of  u(·) to the interval [0, +∞) (and let us still denote by u(·) this continuous extension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' If u(·) is not identically equal to u0 (in other words, if u0 is not a critical  point of V ), then there exists a positive quantity ronce such that, denoting by Ionce the  interval [0, ronce), the following conclusions hold:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' the function ˙u(·) does not vanish on Ionce,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' and, for every r∗ in Ionce and r in [0, +∞),  u(r) = u(r∗) =⇒ r = r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The linearized system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7) at the equilibrium (u0, 0Rdst, 0) reads:  d  dτ  �  �  �  δu  δv  δr  �  �  � =  �  �  �  0  0  0  0  −(dsp − 1)  ∇V (u0)  0  0  1  �  �  �  �  �  �  δu  δv  δr  �  �  � ,  thus the tangent space at (u0, 0Rdst, 0) to W u, 0  V  (u0) (the unstable eigenspace of the matrix  of this system) is spanned by the vector  �0, ∇V (u0)/dsp, 1  �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' it follows that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='6)  ˙u(r) = r  dsp  ∇V (u0)  �1 + or→0+(r)  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Thus, if r0 is a sufficiently small positive quantity, then ˙u(·) does not vanish on (0, r0]  (so that conclusion 1 of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3 holds provided that ronce is not larger than r0), and  the map  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7)  [0, r0] → Rdst ,  r �→ u(r)  is a C1-diffeomorphism onto its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' For r in [0, +∞), let us denote  �u(r), ˙u(r)  � by  U(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' According to the decrease (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2) of the Hamiltonian, there exists a quantity ronce in  (0, r0) such that, for every r∗ in [0, ronce),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='8)  HV  �U(r0)  � < −V  �u(r∗)  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  13  Take r∗ in [0, ronce] and r in [0, +∞), and let us assume that u(r) equals u(r∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' If r was  larger than r0 then it would follow from the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1) of the Hamiltonian, its  decrease (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2), and inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='8) that  −V  �u(r)  � ≤ HV  �U(r)  � ≤ HV  �U(r0)  � < −V  �u(r∗)  � ,  a contradiction with the equality of u(r) and u(r∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Thus r is not larger than r0, and  it follows from the one-to-one property of the function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7) that r must be equal to  r∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' conclusion 2 of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3 thus holds, and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4 Additional properties close to infinity  Let V1 denote a potential function in Ck+1(Rdst, R) and u1,∞ denote a nondegenerate  minimum point of V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' According to the implicit function theorem, there exists a (small)  neighbourhood νrobust(V1, u1,∞) of Vquad-R and a Ck-function V �→ u∞(V ) defined on  νrobust(V1, u1,∞) and with values in Rdst such that u∞(V1) equals u1,∞ and, for every  V in νrobust(V1, u1,∞), u∞(V ) is a local minimum point of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The following proposi-  tion is nothing but the local centre-stable manifold theorem applied to the equilibrium  �u∞(V ), 0Rdst, 0  � of the (autonomous) differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='8), for V close to V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Addi-  tional comments and references concerning local stable/centre/unstable manifolds are  provided in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2 of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4 (local centre-stable manifold at infinity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' There exist a neighbourhood  ν of V1 in Ck+1(Rdst, R), included in νrobust(V1, u1,∞), such that, if ε1 and c1 denote  sufficiently small positive quantities, then, for every V in ν, there exists a Ck-map  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='9)  wcs, ∞  loc, V : BRdst(u1,∞, ε1) × [0, c1] → Rdst ,  (u, c) �→ wcs, ∞  loc, V (u, c) ,  such that, for every (u0, v0, c0) in BRdst(u1,∞, ε1) × Rdst × [0, c1], the following two  statements are equivalent:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' v = wcs, ∞  loc, V (u, c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' the solution r �→  �u(r), v(r), c(r)  � of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='8) with initial condi-  tion (u0, v0, c0) at time r0 = 1/c0 is defined up to +∞, remains in BRdst(u1,∞, ε1)×  Rdst × [0, c1] of all r larger than r0, and goes to  �u∞(V ), 0Rdst, 0  � as r goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  In particular, wcs, ∞  loc, V  �u∞(V ), 0  � is equal to 0Rdst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' In addition, the map  BRdst(u1,∞, ε1) × [0, c1] × ν → Rdst ,  (u, c, V ) �→ wcs, ∞  loc, V (u, c)  is of class Ck (with respect to u and c and V ), and, for every V in ν, the graph of the  differential at  �u∞(V ), 0) of the map (u, c) �→ wcs, ∞  loc, V (u, c) is equal to the centre-stable  subspace of the linearization at  �u∞(V ), 0Rdst, 0  � of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  14  Let us denote by W cs, ∞, +  loc, V, ε1, c1  �u∞(V )  � the graph of the map (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='9) (restricted to positive  values of c), see figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' with symbols,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='10)  W cs, ∞, +  loc, V, ε1, c1  �u∞(V )  � =  ��u, wcs, ∞  loc, V (u, c), c  � : (u, c) ∈ BRdst(u1,∞, ε1) × (0, c1]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  This set defines a local centre-manifold (restricted to positive values of c) for the equilib-  rium  �u∞(V ), 0Rdst, 0  � of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Its uniqueness (for positive values  of c) is ensured by the dynamics of the centre component c, which, according to the  third equation of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='8), decreases to 0 (see figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The global centre-stable  manifold W cs, ∞, +  V  �u∞(V )  � already defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='13) can be redefined as the points of  R2dst ×(0, +∞) that eventually reach the local centre manifold W cs, ∞, +  loc, V, ε1, c1  �u∞(V )  � when  they are transported by the flow of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' If the state dimension dst is equal to 1, then a calculation shows that  wcs, ∞  loc, V (u, c) = −  �u − u∞(V )  � ��  V ′′�u∞(V )  � + dsp − 1  2  c + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  �  ,  where “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' ” stands for higher order terms in u − u∞(V ) and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' In particular the quantity  ∂c∂uwcs, ∞  loc, V  �u∞(V ), 0  � is equal to the (negative) quantity −(dsp − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The display of  the local centre-stable manifold at infinity on figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1 fits with the sign of this quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  3 Tools for genericity  Let  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1)  Vfull = Ck+1(Rdst, R) ,  and, for a positive quantity R, let  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2)  Vquad-R =  �  V ∈ Vfull : for all u in Rd, |u| ≥ R =⇒ V (u) = u2  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Let us recall the notation SV introduced in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' For every positive quantity R and for every potential V in Vquad-R, the  following conclusions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The flow defined by the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2) (governing radially symmetric  stationary solutions of the parabolic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1)) is global (that is, every solution  is defined on (0, +∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' For every u in SV , the following bound holds:  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3)  sup  r∈(0,+∞)  |u(r)| < R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  15  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Let V be in Vquad-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' According to the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2) of Vquad-R, there exists a  positive quantity K such that, for every u in Rdst,  |∇V (u)| ≤ K + |u| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  As a consequence, the following inequalities hold for the right-hand side of the first order  differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5):  ����  �  v, −dsp − 1  r  v + ∇V (u)  ����� ≤ |v| + dsp − 1  r  |v| + K + |u| ≤ K +  �  2 + dsp − 1  r  �  |(u, v)| ,  and this bound prevents the solution from blowing up in finite time, which proves  conclusion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Now, take a function u in SV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Let us still denote by u(·) the continuous extension of  this solution to [0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' For every r in [0, +∞), let  q(r) = u(r)2  2  and  Q(r) = rdsp−1 ˙q(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Then (omitting the dependency on r),  ˙q = u · ˙u  and  ¨q = ˙u2 + u · ¨u = ˙u2 − dsp − 1  r  ˙q + u · ∇V (u) ,  so that  ˙Q = rdsp−1  �  ¨q + dsp − 1  r  ˙q  �  = rdsp−1� ˙u2 + u · ∇V (u)  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  According to the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2) of Vquad-R, there exists a positive quantity δ (sufficiently  small) so that, for every w in Rdst,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4)  |w| ≥ R − δ =⇒ w · ∇V (w) ≥ w2  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Let us proceed by contradiction and assume that supr∈(0,+∞) |u(r)| is not smaller than  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Since u(·) is stable at infinity and since the critical points of V belong to the open  ball BRdst(0, R − δ), it follows that the set  �r ∈ [0, +∞) : |u(r)| ≥ R  �  is nonempty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' let rout denote the minimum of this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' For the same reason, the set  �r ∈ (rout, +∞) : |u(r)| < R − δ  �  is also nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Let rback denote the infimum of this last set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' It follows from these  definitions that rback is larger than rout and that, for every r in (rout, rback), according to  inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5)  ˙Q(r) ≥ rdsp−1  �  ˙u2(r) + u2(r)  2  �  > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  16  If on the one hand rout equals 0 then |u(0)| is not smaller than R and, since Q(0) equals  0, it follows from inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5) that Q(·) is positive on (0, rback), so that the same is  true for ˙q(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Thus q(·) is strictly increasing on [0, rback] and |u(rback)| must be larger  than |u(rout)|, a contradiction with the definition of rback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' If on the other hand rout is  positive, then |u(rout)| is equal to R and ˙q(rout) is nonnegative so that the same is true  for Q(rout), and it again follows from inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5) that Q(·) is positive on (0, rback),  yielding the same contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Conclusion 2 of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' For every positive quantity R and every potential V in Vquad-R, let  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='6)  SV : (0, +∞)2 × R2dst → R2dst ,  �(rinit, r), (uinit, vinit)  � �→ SV  �(rinit, r), (uinit, vinit)  �  denote the (globally defined) flow of the (non-autonomous) differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5) for  this potential V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' In other words, for every rinit in (0, +∞) and (uinit, vinit) in R2dst, the  function  (0, +∞) → R2dst ,  r �→ SV  �(rinit, r1), (uinit, vinit)  �  is the solution of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5) for the initial condition (uinit, vinit) at r  equals rinit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' According to subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3, the flow SV may be extended to the larger set  (0, +∞)2 × R2dst ∪ [0, +∞)2 × Rdst × {0Rdst} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  according to this extension, for every u0 in Rdst, the solution taking its values in the  (one-dimensional) unstable manifold W u, 0, +  V  (u0) reads:  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7)  [0, +∞) → Rdst ,  r �→ SV  �(0, r), (u0, 0Rdst)  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  4 Generic transversality among potentials that are quadratic  past a given radius  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1 Notation and statement  Let us recall the notation SV and SV, u∞ introduced in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' There exists a generic subset of Vquad-R such that, for every potential  V in this subset, every radially symmetric stationary solution stable at infinity of the  parabolic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1) (in other words, every u in SV ) is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2 Reduction to a local statement  Let V1 denote a potential function in Vquad-R and u1,∞ denote a nondegenerate minimum  point of V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' According to the implicit function theorem, there exists a (small) neighbour-  hood νrobust(V1, u1,∞) of Vquad-R and a Ck-function u∞(·) defined on νrobust(V1, u1,∞) and  with values in Rdst such that u∞(V1) equals u1,∞ and, for every V in νrobust(V1, u1,∞),  u∞(V ) is a local minimum point of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The following local generic transversality statement  yields Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1 (as shown below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  17  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' There exists a neighbourhood νV1, u1,∞ of V1 in νrobust(V1, u1,∞) and  a generic subset νV1, u1,∞, gen of νV1, u1,∞ such that, for every V in νV1, u1,∞, gen, every  radially symmetric stationary solution stable close to u∞(V ) at infinity of the parabolic  system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1) (in other words, every u in SV, u∞(V )) is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Proof that Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2 yields Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Let us denote by Vquad-R-Morse the  dense open subset of Vquad-R defined by the Morse property:  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1)  Vquad-R-Morse = {V ∈ Vquad-R : all critical points of V are nondegenerate} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Let V1 denote a potential function in Vquad-R-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' According to the Morse property  its minimum points are isolated and since V1 is in Vquad-R they belong to the open ball  BRd(0, R), so that those minimum points are in finite number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Assume that Proposi-  tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' With the notation of this proposition, let us consider the following two  intersections, at each time over all minimum points u1,∞ of V1:  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2)  νV1 =  �  νV1, u1,∞  and  νV1, gen =  �  νV1, u1,∞, gen .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Since those are finite intersections, νV1 is still a neighbourhood of V1 in Vquad-R and the  set νV1, gen is still a generic subset of νV1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' This shows that the set  {V ∈ Vquad-R-Morse :  every u in SV, u∞(V ) is transverse}  is locally generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3 of [1] as in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2 of this reference shows  that this local genericity implies the global genericity stated in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1, which is  therefore proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3 Proof of the local statement (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1 Setting  For the remaining part of this section, let us fix a potential function V1 in Vquad-R and a  nondegenerate minimum point u1,∞ of V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Let ν be a neighbourhood of V1 in Vquad-R,  included in νrobust(V1, u1,∞), and let ε1 and c1 be positive quantities, with ν and ε1 and  c1 small enough so that the conclusions of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Let  r1 = 1/c1  and  M = Rdst × BRdst(u1,∞, ε1)  and  Λ = ν ,  and  N = (R2dst)2  and  W = {(A, B) ∈ N : A = B}  ,  thus W is the diagonal of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Let N denote an integer not smaller than r1, and let us  consider the functions  Φu : Rdst × Λ → R2dst ,  (u0, V ) �→ SV  �(0, N), (u0, 0Rdst)  � ,  and  Φcs : BRdst(u1,∞, ε1) × Λ → R2dst ,  (uN, V ) �→  �uN, wcs, ∞  loc, V (uN, 1/N)  � ,  and the function  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3)  Φ : M × Λ → N ,  (m, V ) = (u0, uN, V ) �→  �Φu(u0, V ), Φcs(uN, V )  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  18  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2 Equivalent characterizations of transversality  Let us consider the set  SΛ,u1,∞,N =  �(V, u) : V ∈ Λ and u ∈ SV, u∞(V ) and u(N) ∈ BRdst(u1,∞, ε1)  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The map  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4)  Φ−1(W) → SΛ,u1,∞,N ,  (u0, u, V ) �→  �  V, r �→ SV  �(0, r), (u0, 0Rdst  ��  is well defined and one-to-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The image by Φ of a point (u0, uN, V ) of M × Λ belongs to the diagonal W of  N if and only if Φu(u0, V ) equals Φcs(uN, V ), and in this case the function u : r �→  SV  �(0, r), (u0, 0Rdst  � belongs to SV, u∞(V ) and u(N) (which is equal to uN) belongs to  BRdst(u1,∞, ε1), so that (V, u) belongs to SΛ,u1,∞,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The map (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4) above is thus well  defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Now, for every (V, u) in SΛ,u1,∞,N, if we denote by u0 the limit limr→0+ u(r) and by  uN the vector u(N), then (u0, uN, V ) is the only possible antecedent of (V, u) by the map  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' In addition,  SV  �(0, N), (u0, 0Rdst)  � =  �uN, ˙u(N)  � ,  and since u(r) goes to u∞(V ) as r goes to +∞, the vector  �u(N), ˙u(N), 1/N  � must  belong to the centre-stable manifold W cs, ∞, +  V  �u∞(V )  � of u∞(V ), so that, according to  the definition of wcs, ∞  loc, V ,  ˙u(N) = wcs, ∞  loc, V  �u(N), 1/N  � ,  and this yields the equality between Φu(u0, V ) and Φcs(uN, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Thus Φ(V, u) belongs to  W and (u0, uN, V ) belongs to Φ−1(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' For every potential function V in Λ, the following two statements are  equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The image of the function M → N, m �→ Φ(m, V ) is transverse to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Every u in SV, u∞(V ) such that u(N) is in BRdst(u1,∞, ε1) is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' According to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2, for every V in Λ, the constant function r �→ u∞(V ),  which belongs to SV , is already (a priori) known to be transverse, therefore only  nonconstant solutions matter in statement 2 of this proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Let us consider (m2, V2) in M × Λ such that Φ(m2, V2) is in W, let (u2,0, u2,N)  denote the components of m2, and let r �→ u2(r) and r �→ U2(r) denote the functions  satisfying, for all r in [0, +∞),  U2(r) =  �u2(r), ˙u2(r)  � = SV  �(0, r), (u2,0, 0Rdst  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Let us consider the map  ∆Φ : M → R2dst ,  (u0, uN) �→ Φu(u0, V2) − Φcs(uN, V2) ,  19  and let us write, only for this proof, DΦ and DΦu and DΦcs and D(∆Φ) for the  differentials of Φ and Φu and Φcs and ∆Φ at (m2, V2) and with respect to all variables in  M (but not with respect to V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' According to Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5, the transversality of u2 is  defined as the transversality of the intersection W u, 0, +  V2  ∩ ι−1�  W cs, ∞, +  V2  �u∞(V2)  ��  along  the trajectory of U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' This transversality can be considered at a single point, no matter  which, of the trajectory U2  �(0, +∞)  �, in particular at the point Φu(u2,0, V2) which is  equal to Φcs�u2(N), V 2  �, and is equivalent to the transversality of the dst-dimensional  manifolds  W u, 0, +  V2  ∩  �R2dst × {N}  �  and  ι−1�  W cs, ∞, +  V2  �u∞(V2)  ��  ∩  �R2dst × {N}  �  in R2dst ×{N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' It is therefore equivalent to the surjectivity of the map D(∆Φ) (statement  (B) in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' On the other hand, the image of the function M → N,  m �→ Φ(m, V2) is transverse at Φ(m, V2) to the diagonal W of N if and only if the image  of DΦ contains a complementary space of this diagonal (statement (A) in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5  below)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Thus Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4 is a consequence of the next lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The following two statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  (A) The image of DΦ contains a complementary subspace of the diagonal W of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  (B) The map D(∆Φ) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' If statement (A) holds, then, for every (α, β) in N, there exist γ in R2dst and δm  in Tm2M such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5)  (γ, γ) + DΦ · δm = (α, β) ,  so that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='6)  D(∆Φ) · δm = α − β ,  and statement (B) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Conversely, if statement (B) holds, then, for every (α, β) in  N, there exists δm in Tm2M such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='6) holds, and as a consequence, if (δu0, δuN)  denote the components of δm, then α − DΦu(δu0) is equal to β − DΦcs(δuN), and if  this vector is denoted by γ, then equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5) holds, and this shows that statement (A)  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  As explained above, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4 follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5, and is therefore proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3 Checking hypothesis 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2 of [1]  The function Φ is as regular as the flow SV , thus of class Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' It follows from the definitions  of M and N and W that  dim(M) − codim(W) = (dst + dst) − 2dst = 0 ,  so that hypothesis 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2 of [1] is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  20  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4 Checking hypothesis 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2 of [1]  For every V in Vquad-R, let us recall the notation SV introduced in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7) for the  flow of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Take (m2, V2) in the set Φ−1(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Let (u2,0, u2,N)  denote the components of m2, and, for every r in (0, +∞), let us write  U2(r) =  �u2(r), v2(r)  � = SV2  �(0, r), (u2,0, 0Rdst)  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Let us write  DΦ  and  DΦu  and  DΦcs  for the full differentials (with respect to arguments m in M and V in Λ) of the three  functions Φ and Φu and Φcs respectively at the points  �u2,0, u2,N, V2  �,  �u2,0, V2  � and  �u2,N, V2  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Checking hypothesis 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2 of [1] amounts to prove that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7)  im(DΦ) + W = N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  If u2(·) is constant (that is, identically equal to u∞(V2)), then equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7) follows from  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Thus, let us assume that u2(·) is nonconstant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' In this case, equality  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7) is a consequence of the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' For every nonzero vector (φ2, ψ2) in R2dst, there exists a function W in  Ck+1  b  (Rdst, R) such that  supp(W) ⊂ BRd(0, R) ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='8)  and  �DΦu · (0, 0, W)  �� (φ2, ψ2)  � ̸= 0 ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='9)  and  DΦcs · (0, 0, W) = 0R2dst .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='10)  Proof that Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='6 yields equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='9) shows that the orthogonal  complement, in R2dst, of the directions that can be reached by DΦu·(0, 0, W) for potentials  W satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='8) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='10) is reduced to 0R2dst;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' in other words, all directions of R2dst  can be reached by that means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' This shows that  im(DΦ) ⊃ R2dst × {0R2dst} ,  and since the subspace at the right-hand side of this inclusion is transverse to W in  R4dst, this proves equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7) (and shows that hypothesis 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2 of [1] is  fulfilled).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Let (φ2, ψ2) denote a nonzero vector in R2dst, let W be a function  in Ck+1  b  (Rdst, R) satisfying the inclusion  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='11)  supp(W) ⊂ BRd(0, R) \\ BRdst(u1,∞, ε1) ,  and observe that inclusion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='8) and equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='10) follow from this inclusion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Let  us consider the linearization of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2), for the potential V2, around  the solution r �→ U2(r):  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='12)  d  dr  �  δu(r)  δv(r)  �  =  �  0  id  D2V2  �u2(r)  �  −dsp−1  r  � �  δu(r)  δv(r)  �  ,  21  and let T(r, r′) denote the family of evolution operators obtained by integrating this  linearized differential system between r and r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' It follows from the variation of constants  formula that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='13)  DΦu · (0, 0, W) =  � N  −∞  T(r, N)  �  0, ∇W  �u2(r)  ��  dr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  For every r in (0, +∞), let T ∗(r, N) denote the adjoint operator of T(r, N), and let  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='14)  �φ(r), ψ(r)  � = T ∗(r, N) · (φ2, ψ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  According to expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='13), inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='9) reads  � N  −∞  ��  0, ∇W  �u2(r)  �� ��� T ∗(r, N) · (φ2, ψ2)  �  dr ̸= 0 ,  or equivalently  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='15)  � N  −∞  ∇W  �u2(r)  � · ψ(r) dr ̸= 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Due to the expression of the linearized differential system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='12), (φ, ψ) is a solution of  the adjoint linearized system  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='16)  � ˙φ(r)  ˙ψ(r)  �  = −  �  0  D2V2  �u2(r)  �  id  −dsp−1  r  � �  φ(r)  ψ(r)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  According to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3 (and since u2(·) was assumed to be nonconstant), there exists  positive quantity ronce such that, if we denote by Ionce the interval (0, ronce], then ˙u2(·)  does not vanish on Ionce, and, for all r∗ in Ionce and r in R,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='17)  u2(r) = u2(r∗) =⇒ r = r∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  In addition, up to replacing ronce by a smaller positive quantity, it may be assumed that  the following conclusions hold:  u2(Ionce) ∩ BRdst(u1,∞, ε1) = ∅ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  To complete the proof three cases have to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  There exists r∗ in Ionce such that ψ(r∗) is not collinear to ˙u2(r∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  In this case, the construction of a potential function W satisfying inclusion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='11) and  inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='9) (and thus the conclusions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='6) is the same as in the proof of  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7 of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  If case 1 does not occur, then ψ(r) is collinear to ˙u2(r), and since ˙u2(·) does not vanish  on Ionce, there exists a C1-function α : Ionce → R such that, for every r in Ionce,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='18)  ψ(r) = α(r) ˙u2(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  The next cases 2 and 3 differ according to whether the function α(·) is constant or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  22  Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  For every r in Ionce, equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='18) holds for some nonconstant function α(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  In this case there exists r∗ in Ionce such that ˙α(r∗) is nonzero, and again the construction  of a potential function W satisfying inclusion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='11) and inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='9) (and thus the  conclusions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='6) is the same as in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7 of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  For every r in Ionce, ψ(r) = α ˙u2(r) for some real (constant) quantity α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  In this case the quantity α cannot be 0 or else, due to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='16) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='18), both φ(·)  and ψ(·) would identically vanish on Ionce and thus on (0, +∞), a contradiction with the  assumptions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Thus, without loss of generality, we may assume that α is  equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' If supp(W) is included in a sufficiently small neighbourhood of u2,0, then  W(·) vanishes on u2  �[ronce, N]  � and the integral on the left-hand side of inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='15)  reads  � ronce  0  ∇W  �u2(r)  � · ˙u2(r) dr = W  �u2(ronce)  � − W(u2,0) = −W(u2,0) ,  so that inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='15) holds as soon as W(u2,0) is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='6 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' By contrast with the proof of the generic elementarity of standing pulses in  [1], case 3 above cannot be easily precluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Indeed, let us assume that, for every r in  Ionce, ψ(r) is equal to α ˙u2(r) for some nonzero (constant) quantity α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Without loss of  generality, we may assume that α is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Then, it follows from the second equation  of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='16) that, still for every r in Ionce (omitting the dependency on r),  φ = dsp − 1  r  ψ − ˙ψ = dsp − 1  r  ˙u2 − ¨u2 = 2(dsp − 1)  r  ˙u2 − ∇V2(u2) ,  and it follows from the first equation of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='16) that  −D2V2(u2) ˙u2 = ˙φ = −2(dsp − 1)  r2  ˙u2 + 2(dsp − 1)  r  ¨u2 − D2V2(u2) ˙u2 ,  and thus, after simplification,  ¨u2 = 1  r ˙u2 ,  or equivalently  ˙u2 = r  dsp  ∇V (u2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  As illustrated by equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='6), this last equality indeed holds if ∇V2 is constant on the  set u2(Ionce).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Case 3 can therefore not be a priori precluded, and if it may be argued  that this case is “unlikely” (non generic), the direct argument provided above in this  case is simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' By contrast, in [1] for standing pulses in space dimension one (dsp equal  to 1), this case could not occur because ψ was assumed to be nonzero on the symmetry  subspace, defined here as {(v, r) = (0Rdst, 0)}, see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5 Conclusion  As seen in sub-subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3, hypothesis 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2 of [1] is fulfilled for the  function Φ defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3), and since Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='6 yields equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='7), hypothesis 2 of this  23  theorem is also fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The conclusion of this theorem ensures that there exists a generic  subset Λgen, N of Λ such that, for every V in Λgen, N, the image of the function M → N,  m �→ Φ(m, V ) is transverse to the diagonal W of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' According to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4, it  follows that every u in SV, u∞(V ) such that u(N) is in BRdst(u1,∞, ε1) is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' The  set  Λgen =  �  N∈N, N≥r0  Λgen, N  is still a generic subset of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' For every V in Λgen and every u in SV, u∞(V ), since u(r)  goes to u∞(V ) as r goes to +∞, there exists N such that u(N) is in BRdst(u1,∞, ε1), and  according to the previous statements u is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' In other words, the conclusions of  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2 hold with:  νV1, u1,∞ = ν = Λ  and  νV1, u1,∞, gen = Λgen .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  5 Proof of the main results  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1 shows the genericity of the property considered in Theorem 1, but only  inside the space Vquad-R of the potentials that are quadratic past some radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' In this  section, the arguments will be adapted to obtain the genericity of the same property  in the space Vfull (that is Ck+1(Rdst, R)) of all potentials, endowed with the extended  topology (see subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' They are identical to those of section 9 of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Let us recall  the notation SV introduced in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4), and, for every positive quantity R, let us consider  the set  SV,R =  �  u ∈ SV :  sup  r∈[0,+∞)  |u(r)| ≤ R  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Exactly as shown in subsection 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1 of [1], Theorem 1 follows from the next proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' For every positive quantity R, there exists a generic subset Vfull-⋔-S-R  of Vfull such that, for every potential V in this subset, every radially symmetric stationary  solution stable at infinity in SV,R is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Let R denote a positive quantity, let V1 denote a potential function in Vquad-(R+1),  and let u1,∞ denote a nondegenerate minimum point of V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Let us consider the neigh-  bourhood νV1, u1,∞ of V1 in Vquad-(R+1) provided by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2 for these objects,  together with the quantities ε1, c1, and r1 introduced in sub-subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Up to  replacing νV1, u1,∞ by its interior, we may assume that it is open in Vquad-(R+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' As in  sub-subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1, let us consider an integer N not smaller than r1, and the same  function Φ : M × Λ → N as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Here is the sole difference with the setting of sub-subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1: by contrast with the  non-compact set M defining the departure set of Φ, let us consider the compact subset  MN defined as:  MN = BRdst(0Rdst, N) × BRdst(u1,∞, ε1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Thus the integer N now serves two purposes: the “time” (radius) at which the intersection  between unstable and centre-stable manifolds is considered, and the radius of the ball  24  containing the departure points of the unstable manifolds that are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' These  purposes are independent (two different integers instead of the single integer N may as  well be introduced).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Let us consider the set:  OV1,u1,∞,N =  �  V ∈ νV1, u1,∞ : Φ(MN, V ) is transverse to W in N  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  As shown in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4, this set OV1,u1,∞,N is made of the potential functions V in  νV1, u1,∞ such that every u in SV, u∞(V ) such that u(N) is in BRdst(u1,∞, ε1) and u(0) is in  BRdst(0Rdst, N), is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' This set contains the generic subset νV1, u1,∞, gen = Λgen of  νV1, u1,∞ and is therefore generic (thus, in particular, dense) in νV1, u1,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' By comparison  with νV1, u1,∞, gen, the additional feature of this set OV1,u1,∞,N is that it is open: exactly  as in the proof of Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='2 of [1], this openness follows from the intrinsic openness of a  transversality property and the compactness of MN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Let us make the additional assumption that the potential V1 is a Morse function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Then,  the set of minimum points of V1 is finite and depends smoothly on V in a neighbourhood  νrobust(V1) of V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Intersecting the sets νV1, u1,∞ and OV1,u1,∞,N above over all the minimum  points u1,∞ of V1 provides an open neighbourhood νV1 of V1 and an open dense subset  OV1,N of νV1 such that, for all V in νV1, every radially symmetric stationary solution  stable close to a minimum point of V at infinity, and equal at origin to some point of  BRdst(0Rdst, N), is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Denoting by int(A) the interior of a set A and using the notation of subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4 of  [1], let us introduce the sets  ˜νV1 = res−1  R,∞ ◦ resR,(R+1)(νV1) ,  and  ˜OV1,N = res−1  R,∞ ◦ resR,(R+1)(OV1,N) ,  and  ˜Oext  V1,N = ˜OV1,N ⊔ int  �Vfull \\ ˜νV1  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  It follows from these definitions that ˜Oext  V1,N is a dense open subset of Vfull (for more  details, see Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='3 of [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Since Vquad-(R+1) is a separable space, it is second-countable, and can be covered by a  countable number of sets of the form νV1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' With symbols, there exists a countable family  (V1,i)i∈N of potentials of Vquad-(R+1)-Morse so that  Vquad-(R+1)-Morse =  �  i∈N  νV1,i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Let us consider the set  Vfull-⋔-S-R = Vfull-Morse ∩  �  �  �  (i,N)∈N2  ˜Oext  V1,i,N  �  � ,  where Vfull-Morse is the set of potentials in Vfull which are Morse functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' This set is a  countable intersection of dense open subsets of Vfull, and is therefore a generic subset of  Vfull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' And, for every potential V in this set Vfull-⋔-S-R, every radially symmetric stationary  solution stable at infinity in SV,R is transverse (for more details, see Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4 of [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  25  As already mentioned at the beginning of this section, Theorem 1 follows from Proposi-  tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Finally, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='1 follows from Theorem 1 (for more details, see subsection 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='4  of [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Acknowledgements  This paper owes a lot to numerous fruitful discussions with Romain  Joly, about both its content and the content of the companion paper [1] written in  collaboration with him.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  References  [1]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Joly and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Risler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
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+page_content='  arXiv: 2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
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+page_content=' on pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' 3, 5, 7–10, 14, 18, 20–26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  [2]  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Risler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' “Global behaviour of bistable solutions for gradient systems in one  unbounded spatial dimension”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' In: arXiv (2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' 1–91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' arXiv: 1604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='02002  (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  [3]  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Risler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' “Global behaviour of bistable solutions for hyperbolic gradient systems  in one unbounded spatial dimension”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' In: arXiv (2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' 1–75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' arXiv: 1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='01221  (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  [4]  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Risler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' “Global behaviour of radially symmetric solutions stable at infinity for  gradient systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' In: arXiv (2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' 1–52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' arXiv: 1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='02134 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  [5]  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' Risler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' “Global relaxation of bistable solutions for gradient systems in one  unbounded spatial dimension”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' In: arXiv (2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' 1–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' arXiv: 1604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='00804  (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  Emmanuel Risler  Université de Lyon, INSA de Lyon, CNRS UMR 5208, Institut Camille Jordan,  F-69621 Villeurbanne, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='  emmanuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='risler@insa-lyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
+page_content='fr  26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}
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+Low-energy quasi-circular electron correlations with charge order
+wavelength in Bi2Sr2CaCu2O8+δ
+K. Scott,1, 2 E. Kisiel,3 T. J. Boyle,1, 2, 4 R. Basak,3 G. Jargot,5 S. Das,3 S. Agrestini,6
+M. Garcia-Fernandez,6 J. Choi,6 J. Pelliciari,7 J. Li,7 Y. D. Chuang,8 R. D. Zhong,9
+J. A. Schneeloch,9 G. D. Gu,9 F. L´egar´e,5 A. F. Kemper,10 Ke-Jin Zhou,6 V. Bisogni,7
+S. Blanco-Canosa,11, 12 A. Frano,3, 13 F. Boschini,5, 14 and E. H. da Silva Neto1, 2, ∗
+1Department of Physics, Yale University, New Haven, Connecticut 06520, USA
+2Energy Sciences Institute, Yale University, West Haven, Connecticut 06516, USA
+3Department of Physics, University of California San Diego, La Jolla, California 92093, USA
+4Department of Physics and Astronomy, University of California, Davis, California 95616, USA
+5Centre ´Energie Mat´eriaux T´el´ecommunications,
+Institut National de la Recherche Scientifique, Varennes, Qu´ebec J3X 1S2, Canada
+6Diamond Light Source, Harwell Campus, Didcot OX11 0DE, United Kingdom
+7National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, NY 11973, USA
+8Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
+9Condensed Matter Physics and Materials Science, Brookhaven National Laboratory, Upton, NY, USA
+10Department of Physics, North Carolina State University, Raleigh, NC 27695, U.S.A.
+11Donostia International Physics Center, DIPC, 20018 Donostia-San Sebastian, Basque Country, Spain
+12IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain
+13Canadian Institute for Advanced Research, Toronto, ON, M5G 1M1, Canada
+14Quantum Matter Institute, University of British Columbia, Vancouver, BC V6T 1Z4, Canada
+∗ Corresponding Author: eduardo.dasilvaneto@yale.edu
+arXiv:2301.08415v1  [cond-mat.str-el]  20 Jan 2023
+
+2
+ABSTRACT
+In the study of dynamic charge order correlations in the cuprates, most high energy-
+resolution resonant inelastic x-ray scattering (RIXS) measurements have focused on mo-
+menta along the high-symmetry directions of the copper oxide plane. However, electron
+scattering along other in-plane directions should not be neglected as they may contain in-
+formation relevant, for example, to the origin of charge order correlations or to our un-
+derstanding of the isotropic scattering responsible for strange metal behavior in cuprates.
+We report high-resolution resonant inelastic x-ray scattering (RIXS) experiments that re-
+veal the presence of dynamic electron correlations over the qx-qy scattering plane in under-
+doped Bi2Sr2CaCu2O8+δ with Tc = 54 K. We use the softening of the RIXS-measured bond
+stretching phonon line as a marker for the presence of charge-order-related dynamic electron
+correlations. The experiments show that these dynamic correlations exist at energies below
+approximately 70 meV and are centered around a quasi-circular manifold in the qx-qy scat-
+tering plane with radius equal to the magnitude of the charge order wave vector, qCO. We
+also demonstrate how this phonon-tracking procedure provides the necessary experimental
+precision to rule out fluctuations of short-range directional charge order (i.e. centered around
+[qx = ±qCO, qy = 0] and [qx = 0, qy = ±qCO]) as the origin of the observed correlations.
+INTRODUCTION
+Dynamic fluctuations from periodic charge order (CO) pervade the phase diagram of
+cuprate superconductors, perhaps even more than superconductivity itself [1]. The detec-
+tion of these fluctuations over energy and momentum was enabled by several recent advances
+in the energy resolution of resonant inelastic x-ray scattering (RIXS) instruments operat-
+ing in the soft x-ray regime. In the case of YBa2Cu3O6+δ, Cu-L3 RIXS detects dynamic
+correlations at the charge order wavevector, qCO, with a characteristic energy scale of ap-
+proximately 20 meV [2]. It has been proposed that these low-energy short-range dynamic
+charge order correlations are a key ingredient to the strange metal behavior [3, 4] charac-
+terized by linear-in-temperature resistivity [5, 6]. On one hand, this temperature behav-
+ior is often associated with an isotropic scattering rate that depends only on temperature
+in units of energy and Planck’s constant (i.e. ∝ kBT/ℏ, sometimes called the Planckian
+regime) [7–11], as supported by recent angle-dependent magnetoresistance measurements of
+La1.6−xNd0.4SrxCuO4 [12]. On the other hand, combined transport and RIXS studies have
+
+3
+recently shown an unexpected link between linear-in-temperature resistivity and charge or-
+der in YBa2Cu3O6+δ [13, 14]. Combined, these latest results suggest that fluctuations of the
+charge order should somehow result in an effective isotropic scattering. Still, high-resolution
+RIXS experiments have largely focused on the fluctuations along the high-symmetry crystal-
+lographic directions only, leaving the full structure of electron correlations within the copper
+oxide plane unknown.
+Recently, in Bi2Sr2CaCu2O8+δ (Bi-2212), RIXS measurements found the existence of a
+quasi-circular pattern in the qx-qy plane at finite energies and with the same wave vector
+magnitude as that of the observed static charge order peak at q = [qx = ±qCO, qy =
+0] and [qx = 0, qy = ±qCO] – i.e.
+dynamic correlations with charge order wavelength
+along all direction in the CuO2 plane [15]. Although the medium energy-resolution of those
+measurements (∆E ≈ 0.8 eV) precluded a more precise determination of their energy profile,
+the results suggested that these quasi-circular dynamic correlations (QCDCs) appear broad
+over the mid-infrared ranges (defined approximately as 100 to 900 meV). This scattering
+manifold, which may result from combined short- and long-range Coulomb interactions [15–
+17], would provide a large variety of wave vectors for connecting all points of the Fermi
+surface (i.e. an effective isotropic scattering). However, it is not yet experimentally known
+if this manifold extends to electron scattering at lower energies, in the quasi-elastic regime.
+To experimentally investigate this scenario we used high energy-resolution (≈ 37 meV) Cu-L3
+RIXS qx-qy mapping of the electronic correlations in Bi-2212. Using the softening of the bond
+stretching (BS) phonon in RIXS as a marker of charge order correlations, our measurements
+reveal the presence of low-energy quasi-circular dynamic electronic correlations with |q| ≈
+qCO.
+RESULTS
+High-resolution RIXS mapping of dynamic correlations in the qx-qy plane
+We performed measurements at φ = 0◦, 25◦, 30◦, 35◦, 45◦, where φ is defined as the az-
+imuthal angle from the qx axis. For each φ, we acquired RIXS spectra at different values of
+in-plane momentum-transfer q = |q| by varying the incident angle on the sample. Through-
+out the paper, values of q are reported in reciprocal lattice units (r.l.u.), where one r.l.u. is
+defined as 2π/a and a = 3.82 ˚A (the lattice constant along φ = 0◦). In Fig. 1 (A and B), we
+
+4
+show representative spectra obtained at q near qCO for φ = 0◦ and 30◦, and energies below
+1.1 eV. In these two cases, the minimal model that fits the data includes five contributions:
+a quasi-elastic peak, a bond-stretching phonon peak at ≈ 70 meV, a peak at ≈ 135 meV
+(likely from a two-phonon process), a broad paramagnon and a broad background feature
+of unknown origin. A similar assessment can be made regarding all other high-resolution
+spectra acquired in this work. In this type of fitting analysis, the QCDCs are not explicitly
+accounted for and it is generally difficult to disentangle overlapping contributions to the
+RIXS spectra using a fitting model with so many parameters, thus precluding the extraction
+of the exact spectral profile of the QCDCs with any reasonable confidence. Still, we note
+that this high-resolution data is consistent with the previously reported medium-resolution
+data [15], which can be verified by integration of the high-resolution data (see supplementary
+materials, Fig. S7).
+It is likely that the spectral intensity of QCDCs in Bi-2212 is so dilute over energy
+as to preclude the extraction of their spectral structure amidst stronger paramagnon and
+phonon signals. Still, here we develop a different method to detect QCDCs at lower energies,
+by tracking the evolution of the bond-stretching (BS) phonon over the qx-qy plane. This
+method is based on the phenomenology revealed by several recent RIXS measurements of
+the cuprates along qx and qy, which indicate an apparent softening of the BS phonon peak
+at the momentum location of the static CO peak [18–23]. In the case of Bi-2212, it has
+been proposed that the apparent softening of the BS phonon in RIXS is due to an interplay
+between low-energy fluctuations of the charge order and BS phonons that results in a Fano-
+like interference [18, 21, 23, 24]. Another possibility is that the apparent softening is simply
+the result of the phonon peak and a low-energy charge order peak overlapping, as recently
+suggested by measurements of both YBa2Cu3O6+δ and Bi-2212 [25]. In either interpretation
+the location of the phonon softening can be used as a marker for low-energy charge order
+correlations.
+Figure 1 (C and D) shows the spectra acquired as a function of q for φ = 0◦ and 30◦,
+respectively, focusing on the region of the BS phonon.
+At φ = 0◦, it is clear that the
+phonon peak position softens to its lowest energy value at q = qCO ≈ 0.29 r.l.u. (Fig. 1C).
+Careful observation of the spectra taken along φ = 30◦ shows a similar softening effect with
+the lowest phonon energy position occuring for q ≈ qCO (Fig. 1D). Figure 2A shows the
+mapping of the BS phonon mode at φ = 0◦ and 30◦ obtained after subtraction of the fitted
+
+5
+3
+2
+1
+0
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+ϕ=0°
+q=0.27 r.l.u.
+1
+0
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+ϕ=30°
+q=0.27 r.l.u.
+IRIXS (arb. units)
+IRIXS (arb. units)
+Intensity (arb. units)
+Energy Loss (meV)
+A
+B
+C
+Energy Loss (eV)
+10
+8
+6
+4
+2
+120
+80
+40
+Energy Loss (eV)
+ϕ=0°
+D
+ϕ=30°
+120
+80
+40
+q (r.l.u.)
+0.1
+0.12
+0.14
+0.16
+0.17
+0.18
+0.19
+0.2
+0.21
+0.22
+0.23
+0.24
+0.25
+0.26
+0.27
+0.28
+0.29
+0.30
+0.31
+0.32
+0.33
+0.34
+0.35
+0.36
+0.37
+0.39
+0.41
+0.43
+0.45
+0.47
+FIG. 1. RIXS spectra and fitting. (A and B) Examples of spectra at q = 0.27 r.l.u. for φ = 0◦
+and 30◦, respectively (open circles). The red lines are fits to the spectra, composed of a quasi-
+elastic peak (pink), a BS phonon peak at ≈ 70 meV (blue), a peak at ≈ 135 meV (likely from
+a two-phonon process) (purple), a broad paramagnon (orange) and a broad background feature
+of unknown origin (brown). (C and D) RIXS measured BS phonon peak for various values of q
+measured for φ = 0◦ and 30◦, respectively (black circles). The blue lines are the fits to the spectra.
+The vertical orange dashed lines, indicating the lowest phonon peak position at each φ, are shown
+to help the reader observe the phonon dispersions in the raw data.
+elastic line, once again showing the softening of the RIXS phonon even at φ = 30◦. To
+precisely determine the locations of the softening in the qx-qy plane, we fit the spectra to
+extract the dispersion of the BS phonon for each φ (Fig. 2B). We observe a softening of the
+RIXS measured phonon line for all φ, except for φ = 45◦. Remarkably all observed softening
+occurs at a value of q ≈ qCO, precisely as expected for QCDCs at low energies.
+Discriminating QCDCs from short-range directional order
+Dynamic correlations emanating from short range order are bound to be broad in q. It
+is therefore reasonable to ask whether the measured qx-qy profile of the BS phonon could
+
+6
+0.4
+0.3
+0.2
+0.1
+100
+60
+20
+100
+60
+20
+70
+60
+50
+0.4
+0.3
+0.2
+0.1
+ϕ:
+q (r.l.u.)
+q (r.l.u.)
+Energy Loss (meV)
+Phonon Energy (meV)
+1
+2
+1
+2
+3
+A
+B
+ϕ=0°
+ 0°
+ 25°
+ 30°
+ 35°
+ 45°
+ϕ=30°
+FIG. 2. Location of low-energy dynamic correlations extracted from the phonon dis-
+persion. (A) Energy-momentum structure of the excitations at φ = 0◦ and 30◦ after subtraction
+of the elastic line. The image is constructed from RIXS spectra deconvoluted from the energy res-
+olution. (B) Location of the phonon peak obtained by fitting the RIXS spectra deconvoluted from
+energy resolution for different φ (see Materials and Methods and also Supplementary Materials,
+Fig. S3). The solid lines are obtained by fitting the q-dependence of the phonon peak (circles) with
+a negative Lorentzian function plus a linear background. The shaded regions around the solid lines
+are generated from the 95% confidence interval obtained for the various fits to the spectra (see
+Materials and Methods for details). The solid lines for φ = 0◦ and 30◦ in (B) appear as dashed
+white lines in (A).
+simply be the result of diffuse scattering from short-range directional order. The fundamental
+difference between QCDCs and short-range directional order is that the former forms a
+manifold of dynamic correlations centered at q = qCO (similar to Brazovskii-type fluctuations
+[15, 26]), while the the latter results in dynamic correlations around q = [qx = ±qCO, qy = 0]
+and q = [qx = 0, qy = ±qCO] (more details on M1 and M2 are provided in the Materials and
+Methods section). To contrast these scenarios we consider two simple toy models. In both
+cases we start with a flat |q|-independent phonon mode at 72 meV, which is a reasonable
+approximation given the small dispersion of the BS phonon in the absence of charge order
+[25, 27]. In the first model (M1) we construct the QCDCs scenario, where the q-cuts for
+
+7
+-0.5
+-0.25
+0
+0.25
+0.5
+-0.5
+-0.25
+0
+0.25
+0.5
+qx (r.l.u.)
+qy (r.l.u.)
+-0.5
+-0.25
+0
+0.25
+0.5
+40
+45
+50
+55
+60
+65
+70
+Energy (meV)
+qx (r.l.u.)
+0.1
+0.2
+0.3
+0.4
+0.5
+40
+45
+50
+55
+60
+65
+70
+75
+Energy (meV)
+0.1
+0.2
+0.3
+0.4
+0.5
+0.1
+0.25
+0.4
+0°
+90°
+180°
+270°
+φ
+q (r.l.u.)
+A
+B
+C
+D
+E
+T=Tc
+T<Tc
+M1
+M2
+M1
+M2
+qx (r.l.u.)
+qx (r.l.u.)
+45°
+40°
+35°
+30°
+25°
+15°
+0°
+φ:
+FIG. 3. Models of phonon softening for QCDCs and directional order. (A and C) Phonon
+dispersion for M1 and M2, as described in the text. (B and D) Momentum q cuts of the phonon
+dispersion at different φ for the simulated data in (A) and (C) respectively. The dashed orange
+and green lines in (A-D) identify the location of the phonon softening in the qx-qy plane. (E) Polar
+plot contrasting M1, M2 models (orange and green solid lines) and the experimental data (red
+symbols). The error bars in (E) are obtained from the fits to the phonon dispersion in Fig. 2B. See
+Materials and Methods for more details.
+various φ always have a minimum located at q = qCO, Fig. 3B. In the second model (M2) we
+consider the case where dynamic charge order correlations emerge isotropically from static
+peaks at [qx = ±qCO, qy = 0] and [qx = 0, qy = ±qCO]. The corresponding phonon profile is
+shown in Fig. 3 (C and D). To roughly emulate the data we also introduce a φ-dependent
+phonon minimum in M1, which increases from φ = 0◦ to 45◦, Fig. 3A. However, note that the
+magnitude of the softening depends on the φ structure of the electron-phonon coupling, which
+is not known or necessary for discerning the two scenarios. The q-cuts show a qualitatively
+similar behavior in both models: a clear phonon softening at φ = 0 that continues to
+exist even as φ approaches 45◦. However, in M2 the q location of the phonon minima clearly
+decreases with increasing φ from 0◦ to 45◦. This comparison explains our selection of φ values
+for these studies: the experimental ability to differentiate between M1 and M2 is largest in
+the φ = 25◦ to 45◦ range. The polar plot in Fig. 3E summarizes the analysis, comparing the
+
+8
+q-location of the minima for both models to the minima obtained from experiments (red
+markers). Within the error bars, the RIXS measurements are consistent with M1 and rule
+out M2, indicating the quasi-circular nature of the low-energy correlations associated with
+the charge order.
+DISCUSSION
+The experiments presented here provide evidence for the existence of quasi-circular dy-
+namic correlations at low energies in underdoped Bi-2212, which could be a key ingredient
+for models that connect charge order to an effective isotropic scattering. Long-range trans-
+lational symmetry breaking cannot be responsible for this isotropy due to the characteristic
+length scale and directionality of the ordered state. Although short-range electron correla-
+tions from directional order, occupying a much larger region of momentum space, could in
+principle emulate isotropic scattering [3, 4, 28], the QCDCs revealed by our experiments of-
+fer a different scenario. Extending not only around the static charge order wave vectors but
+also in the azimuthal direction, QCDCs might be a more viable platform for isotropic scat-
+tering. To fully understand the impact of QCDCs to electronic properties of the cuprates,
+one requires knowledge of the energy structure of these correlations. Although this might
+still be beyond the current experimental capabilities, our experiments provide some con-
+straints to the low-energy structure of the QCDCs. In particular, for the φ values where
+a softening is detected, QCDCs must exist below ≈ 70 meV (i.e. the approximate energy
+of the bare phonon). Unfortunately the amount of energy softening at q = qCO by itself,
+without knowledge of the φ-dependence of the electron-phonon interaction, does not provide
+more information about the energy structure of the QCDCs. Therefore, it remains possible
+that QCDCs at φ = 45◦ exist below 70 meV but do not significantly interact with the BS
+phonon.
+The quasi-circular shape of the low-energy correlations is similar to the shape obtained
+from the analysis of higher energy correlations (Ref. [15] and Supplementary Materials,
+Fig. S6). This similarity raises the possibility that the QCDCs exist up to much higher
+energies of order of 1 eV. As discussed in Ref. [15], the quasi-circular correlations cannot be
+explained by an instability of the Fermi surface. Instead, it was proposed that the loca-
+tion of the dynamic CO correlations in q-space is determined by the minima of the effective
+Coulomb interaction, which becomes non-monotonic in q due to the inclusion of a long-range
+
+9
+Coulomb interaction. However, this non-monotonic Coulomb interaction by itself failed to
+capture the intensity anisotropy observed at q ≈ qCO. Likewise, here the same proposed
+Coulomb interaction could also explain the most salient feature of our data, namely the
+quasi-circular shape of the low-energy correlations. Recently, a more complete theoretical
+description based on a t-J model with long-range Coulomb interaction shows the presence
+of ring-like charge correlations with the correct intensity anisotropy [17]. The results pre-
+sented here can serve as a guide for future theoretical investigations that also account for
+the apparent decrease of the phonon softening from φ = 0◦ to 45◦.
+Beyond the fact that both the energy-integrated correlations [15] and the low-energy
+dynamic correlations appear to occupy the same quasi-circular scattering manifold, the cur-
+rent RIXS measurements do not provide further experimental evidence to connect these
+two phenomena. Such additional evidence may come from polarimetric RIXS experiments
+that are able to decompose charge and spin excitations in the mid-infrared range, as it
+has been done for electron-doped cuprates [29]. Compared to the energy-integration pro-
+cedure, the phonon tracking method provides larger precision for mapping CO correlations
+in the qx-qy plane, since the large integration ranges required for the former result in very
+broad features in q-space.
+Indeed, we have already performed medium-resolution RIXS
+measurements that detect the presence of similar quasi-circular scattering manifolds in the
+energy-integrated spectrum of optimally and overdoped samples, but the investigation of
+their doping dependence is hindered by the large experimental uncertainty associated with
+the integration method (see Supplementary Materials, Fig. S6). Instead, our new procedure
+to track QCDCs using measurements of the RIXS BS phonon goes beyond demonstrating
+the existence of QCDCs in underdoped Bi-2212 at low energies. It is also a new methodology
+that can be used to detect QCDCs in other cuprates and understand related phenomena
+such as the electron-doped cuprates which also show a quasi-circular scattering [30]. Finally,
+the application of this new method to multiple cuprate families at different dopings and/or
+temperatures will help unveil whether and how QCDCs and the strange metal are related.
+MATERIALS AND METHODS
+RIXS experiments
+High-resolution RIXS experiments were performed at the I21 beamline [31] at Diamond
+
+10
+Light Source, United Kingdom, and at the 2-ID beamline at the National Synchrotron Light
+Source II, Brookhaven National Laboratory, USA. The orientation of the crystal axes of
+the underdoped Bi2Sr2CaCu2O8+δ samples with Tc = 54 K was obtained by x-ray diffraction
+prior to the RIXS experiment. The samples were cleaved in air just moments before inserting
+them into the ultra-high-vacuum chambers. For experiments at I21, the crystal was aligned
+to the scattering geometry in situ from measurements of the 002 Bragg reflection and the
+b-axis superstructure peak. The scattering angle was fixed at 154◦ (I21) and 153◦ (2-ID).
+The incoming light was set to vertical polarization (σ geometry) at the Cu-L3 edge (≈
+931.5 eV). The combined energy resolution (FWHM) was about 37 meV (I21) and 40 meV
+(2-ID), with small variations (±3 meV) over the course of multiple days. In both cases the
+energy resolution was relaxed in a trade-off for intensity. The projection of the momentum
+transfer, q, qx-qy plane was obtained by varying the incident angle on the sample (θ). All
+the measurements were performed at T = 54 K, which is the superconducting transition
+temperature for this sample, except for one measurement performed at T = 25 K below Tc
+(Fig. 3)E and one measurement at 300 K (Supplementary Materials, Figs. S2 and S5).
+Analysis of RIXS spectra
+To ensure the robustness of the extraction of phonon dispersion from the RIXS spectra we
+analyzed the data using multiple methods. Although the overall RIXS cross-section may
+depend on φ, we did not perform any normalization or intensity correction procedure to the
+spectra since the energy location of the phonon does not depend on the overall intensity.
+A comparison between the results for different methods is available in the supplementary
+materials, Fig. S4.
+Method 1: In an effort to maintain an agnostic approach and to not assume particular
+functional forms of the different contributions to the RIXS spectra, we extracted the dis-
+persion by simply tracking the energy positions of the phonon peak maximum in the RIXS
+spectra deconvoluted from the energy resolution. See below for details of the deconvolution
+procedure.
+Method 2: The phonon dispersion shown in Fig. 2B was extracted by fitting the deconvo-
+luted RIXS spectra (see Fig. 2A and supplementary materials, Fig. S3) in the [-30,130] meV
+range to a double Gaussian function plus a second order polynomial background, keeping
+all parameters free. The shaded regions around the solid lines in Fig. 2B were generated by
+fitting the 95% confidence intervals (obtained from the fits) to a polynomial function of q.
+
+11
+Method 3: Following previous works [23, 32], the raw RIXS spectra were fit to a five
+component model that includes a Gaussian (elastic peak of amplitude Ael, position ωel and
+width wel), two anti-Lorentzians (phonon and double-phonon peaks of different amplitude
+Ai and position ωi, and sharing width wph and Fano parameter width qF – note that i=1,2
+indicates the first and second phonon, respectively), a damped harmonic oscillator lineshape
+(paramagnon of amplitude Apm, position ωpm and damping parameter γpm), and an error
+function (smooth background described by an error function with amplitude amplitude ABG,
+position ωBG and width wBG):
+f(ω) = Aele
+− (ω−ωel)2
+w2
+el
++
+�
+i=1,2
+Ai
+2(ω − ωi)/wph + qF
+[2(ω − ωi)/wph]2 + 1+
++Apm
+γpmω
+(ω2 − ω2
+pm)2 + 4γ2
+pmω2 + ABG
+�
+erf
+�ω − ωBG
+wBG
+�
++ 1
+�
+(1)
+The fitting model is convoluted with the RIXS energy resolution (∼37 meV). From this
+analysis we extracted the phonon dispersion for each φ that quantitatively matches the
+phonon dispersion shown in Fig. 2B. All parameters are kept free, except for ωbg, which is
+constrained within a range of [0.2, 0.6] eV.
+Fitting the phonon dispersion
+To obtain a phenomenological form to the dispersions in Fig. 2B, the extracted peak locations
+as a function of q were fit to a linear background plus a negative Lorentzian function. From
+this fit we obtain the q location of the softening (red markers in Fig. 3E). To obtain the
+error bars in Fig. 3E, we follow a conservative approach by taking the average of the two
+q-intercepts of the fitted curve at Emin + 2 meV, where Emin is the lowest energy of the
+dispersion and ±2 meV is the typical amount of scatter observed in the data. For φ = 45◦
+the data is fit to a line.
+Deconvolution procedure
+We employed the Lucy-Richardson deconvolution procedure [33] to deconvolve the energy
+resolution (∼37 meV) from the RIXS spectra (deconvoluted curves for all azimuths are
+displayed in the supplementary materials, Fig. S3). The number of iterations and region
+of interest of the deconvolution procedure were optimized by ensuring that the convolution
+of the deconvoluted curves reproduced the raw data.
+Model simulations
+The toy models M1 and M2 are purely phenomenological. For M1, the phonon dispersion
+
+12
+was modeled as:
+E = E0 − ξ(φ)
+∆
+( q−q0
+Γ )2 + 1
+(2)
+where E0 = 72 meV, ∆ = 30 meV, Γ = 0.065 r.l.u., q0 = 0.29 r.l.u. and ξ(φ) = (| cos(2φ)| +
+0.08)/(1.08). For M2, the phonon dispersion was modeled as:
+E = E0 −
+4
+�
+i=1
+∆
+( q−qi
+Γ )2 + 1
+(3)
+where E0 = 74 meV, ∆ = 30 meV, Γ = 0.065 r.l.u. and qi are the four peaks located at
+[qx = ±qCO, qy = 0] and [qx = 0, qy = ±qCO] (qCO = 0.29 r.l.u.)
+Acknowledgments
+We acknowledge the Diamond Light Source for time on beamline I21-RIXS under propos-
+als MM28523 and MM30146. This research used resources of the Advanced Light Source,
+a DOE Office of Science User Facility under contract no. DE-AC02-05CH11231. This re-
+search used beamline 2-ID of the National Synchrotron Light Source II, a U.S. Department
+of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by
+Brookhaven National Laboratory under Contract No. DE-SC0012704. We especially ac-
+knowledge the incredible work done by the beamline staffs at I-21, 2-ID and at 8.0.1 qRIXS,
+to allow many of these experiments to be performed remotely during the COVID pandemic.
+This work was supported by the Alfred P. Sloan Fellowship (E.H.d.S.N.). E.H.d.S.N ac-
+knowledges support by the National Science Foundation under Grant No. 2034345. A.F.K.
+was supported by the National Science Foundation under grant no. DMR-1752713. F.B.
+acknowledges support from the Fonds de recherche du Qu´ebec – Nature et technologies
+(FRQNT) and the Natural Sciences and Engineering Research Council of Canada (NSERC).
+A.F. was supported by the Research Corporation for Science Advancement via the Cottrell
+Scholar Award (27551) and the CIFAR Azrieli Global Scholars program. This material is
+based upon work supported by the National Science Foundation under Grant No. DMR-
+2145080. The synthesis work at Brookhaven National Laboratory was supported by the US
+Department of Energy, office of Basic Energy Sciences, contract no. DOE-SC0012704.
+
+13
+Supplementary Materials
+12
+10
+8
+6
+4
+2
+0
+0.8
+0.6
+0.4
+0.2
+0.0
+10
+8
+6
+4
+2
+0
+0.8
+0.6
+0.4
+0.2
+0.0
+Intensity (arb. units)
+Energy Loss (eV)
+ϕ=0°
+ϕ=30°
+FIG. S1. Fits to the spectra in Fig. 2 The two panels show the fit of the RIXS spectra over a
+wide energy range using Method (3) as detailed in the Materials and Methods Section of the main
+text.
+
+14
+1.6
+1.4
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+120
+80
+40
+1.6
+1.4
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+Intensity (arb. units)
+Energy Loss (eV)
+120
+80
+40
+ϕ=0°
+ϕ=30°
+70
+60
+50
+40
+0.4
+0.3
+0.2
+ 54 K
+ 300 K
+80
+q (r.l.u.)
+Phonon Dispersion (meV)
+ϕ=30°
+ϕ=0°
+ϕ=45°
+A
+B
+FIG. S2. Data obtained at the 2-ID beamline at NSLS-II. (A) RIXS spectra measured as a
+function of q for different φ at 54 K. The dashed line is a guide to the eye highlighting the phonon
+softening, visible at both φ = 0◦ and φ = 30◦ already from the raw data without need of doing any
+deconvolution. (B) Phonon dispersion obtained by using Method (2) as detailed in the Materials
+and Methods Section of the main text.
+
+15
+10
+8
+6
+4
+2
+0
+0.12
+0.08
+0.04
+0.00
+10
+8
+6
+4
+2
+0
+0.12
+0.08
+0.04
+0.00
+5
+4
+3
+2
+1
+0
+0.12
+0.08
+0.04
+0.00
+16
+14
+12
+10
+8
+6
+4
+2
+0
+0.12
+0.08
+0.04
+0.00
+10
+8
+6
+4
+2
+0
+0.12
+0.08
+0.04
+0.00
+Intensity (arb. units)
+Intensity (arb. units)
+Energy Loss (eV)
+ϕ=0°
+ϕ=45°
+ϕ=35°
+ϕ=30°
+ϕ=25°
+FIG. S3.
+Waterfall plot of the RIXS spectra deconvoluted for energy resolution
+(37 meV) for different φ.
+The softening of the BS phonon is clearly visible for any φ (ex-
+cept φ=45o) by simple visual inspection of the deconvoluted curves (red dash lines are guides to
+the eye).
+
+16
+ϕ:
+q (r.l.u.)
+Phonon Dispersion (meV)
+75
+70
+65
+60
+55
+50
+45
+40
+0.4
+0.3
+0.2
+0.1
+ 0°
+ 25°
+ 30°
+ 35°
+ 45°
+ 
+Fits:
+(I): 
+ 
+(II): 
+(III):
+70
+60
+50
+40
+75
+70
+65
+60
+55
+76
+72
+68
+64
+60
+76
+72
+68
+64
+76
+72
+68
+64
+0.4
+0.3
+0.2
+0.1
+ϕ = 0°
+ϕ = 25°
+ϕ = 30°
+ϕ = 35°
+ϕ = 45°
+q (r.l.u.)
+Phonon Dispersion (meV)
+A
+B
+FIG. S4. Comparison between three different methods for the extraction of the phonon
+dispersion. Methods (1), (2) and (3) are detailed in the Materials and Methods section of the
+main text.
+
+17
+0.25
+0.4
+0°
+90°
+180°
+270°
+φ
+q (r.l.u.)
+0.1
+I21 - T=Tc
+I21 - T<Tc
+M1
+M2
+2-ID - T=Tc
+2-ID - T>Tc
+FIG. S5. Comparison of models to experiments at 2-ID and I21. Polar plot contrasting
+M1, M2 models (orange and green solid lines) and the experimental data (blue and red symbols).
+The error bars are obtained from the fits to the phonon dispersion as described in the Materials and
+Methods section. On the left side, the model was adjusted for a higher value of qCO for comparison
+with the data obtained at 2-ID. The data at 2-ID is consistent with M1 and not with M2. In the
+main text only the data from I21 is shown because for those experiments the sample crystal axes
+could be aligned in situ from structural diffraction peaks by using the photodiode detector in that
+chamber. See Materials and Methods section for details.
+
+18
+Medium resolution RIXS
+In Fig. S6 we show the results of medium resolution RIXS done at the qRIXS endstation
+at the Advanced Light Source in the Lawrence Berkeley National Laboratory. The data
+were obtained by integrating the RIXS spectra over the −0.5 to 0.7 eV energy window and
+normalizing them by spectra integrated over all energies, which allows a comparison between
+the three different dopings. The data was also symmetrized about the high-symmetry φ =
+45◦ direction. In Fig. S6(D-I) the solid lines are fits of a Gaussian function plus a linear
+background to the data. The maps in Fig. S6(A-C) were generated from the fits in Fig. S6(G-
+I), respectively.
+The gray bars in Fig. S6(D-F) are centered at the average radii of the
+correlations, obtained from averaging over φ the peak positions obtained from the fits in
+Fig. S6(G-I). The widths of the grey bars in Fig. S6(D and E) are obtained from the 95%
+confidence intervals obtained from the fits in Fig. S6(G and H), summing them in quadrature
+and taking their square root. The same procedure underestimates the uncertainty for the
+Tc = 54 K. Instead the width of the grey bar in Fig. S6(F) is calculated by taking the lowest
+and largest peak positions over all φ, taking into account the 95% confidence intervals from
+the fits in Fig. S6(I). The width of the grey bar in Fig. S6(F) The data used to generate
+Fig. S6(C, F and I) were used in a previous publication [Boschini et al. Nat. Comm. 12, 1-
+8 2021]. The new data follows the same experimental procedures as the previously published
+data, so we direct the reader to [Boschini et al. Nat. Comm. 12, 1-8 2021] for further details
+of the experimental procedure.
+
+19
+0
+0.2 0.4
+q (r.l.u.)
+20
+30
+40
+50
+60
+70
+80
+90
+100
+0
+0.1 0.2 0.3 0.4
+q (r.l.u.)
+18
+20
+22
+24
+26
+28
+30
+32
+34
+36
+38
+40
+Intensity (a.u.)
+0
+0.2 0.4
+q (r.l.u.)
+20
+30
+40
+50
+60
+70
+80
+90
+100
+0
+0.1 0.2 0.3 0.4
+q (r.l.u.)
+18
+20
+22
+24
+26
+28
+30
+32
+34
+36
+38
+40
+Intensity (a.u.)
+0
+0.2 0.4
+q (r.l.u.)
+20
+30
+40
+50
+60
+70
+80
+90
+100
+0
+0.1 0.2 0.3 0.4
+q (r.l.u.)
+18
+20
+22
+24
+26
+28
+30
+32
+34
+36
+38
+40
+Intensity (a.u.)
+100°
+45°
+-10°
+0.1
+0.2
+0.3
+0.4
+22
+23
+24
+25
+26
+27
+q (r.l.u.)
+Intensity (a.u.)
+100°
+45°
+-10°
+0.1
+0.2
+0.3
+0.4
+22
+23
+24
+25
+26
+27
+q (r.l.u.)
+Intensity (a.u.)
+95°
+45°
+-10°
+0.1
+0.2
+0.3
+0.4
+22
+23
+24
+25
+26
+27
+q (r.l.u.)
+Intensity (a.u.)
+[-10°, 5°]
+[15°, 25°]
+[35°, 55°]
+[-10°, 10°]
+[20°, 30°]
+[40°, 50°]
+[-10°, 10°]
+[15°, 30°]
+[45°]
+-10°
+0°
+5°
+15°
+25°
+35°
+45°
+55°
+65°
+75°
+85°
+90°
+95°
+-10°
+0°
+10°
+20°
+30°
+40°
+50°
+60°
+70°
+80°
+90°
+100°
+-10°
+-5°
+0°
+5°
+10°
+15°
+30°
+45°
+60°
+75°
+80°
+90°
+95°
+85°
+100°
+Overdoped
+Tc = 60K
+Optimally doped
+Tc = 91K
+Underdoped
+Tc = 54K
+A
+B
+C
+D
+E
+F
+G
+H
+I
+FIG. S6. Doping dependence from medium resolution RIXS (A-C) Normalized energy-
+integrated RIXS mapping showing high energy quasi-circular electron correlations in overdoped,
+optimally doped and underdoped samples, respectively. (D-F) q-cuts integrated over different φ
+ranges, as specified in the legends. (G-I) The normalized energy-integrated RIXS data used to
+used to generate (A and D), (B and E) and (C and F).
+
+20
+1.0
+0.8
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+1.0
+0.5
+0.0
+0.8
+0.4
+0.0
+q (r.l.u.)
+Intensity (arb. units)
+∫100 meVIRIXS / ∫IRIXS
+700 meV
+Intensity (arb. units)
+ ϕ=0°
+ 25°
+ 30°
+ 35°
+ 45°
+q (r.l.u.)
+ϕ = 30°
+A
+B
+FIG. S7. Energy-integrated RIXS maps from high resolution RIXS on Bi2212 under-
+doped Tc=54 K at I21. A Energy-Loss RIXS spectrum for (q,φ)=(0.28 rlu, 30o). The orange
+shadow highlights the energy integration window [0.1,0.7] eV. B q-cuts of the RIXS spectra in-
+tegrated over the energy regions highlighted in A and normalized by the total energy-integrated
+RIXS signal. The overall q-dependence and position of the maximum is similar to what observed via
+medium resolution RIXS (see Fig. S6(F and I)). The pink bar reproduces the grey bar in Fig. S6(F),
+which is obtained from the analysis of the correlations observed with medium resolution RIXS for
+underdoped Bi2212 (Tc=54 K).
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+page_content=' North Carolina State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Raleigh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  11Donostia International Physics Center, DIPC, 20018 Donostia-San Sebastian, Basque Country, Spain  12IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain  13Canadian Institute for Advanced Research, Toronto, ON, M5G 1M1, Canada  14Quantum Matter Institute, University of British Columbia, Vancouver, BC V6T 1Z4, Canada  ∗ Corresponding Author: eduardo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='dasilvaneto@yale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='edu  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='08415v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='str-el]  20 Jan 2023  2  ABSTRACT  In the study of dynamic charge order correlations in the cuprates, most high energy-  resolution resonant inelastic x-ray scattering (RIXS) measurements have focused on mo-  menta along the high-symmetry directions of the copper oxide plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' However, electron  scattering along other in-plane directions should not be neglected as they may contain in-  formation relevant, for example, to the origin of charge order correlations or to our un-  derstanding of the isotropic scattering responsible for strange metal behavior in cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  We report high-resolution resonant inelastic x-ray scattering (RIXS) experiments that re-  veal the presence of dynamic electron correlations over the qx-qy scattering plane in under-  doped Bi2Sr2CaCu2O8+δ with Tc = 54 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' We use the softening of the RIXS-measured bond  stretching phonon line as a marker for the presence of charge-order-related dynamic electron  correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The experiments show that these dynamic correlations exist at energies below  approximately 70 meV and are centered around a quasi-circular manifold in the qx-qy scat-  tering plane with radius equal to the magnitude of the charge order wave vector, qCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' We  also demonstrate how this phonon-tracking procedure provides the necessary experimental  precision to rule out fluctuations of short-range directional charge order (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' centered around  [qx = ±qCO, qy = 0] and [qx = 0, qy = ±qCO]) as the origin of the observed correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  INTRODUCTION  Dynamic fluctuations from periodic charge order (CO) pervade the phase diagram of  cuprate superconductors, perhaps even more than superconductivity itself [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The detec-  tion of these fluctuations over energy and momentum was enabled by several recent advances  in the energy resolution of resonant inelastic x-ray scattering (RIXS) instruments operat-  ing in the soft x-ray regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' In the case of YBa2Cu3O6+δ, Cu-L3 RIXS detects dynamic  correlations at the charge order wavevector, qCO, with a characteristic energy scale of ap-  proximately 20 meV [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' It has been proposed that these low-energy short-range dynamic  charge order correlations are a key ingredient to the strange metal behavior [3, 4] charac-  terized by linear-in-temperature resistivity [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' On one hand, this temperature behav-  ior is often associated with an isotropic scattering rate that depends only on temperature  in units of energy and Planck’s constant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' ∝ kBT/ℏ, sometimes called the Planckian  regime) [7–11], as supported by recent angle-dependent magnetoresistance measurements of  La1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='6−xNd0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='4SrxCuO4 [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' On the other hand, combined transport and RIXS studies have  3  recently shown an unexpected link between linear-in-temperature resistivity and charge or-  der in YBa2Cu3O6+δ [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Combined, these latest results suggest that fluctuations of the  charge order should somehow result in an effective isotropic scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Still, high-resolution  RIXS experiments have largely focused on the fluctuations along the high-symmetry crystal-  lographic directions only, leaving the full structure of electron correlations within the copper  oxide plane unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  Recently, in Bi2Sr2CaCu2O8+δ (Bi-2212), RIXS measurements found the existence of a  quasi-circular pattern in the qx-qy plane at finite energies and with the same wave vector  magnitude as that of the observed static charge order peak at q = [qx = ±qCO, qy =  0] and [qx = 0, qy = ±qCO] – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  dynamic correlations with charge order wavelength  along all direction in the CuO2 plane [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Although the medium energy-resolution of those  measurements (∆E ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='8 eV) precluded a more precise determination of their energy profile,  the results suggested that these quasi-circular dynamic correlations (QCDCs) appear broad  over the mid-infrared ranges (defined approximately as 100 to 900 meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' This scattering  manifold, which may result from combined short- and long-range Coulomb interactions [15–  17], would provide a large variety of wave vectors for connecting all points of the Fermi  surface (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' an effective isotropic scattering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' However, it is not yet experimentally known  if this manifold extends to electron scattering at lower energies, in the quasi-elastic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  To experimentally investigate this scenario we used high energy-resolution (≈ 37 meV) Cu-L3  RIXS qx-qy mapping of the electronic correlations in Bi-2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Using the softening of the bond  stretching (BS) phonon in RIXS as a marker of charge order correlations, our measurements  reveal the presence of low-energy quasi-circular dynamic electronic correlations with |q| ≈  qCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  RESULTS  High-resolution RIXS mapping of dynamic correlations in the qx-qy plane  We performed measurements at φ = 0◦, 25◦, 30◦, 35◦, 45◦, where φ is defined as the az-  imuthal angle from the qx axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' For each φ, we acquired RIXS spectra at different values of  in-plane momentum-transfer q = |q| by varying the incident angle on the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Through-  out the paper, values of q are reported in reciprocal lattice units (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' ), where one r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' is  defined as 2π/a and a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='82 ˚A (the lattice constant along φ = 0◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 1 (A and B), we  4  show representative spectra obtained at q near qCO for φ = 0◦ and 30◦, and energies below  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' In these two cases, the minimal model that fits the data includes five contributions:  a quasi-elastic peak, a bond-stretching phonon peak at ≈ 70 meV, a peak at ≈ 135 meV  (likely from a two-phonon process), a broad paramagnon and a broad background feature  of unknown origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' A similar assessment can be made regarding all other high-resolution  spectra acquired in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' In this type of fitting analysis, the QCDCs are not explicitly  accounted for and it is generally difficult to disentangle overlapping contributions to the  RIXS spectra using a fitting model with so many parameters, thus precluding the extraction  of the exact spectral profile of the QCDCs with any reasonable confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Still, we note  that this high-resolution data is consistent with the previously reported medium-resolution  data [15], which can be verified by integration of the high-resolution data (see supplementary  materials, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  It is likely that the spectral intensity of QCDCs in Bi-2212 is so dilute over energy  as to preclude the extraction of their spectral structure amidst stronger paramagnon and  phonon signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Still, here we develop a different method to detect QCDCs at lower energies,  by tracking the evolution of the bond-stretching (BS) phonon over the qx-qy plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' This  method is based on the phenomenology revealed by several recent RIXS measurements of  the cuprates along qx and qy, which indicate an apparent softening of the BS phonon peak  at the momentum location of the static CO peak [18–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' In the case of Bi-2212, it has  been proposed that the apparent softening of the BS phonon in RIXS is due to an interplay  between low-energy fluctuations of the charge order and BS phonons that results in a Fano-  like interference [18, 21, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Another possibility is that the apparent softening is simply  the result of the phonon peak and a low-energy charge order peak overlapping, as recently  suggested by measurements of both YBa2Cu3O6+δ and Bi-2212 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' In either interpretation  the location of the phonon softening can be used as a marker for low-energy charge order  correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  Figure 1 (C and D) shows the spectra acquired as a function of q for φ = 0◦ and 30◦,  respectively, focusing on the region of the BS phonon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  At φ = 0◦, it is clear that the  phonon peak position softens to its lowest energy value at q = qCO ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='29 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 1C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  Careful observation of the spectra taken along φ = 30◦ shows a similar softening effect with  the lowest phonon energy position occuring for q ≈ qCO (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 1D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Figure 2A shows the  mapping of the BS phonon mode at φ = 0◦ and 30◦ obtained after subtraction of the fitted  5  3  2  1  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='0  ϕ=0°  q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='27 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  1  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='0  ϕ=30°  q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='27 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  IRIXS (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' units)  IRIXS (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' units)  Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' units)  Energy Loss (meV)  A  B  C  Energy Loss (eV)  10  8  6  4  2  120  80  40  Energy Loss (eV)  ϕ=0°  D  ϕ=30°  120  80  40  q (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=')  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content='23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content='43  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='47  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' RIXS spectra and fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' (A and B) Examples of spectra at q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='27 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' for φ = 0◦  and 30◦, respectively (open circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The red lines are fits to the spectra, composed of a quasi-  elastic peak (pink), a BS phonon peak at ≈ 70 meV (blue), a peak at ≈ 135 meV (likely from  a two-phonon process) (purple), a broad paramagnon (orange) and a broad background feature  of unknown origin (brown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' (C and D) RIXS measured BS phonon peak for various values of q  measured for φ = 0◦ and 30◦, respectively (black circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The blue lines are the fits to the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  The vertical orange dashed lines, indicating the lowest phonon peak position at each φ, are shown  to help the reader observe the phonon dispersions in the raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  elastic line, once again showing the softening of the RIXS phonon even at φ = 30◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' To  precisely determine the locations of the softening in the qx-qy plane, we fit the spectra to  extract the dispersion of the BS phonon for each φ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 2B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' We observe a softening of the  RIXS measured phonon line for all φ, except for φ = 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Remarkably all observed softening  occurs at a value of q ≈ qCO, precisely as expected for QCDCs at low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  Discriminating QCDCs from short-range directional order  Dynamic correlations emanating from short range order are bound to be broad in q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' It  is therefore reasonable to ask whether the measured qx-qy profile of the BS phonon could  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='1  100  60  20  100  60  20  70  60  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='1  ϕ:  q (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=')  q (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=')  Energy Loss (meV)  Phonon Energy (meV)  1  2  1  2  3  A  B  ϕ=0°  0°  25°  30°  35°  45°  ϕ=30°  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Location of low-energy dynamic correlations extracted from the phonon dis-  persion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' (A) Energy-momentum structure of the excitations at φ = 0◦ and 30◦ after subtraction  of the elastic line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The image is constructed from RIXS spectra deconvoluted from the energy res-  olution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' (B) Location of the phonon peak obtained by fitting the RIXS spectra deconvoluted from  energy resolution for different φ (see Materials and Methods and also Supplementary Materials,  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The solid lines are obtained by fitting the q-dependence of the phonon peak (circles) with  a negative Lorentzian function plus a linear background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The shaded regions around the solid lines  are generated from the 95% confidence interval obtained for the various fits to the spectra (see  Materials and Methods for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The solid lines for φ = 0◦ and 30◦ in (B) appear as dashed  white lines in (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  simply be the result of diffuse scattering from short-range directional order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The fundamental  difference between QCDCs and short-range directional order is that the former forms a  manifold of dynamic correlations centered at q = qCO (similar to Brazovskii-type fluctuations  [15, 26]), while the the latter results in dynamic correlations around q = [qx = ±qCO, qy = 0]  and q = [qx = 0, qy = ±qCO] (more details on M1 and M2 are provided in the Materials and  Methods section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' To contrast these scenarios we consider two simple toy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' In both  cases we start with a flat |q|-independent phonon mode at 72 meV, which is a reasonable  approximation given the small dispersion of the BS phonon in the absence of charge order  [25, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' In the first model (M1) we construct the QCDCs scenario, where the q-cuts for  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='25  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content='25  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='5  qx (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=')  qy (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=')  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='25  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='5  40  45  50  55  60  65  70  Energy (meV)  qx (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=')  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='5  40  45  50  55  60  65  70  75  Energy (meV)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='4  0°  90°  180°  270°  φ  q (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=')  A  B  C  D  E  T=Tc  T<Tc  M1  M2  M1  M2  qx (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=')  qx (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=')  45°  40°  35°  30°  25°  15°  0°  φ:  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Models of phonon softening for QCDCs and directional order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' (A and C) Phonon  dispersion for M1 and M2, as described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' (B and D) Momentum q cuts of the phonon  dispersion at different φ for the simulated data in (A) and (C) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The dashed orange  and green lines in (A-D) identify the location of the phonon softening in the qx-qy plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' (E) Polar  plot contrasting M1, M2 models (orange and green solid lines) and the experimental data (red  symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The error bars in (E) are obtained from the fits to the phonon dispersion in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' See  Materials and Methods for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  various φ always have a minimum located at q = qCO, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 3B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' In the second model (M2) we  consider the case where dynamic charge order correlations emerge isotropically from static  peaks at [qx = ±qCO, qy = 0] and [qx = 0, qy = ±qCO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The corresponding phonon profile is  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 3 (C and D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' To roughly emulate the data we also introduce a φ-dependent  phonon minimum in M1, which increases from φ = 0◦ to 45◦, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 3A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' However, note that the  magnitude of the softening depends on the φ structure of the electron-phonon coupling, which  is not known or necessary for discerning the two scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The q-cuts show a qualitatively  similar behavior in both models: a clear phonon softening at φ = 0 that continues to  exist even as φ approaches 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' However, in M2 the q location of the phonon minima clearly  decreases with increasing φ from 0◦ to 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' This comparison explains our selection of φ values  for these studies: the experimental ability to differentiate between M1 and M2 is largest in  the φ = 25◦ to 45◦ range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The polar plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 3E summarizes the analysis, comparing the  8  q-location of the minima for both models to the minima obtained from experiments (red  markers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Within the error bars, the RIXS measurements are consistent with M1 and rule  out M2, indicating the quasi-circular nature of the low-energy correlations associated with  the charge order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  DISCUSSION  The experiments presented here provide evidence for the existence of quasi-circular dy-  namic correlations at low energies in underdoped Bi-2212, which could be a key ingredient  for models that connect charge order to an effective isotropic scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Long-range trans-  lational symmetry breaking cannot be responsible for this isotropy due to the characteristic  length scale and directionality of the ordered state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Although short-range electron correla-  tions from directional order, occupying a much larger region of momentum space, could in  principle emulate isotropic scattering [3, 4, 28], the QCDCs revealed by our experiments of-  fer a different scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Extending not only around the static charge order wave vectors but  also in the azimuthal direction, QCDCs might be a more viable platform for isotropic scat-  tering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' To fully understand the impact of QCDCs to electronic properties of the cuprates,  one requires knowledge of the energy structure of these correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Although this might  still be beyond the current experimental capabilities, our experiments provide some con-  straints to the low-energy structure of the QCDCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' In particular, for the φ values where  a softening is detected, QCDCs must exist below ≈ 70 meV (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' the approximate energy  of the bare phonon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Unfortunately the amount of energy softening at q = qCO by itself,  without knowledge of the φ-dependence of the electron-phonon interaction, does not provide  more information about the energy structure of the QCDCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Therefore, it remains possible  that QCDCs at φ = 45◦ exist below 70 meV but do not significantly interact with the BS  phonon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  The quasi-circular shape of the low-energy correlations is similar to the shape obtained  from the analysis of higher energy correlations (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' [15] and Supplementary Materials,  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' This similarity raises the possibility that the QCDCs exist up to much higher  energies of order of 1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' As discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' [15], the quasi-circular correlations cannot be  explained by an instability of the Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Instead, it was proposed that the loca-  tion of the dynamic CO correlations in q-space is determined by the minima of the effective  Coulomb interaction, which becomes non-monotonic in q due to the inclusion of a long-range  9  Coulomb interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' However, this non-monotonic Coulomb interaction by itself failed to  capture the intensity anisotropy observed at q ≈ qCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Likewise, here the same proposed  Coulomb interaction could also explain the most salient feature of our data, namely the  quasi-circular shape of the low-energy correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Recently, a more complete theoretical  description based on a t-J model with long-range Coulomb interaction shows the presence  of ring-like charge correlations with the correct intensity anisotropy [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The results pre-  sented here can serve as a guide for future theoretical investigations that also account for  the apparent decrease of the phonon softening from φ = 0◦ to 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  Beyond the fact that both the energy-integrated correlations [15] and the low-energy  dynamic correlations appear to occupy the same quasi-circular scattering manifold, the cur-  rent RIXS measurements do not provide further experimental evidence to connect these  two phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Such additional evidence may come from polarimetric RIXS experiments  that are able to decompose charge and spin excitations in the mid-infrared range, as it  has been done for electron-doped cuprates [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Compared to the energy-integration pro-  cedure, the phonon tracking method provides larger precision for mapping CO correlations  in the qx-qy plane, since the large integration ranges required for the former result in very  broad features in q-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  Indeed, we have already performed medium-resolution RIXS  measurements that detect the presence of similar quasi-circular scattering manifolds in the  energy-integrated spectrum of optimally and overdoped samples, but the investigation of  their doping dependence is hindered by the large experimental uncertainty associated with  the integration method (see Supplementary Materials, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Instead, our new procedure  to track QCDCs using measurements of the RIXS BS phonon goes beyond demonstrating  the existence of QCDCs in underdoped Bi-2212 at low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' It is also a new methodology  that can be used to detect QCDCs in other cuprates and understand related phenomena  such as the electron-doped cuprates which also show a quasi-circular scattering [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Finally,  the application of this new method to multiple cuprate families at different dopings and/or  temperatures will help unveil whether and how QCDCs and the strange metal are related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  MATERIALS AND METHODS  RIXS experiments  High-resolution RIXS experiments were performed at the I21 beamline [31] at Diamond  10  Light Source, United Kingdom, and at the 2-ID beamline at the National Synchrotron Light  Source II, Brookhaven National Laboratory, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The orientation of the crystal axes of  the underdoped Bi2Sr2CaCu2O8+δ samples with Tc = 54 K was obtained by x-ray diffraction  prior to the RIXS experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The samples were cleaved in air just moments before inserting  them into the ultra-high-vacuum chambers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' For experiments at I21, the crystal was aligned  to the scattering geometry in situ from measurements of the 002 Bragg reflection and the  b-axis superstructure peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The scattering angle was fixed at 154◦ (I21) and 153◦ (2-ID).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  The incoming light was set to vertical polarization (σ geometry) at the Cu-L3 edge (≈  931.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='5 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The combined energy resolution (FWHM) was about 37 meV (I21) and 40 meV  (2-ID), with small variations (±3 meV) over the course of multiple days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' In both cases the  energy resolution was relaxed in a trade-off for intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The projection of the momentum  transfer, q, qx-qy plane was obtained by varying the incident angle on the sample (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' All  the measurements were performed at T = 54 K, which is the superconducting transition  temperature for this sample, except for one measurement performed at T = 25 K below Tc  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 3)E and one measurement at 300 K (Supplementary Materials, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S2 and S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  Analysis of RIXS spectra  To ensure the robustness of the extraction of phonon dispersion from the RIXS spectra we  analyzed the data using multiple methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Although the overall RIXS cross-section may  depend on φ, we did not perform any normalization or intensity correction procedure to the  spectra since the energy location of the phonon does not depend on the overall intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  A comparison between the results for different methods is available in the supplementary  materials, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  Method 1: In an effort to maintain an agnostic approach and to not assume particular  functional forms of the different contributions to the RIXS spectra, we extracted the dis-  persion by simply tracking the energy positions of the phonon peak maximum in the RIXS  spectra deconvoluted from the energy resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' See below for details of the deconvolution  procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  Method 2: The phonon dispersion shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 2B was extracted by fitting the deconvo-  luted RIXS spectra (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 2A and supplementary materials, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S3) in the [-30,130] meV  range to a double Gaussian function plus a second order polynomial background, keeping  all parameters free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The shaded regions around the solid lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 2B were generated by  fitting the 95% confidence intervals (obtained from the fits) to a polynomial function of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  11  Method 3: Following previous works [23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 32],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' the raw RIXS spectra were fit to a five  component model that includes a Gaussian (elastic peak of amplitude Ael,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' position ωel and  width wel),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' two anti-Lorentzians (phonon and double-phonon peaks of different amplitude  Ai and position ωi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' and sharing width wph and Fano parameter width qF – note that i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='2  indicates the first and second phonon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' respectively),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' a damped harmonic oscillator lineshape  (paramagnon of amplitude Apm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' position ωpm and damping parameter γpm),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' and an error  function (smooth background described by an error function with amplitude amplitude ABG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  position ωBG and width wBG):  f(ω) = Aele  − (ω−ωel)2  w2  el  +  �  i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='2  Ai  2(ω − ωi)/wph + qF  [2(ω − ωi)/wph]2 + 1+  +Apm  γpmω  (ω2 − ω2  pm)2 + 4γ2  pmω2 + ABG  �  erf  �ω − ωBG  wBG  �  + 1  �  (1)  The fitting model is convoluted with the RIXS energy resolution (∼37 meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' From this  analysis we extracted the phonon dispersion for each φ that quantitatively matches the  phonon dispersion shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' All parameters are kept free, except for ωbg, which is  constrained within a range of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='6] eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  Fitting the phonon dispersion  To obtain a phenomenological form to the dispersions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 2B, the extracted peak locations  as a function of q were fit to a linear background plus a negative Lorentzian function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' From  this fit we obtain the q location of the softening (red markers in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 3E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' To obtain the  error bars in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 3E, we follow a conservative approach by taking the average of the two  q-intercepts of the fitted curve at Emin + 2 meV, where Emin is the lowest energy of the  dispersion and ±2 meV is the typical amount of scatter observed in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' For φ = 45◦  the data is fit to a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  Deconvolution procedure  We employed the Lucy-Richardson deconvolution procedure [33] to deconvolve the energy  resolution (∼37 meV) from the RIXS spectra (deconvoluted curves for all azimuths are  displayed in the supplementary materials, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The number of iterations and region  of interest of the deconvolution procedure were optimized by ensuring that the convolution  of the deconvoluted curves reproduced the raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  Model simulations  The toy models M1 and M2 are purely phenomenological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' For M1, the phonon dispersion  12  was modeled as:  E = E0 − ξ(φ)  ∆  ( q−q0  Γ )2 + 1  (2)  where E0 = 72 meV, ∆ = 30 meV, Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='065 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=', q0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='29 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' and ξ(φ) = (| cos(2φ)| +  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='08)/(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='08).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' For M2, the phonon dispersion was modeled as:  E = E0 −  4  �  i=1  ∆  ( q−qi  Γ )2 + 1  (3)  where E0 = 74 meV, ∆ = 30 meV, Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='065 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' and qi are the four peaks located at  [qx = ±qCO, qy = 0] and [qx = 0, qy = ±qCO] (qCO = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='29 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=')  Acknowledgments  We acknowledge the Diamond Light Source for time on beamline I21-RIXS under propos-  als MM28523 and MM30146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' This research used resources of the Advanced Light Source,  a DOE Office of Science User Facility under contract no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' DE-AC02-05CH11231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' This re-  search used beamline 2-ID of the National Synchrotron Light Source II, a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Department  of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by  Brookhaven National Laboratory under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' DE-SC0012704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' We especially ac-  knowledge the incredible work done by the beamline staffs at I-21, 2-ID and at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='1 qRIXS,  to allow many of these experiments to be performed remotely during the COVID pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  This work was supported by the Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Sloan Fellowship (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='N ac-  knowledges support by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 2034345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content=' DMR-1752713.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content='  acknowledges support from the Fonds de recherche du Qu´ebec – Nature et technologies  (FRQNT) and the Natural Sciences and Engineering Research Council of Canada (NSERC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' was supported by the Research Corporation for Science Advancement via the Cottrell  Scholar Award (27551) and the CIFAR Azrieli Global Scholars program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' This material is  based upon work supported by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' DMR-  2145080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The synthesis work at Brookhaven National Laboratory was supported by the US  Department of Energy, office of Basic Energy Sciences, contract no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' DOE-SC0012704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  13  Supplementary Materials  12  10  8  6  4  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='0  Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' units)  Energy Loss (eV)  ϕ=0°  ϕ=30°  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Fits to the spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 2 The two panels show the fit of the RIXS spectra over a  wide energy range using Method (3) as detailed in the Materials and Methods Section of the main  text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  14  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='2  120  80  40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='2  Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' units)  Energy Loss (eV)  120  80  40  ϕ=0°  ϕ=30°  70  60  50  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='2  54 K  300 K  80  q (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content=')  Phonon Dispersion (meV)  ϕ=30°  ϕ=0°  ϕ=45°  A  B  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Data obtained at the 2-ID beamline at NSLS-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' (A) RIXS spectra measured as a  function of q for different φ at 54 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The dashed line is a guide to the eye highlighting the phonon  softening, visible at both φ = 0◦ and φ = 30◦ already from the raw data without need of doing any  deconvolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' (B) Phonon dispersion obtained by using Method (2) as detailed in the Materials  and Methods Section of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  15  10  8  6  4  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content='00  5  4  3  2  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='00  16  14  12  10  8  6  4  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='00  10  8  6  4  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='00  Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' units)  Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' units)  Energy Loss (eV)  ϕ=0°  ϕ=45°  ϕ=35°  ϕ=30°  ϕ=25°  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  Waterfall plot of the RIXS spectra deconvoluted for energy resolution  (37 meV) for different φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  The softening of the BS phonon is clearly visible for any φ (ex-  cept φ=45o) by simple visual inspection of the deconvoluted curves (red dash lines are guides to  the eye).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  16  ϕ:  q (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=')  Phonon Dispersion (meV)  75  70  65  60  55  50  45  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='1  0°  25°  30°  35°  45°  Fits:  (I):  (II):  (III):  70  60  50  40  75  70  65  60  55  76  72  68  64  60  76  72  68  64  76  72  68  64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='1  ϕ = 0°  ϕ = 25°  ϕ = 30°  ϕ = 35°  ϕ = 45°  q (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=')  Phonon Dispersion (meV)  A  B  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Comparison between three different methods for the extraction of the phonon  dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Methods (1), (2) and (3) are detailed in the Materials and Methods section of the  main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='4  0°  90°  180°  270°  φ  q (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=')  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='1  I21 - T=Tc  I21 - T<Tc  M1  M2  2-ID - T=Tc  2-ID - T>Tc  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Comparison of models to experiments at 2-ID and I21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Polar plot contrasting  M1, M2 models (orange and green solid lines) and the experimental data (blue and red symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  The error bars are obtained from the fits to the phonon dispersion as described in the Materials and  Methods section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' On the left side, the model was adjusted for a higher value of qCO for comparison  with the data obtained at 2-ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The data at 2-ID is consistent with M1 and not with M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' In the  main text only the data from I21 is shown because for those experiments the sample crystal axes  could be aligned in situ from structural diffraction peaks by using the photodiode detector in that  chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' See Materials and Methods section for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  18  Medium resolution RIXS  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S6 we show the results of medium resolution RIXS done at the qRIXS endstation  at the Advanced Light Source in the Lawrence Berkeley National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The data  were obtained by integrating the RIXS spectra over the −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='5 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='7 eV energy window and  normalizing them by spectra integrated over all energies, which allows a comparison between  the three different dopings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The data was also symmetrized about the high-symmetry φ =  45◦ direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S6(D-I) the solid lines are fits of a Gaussian function plus a linear  background to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The maps in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S6(A-C) were generated from the fits in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S6(G-  I), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  The gray bars in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S6(D-F) are centered at the average radii of the  correlations, obtained from averaging over φ the peak positions obtained from the fits in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S6(G-I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The widths of the grey bars in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S6(D and E) are obtained from the 95%  confidence intervals obtained from the fits in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S6(G and H), summing them in quadrature  and taking their square root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The same procedure underestimates the uncertainty for the  Tc = 54 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Instead the width of the grey bar in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S6(F) is calculated by taking the lowest  and largest peak positions over all φ, taking into account the 95% confidence intervals from  the fits in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S6(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The width of the grey bar in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S6(F) The data used to generate  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' S6(C, F and I) were used in a previous publication [Boschini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' 12, 1-  8 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' The new data follows the same experimental procedures as the previously published  data, so we direct the reader to [Boschini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content=' 12, 1-8 2021] for further details  of the experimental procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  19  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content=')  18  20  22  24  26  28  30  32  34  36  38  40  Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content=')  Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=')  [-10°, 5°]  [15°, 25°]  [35°, 55°]  [-10°, 10°]  [20°, 30°]  [40°, 50°]  [-10°, 10°]  [15°, 30°]  [45°]  10°  0°  5°  15°  25°  35°  45°  55°  65°  75°  85°  90°  95°  10°  0°  10°  20°  30°  40°  50°  60°  70°  80°  90°  100°  10°  5°  0°  5°  10°  15°  30°  45°  60°  75°  80°  90°  95°  85°  100°  Overdoped  Tc = 60K  Optimally doped  Tc = 91K  Underdoped  Tc = 54K  A  B  C  D  E  F  G  H  I  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content=' units)  ∫100 meVIRIXS / ∫IRIXS  700 meV  Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' units)  ϕ=0°  25°  30°  35°  45°  q (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content=' A Energy-Loss RIXS spectrum for (q,φ)=(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content=' B q-cuts of the RIXS spectra in-  tegrated over the energy regions highlighted in A and normalized by the total energy-integrated  RIXS signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content=', Di Castro, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=', Grilli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=',  Ghiringhelli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' & Caprara, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Strange metal behaviour from charge density fluctuations in  cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Communications Physics 4, 1–6 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content='  [4] Caprara, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=', Castro, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content=', Seibold, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' & Grilli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Dissipation-driven strange  metal behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Communications Physics 5, 1–7 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content='  [5] Gurvitch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' & Fiory, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Resistivity of La1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='825Sr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='175CuO4 and YBa2Cu3O7 to 1100 K:  Absence of saturation and its implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content='  Phenomenology of the normal state of Cu-O high-temperature superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content=' Theory of the Quantum Critical Fluctuations in Cuprate Supercon-  ductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content=' Advances in Physics 52, 119–218  (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content='  nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='com/articles/s41467-022-30918-z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='  [33] Yang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
+page_content='com/articles/nature07400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfDRyP/content/2301.08415v1.pdf'}
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+Look, Listen, and Attack: Backdoor Attacks Against Video Action Recognition
+Hasan Abed Al Kader Hammoud1
+Shuming Liu1
+Mohammad Alkhrasi2
+Fahad AlBalawi2
+Bernard Ghanem1
+1 King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia
+2 Saudi Data and Artificial Intelligence Authority (SDAIA), Riyadh, Saudi Arabia
+{hasanabedalkader.hammoud,shuming.liu,bernard.ghanem} @kaust.edu.sa
+{mkhrashi,falbalawi} @sdaia.gov.sa
+Abstract
+Deep neural networks (DNNs) are vulnerable to a class
+of attacks called “backdoor attacks”, which create an as-
+sociation between a backdoor trigger and a target label the
+attacker is interested in exploiting. A backdoored DNN per-
+forms well on clean test images, yet persistently predicts an
+attacker-defined label for any sample in the presence of the
+backdoor trigger. Although backdoor attacks have been ex-
+tensively studied in the image domain, there are very few
+works that explore such attacks in the video domain, and
+they tend to conclude that image backdoor attacks are less
+effective in the video domain. In this work, we revisit the
+traditional backdoor threat model and incorporate addi-
+tional video-related aspects to that model. We show that
+poisoned-label image backdoor attacks could be extended
+temporally in two ways, statically and dynamically, leading
+to highly effective attacks in the video domain. In addition,
+we explore natural video backdoors to highlight the seri-
+ousness of this vulnerability in the video domain. And, for
+the first time, we study multi-modal (audiovisual) backdoor
+attacks against video action recognition models, where we
+show that attacking a single modality is enough for achiev-
+ing a high attack success rate.
+1. Introduction
+A fundamental requirement for the deployment of deep
+neural networks (DNNs) in real-world tasks is their safety
+and robustness against possible vulnerabilities and security
+breaches. This requirement is, in essence, the motivation
+behind exploring adversarial attacks. One particularly in-
+teresting adversarial attack is “backdoor attacks”. Backdoor
+attacks or neural trojan attacks explore the scenario in which
+a user with limited computational capabilities downloads
+pretrained DNNs from an untrusted party or outsources the
+training procedure to such a party that we refer to as the ad-
+versary. The adversary provides the user with a model that
+performs well on an unseen validation set, but produces a
+pre-defined class label in the presence of an attacker-defined
+trigger called the backdoor trigger. The association between
+the backdoor trigger and the attacker-specified label is cre-
+ated by training the DNN on poisoned training samples,
+which are samples polluted by the attacker’s trigger [39].
+In poisoned-label attacks, unlike clean-label attacks, the at-
+tacker also switches the label of the poisoned samples to the
+intended target label.
+Considerable attention has been paid to explore back-
+door attacks and defenses for 2D image classification mod-
+els [5,22,25]. However, little attention has been paid to ex-
+ploring backdoor attacks and defenses against video action
+recognition models. The disappointing conclusion uncov-
+ered by [87] regarding the limited effectiveness of image
+backdoor attacks on videos stunted further development of
+video backdoor attacks. Unfortunately, the attacks consid-
+ered in [87] were limited to only visible patch-based clean-
+label attacks. Moreover, [87] directly adopted the 2D back-
+door attack threat model without incorporating important
+video-specific considerations.
+To this end, and as opposed to [87]. we first revisit and
+revise the commonly adopted 2D poisoned-label backdoor
+threat model by incorporating additional constraints that are
+inherently imposed by video systems.
+These constraints
+arise due to the presence of the temporal dimension. We
+then explore two ways to extend image backdoor attacks to
+incorporate the temporal dimension into the attack to enable
+more video-specific backdoor attacks. In particular, image
+backdoor attacks could be either extended statically by ap-
+plying the same attack to each frame of the video or dynam-
+ically by adjusting the attack parameters differently for each
+frame. Then, three novel natural video backdoor attacks are
+presented to highlight the seriousness of the risks associ-
+ated with backdoor attacks in the video domain. We then
+test the attacked models against three 2D backdoor defenses
+and discuss the reason behind the failure of those methods.
+arXiv:2301.00986v1  [cs.CV]  3 Jan 2023
+
+We also study, for the first time, audiovisual backdoor at-
+tacks, where we ablate the importance and contribution of
+each modality on the performance of the attack for both late
+and early fusion settings. We show that attacking a single
+modality is enough to achieve a high attack success rate.
+Contributions. Our contributions are twofold. (1) We re-
+visit the traditional backdoor attack threat model and incor-
+porate video-related aspects, such as video subsampling and
+spatial cropping, into the model. We also extend existing
+image backdoor attacks to the video domain in two differ-
+ent ways, statically and dynamically, after which we pro-
+pose three novel natural video backdoor attacks. Through
+extensive experiments, we provide evidence that the previ-
+ous perception of image backdoor attacks in the video do-
+main is not necessarily true, especially in the poisoned-label
+attack setup. (2) To the best of our knowledge, this work is
+the first to investigate audiovisual backdoor attacks against
+video action recognition models.
+2. Related Work
+Backdoor Attacks. Backdoor attacks were first introduced
+in [22]. The attack, called BadNet, was based on adding a
+patch to the corner of a subset of training images to create a
+backdoor that could be triggered by the attacker at will. Fol-
+lowing BadNet, [44] proposed optimizing for the values of
+the patch to obtain a more effective backdoor attack. Shortly
+after the development of patch-based backdoor attacks, the
+community realized the importance of adding an invisibility
+constraint to the design of backdoor triggers to bypass any
+human inspection. Works such as [9] proposed blending
+the backdoor trigger with the image rather than stamping
+it. [37] generated backdoor attacks using the least signifi-
+cant bit algorithm. [52] generated warping fields to warp the
+image content as a backdoor trigger. [14] went one step fur-
+ther and designed learnable transformations to generate op-
+timal backdoor triggers. After many attacks were proposed
+in the spatial domain [10, 37, 40, 45, 56, 57, 67, 72, 75], and
+others in the latent representation domain [13,53,80,88,91],
+[19, 25, 71, 82, 84] proposed to switch attention to the fre-
+quency domain. [25] utilized frequency heatmaps proposed
+in [81] to create backdoor attacks that target the most sen-
+sitive frequency components of the network. [19] proposed
+blending low frequency content from a trigger image with
+training images as a poisoning technique. In our work, we
+extend the 2D backdoor threat model to the video domain
+by incorporating video-related aspects into it. We also ex-
+tend five image backdoor attacks into the video domain and
+propose three natural video backdoor attacks.
+Backdoor Defenses. Backdoor attack literature was im-
+mediately opposed by various defenses.
+Backdoor de-
+fenses are generally of five types:
+preprocessing-based
+[12,47,55], model reconstruction-based [38,42,74,83,89],
+trigger synthesis-based [23,24,29,43,54,58,62,68], model
+diagonsis-based [15, 34, 46, 76, 90], and sample-filtering
+based [8, 20, 26, 30, 61, 63]. Early backdoor defenses such
+as [68] hypothesized that backdoor attacks create a short-
+cut between all samples and the poisoned class. Based on
+that, they solved an optimization problem to find whether a
+trigger of an abnormally small norm exists that would flip
+all samples to one label. Later, multiple improved itera-
+tions of this method were proposed, such as [23, 43, 83].
+Fine pruning [42] suggested that the backdoor is triggered
+by particular neurons that are dormant in the absence of
+the trigger. Therefore, the authors proposed pruning the
+least active neurons on clean samples. STRIP [20] showed
+that blending clean samples with other clean samples would
+yield a higher entropy compared to when clean images are
+blended with poisoned samples. Activation clustering [8]
+uses KMeans to cluster the activations of an inspection, a
+potentially poisoned data set, into two clusters. A large sil-
+houette distance between the two clusters would uncover
+the poisoned samples. In our work, we show that current
+image backdoor attacks have limited effectiveness in de-
+fending against backdoor attacks in the video domain, es-
+pecially against the proposed natural video attacks.
+Video Action Recognition. Video action recognition mod-
+els, which only leverage the raw frames of a video, can be
+categorized into two categories, CNN-based networks and
+transformer-based networks. 2D CNN-based methods are
+built on top of pretrained image recognition networks with
+well-designed modules to capture the temporal relationship
+between multiple frames [41,49,69,70]. Those methods are
+computationally efficient as they use 2D convolutional ker-
+nels. To learn stronger spatial-temporal representations, 3D
+CNN-based methods were proposed. These methods utilize
+3D kernels to jointly leverage the spatio-temporal context
+within a video clip [17, 18, 64, 65]. To better initialize the
+network, I3D [7] inflated the weights of 2D pretrained im-
+age recognition models to adapt them to 3D CNNs. Real-
+izing the importance of computational efficiency, S3D [78]
+and R(2+1)D [66] proposed to disentangle spatial and tem-
+poral convolutions to reduce computational cost. Recently,
+transformer-based action recognition models were able to
+achieve better performance in large training data sets com-
+pared to CNN-based models, e.g. [4,6,16,48]. In this work,
+we test backdoor attacks against three action recognition
+architectures, namely I3D, SlowFast, and TSM.
+Audiovisual Action Recognition. In addition to frames, a
+line of action recognition models [1,27,28,51] has used the
+accompanying audio to better understand activities such as
+“playing music” or “washing dishes”. To take advantage of
+existing CNN and transformer-based models, the Log-Mel
+spectrogram was introduced to convert audio data from a
+non-structured signal into a 2D representation in time and
+frequency usable by these models [2,3,35,77]. Current au-
+diovisual action recognition methods are divided into two
+
+Figure 1. Traditional Backdoor Attack Pipeline. After selecting a backdoor trigger and a target label, the attacker poisons a subset of
+the training data referred to as the poisoned dataset (Dp). The label of the poisoned dataset is fixed to a target poisoning label specified by
+the attacker. The attacker trains jointly on clean (non-poisoned) samples (Dc) and poisoned samples leading to a backdoored model, which
+outputs the target label in the presence of the backdoor trigger.
+categories based on when the audio and visual signals are
+merged in the recognition pipeline: early fusion and late fu-
+sion. Early fusion combines features before classification,
+which can better capture features [32,77]. The disadvantage
+of early fusion is that there is a higher risk of overfitting to
+the training data [59]. Late fusion, on the other hand, treats
+the video and audio networks separately, and the predictions
+of each network are carried out independently, after which
+the logits are aggregated to make a final prediction [21]. For
+the first time, we test backdoor attacks against audiovisual
+action recognition networks in both late and early fusion
+setups.
+3. Video Backdoor Attacks
+3.1. The Traditional Threat Model
+The commonly adopted threat model for backdoor at-
+tacks dates back to the works that studied those attacks
+against 2D image classification models [22]. The victim
+outsources the training process to a trainer who is given ac-
+cess to both the victim’s training data and the network ar-
+chitecture. The victim only accepts the model provided by
+the trainer if it performs well on the victim’s private val-
+idation set. The attacker aims to maximize the effective-
+ness of the embedded backdoor attack [39]. We refer to
+the model’s performance on the validation set as clean data
+accuracy (CDA). The effectiveness of the backdoor attack
+is measured by the attack success rate (ASR), which is de-
+fined as the percentage of test examples not labeled as the
+target class that are classified as the target class when the
+backdoor pattern is applied. To achieve this goal, the at-
+tacker applies a backdoor trigger to a subset of the train-
+ing images and then, in the poisoned-label setup, switches
+the labels of those images to a target class of choice before
+training begins. A more powerful backdoor attack is one
+that is visually imperceptible (usually measured in terms of
+ℓ2/ℓ∞-norm, PSNR, SSIM, or LPIPS) but achieves both a
+high CDA and a high ASR. This is summarized in Figure 1.
+More formally, we denote the classifier which is param-
+eterized by θ as fθ : X → Y. It maps the input x ∈ X,
+such as images or videos, to class labels y ∈ Y.
+Let
+Gη : X → X indicate an attacker-specific poisoned im-
+age generator that is parameterized by some trigger-specific
+parameters η. The generator may be image-dependent. Fi-
+nally, let S : Y → Y be an attacker-specified label shifting
+function. In our case, we consider the scenario in which the
+attacker is trying to flip all the labels into one particular la-
+bel, i.e. S : Y → t, where t ∈ Y is an attacker-specified
+label that will be activated in the presence of the backdoor
+trigger. Let D = {(xi, yi)}N
+i=1 indicate the training dataset.
+The attacker splits D into two subsets, a clean subset Dc and
+a poisoned subset Dp, whose images are poisoned by Gη
+and labels are poisoned by S. The poisoning rate is the ra-
+tio α = |Dp|
+|D| , generally a lower poisoning rate is associated
+with a higher clean data accuracy. The attacker typically
+trains the network by minimizing the cross-entropy loss on
+Dc∪Dp, i.e. minimizes E(x,y)∼Dc∪Dp[LCE(fθ(x), y)]. The
+attacker aims to achieve high accuracy on the user’s valida-
+tion set Dval while being able to trigger the poisoned-label,
+t, in the presence of the backdoor trigger, i.e. fθ(Gη(x)) =
+t, ∀x ∈ X (ideally).
+3.2. From Images to Videos
+Unlike images, videos have an additional dimension, the
+temporal dimension. This dimension introduces new rules
+to the game between the attacker and the victim.
+More
+
+Settings
+Training Stage
+Inference Stage
+"Eat"
+"Jump'
+fe
+fe
+Poisoned
+Network
+Target Label = "Eat"
+Backdoor Trigger
+Clean Samples (Dc)
+Poisoned Video (Gn(α))
+(α)
+Label = "Eat"
+Clean Video
+Ground Truth LabelFigure 2. Static vs Dynamic Backdoor Attacks. Static backdoor attacks apply the same trigger across all frames along the temporal
+dimension. On the other hand, dynamic attacks apply a different trigger per frame along the temporal dimension.
+precisely, the attacker now has an additional dimension to
+hide the backdoor trigger, leading to a higher level of im-
+perceptibility.
+The backdoor attack could be applied to
+all the frames or a subset of the frames statically, i.e. the
+same trigger is applied to each frame, or dynamically, i.e.
+a different trigger is applied to each frame. On the other
+hand, the testing pipeline now imposes harsher conditions
+against the backdoor attack. Video recognition models tend
+to test the model on multiple sub-sampled clips with vari-
+ous crops [7,18,41] which might, in turn, destroy the back-
+door trigger.
+For example, if the trigger is applied to a
+single frame, it might not be sampled, and if the trigger
+is applied to the corner of the image, it might be cropped
+out. The threat model presented in Subsection 3.1 was di-
+rectly adopted in [87], which to the best of our knowledge,
+is the only previous work that considered backdoor attacks
+for video action recognition.
+Our work sheds light on the aforementioned video-
+related aspects. In Section 4.2, we show the effect of the
+number of frames poisoned on CDA and ASR. We also
+show how existing 2D methods could be extended both stat-
+ically and dynamically to suit the video domain. For exam-
+ple, BadNet [22] applies a fixed patch as a backdoor trigger.
+The patch could be applied statically using the same pixel
+values and the same position along the temporal dimension
+or applied dynamically by changing the position and possi-
+bly the pixel values of the patch for each frame. Figure 2
+shows a BadNet attack when applied in a static and dynamic
+way. Additionally, we show how simple yet natural video
+“artifacts” could be used as backdoor triggers. More specif-
+ically, lag in a video, motion blur, and compression glitches
+could all be used as naturally occurring backdoor triggers.
+3.3. Audiovisual Backdoor Attacks
+Videos are naturally accompanied by audio signals. Sim-
+ilarly to how the video modality could be attacked, the audio
+signal could also be attacked. The interesting question that
+arises is how backdoor attacks would perform in a multi-
+modal setup. In the experiments of Section 4.4, we answer
+the following questions: (1) What is the effect of having two
+attacked modalities on CDA and ASR?; (2) What happens
+if only one modality is attacked and the other is left clean?;
+(3) What is the difference in performance between late and
+early fusion in terms of CDA and ASR?
+4. Experiments
+4.1. Experimental Settings
+Datasets. We consider three standard benchmark datasets
+used in video action recognition: UCF-101 [60], HMDB-
+51 [36], and Kinetics-Sounds [31]. Kinetics-Sounds is a
+subset of Kinetics400 that contains classes that can be clas-
+sified from the audio signal, i.e. classes where audio is use-
+ful for action recognition [2]. Kinetics-Sounds is particu-
+larly interesting for Sections 4.3 and 4.4, where we explore
+backdoor attacks against audio and audiovisual classifiers.
+Network Architectures. Following common practice, for
+the visual modality, we use a dense sampling strategy to
+sub-sample 32 frames per video to fine-tune a pretrained
+I3D network on the target dataset [7]. In Section 4.2, we
+also show results using TSM [41] and SlowFast [18] net-
+works. All three models adopt ResNet-50 as the backbone
+and are pretrained on Kinetics-400. Similarly to [2], for
+the audio modality, a ResNet-18 is trained from scratch on
+Mel-Spectrograms composed of 80 Mel bands sub-sampled
+temporally to a fixed length of 256.
+Attack Setting.
+For the video modality, we study and
+extend the following image-based backdoor attacks to the
+video domain: BadNet [22], Blend [9], SIG [5], WaNet
+[52], and FTrojan [71]. We also explore three additional
+natural video backdoor attacks.
+For the audio modality,
+we consider two attacks: sine attack and high-frequency
+noise attack, both of which we explain later.
+Following
+[22,25,52], the target class is arbitrarily set to the first class
+
+t=0
+t=1
+t=12
+t=13
+t=14
+t=N-1
+t=N
+Static
+DynamicFigure 3. Visualization of 2D Backdoor Attacks. Image backdoor attacks mainly differ according to the backdoor trigger used to poison
+the training samples. They could be extended either statically or dynamically based on how the attack is applied across the frames.
+of each data set (class 0), and the poisoning rate is set to
+10%. Unless otherwise stated, the considered image back-
+door attacks poison all frames of the sampled clips during
+training and evaluation.
+Evaluation Metrics. As is commonly done in the back-
+door literature, we evaluate the performance of the model
+using clean data accuracy (CDA) and attack success rate
+(ASR) explained in Section 3. CDA represents the usual
+validation/test accuracy on an unseen dataset hence mea-
+suring the generalizability of the model. On the other hand,
+ASR measures the effectiveness of the attack when the poi-
+son is applied to the validation/test set.
+In addition, we
+test the attacked models against some of the early 2D back-
+door defenses, more precisely against activation clustering
+(AC) [8], STRIP [20], and pruning [42].
+Implementation Details. Our method is built on MMAc-
+tion2 library [11], and follows their default training config-
+urations and testing protocols, except for the learning rate
+and the number of training epochs (check Supplementary).
+All experiments were run using 4 NVIDIA A100 GPUs.
+4.2. Video Backdoor Attacks
+Extending Image Backdoor Attacks to the Video Do-
+main. As mentioned in Section 3.2, image backdoor attacks
+could be extended either statically by applying an attack in
+the same way across all frames or dynamically by adjusting
+the attack parameters for different frames. We consider five
+attacks that differ according to the applied backdoor trig-
+ger. BadNet applies a patch as a trigger, Blend blends a
+trigger image to the original image, SIG superimposes a si-
+nusoidal trigger to the image, WaNet warps the content of
+the image, and FTrojan poisons a high- and mid- frequency
+component in the discrete cosine transform (DCT). Figure 3
+visualizes all five attacks on the same video frame. Each of
+the considered methods could be extended dynamically as
+follows: BadNet: change the patch location for each frame;
+Blend: blend a uniform noise that is different per frame;
+SIG: change the frequency of the sine component superim-
+posed with each frame; WaNet: generate a different warp-
+ing field for each frame; FTrojan: select a different DCT
+basis to perturb at each frame. Note that Blend and FTro-
+jan are generally imperceptible. Visualizations and saliency
+UCF101
+HMDB51
+KineticsSound
+CDA(%)
+ASR(%)
+CDA(%)
+ASR(%)
+CDA(%)
+ASR(%)
+Baseline
+93.95
+-
+69.59
+-
+81.41
+-
+BadNet
+93.95
+99.63
+69.35
+98.89
+82.97
+99.09
+Blend
+94.29
+99.26
+68.37
+86.73
+82.12
+97.54
+SIG
+93.97
+99.97
+68.50
+99.80
+82.84
+99.87
+WaNet
+94.05
+99.84
+68.95
+99.61
+82.38
+99.09
+FTrojan
+94.16
+99.34
+68.10
+97.52
+82.45
+97.86
+Table 1. Statically Extended 2D Backdoor Attacks. Statically
+extending 2D backdoor attacks to the video domain leads to high
+CDA and ASR across all three considered datasets.
+UCF101
+HMDB51
+KineticsSound
+CDA(%)
+ASR(%)
+CDA(%)
+ASR(%)
+CDA(%)
+ASR(%)
+Baseline
+93.95
+-
+69.59
+-
+81.41
+-
+BadNet
+94.11
+99.97
+69.08
+99.54
+82.25
+99.74
+Blend
+94.21
+99.44
+67.03
+95.95
+81.67
+95.79
+SIG
+94.24
+100.00
+68.63
+100.00
+82.84
+100.00
+WaNet
+94.29
+99.79
+69.22
+99.80
+82.25
+99.61
+FTrojan
+94.16
+99.34
+67.19
+98.69
+82.25
+95.27
+Table 2. Dynamically Extended 2D Backdoor Attacks. Dynam-
+ically extending 2D backdoor attacks to the video domain leads to
+high CDA and ASR across all three considered datasets.
+maps for each attack are found in the Supplementary.
+Tables 1 and 2 show the CDA and ASR of the I3D mod-
+els attacked using various backdoor attacks on UCF-101,
+HMDB-51, and Kinetics-Sounds. Contrary to the conclu-
+sion presented in [87], we find that backdoor attacks are
+actually highly effective in the video domain. The CDA
+of the attacked models is very similar to that of the clean
+unattacked model (baseline), surpassing it in some cases.
+Extending attacks dynamically, almost always, improves
+CDA and ASR compared to extending them statically.
+Natural Video Backdoors. A more interesting attack is
+one that seems natural and could bypass human inspec-
+tion [50, 73, 79, 86]. There are several natural “glitches”
+that occur in the video domain and that one could exploit
+to design a natural backdoor attack. For example, videos
+might contain some frame lag, motion blur, video compres-
+sion corruptions, camera focus/defocus, etc. In Table 3, we
+report the CDA and ASR of three natural backdoor attacks:
+
+Clean
+BadNet
+Blend
+SIG
+WaNet
+FTrojanUCF101
+HMDB51
+KineticsSound
+CDA(%)
+ASR(%)
+CDA(%)
+ASR(%)
+CDA(%)
+ASR(%)
+Baseline
+93.95
+-
+69.59
+-
+81.41
+-
+Frame Lag
+92.94
+97.20
+68.04
+98.76
+82.51
+98.19
+Video Corrupt.
+94.26
+99.87
+69.22
+99.22
+81.74
+98.51
+Motion Blur
+93.97
+99.92
+68.17
+97.52
+82.19
+99.22
+Table 3.
+Natural Video Backdoor Attacks.
+Natural attacks
+against video action recognition models could achieve high CDA
+and ASR while looking completely natural to human inspection.
+SlowFast
+TSM
+CDA(%)
+ASR(%)
+CDA(%)
+ASR(%)
+Baseline
+96.72
+-
+94.77
+-
+BadNet
+96.64
+99.47
+94.69
+97.78
+SIG
+96.70
+99.97
+94.77
+99.47
+FTrojan
+96.25
+98.52
+94.21
+100.00
+Frame Lag
+96.43
+99.97
+94.63
+97.96
+Video Corruption
+96.54
+99.76
+95.08
+98.97
+Motion Blur
+96.46
+99.55
+94.50
+99.39
+Table 4.
+Video Backdoor Attacks Against Different Archi-
+tectures (UCF-101). When tested against network architectures
+other than I3D such as TSM and SlowFast, both image and natural
+backdoor attacks can still achieve high CDA and high ASR.
+frame lag (lagging video), video compression glitch (which
+we refer to as Video Corruption), and motion blur. Interest-
+ingly, these attacks could achieve both high clean data ac-
+curacy and high attack success rate. It is worth noting that
+for frame lag, a two-frame lag is used for UCF-101 and a
+three-frame lag is used for HMDB-51 and Kinetics-Sounds.
+More details are provided in the Supplementary.
+Attacks Against Different Architectures. So far, all at-
+tacks have been experimented with against an I3D network.
+To further explore the behavior of backdoor attacks against
+other video recognition models, we test a subset of the con-
+sidered attacks against a 2D based model, TSM, and another
+3D based model, SlowFast, on UCF-101. Table 4 shows
+that all the aforementioned backdoor attacks perform sig-
+nificantly well in terms of CDA and ASR against both TSM
+and SlowFast architectures. Note that even though TSM is
+a 2D based model, our proposed natural video backdoor at-
+tacks still succeed in attacking it.
+Recommendations for Video Backdoor Attacks. As men-
+tioned in Section 3.2, the attacker must select a number of
+frames to poison per video, keeping in mind that the video
+will be sub-sampled and randomly cropped during evalua-
+tion. Since the attacker is the one who trained the network in
+the first place, he/she has access to the processing pipeline
+and could exploit this during the attack. For example, if
+video processing involves sub-sampling the video into clips
+of 32 frames and cropping the frames into 224×224 crops,
+the attacker could pass to the network an attacked video of
+a temporal length of 32 frames and a spatial size 224×224,
+Figure 4. Effect of the Number of Poisoned Frames (UCF-101).
+Different colors refer to different number of frames poisoned dur-
+ing the training of the attacked model. Training the model with
+a single poisoned frame performs best for various choices of the
+number of frames poisoned during evaluation.
+Frame
+Lag
+Motion
+Blur
+SIG
+BadNet
+FTrojan
+Elimination Rate(%)
+0.00
+0.00
+34.21
+33.77
+34.12
+Sacrifice Rate(%)
+13.08
+12.82
+15.17
+14.25
+13.00
+Table 5. Activation Clustering Defense (UCF-101). Whereas
+Activation Clustering provides partial success in defending against
+image backdoor attacks, it fails completely against natural attacks.
+hence bypassing sub-sampling and cropping. However, a
+system could force the user to input a video of a partic-
+ular length, possibly greater than the length of the sub-
+sampled clips. This raises an important question regarding
+how many frames the attacker should poison. Clearly, the
+smaller the number of frames the attacker poisons, the less
+detectable the attack is, but does the attack remain effective?
+In Figure 4, we show the attack success rate of backdoor-
+attacked models trained on clips of 1, 8, 16, and 32 frames,
+and a randomly sampled number of poisoned frames (out of
+32 total frames) when evaluated on clips of 1, 8, 16, and 32
+poisoned frames (out of 32 total frames). Random refers to
+training on a varying number of poisoned frames per clip.
+Note that training the model against the worst-case scenario
+(single frame), which mimics the case where only one of
+the poisoned frames is sub-sampled, provides the best guar-
+antees for achieving a high attack success rate.
+Defenses Against Video Backdoor Attacks. We explore
+the effect of extending some of the existing 2D backdoor
+defenses against video backdoor attacks.
+Optimization-
+based defenses are extremely costly when extended to the
+video domain. For example, Neural Cleanse (NC) [68], I-
+BAU [83], and TABOR [23] involve a trigger reconstruc-
+tion phase. The trigger space is now bigger in the presence
+of the temporal dimension, and therefore, instead of opti-
+mizing for a 224×224×3 trigger, the defender has to search
+for a 32×224×224×3 trigger (assuming 32 frame clips are
+used), which is both costly and hard to solve. The attacker
+has the spatial and temporal dimensions to design and em-
+
+BadNet
+SIG
+100
+75
+# Poisoned Frames
+ASR(%)
+(Training)
+1
+50
+8
+16
+25
+32
+Random
+0
+8
+16
+32 1
+1
+8
+16
+32
+# Poisoned Frames
+(EvaluationFigure 5. STRIP Defense (UCF-101). Whereas the entropy of
+image backdoor attacks is very low compared to that of clean sam-
+ples, the proposed natural backdoor attacks have a natural distri-
+bution of entropies similar to that of clean samples.
+Figure 6. Pruning Defense (Kinetics-Sounds). Pruning is com-
+pletely ineffective against image backdoor attacks extended to the
+video domain and natural video backdoor attacks. Even though
+the clean accuracy has dropped to random, the attack success rate
+is maintained at very high levels.
+bed their attack in, and, therefore, reverse engineering the
+trigger is quite hard.
+We consider three well-known defenses that introduce no
+computational overhead when adopted to the video domain,
+namely Activation Cluster (AC) [8], STRIP [20], and prun-
+ing [42]. AC computes the activations of a neural network
+on clean samples (from the test set) and an inspection set
+of interest which may be poisoned. AC then applies PCA
+to reduce the dimension of the activations, after which the
+projected activations are clustered into two classes and com-
+pared to the activations of the clean set. STRIP blends clean
+samples with the samples of a possibly poisoned inspec-
+tion set. The entropy of the predicted probabilities is then
+checked for any abnormalities. Unlike clean samples, poi-
+soned samples tend to have a low entropy. Pruning suggests
+that the backdoor is usually embedded in particular neurons
+Baseline
+Sine Attack
+High Frequency Attack
+CDA(%)
+49.21
+47.21
+47.61
+ASR(%)
+-
+96.36
+95.96
+Table 6. Audio Backdoor Attacks (Kinetics-Sounds). Both sine
+attack and the high-frequency band attack perform similarly to
+baseline in terms of CDA while being able to achieve high ASR.
+in the network that are only activated in the presence of the
+trigger. Therefore, those neurons are supposed to be dor-
+mant as far as the test set samples, i.e. clean samples, are
+concerned. This allows us to detect and prune those dor-
+mant neurons to eliminate the backdoor. Table 5 shows the
+elimination and sacrifice rates of AC when applied against
+some of the considered attacks. The elimination rate refers
+to the ratio of poisoned samples correctly detected as poi-
+soned to the total number of poisoned samples, whereas the
+sacrifice rate refers to the ratio of clean samples incorrectly
+detected as poisoned to the total number of clean samples.
+Whereas AC has partial success in defending against image
+backdoor attacks, it fails completely against the proposed
+natural backdoor attacks. Figure 5 shows that the entropy
+of the clean and poisoned samples of the proposed natural
+attacks is very similar and therefore could evade the STRIP
+defense, while BadNet and FTrojan are detectable. Finally,
+Figure 6 shows that pruning the least active neurons causes
+a reduction in CDA without reducing ASR. This is observed
+not only for the natural attacks, but also for the extended im-
+age backdoor attacks, hinting that image backdoor defenses
+are not effective in the video domain.
+4.3. Audio Backdoor Attacks
+Attacks proposed against audio networks have been lim-
+ited to adding a low-volume one-hot-spectrum noise in the
+frequency domain, which leaves highly visible artifacts in
+the spectrogram [85] or adding a human non-audible com-
+ponent [33], f < 20Hz or f > 20kHz, which is non-
+realistic, since spectrograms usually filter out those frequen-
+cies. We consider two attacks against the Kinetics-Sounds
+dataset; the first is to add a low-amplitude sine wave com-
+ponent with f = 800Hz to the audio signal, and the second
+is to add band-limited noise 5kHz < f < 6kHz. The spec-
+trograms and the absolute difference between the attacked
+spectrograms and the clean spectrogram are shown in Fig-
+ure 7. Since no clear artifacts are observed in the spectro-
+grams, human inspection fails to label the spectrograms as
+attacked. The CDA and ASR rates of the backdoor-attacked
+models for both attacks are shown in Table 6. The attacks
+achieve a relatively high ASR.
+4.4. Audiovisual Backdoor Attacks
+Now, we combine video and audio attacks to build a
+multi-modal audiovisual backdoor attack. The way we do
+
+BadNet
+FTrojan
+1500
+1500
+Poisoned
+Clean
+1000
+1000
+500
+500
+0
+0
+2
+4
+0
+2
+4
+Frame Lag
+Motion Blur
+600
+600
+400
+400
+200
+200
+0
+0
+0
+2
+4
+0
+2
+4
+EntropyBadNet
+Frame Lag
+100
+ASR
+ASR
+CDA
+CDA
+Accuracy(%)
+75
+50
+25
+0
+0
+25
+50
+75
+100
+0
+25
+50
+75
+100
+Percentage Pruned (%Late Fusion
+Early Fusion
+Clean Audio
+Sine Attack
+High Freq. Attack
+Clean Audio
+Sine Attack
+High Freq. Attack
+Clean Video
+80.25 / -
+81.74 / 70.98
+80.96 / 77.91
+84.72 / -
+83.48 / 92.23
+83.94 / 93.72
+BadNet
+77.33 / 66.97
+78.63 / 99.74
+77.33 / 99.87
+87.50 / 99.29
+85.10 / 99.87
+85.75 / 100.00
+Blend
+79.60 / 75.06
+80.76 / 99.68
+79.08 / 99.61
+86.08 / 98.19
+83.55 / 99.81
+85.43 / 99.87
+SIG
+78.50 / 68.33
+80.12 / 99.87
+79.02 / 100.00
+86.92 / 99.81
+84.97 / 100.00
+85.95 / 100.00
+WaNet
+77.66 / 68.39
+79.79 / 99.94
+79.02 / 99.94
+86.46 / 98.96
+84.97 / 100.00
+85.88 / 100.00
+FTrojan
+79.66 / 67.16
+80.76 / 99.48
+79.99 / 99.29
+86.08 / 98.58
+84.65 / 99.94
+85.49 / 100.00
+Frame Lag
+79.08 / 63.41
+80.57 / 99.74
+79.47 / 99.87
+86.08 / 98.19
+84.59 / 99.94
+84.65 / 100.00
+Video Corruption
+78.11 / 64.57
+78.24 / 99.68
+77.66 / 99.94
+86.59 / 99.29
+84.59 / 100.00
+85.43 / 100.00
+Motion Blur
+79.79 / 69.24
+80.70 / 99.68
+79.86 / 99.94
+86.40 / 98.58
+84.65 / 100.00
+85.62 / 100.00
+Table 7. Audiovisual Backdoor Attacks (Kinetics-Sounds). The entries in the table report the CDA(%)/ASR(%) of attacking late and
+early fused audiovisual networks. When a single modality is attacked, late fusion has a low ASR compared to early fusion. When both
+modalities are attacked, the ASR of both late and early fusion are high.
+Figure 7. Clean and Attacked Audio Spectrograms. The uti-
+lized audio backdoor attacks are not only audibly imperceptible
+but also leave no perceptible artifacts in the Mel spectrogram. The
+spectrogram of each attack is followed by the absolute difference
+of the attacked spectrogram with the clean one.
+it is by taking our attacked models from Sections 4.2 and
+4.3 and applying early or late fusion. For early fusion, we
+extract video and audio features using our trained audio and
+video backbones, and we then train a classifier on the con-
+catenation of the features. In late fusion, the video and au-
+dio networks predict independently on the input, and then
+the individual logits are aggregated to produce the final pre-
+diction. To answer the three questions posed in Section 3.3,
+we run experiments in which both modalities are attacked
+and others in which only a single modality is attacked for
+both early and late fusion setups (Table 7). We summarize
+the results as follows. (1) Attacking two modalities con-
+sistently improves ASR and even CDA in some cases. (2)
+Attacking a single modality is good enough to achieve a
+high ASR in the case of early fusion but not late fusion.
+(3) Early fusion enables the best of both worlds for the at-
+tacker, namely, a high CDA and an almost perfect ASR. On
+the other hand, late fusion experiences some serious drops
+in ASR in the unimodal attack setup. An interesting find-
+ing in these experiments is the following: if the outsourcer
+has the option to outsource the most expensive modality,
+training wise, while training other modalities in-house, ap-
+plying late fusion could be used as a defense mechanism,
+especially in the presence of more clean modalities.
+5. Conclusion
+Backdoor attacks present a serious and exploitable vul-
+nerability against both unimodal and multi-modal video
+action recognition models. We showed how existing im-
+age backdoor attacks could be extended either statically
+or dynamically to develop powerful backdoor attacks that
+achieve both a high clean data accuracy and a high attack
+success rate. Besides existing image backdoor attacks, there
+exists a set of natural video backdoor attacks, such as mo-
+tion blur and frame lag, that are resilient to existing image
+backdoor defenses. Given that videos are usually accom-
+panied by audio, we showed two ways in which one could
+attack audio classifiers in a human inaudible manner. The
+
+Clean Spectrogram
+80
+40
+0
+Sine Attack
+80
+40
+0
+Mel Frequency
+80
+40
+0
+High Frequency Attack
+80
+40
+0
+80
+40
+0
+0
+50
+100
+150
+200
+250
+300
+Frameattacked video and audio models are then used to train an
+audiovisual action recognition model by applying both early
+and late fusion. Different combinations of poisoned modal-
+ities are tested, concluding that: (1) poisoning two modal-
+ities could achieve extremely high attack success rates in
+both late and early fusion settings, and (2) if a single modal-
+ity is poisoned, unlike early fusion, late fusion could reduce
+the effectiveness of the backdoor. We hope that our work
+reignites the attention of the community towards exploring
+backdoor attacks and defenses in the video domain.
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf,len=889
+page_content='Look, Listen, and Attack: Backdoor Attacks Against Video Action Recognition  Hasan Abed Al Kader Hammoud1  Shuming Liu1  Mohammad Alkhrasi2  Fahad AlBalawi2  Bernard Ghanem1  1 King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia  2 Saudi Data and Artificial Intelligence Authority (SDAIA), Riyadh, Saudi Arabia  {hasanabedalkader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='hammoud,shuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='liu,bernard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='ghanem} @kaust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='sa  {mkhrashi,falbalawi} @sdaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='sa  Abstract  Deep neural networks (DNNs) are vulnerable to a class  of attacks called “backdoor attacks”, which create an as-  sociation between a backdoor trigger and a target label the  attacker is interested in exploiting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' A backdoored DNN per-  forms well on clean test images, yet persistently predicts an  attacker-defined label for any sample in the presence of the  backdoor trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Although backdoor attacks have been ex-  tensively studied in the image domain, there are very few  works that explore such attacks in the video domain, and  they tend to conclude that image backdoor attacks are less  effective in the video domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' In this work, we revisit the  traditional backdoor threat model and incorporate addi-  tional video-related aspects to that model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' We show that  poisoned-label image backdoor attacks could be extended  temporally in two ways, statically and dynamically, leading  to highly effective attacks in the video domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' In addition,  we explore natural video backdoors to highlight the seri-  ousness of this vulnerability in the video domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' And, for  the first time, we study multi-modal (audiovisual) backdoor  attacks against video action recognition models, where we  show that attacking a single modality is enough for achiev-  ing a high attack success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Introduction  A fundamental requirement for the deployment of deep  neural networks (DNNs) in real-world tasks is their safety  and robustness against possible vulnerabilities and security  breaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' This requirement is, in essence, the motivation  behind exploring adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' One particularly in-  teresting adversarial attack is “backdoor attacks”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Backdoor  attacks or neural trojan attacks explore the scenario in which  a user with limited computational capabilities downloads  pretrained DNNs from an untrusted party or outsources the  training procedure to such a party that we refer to as the ad-  versary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The adversary provides the user with a model that  performs well on an unseen validation set, but produces a  pre-defined class label in the presence of an attacker-defined  trigger called the backdoor trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The association between  the backdoor trigger and the attacker-specified label is cre-  ated by training the DNN on poisoned training samples,  which are samples polluted by the attacker’s trigger [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  In poisoned-label attacks, unlike clean-label attacks, the at-  tacker also switches the label of the poisoned samples to the  intended target label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Considerable attention has been paid to explore back-  door attacks and defenses for 2D image classification mod-  els [5,22,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' However, little attention has been paid to ex-  ploring backdoor attacks and defenses against video action  recognition models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The disappointing conclusion uncov-  ered by [87] regarding the limited effectiveness of image  backdoor attacks on videos stunted further development of  video backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Unfortunately, the attacks consid-  ered in [87] were limited to only visible patch-based clean-  label attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Moreover, [87] directly adopted the 2D back-  door attack threat model without incorporating important  video-specific considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  To this end, and as opposed to [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' we first revisit and  revise the commonly adopted 2D poisoned-label backdoor  threat model by incorporating additional constraints that are  inherently imposed by video systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  These constraints  arise due to the presence of the temporal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' We  then explore two ways to extend image backdoor attacks to  incorporate the temporal dimension into the attack to enable  more video-specific backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' In particular, image  backdoor attacks could be either extended statically by ap-  plying the same attack to each frame of the video or dynam-  ically by adjusting the attack parameters differently for each  frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Then, three novel natural video backdoor attacks are  presented to highlight the seriousness of the risks associ-  ated with backdoor attacks in the video domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' We then  test the attacked models against three 2D backdoor defenses  and discuss the reason behind the failure of those methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='00986v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='CV]  3 Jan 2023  We also study, for the first time, audiovisual backdoor at-  tacks, where we ablate the importance and contribution of  each modality on the performance of the attack for both late  and early fusion settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' We show that attacking a single  modality is enough to achieve a high attack success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Our contributions are twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' (1) We re-  visit the traditional backdoor attack threat model and incor-  porate video-related aspects, such as video subsampling and  spatial cropping, into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' We also extend existing  image backdoor attacks to the video domain in two differ-  ent ways, statically and dynamically, after which we pro-  pose three novel natural video backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Through  extensive experiments, we provide evidence that the previ-  ous perception of image backdoor attacks in the video do-  main is not necessarily true, especially in the poisoned-label  attack setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' (2) To the best of our knowledge, this work is  the first to investigate audiovisual backdoor attacks against  video action recognition models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Related Work  Backdoor Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Backdoor attacks were first introduced  in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The attack, called BadNet, was based on adding a  patch to the corner of a subset of training images to create a  backdoor that could be triggered by the attacker at will.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Fol-  lowing BadNet, [44] proposed optimizing for the values of  the patch to obtain a more effective backdoor attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Shortly  after the development of patch-based backdoor attacks, the  community realized the importance of adding an invisibility  constraint to the design of backdoor triggers to bypass any  human inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Works such as [9] proposed blending  the backdoor trigger with the image rather than stamping  it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' [37] generated backdoor attacks using the least signifi-  cant bit algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' [52] generated warping fields to warp the  image content as a backdoor trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' [14] went one step fur-  ther and designed learnable transformations to generate op-  timal backdoor triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' After many attacks were proposed  in the spatial domain [10, 37, 40, 45, 56, 57, 67, 72, 75], and  others in the latent representation domain [13,53,80,88,91],  [19, 25, 71, 82, 84] proposed to switch attention to the fre-  quency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' [25] utilized frequency heatmaps proposed  in [81] to create backdoor attacks that target the most sen-  sitive frequency components of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' [19] proposed  blending low frequency content from a trigger image with  training images as a poisoning technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' In our work, we  extend the 2D backdoor threat model to the video domain  by incorporating video-related aspects into it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' We also ex-  tend five image backdoor attacks into the video domain and  propose three natural video backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Backdoor Defenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Backdoor attack literature was im-  mediately opposed by various defenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Backdoor de-  fenses are generally of five types:  preprocessing-based  [12,47,55], model reconstruction-based [38,42,74,83,89],  trigger synthesis-based [23,24,29,43,54,58,62,68], model  diagonsis-based [15, 34, 46, 76, 90], and sample-filtering  based [8, 20, 26, 30, 61, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Early backdoor defenses such  as [68] hypothesized that backdoor attacks create a short-  cut between all samples and the poisoned class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Based on  that, they solved an optimization problem to find whether a  trigger of an abnormally small norm exists that would flip  all samples to one label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Later, multiple improved itera-  tions of this method were proposed, such as [23, 43, 83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Fine pruning [42] suggested that the backdoor is triggered  by particular neurons that are dormant in the absence of  the trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Therefore, the authors proposed pruning the  least active neurons on clean samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' STRIP [20] showed  that blending clean samples with other clean samples would  yield a higher entropy compared to when clean images are  blended with poisoned samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Activation clustering [8]  uses KMeans to cluster the activations of an inspection, a  potentially poisoned data set, into two clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' A large sil-  houette distance between the two clusters would uncover  the poisoned samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' In our work, we show that current  image backdoor attacks have limited effectiveness in de-  fending against backdoor attacks in the video domain, es-  pecially against the proposed natural video attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Video Action Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Video action recognition mod-  els, which only leverage the raw frames of a video, can be  categorized into two categories, CNN-based networks and  transformer-based networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' 2D CNN-based methods are  built on top of pretrained image recognition networks with  well-designed modules to capture the temporal relationship  between multiple frames [41,49,69,70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Those methods are  computationally efficient as they use 2D convolutional ker-  nels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' To learn stronger spatial-temporal representations, 3D  CNN-based methods were proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' These methods utilize  3D kernels to jointly leverage the spatio-temporal context  within a video clip [17, 18, 64, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' To better initialize the  network, I3D [7] inflated the weights of 2D pretrained im-  age recognition models to adapt them to 3D CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Real-  izing the importance of computational efficiency, S3D [78]  and R(2+1)D [66] proposed to disentangle spatial and tem-  poral convolutions to reduce computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Recently,  transformer-based action recognition models were able to  achieve better performance in large training data sets com-  pared to CNN-based models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' [4,6,16,48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' In this work,  we test backdoor attacks against three action recognition  architectures, namely I3D, SlowFast, and TSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Audiovisual Action Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' In addition to frames, a  line of action recognition models [1,27,28,51] has used the  accompanying audio to better understand activities such as  “playing music” or “washing dishes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' To take advantage of  existing CNN and transformer-based models, the Log-Mel  spectrogram was introduced to convert audio data from a  non-structured signal into a 2D representation in time and  frequency usable by these models [2,3,35,77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Current au-  diovisual action recognition methods are divided into two  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Traditional Backdoor Attack Pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' After selecting a backdoor trigger and a target label, the attacker poisons a subset of  the training data referred to as the poisoned dataset (Dp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The label of the poisoned dataset is fixed to a target poisoning label specified by  the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The attacker trains jointly on clean (non-poisoned) samples (Dc) and poisoned samples leading to a backdoored model, which  outputs the target label in the presence of the backdoor trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  categories based on when the audio and visual signals are  merged in the recognition pipeline: early fusion and late fu-  sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Early fusion combines features before classification,  which can better capture features [32,77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The disadvantage  of early fusion is that there is a higher risk of overfitting to  the training data [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Late fusion, on the other hand, treats  the video and audio networks separately, and the predictions  of each network are carried out independently, after which  the logits are aggregated to make a final prediction [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' For  the first time, we test backdoor attacks against audiovisual  action recognition networks in both late and early fusion  setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Video Backdoor Attacks  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The Traditional Threat Model  The commonly adopted threat model for backdoor at-  tacks dates back to the works that studied those attacks  against 2D image classification models [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The victim  outsources the training process to a trainer who is given ac-  cess to both the victim’s training data and the network ar-  chitecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The victim only accepts the model provided by  the trainer if it performs well on the victim’s private val-  idation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The attacker aims to maximize the effective-  ness of the embedded backdoor attack [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' We refer to  the model’s performance on the validation set as clean data  accuracy (CDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The effectiveness of the backdoor attack  is measured by the attack success rate (ASR), which is de-  fined as the percentage of test examples not labeled as the  target class that are classified as the target class when the  backdoor pattern is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' To achieve this goal, the at-  tacker applies a backdoor trigger to a subset of the train-  ing images and then, in the poisoned-label setup, switches  the labels of those images to a target class of choice before  training begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' A more powerful backdoor attack is one  that is visually imperceptible (usually measured in terms of  ℓ2/ℓ∞-norm, PSNR, SSIM, or LPIPS) but achieves both a  high CDA and a high ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' This is summarized in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  More formally, we denote the classifier which is param-  eterized by θ as fθ : X → Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' It maps the input x ∈ X,  such as images or videos, to class labels y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Let  Gη : X → X indicate an attacker-specific poisoned im-  age generator that is parameterized by some trigger-specific  parameters η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The generator may be image-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Fi-  nally, let S : Y → Y be an attacker-specified label shifting  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' In our case, we consider the scenario in which the  attacker is trying to flip all the labels into one particular la-  bel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' S : Y → t, where t ∈ Y is an attacker-specified  label that will be activated in the presence of the backdoor  trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Let D = {(xi, yi)}N  i=1 indicate the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  The attacker splits D into two subsets, a clean subset Dc and  a poisoned subset Dp, whose images are poisoned by Gη  and labels are poisoned by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The poisoning rate is the ra-  tio α = |Dp|  |D| , generally a lower poisoning rate is associated  with a higher clean data accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The attacker typically  trains the network by minimizing the cross-entropy loss on  Dc∪Dp, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' minimizes E(x,y)∼Dc∪Dp[LCE(fθ(x), y)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The  attacker aims to achieve high accuracy on the user’s valida-  tion set Dval while being able to trigger the poisoned-label,  t, in the presence of the backdoor trigger, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' fθ(Gη(x)) =  t, ∀x ∈ X (ideally).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' From Images to Videos  Unlike images, videos have an additional dimension, the  temporal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' This dimension introduces new rules  to the game between the attacker and the victim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  More  Settings  Training Stage  Inference Stage  "Eat"  "Jump\'  fe  fe  Poisoned  Network  Target Label = "Eat"  Backdoor Trigger  Clean Samples (Dc)  Poisoned Video (Gn(α))  (α)  Label = "Eat"  Clean Video  Ground Truth LabelFigure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Static vs Dynamic Backdoor Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Static backdoor attacks apply the same trigger across all frames along the temporal  dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' On the other hand, dynamic attacks apply a different trigger per frame along the temporal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  precisely, the attacker now has an additional dimension to  hide the backdoor trigger, leading to a higher level of im-  perceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  The backdoor attack could be applied to  all the frames or a subset of the frames statically, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' the  same trigger is applied to each frame, or dynamically, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  a different trigger is applied to each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' On the other  hand, the testing pipeline now imposes harsher conditions  against the backdoor attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Video recognition models tend  to test the model on multiple sub-sampled clips with vari-  ous crops [7,18,41] which might, in turn, destroy the back-  door trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  For example, if the trigger is applied to a  single frame, it might not be sampled, and if the trigger  is applied to the corner of the image, it might be cropped  out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The threat model presented in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='1 was di-  rectly adopted in [87], which to the best of our knowledge,  is the only previous work that considered backdoor attacks  for video action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Our work sheds light on the aforementioned video-  related aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='2, we show the effect of the  number of frames poisoned on CDA and ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' We also  show how existing 2D methods could be extended both stat-  ically and dynamically to suit the video domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' For exam-  ple, BadNet [22] applies a fixed patch as a backdoor trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  The patch could be applied statically using the same pixel  values and the same position along the temporal dimension  or applied dynamically by changing the position and possi-  bly the pixel values of the patch for each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Figure 2  shows a BadNet attack when applied in a static and dynamic  way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Additionally, we show how simple yet natural video  “artifacts” could be used as backdoor triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' More specif-  ically, lag in a video, motion blur, and compression glitches  could all be used as naturally occurring backdoor triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Audiovisual Backdoor Attacks  Videos are naturally accompanied by audio signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Sim-  ilarly to how the video modality could be attacked, the audio  signal could also be attacked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The interesting question that  arises is how backdoor attacks would perform in a multi-  modal setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' In the experiments of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='4, we answer  the following questions: (1) What is the effect of having two  attacked modalities on CDA and ASR?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' (2) What happens  if only one modality is attacked and the other is left clean?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  (3) What is the difference in performance between late and  early fusion in terms of CDA and ASR?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Experiments  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Experimental Settings  Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' We consider three standard benchmark datasets  used in video action recognition: UCF-101 [60], HMDB-  51 [36], and Kinetics-Sounds [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Kinetics-Sounds is a  subset of Kinetics400 that contains classes that can be clas-  sified from the audio signal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' classes where audio is use-  ful for action recognition [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Kinetics-Sounds is particu-  larly interesting for Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='4, where we explore  backdoor attacks against audio and audiovisual classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Network Architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Following common practice, for  the visual modality, we use a dense sampling strategy to  sub-sample 32 frames per video to fine-tune a pretrained  I3D network on the target dataset [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='2, we  also show results using TSM [41] and SlowFast [18] net-  works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' All three models adopt ResNet-50 as the backbone  and are pretrained on Kinetics-400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Similarly to [2], for  the audio modality, a ResNet-18 is trained from scratch on  Mel-Spectrograms composed of 80 Mel bands sub-sampled  temporally to a fixed length of 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Attack Setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  For the video modality, we study and  extend the following image-based backdoor attacks to the  video domain: BadNet [22], Blend [9], SIG [5], WaNet  [52], and FTrojan [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' We also explore three additional  natural video backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  For the audio modality,  we consider two attacks: sine attack and high-frequency  noise attack, both of which we explain later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Following  [22,25,52], the target class is arbitrarily set to the first class  t=0  t=1  t=12  t=13  t=14  t=N-1  t=N  Static  DynamicFigure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Visualization of 2D Backdoor Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Image backdoor attacks mainly differ according to the backdoor trigger used to poison  the training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' They could be extended either statically or dynamically based on how the attack is applied across the frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  of each data set (class 0), and the poisoning rate is set to  10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Unless otherwise stated, the considered image back-  door attacks poison all frames of the sampled clips during  training and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' As is commonly done in the back-  door literature, we evaluate the performance of the model  using clean data accuracy (CDA) and attack success rate  (ASR) explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' CDA represents the usual  validation/test accuracy on an unseen dataset hence mea-  suring the generalizability of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' On the other hand,  ASR measures the effectiveness of the attack when the poi-  son is applied to the validation/test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  In addition, we  test the attacked models against some of the early 2D back-  door defenses, more precisely against activation clustering  (AC) [8], STRIP [20], and pruning [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Our method is built on MMAc-  tion2 library [11], and follows their default training config-  urations and testing protocols, except for the learning rate  and the number of training epochs (check Supplementary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  All experiments were run using 4 NVIDIA A100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Video Backdoor Attacks  Extending Image Backdoor Attacks to the Video Do-  main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' As mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='2, image backdoor attacks  could be extended either statically by applying an attack in  the same way across all frames or dynamically by adjusting  the attack parameters for different frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' We consider five  attacks that differ according to the applied backdoor trig-  ger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' BadNet applies a patch as a trigger, Blend blends a  trigger image to the original image, SIG superimposes a si-  nusoidal trigger to the image, WaNet warps the content of  the image, and FTrojan poisons a high- and mid- frequency  component in the discrete cosine transform (DCT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Figure 3  visualizes all five attacks on the same video frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Each of  the considered methods could be extended dynamically as  follows: BadNet: change the patch location for each frame;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Blend: blend a uniform noise that is different per frame;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  SIG: change the frequency of the sine component superim-  posed with each frame;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' WaNet: generate a different warp-  ing field for each frame;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' FTrojan: select a different DCT  basis to perturb at each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Note that Blend and FTro-  jan are generally imperceptible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Visualizations and saliency  UCF101  HMDB51  KineticsSound  CDA(%)  ASR(%)  CDA(%)  ASR(%)  CDA(%)  ASR(%)  Baseline  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='95 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='59 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='41 BadNet  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='95  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='63  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='35  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='89  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='97  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='09  Blend  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='29  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='26  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='37  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='73  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='12  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='54  SIG  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='97  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='97  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='50  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='80  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='84  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='87  WaNet  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='05  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='84  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='95  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='61  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='38  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='09  FTrojan  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='16  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='34  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='10  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='52  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='45  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='86  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Statically Extended 2D Backdoor Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Statically  extending 2D backdoor attacks to the video domain leads to high  CDA and ASR across all three considered datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  UCF101  HMDB51  KineticsSound  CDA(%)  ASR(%)  CDA(%)  ASR(%)  CDA(%)  ASR(%)  Baseline  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='95 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='59 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='41 BadNet  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='11  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='97  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='08  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='54  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='25  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='74  Blend  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='21  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='44  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='03  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='95  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='67  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='79  SIG  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='24  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='00  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='63  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='00  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='84  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='00  WaNet  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='29  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='79  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='22  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='80  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='25  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='61  FTrojan  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='16  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='34  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='19  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='69  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='25  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='27  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Dynamically Extended 2D Backdoor Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Dynam-  ically extending 2D backdoor attacks to the video domain leads to  high CDA and ASR across all three considered datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  maps for each attack are found in the Supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Tables 1 and 2 show the CDA and ASR of the I3D mod-  els attacked using various backdoor attacks on UCF-101,  HMDB-51, and Kinetics-Sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Contrary to the conclu-  sion presented in [87], we find that backdoor attacks are  actually highly effective in the video domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The CDA  of the attacked models is very similar to that of the clean  unattacked model (baseline), surpassing it in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Extending attacks dynamically, almost always, improves  CDA and ASR compared to extending them statically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Natural Video Backdoors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' A more interesting attack is  one that seems natural and could bypass human inspec-  tion [50, 73, 79, 86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' There are several natural “glitches”  that occur in the video domain and that one could exploit  to design a natural backdoor attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' For example, videos  might contain some frame lag, motion blur, video compres-  sion corruptions, camera focus/defocus, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' In Table 3, we  report the CDA and ASR of three natural backdoor attacks:  Clean  BadNet  Blend  SIG  WaNet  FTrojanUCF101  HMDB51  KineticsSound  CDA(%)  ASR(%)  CDA(%)  ASR(%)  CDA(%)  ASR(%)  Baseline  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='95 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='59 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='41 Frame Lag  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='94  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='20  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='04  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='76  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='51  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='19  Video Corrupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='26  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='87  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='22  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='22  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='74  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='51  Motion Blur  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='97  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='92  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='17  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='52  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='19  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='22  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Natural Video Backdoor Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Natural attacks  against video action recognition models could achieve high CDA  and ASR while looking completely natural to human inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  SlowFast  TSM  CDA(%)  ASR(%)  CDA(%)  ASR(%)  Baseline  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='72 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='77 BadNet  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='64  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='47  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='69  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='78  SIG  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='70  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='97  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='77  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='47  FTrojan  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='25  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='52  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='21  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='00  Frame Lag  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='43  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='97  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='63  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='96  Video Corruption  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='54  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='76  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='08  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='97  Motion Blur  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='46  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='55  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='50  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='39  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Video Backdoor Attacks Against Different Archi-  tectures (UCF-101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' When tested against network architectures  other than I3D such as TSM and SlowFast, both image and natural  backdoor attacks can still achieve high CDA and high ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  frame lag (lagging video), video compression glitch (which  we refer to as Video Corruption), and motion blur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Interest-  ingly, these attacks could achieve both high clean data ac-  curacy and high attack success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' It is worth noting that  for frame lag, a two-frame lag is used for UCF-101 and a  three-frame lag is used for HMDB-51 and Kinetics-Sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  More details are provided in the Supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Attacks Against Different Architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' So far, all at-  tacks have been experimented with against an I3D network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  To further explore the behavior of backdoor attacks against  other video recognition models, we test a subset of the con-  sidered attacks against a 2D based model, TSM, and another  3D based model, SlowFast, on UCF-101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Table 4 shows  that all the aforementioned backdoor attacks perform sig-  nificantly well in terms of CDA and ASR against both TSM  and SlowFast architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Note that even though TSM is  a 2D based model, our proposed natural video backdoor at-  tacks still succeed in attacking it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Recommendations for Video Backdoor Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' As men-  tioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='2, the attacker must select a number of  frames to poison per video, keeping in mind that the video  will be sub-sampled and randomly cropped during evalua-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Since the attacker is the one who trained the network in  the first place, he/she has access to the processing pipeline  and could exploit this during the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' For example, if  video processing involves sub-sampling the video into clips  of 32 frames and cropping the frames into 224×224 crops,  the attacker could pass to the network an attacked video of  a temporal length of 32 frames and a spatial size 224×224,  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Effect of the Number of Poisoned Frames (UCF-101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Different colors refer to different number of frames poisoned dur-  ing the training of the attacked model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Training the model with  a single poisoned frame performs best for various choices of the  number of frames poisoned during evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Frame  Lag  Motion  Blur  SIG  BadNet  FTrojan  Elimination Rate(%)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='00  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='21  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='77  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='12  Sacrifice Rate(%)  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='08  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='82  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='17  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='25  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='00  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Activation Clustering Defense (UCF-101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Whereas  Activation Clustering provides partial success in defending against  image backdoor attacks, it fails completely against natural attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  hence bypassing sub-sampling and cropping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' However, a  system could force the user to input a video of a partic-  ular length, possibly greater than the length of the sub-  sampled clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' This raises an important question regarding  how many frames the attacker should poison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Clearly, the  smaller the number of frames the attacker poisons, the less  detectable the attack is, but does the attack remain effective?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  In Figure 4, we show the attack success rate of backdoor-  attacked models trained on clips of 1, 8, 16, and 32 frames,  and a randomly sampled number of poisoned frames (out of  32 total frames) when evaluated on clips of 1, 8, 16, and 32  poisoned frames (out of 32 total frames).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Random refers to  training on a varying number of poisoned frames per clip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Note that training the model against the worst-case scenario  (single frame), which mimics the case where only one of  the poisoned frames is sub-sampled, provides the best guar-  antees for achieving a high attack success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Defenses Against Video Backdoor Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' We explore  the effect of extending some of the existing 2D backdoor  defenses against video backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Optimization-  based defenses are extremely costly when extended to the  video domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' For example, Neural Cleanse (NC) [68], I-  BAU [83], and TABOR [23] involve a trigger reconstruc-  tion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The trigger space is now bigger in the presence  of the temporal dimension, and therefore, instead of opti-  mizing for a 224×224×3 trigger, the defender has to search  for a 32×224×224×3 trigger (assuming 32 frame clips are  used), which is both costly and hard to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The attacker  has the spatial and temporal dimensions to design and em-  BadNet  SIG  100  75  # Poisoned Frames  ASR(%)  (Training)  1  50  8  16  25  32  Random  0  8  16  32 1  1  8  16  32  # Poisoned Frames  (EvaluationFigure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' STRIP Defense (UCF-101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Whereas the entropy of  image backdoor attacks is very low compared to that of clean sam-  ples, the proposed natural backdoor attacks have a natural distri-  bution of entropies similar to that of clean samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Pruning Defense (Kinetics-Sounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Pruning is com-  pletely ineffective against image backdoor attacks extended to the  video domain and natural video backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Even though  the clean accuracy has dropped to random, the attack success rate  is maintained at very high levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  bed their attack in, and, therefore, reverse engineering the  trigger is quite hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  We consider three well-known defenses that introduce no  computational overhead when adopted to the video domain,  namely Activation Cluster (AC) [8], STRIP [20], and prun-  ing [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' AC computes the activations of a neural network  on clean samples (from the test set) and an inspection set  of interest which may be poisoned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' AC then applies PCA  to reduce the dimension of the activations, after which the  projected activations are clustered into two classes and com-  pared to the activations of the clean set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' STRIP blends clean  samples with the samples of a possibly poisoned inspec-  tion set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The entropy of the predicted probabilities is then  checked for any abnormalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Unlike clean samples, poi-  soned samples tend to have a low entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Pruning suggests  that the backdoor is usually embedded in particular neurons  Baseline  Sine Attack  High Frequency Attack  CDA(%)  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='21  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='21  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='61  ASR(%) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='36  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='96  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Audio Backdoor Attacks (Kinetics-Sounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Both sine  attack and the high-frequency band attack perform similarly to  baseline in terms of CDA while being able to achieve high ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  in the network that are only activated in the presence of the  trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Therefore, those neurons are supposed to be dor-  mant as far as the test set samples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' clean samples, are  concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' This allows us to detect and prune those dor-  mant neurons to eliminate the backdoor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Table 5 shows the  elimination and sacrifice rates of AC when applied against  some of the considered attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The elimination rate refers  to the ratio of poisoned samples correctly detected as poi-  soned to the total number of poisoned samples, whereas the  sacrifice rate refers to the ratio of clean samples incorrectly  detected as poisoned to the total number of clean samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Whereas AC has partial success in defending against image  backdoor attacks, it fails completely against the proposed  natural backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Figure 5 shows that the entropy  of the clean and poisoned samples of the proposed natural  attacks is very similar and therefore could evade the STRIP  defense, while BadNet and FTrojan are detectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Finally,  Figure 6 shows that pruning the least active neurons causes  a reduction in CDA without reducing ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' This is observed  not only for the natural attacks, but also for the extended im-  age backdoor attacks, hinting that image backdoor defenses  are not effective in the video domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Audio Backdoor Attacks  Attacks proposed against audio networks have been lim-  ited to adding a low-volume one-hot-spectrum noise in the  frequency domain, which leaves highly visible artifacts in  the spectrogram [85] or adding a human non-audible com-  ponent [33], f < 20Hz or f > 20kHz, which is non-  realistic, since spectrograms usually filter out those frequen-  cies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' We consider two attacks against the Kinetics-Sounds  dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' the first is to add a low-amplitude sine wave com-  ponent with f = 800Hz to the audio signal, and the second  is to add band-limited noise 5kHz < f < 6kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The spec-  trograms and the absolute difference between the attacked  spectrograms and the clean spectrogram are shown in Fig-  ure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Since no clear artifacts are observed in the spectro-  grams, human inspection fails to label the spectrograms as  attacked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The CDA and ASR rates of the backdoor-attacked  models for both attacks are shown in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The attacks  achieve a relatively high ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Audiovisual Backdoor Attacks  Now, we combine video and audio attacks to build a  multi-modal audiovisual backdoor attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The way we do  BadNet  FTrojan  1500  1500  Poisoned  Clean  1000  1000  500  500  0  0  2  4  0  2  4  Frame Lag  Motion Blur  600  600  400  400  200  200  0  0  0  2  4  0  2  4  EntropyBadNet  Frame Lag  100  ASR  ASR  CDA  CDA  Accuracy(%)  75  50  25  0  0  25  50  75  100  0  25  50  75  100  Percentage Pruned (%Late Fusion  Early Fusion  Clean Audio  Sine Attack  High Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Attack  Clean Audio  Sine Attack  High Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Attack  Clean Video  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='25 / -  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='74 / 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='98  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='96 / 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='91  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='72 / -  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='48 / 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='23  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='94 / 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='72  BadNet  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='33 / 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='97  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='63 / 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='74  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='33 / 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='87  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='50 / 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='29  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='10 / 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
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+page_content='00  Video Corruption  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
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+page_content='00  Motion Blur  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
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+page_content='58  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='65 / 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='00  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
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+page_content='00  Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Audiovisual Backdoor Attacks (Kinetics-Sounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The entries in the table report the CDA(%)/ASR(%) of attacking late and  early fused audiovisual networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' When a single modality is attacked, late fusion has a low ASR compared to early fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' When both  modalities are attacked, the ASR of both late and early fusion are high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Clean and Attacked Audio Spectrograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The uti-  lized audio backdoor attacks are not only audibly imperceptible  but also leave no perceptible artifacts in the Mel spectrogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The  spectrogram of each attack is followed by the absolute difference  of the attacked spectrogram with the clean one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  it is by taking our attacked models from Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='2 and  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='3 and applying early or late fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' For early fusion, we  extract video and audio features using our trained audio and  video backbones, and we then train a classifier on the con-  catenation of the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' In late fusion, the video and au-  dio networks predict independently on the input, and then  the individual logits are aggregated to produce the final pre-  diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' To answer the three questions posed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='3,  we run experiments in which both modalities are attacked  and others in which only a single modality is attacked for  both early and late fusion setups (Table 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' We summarize  the results as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' (1) Attacking two modalities con-  sistently improves ASR and even CDA in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' (2)  Attacking a single modality is good enough to achieve a  high ASR in the case of early fusion but not late fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  (3) Early fusion enables the best of both worlds for the at-  tacker, namely, a high CDA and an almost perfect ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' On  the other hand, late fusion experiences some serious drops  in ASR in the unimodal attack setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' An interesting find-  ing in these experiments is the following: if the outsourcer  has the option to outsource the most expensive modality,  training wise, while training other modalities in-house, ap-  plying late fusion could be used as a defense mechanism,  especially in the presence of more clean modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Conclusion  Backdoor attacks present a serious and exploitable vul-  nerability against both unimodal and multi-modal video  action recognition models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' We showed how existing im-  age backdoor attacks could be extended either statically  or dynamically to develop powerful backdoor attacks that  achieve both a high clean data accuracy and a high attack  success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Besides existing image backdoor attacks, there  exists a set of natural video backdoor attacks, such as mo-  tion blur and frame lag, that are resilient to existing image  backdoor defenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Given that videos are usually accom-  panied by audio, we showed two ways in which one could  attack audio classifiers in a human inaudible manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' The  Clean Spectrogram  80  40  0  Sine Attack  80  40  0  Mel Frequency  80  40  0  High Frequency Attack  80  40  0  80  40  0  0  50  100  150  200  250  300  Frameattacked video and audio models are then used to train an  audiovisual action recognition model by applying both early  and late fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Different combinations of poisoned modal-  ities are tested, concluding that: (1) poisoning two modal-  ities could achieve extremely high attack success rates in  both late and early fusion settings, and (2) if a single modal-  ity is poisoned, unlike early fusion, late fusion could reduce  the effectiveness of the backdoor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' We hope that our work  reignites the attention of the community towards exploring  backdoor attacks and defenses in the video domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  References  [1] Humam Alwassel, Dhruv Kumar Mahajan, Lorenzo Tor-  resani, Bernard Ghanem, and Du Tran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
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+page_content=' Data-  free backdoor removal based on channel lipschitzness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' In  European Conference on Computer Vision, pages 175–191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content='  Springer, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' 2  [90] Songzhu Zheng, Yikai Zhang, Hubert Wagner, Mayank  Goswami, and Chao Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Topological detection of trojaned  neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Advances in Neural Information Process-  ing Systems, 34:17258–17272, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' 2  [91] Nan Zhong, Zhenxing Qian, and Xinpeng Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' Impercep-  tible backdoor attack: From input space to feature represen-  tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' In IJCAI, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
+page_content=' 2' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfD_qi/content/2301.00986v1.pdf'}
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+A Safety Framework for Flow Decomposition Problems via
+Integer Linear Programming
+Fernando H. C. Dias1,⋆[0000−0002−6398−919X], Manuel C´aceres1,⋆[0000−0003−0235−6951], Lucia
+Williams2,⋆[0000−0003−3785−0247], Brendan Mumey2,⋆⋆[0000−0001−7151−2124], and
+Alexandru I. Tomescu1,⋆⋆[0000−0002−5747−8350]
+1 Department of Computer Science, University of Helsinki, Finland
+{fernando.cunhadias,manuel.caceres,alexandru.tomescu}@helsinki.fi
+2 School of Computing, Montana State University, Bozeman, MT, USA
+{lucia.williams,brendan.mumey}@montana.edu
+Abstract. Many important problems in Bioinformatics (e.g., assembly or multi-assembly) admit mul-
+tiple solutions, while the final objective is to report only one. A common approach to deal with this
+uncertainty is finding safe partial solutions (e.g., contigs) which are common to all solutions. Previous
+research on safety has focused on polynomially-time solvable problems, whereas many successful and
+natural models are NP-hard to solve, leaving a lack of “safety tools” for such problems. We propose
+the first method for computing all safe solutions for an NP-hard problem, minimum flow decomposi-
+tion. We obtain our results by developing a “safety test” for paths based on a general Integer Linear
+Programming (ILP) formulation. Moreover, we provide implementations with practical optimizations
+aimed to reduce the total ILP time, the most efficient of these being based on a recursive group-testing
+procedure.
+Results: Experimental results on the transcriptome datasets of Shao and Kingsford (TCBB, 2017)
+show that all safe paths for minimum flow decompositions correctly recover up to 90% of the full RNA
+transcripts, which is at least 25% more than previously known safe paths, such as (C´aceres et al. TCBB,
+2021), (Zheng et al., RECOMB 2021), (Khan et al., RECOMB 2022, ESA 2022). Moreover, despite the
+NP-hardness of the problem, we can report all safe paths for 99.8% of the over 27,000 non-trivial graphs
+of this dataset in only 1.5 hours. Our results suggest that, on perfect data, there is less ambiguity than
+thought in the notoriously hard RNA assembly problem.
+Availability: https://github.com/algbio/mfd-safety
+Contact: alexandru.tomescu@helsinki.fi
+Keywords: RNA assembly · Network flow · Flow decomposition · Integer linear programming · Safety
+⋆ Shared first-author contribution
+⋆⋆ Shared last-author contribution
+arXiv:2301.13245v1  [cs.DS]  30 Jan 2023
+
+1
+Introduction
+In real-world scenarios where an unknown object needs to be discovered from the input data, we would like
+to formulate a computational problem loosely enough so that the unknown object is indeed a solution to
+the problem, but also tightly enough so that the problem does not admit many other solutions. However,
+this goal is difficult in practice, and indeed, various commonly used problem formulations in Bioinformatics
+still admit many solutions. While a naive approach is to just exhaustively enumerate all these solutions, a
+more practical approach is to report only those sub-solutions (or partial solutions) that are common to all
+solutions to the problem.
+In the graph theory community such sub-solutions have been called persistent [14,21], and in the Bioin-
+formatics community reliable [54], or more recently, safe [51]. The study of safe sub-solutions started in
+Bioinformatics in the 1990’s [54,11,37] with those amino-acid pairs that are common to all optimal and
+suboptimal alignments of two protein sequences.
+In the genome assembly community, the notion of contig, namely a string that is guaranteed to appear in
+any possible assembly of the reads, is at the core of most genome assemblers. This approach originated in 1995
+with the notion of unitigs [25] (non-branching paths in an assembly graph), which were progressively [42,6]
+generalized to paths made up of a prefix of nodes with in-degree one followed by nodes with out-degree
+one [35,24,29] (also called extended unitigs, or Y-to-V contigs).
+Later, [51] formalized all such types of contigs as those safe strings that appear in all solutions to a
+genome assembly problem formulation, expressed as a certain type of walk in a graph. [10,9] proposed more
+efficient and unifying safety algorithms for several types of graph walks. [45] recently studied the safety of
+contigs produced by state-of-the-art genome assemblers on real data.
+Analogous studies were recently made also for multi-assembly problems, where several related genomic
+sequences need to be assembled from a sample of mixed reads. [8] studied safe paths that appear in all
+constrained path covers of a directed acyclic graph (DAG). Zheng, Ma and Kingsford studied the more
+practical setting of a network flow in a DAG by finding those paths that appear in any flow decomposition
+of the given network flow, under a probabilistic framework [34], or a combinatorial framework [58].3 [27]
+presented a simple characterization of safe paths appearing in any flow decomposition of a given acyclic
+network flow, leading to a more efficient algorithm than the one of [58], and further optimized by [28].
+Motivation. Despite the significant progress in obtaining safe algorithms for a range of different appli-
+cations, current safe algorithms are limited to problems where computing a solution itself is achievable in
+polynomial time. However, many natural problems are NP-hard, and safe algorithms for such problems are
+fully missing. Apart from the theoretical interest, usually such NP-hard problems correspond to restrictions
+of easier (polynomially-computable) problems, and thus by definition, also have longer safe sub-solutions.
+As such, current safety algorithms miss data that could be reported as correct, just because they do not
+constrain the solution space enough. A major reason for this lack of progress is that if a problem is NP-hard,
+then its safety version is likely to be hard too. This phenomenon can be found both in classically studied NP-
+hard problems — for example, computing nodes present in all maximum independent sets of an undirected
+graph is NP-hard [21] — as well as in NP-hard problems studied for their application to Bioinformatics, as
+we discuss further in the appendix.
+We introduce our results by focusing on the flow decomposition problem. This is a classical model at the
+core of multi-assembly software for RNA transcripts [33,31,5,50] and viral quasi-species genomes [3,2,44,12],
+and also a standard problem with applications in other fields, such as networking [36,22,13,23] or transporta-
+tion [39,38]. In its most basic optimization form, minimum flow decomposition (MFD), we are given a flow
+in a graph, and we need to decompose it into a minimum number of paths with associated weights, such
+that the superposition of these weighted paths gives the original flow. This is an NP-hard problem, even
+when restricted to DAGs [53,22]. Various approaches have been proposed to tackle the problem, including
+fixed-parameter tractable algorithms [30], approximation algorithms [36,7] and Integer Linear Programming
+formulations [15,46].
+3 The problem AND-Quant from [58] actually handles a more general version of this problem.
+1
+
+In Bioinformatics applications, reads or contigs originating from a mixed sample of genomic sequences
+with different abundances are aligned to a reference. A graph model, such as a splice graph or a variation
+graph, is built from these alignments. Read abundances assigned to the nodes and edges of this graph
+then correspond to a flow in case of perfect data. If this is not the case, the abundance values can either
+be minimally corrected to become a flow, or one can consider variations of the problem where e.g., the
+superposition of the weighted paths is closest (or within a certain range) to the edge abundances [50,5].
+Current safety algorithms for flow decompositions such as [58,27,26,28] compute paths appearing in all
+possible flow decompositions (of any size), even though decompositions of minimum size are assumed to
+better model the RNA assembly problem [30,48,55]. Even dropping the minimality constraint, but adding
+other simple constraints easily renders the problem NP-hard (see e.g., [56]), motivating further study of
+practical safe algorithms for NP-hard problems.
+Contributions. Integer Linear Programming (ILP) is a general and flexible method that has been suc-
+cessfully applied to solve NP-hard problems, including in Bioinformatics. In this paper, we consider graph
+problems whose solution consists of a set of paths (i.e., not repeating nodes) that can be formulated in
+ILP. We introduce a technique that, given an ILP formulation of such a graph problem, can enhance it
+with additional variables and constraints in order to test the safety of a given set of paths. An obvious first
+application of this safety test is to use it with a single path in a straightforward avoid-and-test approach,
+using a standard two-pointer technique that has been used previously to find safe paths for flow decomposi-
+tion. However, we find that a top-down recursive approach that uses the group-testing capability halves the
+number of computationally-intensive ILP calls, resulting in a 3x speedup over the straightforward approach.
+Additionally, we prove that computing all the safe paths for MFDs is an intractable problem, confirming
+the above intuitive claim that if a problem is hard, then also its safety version is hard. We give this proof
+in the appendix by showing that the NP-hardness reduction for MFD by [22] can be modified into a Turing
+reduction from the UNIQUE 3SAT problem.
+On the dataset [47] containing splice graphs from human, zebrafish and mouse transcriptomes, safe
+paths for MFDs (SafeMFD) correctly recover up to 90% of the full RNA transcripts while maintaining a
+99% precision, outperforming, by a wide margin (25% increase), state-of-the-art safety approaches, such as
+extended unitigs [35,24,29], safe paths for constrained path covers of the edges [8], and safe paths for all
+flow decompositions [28,27,26,58]. On the harder dataset by [26], SafeMFD also dominates in a significant
+proportion of splice graphs (built from t ≤ 15 RNA transcripts), recovering more than 95% of the full
+transcripts while maintaining a 98% precision. For larger t, precision drastically drops (91% precision in the
+entire dataset), suggesting that in more complex splice graphs smaller solutions are introduced as an artifact
+of the combinatorial nature of the splice graph, and the minimality condition [30,48,55] is thus incorrect in
+this domain.
+2
+Methods
+2.1
+Preliminaries
+ILP models. In this paper we use ILP models as blackboxes, with as few assumptions as possible to further
+underline the generality of our approach. Let M(V, C) be an ILP model consisting of a set V of variables
+and a set C of constraints on these variables, built from an input graph G = (V, E). We make only two
+assumptions on M. First, that a solution to this model consists of a given number k ≥ 1 of paths P1, . . . , Pk
+in G (in this paper, paths do not repeat vertices). Second, we assume that the k paths are modeled via
+binary edge variables xuvi, for all (u, v) ∈ E and for all i ∈ {1, . . . , k}. More specifically, for all i ∈ {1, . . . , k},
+we require that the edges (u, v) ∈ E for which the corresponding variable xuvi equals 1 induce a path in G.
+For example, if G is a DAG, it is a standard fact (see e.g., [49]) that a path from a given s ∈ V to a given
+2
+
+t ∈ V (an s-t path) can be expressed with the following constraints:
+�
+(u,v)∈E
+xuvi −
+�
+(v,u)∈E
+xvui =
+�
+�
+�
+�
+�
+0,
+if v ∈ V \ {s, t},
+1,
+if v = t,
+−1,
+if v = s.
+(1)
+If G is not a DAG, there are other types of constraints that can be added to the xuvi variables to ensure
+that they induce a path; see, for example, the many formulations in [49]. We will assume that such constraints
+are part of the set C of constraints of M(V, C), but their exact formulation is immaterial for our approach. In
+fact, one could even add additional constraints to C to further restrict the solution space. For example, some
+ILP models from [15,46] handle the case when the input also contains a set of paths (subpath constraints)
+that must appear in at least one of the k solution paths.
+Flow decomposition. In the flow decomposition problem we are given a flow network (V, E, f), where
+G = (V, E) is a (directed) graph with unique source s ∈ V and unique sink t ∈ V , and f assigns a positive
+integer flow value fuv to every edge (u, v) ∈ E. Flow conservation must hold for every node different from s
+and t, namely, the sum of the flow values entering the node must equal the sum of the flow values exiting the
+node. See Figure 1(a) for an example. We say that k s-t paths P1, . . . , Pk, with associated positive integer
+weights w1, . . . , wk, are a flow decomposition (FD) if their superposition equals the flow f. Formally, for
+every (u, v) ∈ E it must hold that
+�
+i∈{1,...,k} s.t.
+(u,v)∈Pi
+wi = fuv.
+(2)
+See Figures 1(b) and 1(c) for two examples. The number k of paths is also called the size of the flow
+decomposition. In the minimum flow decomposition (MFD) problem, we need to find a flow decomposition
+of minimum size.4 On DAGs, a flow decomposition into paths always exists [1], but in general graphs, cycles
+may be necessary to decompose the flow (see e.g. [16] for different possible formulations of the problem).
+For concreteness, we now describe the ILP models from [15] for finding a flow decomposition into k
+weighted paths in a DAG. They consist of (i) modeling the k paths via the xuvi variables (with constraints
+(1)), (ii) adding path-weight variables w1, . . . , wk, and (iii) requiring that these weighted paths form a flow
+decomposition, via the following (non-linear) constraint:
+�
+i∈{1,...,k}
+xuviwi = fuv,
+∀(u, v) ∈ E.
+(3)
+This constraint can then be easily linearized by introducing additional variables and constraints; see e.g. [15]
+for these technical details. However, as mentioned above, the precise formulation of the ILP model M for a
+problem is immaterial for our method. Only the two assumptions on M made above matter for obtaining
+our results.
+Safety. Given a problem on a graph G whose solutions consist of k paths in G, we say that a path P is safe
+if for any solution P1, . . . , Pk to the problem, there exists some i ∈ {1, . . . , k} such that P is a subpath of Pi.
+If the problem is given as an ILP model M, we also say that P is safe for M. We say that P is a maximal
+safe path, if P is a safe path and there is no larger safe path containing P as subpath. [27] characterized safe
+paths for all FDs (not necessarily of minimum size) using the excess flow fP of a path P, defined as the flow
+on the first edge of P minus the flow on the edges out-going from the internal nodes of P, and different from
+the edges of P (see Figure 1(d) for an example). It holds that P is safe for all FDs if and only if fP > 0 [27].
+4 In this paper we work only with integer flow values and weights for simplicity and since this is the most studied
+version of the problem, see e.g., [30]. However, the problem can also be defined with fractional weights [41], and
+in this case the two problems can have different minima on the same input [53]. This fractional case can also be
+modeled by ILP [15], and all the results from our paper also immediately carry over to this variant.
+3
+
+ 
+8
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+9
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+t
+b
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+e
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+ c
+10
+10
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+5
+3
+2
+7
+3
+4
+1
+9
+3
+3
+Figure 1
+Figure 2
+(a) A flow network with source s and sink t.
+ 
+8
+9
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+3
+5
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+s
+t
+b
+a
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+e
+7
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+ 
+s
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+ c
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+ 
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+e
+ c
+f
+5
+3
+2
+7
+3
+4
+1
+9
+3
+3
+Figure 1
+Figure 2
+ 
+8
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+7
+s
+t
+b
+a
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+e
+7
+ c
+10
+10
+f
+3
+3
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+xsai = 1
+xabi = 1
+xbci = 1
+xcdi = 1
+xdfi = 1
+Figure 3
+xadi = 0
+xeti = 0
+xdei = 0
+xfti = 1
+xsbi = 0
+xbdi = 0
+(b) An MFD into 4 paths of weights 5,3,7,2, respec-
+tively. The green dashed path is a subpath of the
+orange path.
+ 
+8
+9
+9
+3
+5
+7
+s
+t
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+a
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+e
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+ c
+10
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+ 
+s
+t
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+e
+ c
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+5
+3
+2
+7
+3
+4
+1
+9
+3
+3
+Figure 1
+Figure 2
+ 
+8
+9
+9
+3
+5
+7
+s
+t
+b
+a
+d
+e
+7
+ c
+10
+10
+f
+3
+3
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+xsai = 1
+xabi = 1
+xbci = 1
+xcdi = 1
+xdfi = 1
+Figure 3
+xadi = 0
+xeti = 0
+xdei = 0
+xfti = 1
+xsbi = 0
+xbdi = 0
+(c) An MFD into 4 paths of weights 3,4,1,9, respec-
+tively. The green dashed path is a subpath of the pink
+path.
+ 
+8
+9
+9
+3
+5
+7
+s
+t
+b
+a
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+e
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+ c
+10
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+ 
+s
+t
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+a
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+e
+ c
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+5
+3
+2
+7
+3
+4
+1
+9
+3
+3
+Figure 1
+Figure 2
+ 
+8
+9
+9
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+5
+7
+s
+t
+b
+a
+d
+e
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+ c
+10
+10
+f
+3
+3
+ 
+s
+t
+b
+a
+d
+e
+7
+ c
+10
+f
+3
+xsai = 1
+xabi = 1
+xbci = 1
+xcdi = 1
+xdfi = 1
+(d) The two subpaths (red and blue) of the green
+dashed path that are maximal safe paths for all FDs.
+Fig. 1: Flow decompositions and safe paths. The flow network in (a) admits different MFDs, in (b) and in (c).
+The path (s, a, b, c, d) (dashed green) is a maximal safe path for MFDs, i.e., it is a subpath of some path
+of all MFDs and it cannot be extended without losing this property. However, the path (s, a, b, c, d) is not
+safe for all FDs. Indeed, its two subpaths (s, a, b) (dashed red in (d)) and (b, c, d) (dashed blue in (d)) are
+maximal safe paths for all FDs. To see this, note that the excess flow of (s, a, b) is 3, while the excess flow of
+(s, a, b, c) (and of (s, a, b, c, d)) is −6.
+4
+
+The excess flow can be computed in time linear in the length of P (assuming we have pre-computed the flow
+outgoing from every node), giving thus a linear-time verification of whether P is safe.
+A basic property of safe solutions is that any sub-solution of them is also safe. Computing safe paths
+for MFDs can thus potentially lead to joining several safe paths for FDs, obtaining longer paths from the
+unknown sequences we are trying to assemble. See Figure 1 for an example of a maximal safe path for MFDs
+and two maximal subpaths of it that are safe for FDs.
+2.2
+Finding Maximal Safe Paths for MFD via ILP
+We now present a method for finding all maximal safe paths for MFD via ILP. The basic idea is to define an
+inner “safety test” which can be repeatedly called as part of an outer algorithm over the entire instance to
+find all maximal safe paths. Because calls to the ILP solver are expensive, the guiding choice for our overall
+approach is to minimize the number of ILP calls. This inspires us to test the safety of a group of paths as the
+inner safety test, which we achieve by augmenting our ILP model so that it can give us information about
+the safety of the paths in the set. We use this to define a recursive algorithm to fully determine the safety
+status of each path in a group of paths. We can then structure the safety test in either a top-down manner
+(starting with long unsafe paths and shrinking them until they are safe) or a bottom-up manner (starting
+with short safe paths and lengthening them until they become unsafe).
+Safety test (inner algorithm) Let M(V, C) be an ILP model as discussed in Section 2.1; namely, its k
+solution paths are modeled by binary variables xuvi for each (u, v) ∈ E and each i ∈ {1, . . . , k}. We assume
+that M(V, C) is feasible (i.e., the problem admits at least one solution). We first show how to modify the
+ILP model so that, for a given set of paths, it can tell us one of the following: (1) a set of paths that are not
+safe (the remaining being of unknown status), or (2) that all paths are safe. The idea is to maximize the
+number of paths that can be simultaneously avoided from the given set of paths.
+Let P be a set of paths. For each path P ∈ P, we create an auxiliary binary variable γP that indicates:
+γP ≡
+�
+1
+if P was avoided in the solution,
+0
+otherwise.
+(4)
+Since the model solutions are paths (i.e., not repeating nodes), we can encode whether P appears in the
+solution by whether all of the ℓ − 1 edges of P appear simultaneously. Using this fact, we add a new set of
+constraints R(P) that include the γP indicator variables for each path P ∈ P:
+R(P) := {xv1v2i + xv2v3i + · · · + xvℓ−1vℓi
+≤ ℓ − 1 − γP : ∀i ∈ {1, . . . , k}, ∀P ∈ P}.
+(5)
+Next, as the objective function of the ILP model, we require that it should maximize the number of
+avoided paths from P, i,e., the sum of the γP variables:
+max
+�
+P ∈P
+γP .
+(6)
+All paths P such that γP = 1 are unsafe, since they were avoided in some minimum flow decomposition.
+Conversely, if the objective value of Eq. (6) was 0, then γP = 0 for all paths in P, and it must be that all
+paths in P are safe (if not, at least one path could be avoided and increase the objective). We encapsulate
+this group testing ILP in a function GroupTest(M, P) that returns a set N ⊆ P with the properties that:
+(1) if N = ∅, then all paths in P are safe, and (2) if N ̸= ∅, then all paths in N are unsafe (and |N| is
+maximized).
+We employ GroupTest(M, P) to construct a recursive procedure GetSafe(M, P) that determines all safe
+paths in P, as shown in Algorithm 1.
+5
+
+ 
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+2
+7
+3
+4
+1
+9
+3
+3
+Figure 1
+Figure 2
+ 
+8
+9
+9
+3
+5
+7
+s
+t
+b
+a
+d
+e
+7
+ c
+10
+10
+f
+3
+3
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+xsai = 1
+xabi = 1
+xbci = 1
+xcdi = 1
+xdfi = 0
+Figure 3
+xadi = 0
+xeti = 1
+xdei = 1
+xfti = 0
+xsbi = 0
+xbdi = 0
+Fig. 2: Illustration of modeling a solution path and a tested path via binary edge variables and safety
+verification constraints. The ith solution path Pi is shown in orange, and a tested path P is shown in dashed
+green. Constraint (5) includes xsai +xabi +xbci +xcdi +xdei ≤ 5−γP . This simplifies to γP ≤ 0, thus forcing
+γP = 0, which indicates P was not avoided in the solution.
+Algorithm 1: Testing a set of paths P for safety.
+Input: A feasible ILP model M(V, C), and a set of paths P
+Output: Those paths P ∈ P that are safe for M(V, C)
+1 Procedure GetSafe(M, P)
+2
+N = GroupTest(M, P) if N = ∅ then
+3
+return P
+4
+else
+5
+return GetSafe(M, P \ N)
+We note that in the special case that |P| = 1, GetSafe(M, P) makes only a single call to the ILP via
+GroupTest(M, P) to determine whether not the given path is safe. With this safety test for a single path, we
+can easily adapt a standard two-pointer approach as the outer algorithm to find all maximal safe paths for
+MFD by starting with some MFD solution P1, . . . , Pk of M(V, C). This same procedure was used in [26] to
+find all maximal safe paths for FD, using an excess flow check as the inner safety algorithm.
+Find all maximal safe paths (outer algorithm) We give two algorithms for finding all maximal safe
+paths. Both algorithms use a similar approach, however the first uses a top-down approach starting from the
+original full solution paths and reports all safe paths (these again must be maximal safe), and then trims all
+the unsafe paths to find new maximal safe paths. The second is bottom-up in that it tries to extend known
+safe subpaths until they cannot be further extended (and at this point must be maximal safe). We present
+the first algorithm in detail and defer discussion of the second to the appendix.
+We say a set of subpaths T = {Pi[li, ri]} is a trimming core provided that for any unreported maximal
+safe path P = Pi[l, r], there is a Pi[li, ri] ∈ T , where li ≤ l ≤ r ≤ ri.
+We will use the original k solution paths {Pi} as our initial trimming core; the complete algorithm is
+given in Algorithm 2. See Fig. 3 in the appendix for an illustration of the algorithm’s initial steps. The
+algorithm first checks if any of the paths in T are safe; if so, these are reported as maximal safe. For those
+paths that were unsafe, it then considers trimming one vertex from the left and one vertex from the right
+to create new subpaths. Of these subpaths, some may be contained in a safe path in T ; these subpaths
+can be ignored as they are not maximal safe. The algorithm recurses on those subpaths whose safety status
+cannot be determined (lines 6–10). In this way, the algorithm maintains the invariant that no paths in T are
+properly contained in a safe path; thus paths reported in line 4 must be maximal safe.
+6
+
+Algorithm 2: An algorithm to compute all maximal safe paths that can be trimmed from a
+trimming core set T .
+Input: An ILP model M and a trimming core set T
+Output: All maximal safe paths for M that are trimmed subpaths of T
+1 Procedure AllMaxSafe-TopDown(M, T )
+2
+S = GetSafe(M, T ) for Pi[li, ri] ∈ S do
+3
+output Pi[li, ri]
+4
+U = T \ S L = {Pi[li + 1, ri] : Pi[li, ri] ∈ U, (ri = |Pi| or Pi[li + 1, ri + 1] ∈ U)}
+R = {Pi[li, ri − 1] : Pi[li, ri] ∈ U, (li = 1 or Pi[li − 1, ri − 1] ∈ U)} P = L ∪ R if P ̸= ∅ then
+5
+AllMaxSafe-TopDown(M, P)
+3
+Experiments
+To test the performance of our methods, we computed safe paths using different safety approaches and re-
+ported the quality and running time performances as described below. Additional details on the experimental
+setup are given in the appendix.
+Implementation details – SafeMFD. We implemented the previously described algorithms to compute
+all maximal safe paths for minimum flow decompositions in Python. The implementation, SafeMFD, uses
+the package NetworkX [20] for graph processing and the package gurobipy [19] to model and solve the ILPs
+and it is openly available5. Our fastest variant (see Table 2 in the appendix for a comparison of running
+times) implements Algorithm 2 using the group testing in Algorithm 1. We used this variant to compare
+against other safety approaches. All tested variants of SafeMFD implement the following two optimizations:
+1.
+Before processing an input flow graph we contract it using Y-to-V contraction [51], which is known [30]
+to maintain (M)FD solution paths. Moreover, since edges in the contracted graph correspond to extended
+unitigs [35,24,29], source-to-sink edges are further removed from the contracted graph and reported as
+safe. As such, our algorithms compute all maximal safe paths for funnels [17,26] without using the ILP.
+2.
+Before testing the safety of a path we check if its excess-flow [26] is positive. If this is the case, the
+path is removed from the corresponding test. Having positive excess flow implies safety for all flow
+decomposition and thus also safety for minimum flow decompositions.
+Safety approaches tested. We compare the following state-of-the-art safety approaches:
+EUnitigs:
+Maximal paths made up of a prefix of nodes with in-degree one followed by nodes with out-
+degree one; also called extended unitigs [51,35,24,29]. We use the C++ implementation provided by Khan
+et al. [26] (which computes only the extended unitigs contained in FD paths).
+SafeFlow:
+Maximal safe paths for all flow decompositions [26]. We use the C++ implementation provided
+by Khan et al. [26].
+SafeMFD:
+Maximal safe paths for all minimum flow decompositions, as proposed in this work. Every
+flow graph processed is given a time budget of 2 minutes. If a flow graph consumes its time budget, the
+solution of SafeFlow is output instead.
+SafeEPC:
+Maximal safe paths for all constrained path covers of edges. Previous authors [8,26] have con-
+sidered safe path covers of the nodes, but for a more fair comparison, we instead use path covers of edges.
+To this end, we transform the input graphs by splitting every edge by adding a node in the middle and
+run the C++ implementation provided by the authors of [8]. Since flow decompositions are path covers
+of edges, safe paths for all edge path covers are subpaths of safe paths for MFD. However, we restrict
+the path covers to those of minimum size and minimum size plus one, as recommended by the authors
+of [8] to obtain good coverage results while maintaining high precision.
+5 https://github.com/algbio/mfd-safety
+7
+
+All safety approaches require a post processing step for removing duplicates, prefixes and suffixes. We
+use the C++ implementation provided by [26] for this purpose.
+Datasets. We use two datasets of flow graphs inspired by RNA transcript assembly. The datasets were
+created by simulating abundances on a set of transcripts and then perfectly superposing them into a splice
+graphs that are guaranteed to respect flow conservation. As such, the ground truth corresponds to a flow
+decomposition (not necessarily minimum). To avoid a skewed picture of our results we filtered out trivial
+instances with a unique flow decomposition (or funnels, see [17,26]) from the two datasets.6
+Catfish:
+Created by [48], it includes 100 simulated human, mouse and zebrafish transcriptomes using Flux-
+Simulator [18] as well as 1,000 experiments from the Sequence Read Archive simulating abundances using
+Salmon [40]. We took one experiment per dataset, which corresponds to 27,696 non-trivial flow graphs.
+RefSim:
+Created by [8] from the Ensembl [57] annotated transcripts of GRCh.104 homo sapiens reference
+genome, and later augmented by Khan et al. [26] with simulated abundances using the RNASeqRead-
+Simulator [32]. This dataset has 10,323 non-trivial graphs.
+Quality metrics. We use the same quality metrics employed by previous multi-assembly safety approaches [8,26].
+We provide a high-level description of them for completeness.
+Weighted precision of reported paths:
+As opposed to normal precision, the weighted version considers
+the length of the reported subpaths. It is computed as the total length of the correctly reported subpaths
+divided by the total length of all reported subpaths. A reported subpath is considered correct if and only
+if it is a subpath of some path in the ground truth (exact alignment of exons/nodes).
+Maximum coverage of a ground truth path P:
+The longest segment of P covered by some reported
+subpath (exact alignment of exons/nodes), divided by |P|.
+We compute the weighted precision of a graph as the average weighted precision over all reported
+paths in the graph, and the maximum coverage of a graph as the average maximum coverage over all
+ground truth paths in the graph.
+F-Score of a graph:
+Harmonic mean between weighted precision and maximum coverage of a graph,
+which assigns a global score to the corresponding approach on the graph.
+These metrics are computed per flow graph and reported as an average. In the case of the Catfish dataset
+the metrics are computed in terms of exons (nodes), since genomic coordinates of exons are missing, whereas
+in the case of the RefSim dataset the metrics are computed in terms of genomic positions, as this information
+is present in the input.
+4
+Results and Discussion
+In the Catfish dataset, EUnitigs and SafeFlow ran in less than a second, while SafeEPC took approximately
+30 seconds to compute. On the other hand, solving a harder problem, SafeMFD took approximately 1.5
+hours to compute in the rest of the dataset, timing out in only 54 graphs (we use a cutoff of 2 minutes), i.e.,
+only 0.2% of the entire dataset. This equates to only 0.2 seconds on average per solved graph, underlying
+the scalability of our approach.
+Table 1 shows that SafeMFD, on average, covers close to 90% of the ground truth paths, while maintaining
+a high precision (99%). This corresponds to an increase of approximately 25% in coverage against its closest
+competitor SafeFlow. SafeMFD also dominates in the combined metric of F-Score, being the only safe
+approach with F-Score over 90%. Figure 4 in the appendix shows the metrics on graphs grouped by number
+t of ground truth paths, indicating the dominance in coverage and F-Score of SafeMFD across all values of
+t, and indicating that the decrease in precision appears for large values of t (t ≥ 12).
+6 The exact datasets used in our experiments can be found at https://zenodo.org/record/7182096.
+8
+
+Table 1: Summary of quality metrics for both datasets. For Catfish, the metrics are computed in terms of
+nodes/exons and for RefSim in terms of genomic positions; t is the number of ground truth paths.
+Dataset
+Graphs
+Algorithm Max. Coverage Wt. Precision F-Score
+Catfish
+All
+(100%)
+EUnitigs
+0.60
+1.00
+0.74
+SafeEPC
+0.60
+0.99
+0.74
+SafeFlow
+0.71
+1.00
+0.82
+SafeMFD
+0.88
+0.99
+0.93
+RefSim
+t ≤ 10
+(68%)
+EUnitigs
+0.72
+1.00
+0.83
+SafeEPC
+0.73
+1.00
+0.84
+SafeFlow
+0.84
+1.00
+0.91
+SafeMFD
+0.97
+0.99
+0.98
+t ≤ 15
+(84%)
+EUnitigs
+0.70
+1.00
+0.82
+SafeEPC
+0.71
+1.00
+0.83
+SafeFlow
+0.83
+1.00
+0.90
+SafeMFD
+0.96
+0.98
+0.97
+All
+(100%)
+EUnitigs
+0.68
+1.00
+0.80
+SafeEPC
+0.69
+0.99
+0.81
+SafeFlow
+0.81
+1.00
+0.89
+SafeMFD
+0.93
+0.91
+0.90
+In the harder RefSim dataset, EUnitigs and SafeFlow also ran in less than a second, while SafeEPC
+took approximately 2 minutes. In this case, SafeMFD ran out of time in 1,562 graphs (15% of the entire
+dataset); however, recall that in these experiments we allow a time budget of only 2 minutes. In the rest of
+the dataset, it took approximately 7.5 hours in total, corresponding to only 3 seconds on average per graph,
+again underlying that our method, even though it solves many NP-hard problems for each input graph,
+overall scales sufficiently well.
+Table 1 shows that again SafeMFD dominates in coverage, being the only approach obtaining coverage
+over 90%, with is a 15% improvement over SafeFlow. This time its precision drops to close to 90%, and
+obtaining an F-Score of 90%, very similar to its closest competitor, SafeFlow. However, recall that coverage
+is computed only from correctly aligned paths, thus the drop in precision comes only from safe paths not
+counting in the coverage metric. If we restrict the metrics to graphs with at most 15 ground truth paths,
+which is still a significant proportion (84%) of the entire dataset, then SafeMFD has a very high precision
+(98%) while improving coverage by 15% with respect to SafeFlow. Thus, the drop in precision occurs in
+graphs with a large number of ground truth paths, which can also be corroborated by Figure 5 in the
+appendix.
+These drops in precision (both in RefSim and Catfish) for large t can be explained by the fact that a
+larger number of ground truth paths produces more complex splice graphs and introduces more artificial
+solutions of potentially smaller size. As such, the larger t, the less likely that the ground truth is a minimum
+flow decomposition of the graph, and thus the more likely that SafeMFD reports incorrect solutions. This
+motivates future work on safety not only on minimum flow decompositions but also in flow decompositions
+of at most a certain size, analogously to how it is done for SafeEPC. This is still easily achievable with
+our framework by just changing the ILP blackbox, and keeping everything else unchanged (e.g., the inner
+and outer algorithms). Namely, instead of formulating the ILP model M(V, C) to admit solutions of exactly
+optimal k paths, it can be changed to allow solutions of at most some k′ paths, with k′ greater than the
+optimal k. If k′ is also greater than the number of ground truth paths in these complex graphs, then safe
+paths are fully correct, meaning that we overall increase precision.
+9
+
+5
+Conclusion
+RNA assembly is a difficult problem in practice, with even the top tools reporting low precision values. While
+there are still many issues that can introduce uncertainty in practice, we can now provide a major source
+of additional information during the process: which RNA fragments must be included in any parsimonious
+explanation of the data? Though others have considered RNA assembly in the safety framework [58,26], we
+are the first to show that safety can be practically used even when we look for optimal (i.e., minimum) size
+solutions. Our experimental results show that safe paths for MFD clearly outperform other safe approaches
+for the Catfish dataset, commonly used in this field. On a significant proportion of the second dataset, safe
+paths for MFD still significantly outperforms other safe methods.
+More generally, this is the first work to show that the safety framework can be practically applied to
+NP-hard problems, where the inner algorithm is an efficient test of safety of a group of paths, and the outer
+algorithm guides the applications of this test. Because our method was very successful on our test data set,
+there is strong motivation to try the approach to on other NP-hard graph problems whose solutions are
+sets of paths. For example, we could study other variations on MFD, such as finding flow decompositions
+minimizing the longest path (NP-hard when flow values are integer [4,43]). The approach given in this paper
+can also be directly extended to find decompositions into both cycles and paths [16], though not trails and
+walks, because they repeat edges. We could also formulate a safety test for classic NP-hard graph problems
+like Hamiltonian path.
+Acknowledgements This work was partially funded by the European Research Council (ERC) under
+the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 851093,
+SAFEBIO), partially by the Academy of Finland (grants No. 322595, 352821, 346968), and partially by
+the US National Science Foundation (NSF) (grants No. 1759522, 1920954).
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+Leibniz-Zentrum f¨ur Informatik, 2021.
+56. Lucia Williams, Alexandru I. Tomescu, and Brendan Mumey. Flow decomposition with subpath constraints.
+IEEE/ACM Transactions on Computational Biology and Bioinformatics, 2022. Accepted.
+57. Andrew D Yates, Premanand Achuthan, Wasiu Akanni, James Allen, Jamie Allen, Jorge Alvarez-Jarreta, M Rid-
+wan Amode, Irina M Armean, Andrey G Azov, Ruth Bennett, et al. Ensembl 2020. Nucleic acids research,
+48(D1):D682–D688, 2020.
+58. Hongyu Zheng, Cong Ma, and Carl Kingsford. Deriving ranges of optimal estimated transcript expression due
+to nonidentifiability. Journal of Computational Biology, 29(2):121–139, 2022. Presented at RECOMB 2021.
+12
+
+A
+Additional Figures
+P1
+P2
+P3
+P4
+P1
+P2
+P3
+P4
+a) Group Testing - Initial Iteration
+c) Group Testing -  Second Iteration
+b) Identifying Safe Paths
+P2
+P2
+P4
+P4
+(a) First group test
+P1
+P2
+P3
+P4
+P1
+P2
+P3
+P4
+a) Group Testing - Initial Iteration
+c) Group Testing -  Second Iteration
+b) Identifying Safe Paths
+P2
+P2
+P4
+P4
+(b) Result: {P1, P3} are safe, {P2, P4}
+are unsafe
+P1
+P2
+P3
+P4
+P1
+P2
+P3
+P4
+a) Group Testing - Initial Iteration
+c) Group Testing -  Second Iteration
+b) Identifying Safe Paths
+P2
+P2
+P4
+P4
+(c) Second group test
+Fig. 3: Illustration of the initial group tests performed by Algorithm 2. Fig. 3(a) shows the first group test
+(using Algorithm 1) on MFD solution paths {P1, P2, P3, P4}; suppose {P1, P3} were safe (Fig. 3(b)); these
+are then reported as maximal safe. In this case we trim {P2, P4} on both the left and right and make the
+next group test shown in Fig. 3(c).
+13
+
+(a) Weighted Precision
+(b) Maximum Coverage
+(c) F-Score
+Fig. 4: Quality metrics on graphs distributed by number of paths in the ground truth for the Catfish dataset.
+The metrics are computed in terms of exons/nodes.
+(a) Weighted Precision
+(b) Maximum Coverage
+(c) F-Score
+Fig. 5: Quality metrics on graphs distributed by number of paths in the ground truth for the RefSim dataset.
+The metrics are computed in terms of genomic positions.
+B
+Additional Algorithms and Experimental Results
+B.1
+The bottom-up algorithm
+Algorithm 3, detailed below, uses a bottom-up group-testing strategy to find all maximal safe paths.
+Definition 1. We say a set of subpaths E = {Pi[li, ri]} is an extending core provided all paths in E are safe
+and for any unreported maximal safe path P = Pi[l, r], there is a Pi[li, ri] ∈ E, where l ≤ li ≤ ri ≤ r.
+Note that maximal FD-safe subpaths provide an extending core (as well just the set of all single-edge
+subpaths in each path). Algorithm 3 provides an algorithm to find all maximal safe paths based on group
+testing, starting from an extending core. The idea is to try both left-extending (by one) and right-extending
+(by one) each subpath in the core; if neither of these extensions are safe, then we know that that core subpath
+must be maximal safe. Testing all extensions can done quickly using Algorithm 1. We then recurse on a new
+core set consisting of those extensions that were found to be safe.
+14
+
+1.0
+0.8
+0.6
+0.4
+0.2
+EUnitigs
+SafeEPC
+SafeFlow
+SafeMFD
+0.0
+3
+4
+5
+6
+7
+8
+9
+10 11 12
+13
+14
+15
+# Ground truth paths1.0
+0.8
+0.6
+0.4
+0.2
+EUnitigs
+SafeEPC
+SafeFlow
+0.0
+SafeMFD
+3
+4
+5
+6
+7
+8
+9
+1011.12
+13
+14
+15
+# Ground truth paths1.0
+0.8
+0.6
+0.4
+0.2
+EUnitigs
+SafeEPC
+SafeFlow
+0.0
+SafeMFD
+3
+4
+5
+6
+7
+8
+9
+1011.12
+13
+14
+15
+# Ground truth paths1.0
+0.8
+0.6
+0.4
+0.2
+EUnitigs
+SafeEPC
+SafeFlow
+SafeMFD
+0.0
+3
+4
+5
+6
+7
+8
+9
+10 11 12
+13
+14
+15
+# Ground truth paths1.0
+0.8
+0.6
+0.4
+0.2
+EUnitigs
+SafeEPC
+SafeFlow
+0.0
+SafeMFD
+3
+4
+5
+6
+7
+8
+9
+10.11.12
+13
+14
+15
+# Ground truth paths1.0
+0.8
+0.6
+0.4
+0.2
+EUnitigs
+SafeEPC
+SafeFlow
+0.0
+SafeMFD
+3
+4
+5
+6
+7
+8
+9
+1011.12
+13
+14
+15
+# Ground truth pathsAlgorithm 3: An algorithm to output all maximal safe subpaths that can be extended from an
+extending core set E.
+Input: An ILP model M and an extending core set E
+Output: All maximal safe paths for M that extend some path from E
+1 Procedure AllMaxSafe-BottomUp(M, E)
+2
+L = {Pi[li − 1, ri] : Pi[li, ri] ∈ E, li > 1} R = {Pi[li, ri + 1] : Pi[li, ri] ∈ E, ri < |Pi|} P = L ∪ R S
+= GetSafe(M, P) for Pi[li, ri] ∈ E do
+3
+if Pi[li − 1, ri] /∈ S and Pi[li, ri + 1] /∈ S then
+4
+output Pi[li, ri]
+5
+if S ̸= ∅ then
+6
+AllMaxSafe-BottomUp(M, S)
+B.2
+The two-pointer algorithm
+As we observed in Section 2.2, we can test whether a single path P is safe using one ILP call. We will
+assume that this test is encapsulated as a procedure IsSafe(M, P). Once we can test whether a single path
+is safe for M(V, C), we can adopt a standard approach to compute all maximal safe paths. Namely, we start
+by computing one solution of M(V, C), P1, . . . , Pk and then compute maximal safe paths by a two-pointer
+technique that for each path Pi, finds all maximal safe paths by just a linear number of calls to the procedure
+IsSafe [26].
+This works as follows. We use two pointers, a left pointer L, and a right pointer R. Initially, L points to
+the first node of path Pi and R to the second node. As long as the subpath of Pi between L and R is safe,
+we move the right pointer to the next node on Pi. When this subpath is not safe, we output the subpath
+between L and the previous location of R as a maximal safe path, and we start moving the left pointer to
+the next node on Pi, until the subpath between L and R is safe. We stop the procedure once we reach the
+end of Pi. We summarize this procedure as Algorithm 4; see also Figure 6 for an example.
+ 
+1
+8
+3
+2
+5
+6
+ 4
+7
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+xsai + xabi + xbci + xcdi + xdfi ≤ 4
+xsai + xabi + xbci + xcdi ≤ 3
+xabi + xbci + xcdi + xdfi ≤ 3
+R
+L
+R
+∀i ∈ {1,…, k}
+∀i ∈ {1,…, k}
+∀i ∈ {1,…, k}
+L
+L
+R
+(a) Current iteration
+ 
+1
+8
+3
+2
+5
+6
+ 4
+7
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+xsai + xabi + xbci + xcdi + xdfi ≤ 4
+xsai + xabi + xbci + xcdi ≤ 3
+xabi + xbci + xcdi + xdfi ≤ 3
+R
+L
+R
+∀i ∈ {1,…, k}
+∀i ∈ {1,…, k}
+∀i ∈ {1,…, k}
+L
+L
+R
+(b) Right pointer movement
+ 
+1
+8
+3
+2
+5
+6
+ 4
+7
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+xsai + xabi + xbci + xcdi + xdfi ≤ 4
+xsai + xabi + xbci + xcdi ≤ 3
+xabi + xbci + xcdi + xdfi ≤ 3
+R
+L
+R
+∀i ∈ {1,…, k}
+∀i ∈ {1,…, k}
+∀i ∈ {1,…, k}
+L
+L
+R
+(c) Left pointer movement
+Fig. 6: Illustration of the two-pointer algorithm applied on a flow decomposition path Pi (in orange). In each
+sub-figure, the subpath P (dashed green) between the nodes pointed by the left pointer L and the right
+pointer R is tested for safety, by adding constraints S(P). In (a), IsSafe(M, P) returns True, and the right
+pointer advances on Pi. In (b), IsSafe(M, P) returns False, and the previous subpath from (a) is output as
+a maximal safe path. In (c), the left pointer has advanced, and the new path P is tested for safety.
+15
+
+Algorithm 4: The two-pointer algorithm applied to compute all maximal subpaths of a given
+solution path Pi
+Input: An ILP model M and one of its k solution paths, Pi = (v1, . . . , vt), t ≥ 2
+Output: All maximal safe subpaths of Pi for M
+1 Procedure AllMaxSafe-TwoPointer(M, Pi)
+2
+L ← 1, R ← 2 while True do
+3
+while IsSafe(M, Pi[L, R]) and R ≤ t do
+4
+R ← R + 1
+5
+output Pi[L, R − 1] if R > t then return;
+6
+while not IsSafe(M, Pi[L, R]) do
+7
+L ← L + 1
+Dataset (# Graphs)
+Variant
+Time (hh:mm:ss) # ILP calls
+Catfish
+(27,613)
+TopDown
+01:13:27
+124,676
+BottomUp
+03:22:13
+212,774
+TwoPointer
+04:21:44
+226,365
+TwoPointerBin
+03:31:57
+216,540
+RefSim
+(5,808)
+TopDown
+04:38:41
+55,450
+BottomUp
+11:55:20
+76,837
+TwoPointer
+13:48:00
+127,352
+TwoPointerBin
+11:34:02
+119,218
+Table 2: Running times and number of ILP calls in four different variants of SafeMFD.
+B.3
+Running time experiments among different variants proposed
+We conducted the experiments on an isolated Linux server with AMD Ryzen Threadripper PRO 3975WX
+CPU with 32 cores (64 virtual) and 504GB of RAM. Time and peak memory usage of each program were
+measured with the GNU time command. SafeMFD was allowed to run Gurobi with 12 threads. All C++
+implementations were compiled with optimization level 3 (-O3 flag). Running time and peak memory is
+computed and reported per dataset.
+SafeMFD includes the following four variants computing maximal safe paths:
+TopDown : Implements Algorithm 2 using the group testing in Algorithm 1.
+BottomUp : Implements Algorithm 3 (Appendix B.1) using the group testing in Algorithm 1.
+TwoPointer : Implements Algorithm 4 (Appendix B.2), the traditional two-pointer algorithm [26].
+TwoPointerBin : Same as previous variant, but it additionally replaces the linear scan employed to extend
+and reduce the currently processed safe path by a binary search7.
+To compare between our four different variants we first run them all on every dataset, and then filter
+out those graphs that ran out of time in some variant. This way we ensure that no variant consumes its
+time budget and thus our running time measurements are not skewed by the unsuccessful inputs’ timeouts.
+Applying this filter we removed 83 graphs from the Catfish dataset (0.3%) and 4,515 graphs from the RefSim
+dataset (43.74%).
+Table 2 shows the running times and number of ILP calls of the different variants on both datasets.
+TopDown clearly outperforms the rest, being at least twice as fast, and performing (roughly) half many ILP
+calls. While BottomUp is analogous to TopDown, the superiority of the latter can be explained by the length
+maximal safe paths. Since maximal safe paths are long it is faster to obtain them by reducing unsafe paths
+7 The binary search is only applied if the search space is larger than a constant threshold set experimentally.
+16
+
+(TopDown) than by extending safe paths (BottomUp and both TwoPointer variants). On the other hand,
+TwoPointer is the slowest variant and BottomUp and TwoPointerBin obtain similar improvements (over
+TwoPointer) by following different strategies. While BottomUp reduces the number of ILP calls more than
+TwoPointerBin (better appreciated in the RefSim dataset), the ILP calls of BottomUp take longer (since
+BottomUp tests several paths at the same time and TwoPointerBin only one), and thus the total running
+times of both is similar. This motivates future work on combining both approaches, while processing the
+paths starting from unsafe (as in TopDown) for better performance.
+C
+Hardness of Testing MFD Safety
+In this section we give a Turing-reduction from the UNIQUE 3SAT problem (U3SAT) to the problem of
+determining if a given path P in a flow network G is safe for minimum flow decomposition (call this problem
+MFD-SAFETY ). A 3SAT instance belongs to U3SAT if and only if it has exactly one satisfying assignment.
+U3SAT has been shown to be NP-hard under randomized reductions [52], but it is open as to whether it is
+NP-hard in general.
+The reduction leverages the construction in [22] that reduces 3SAT to minimum flow decomposition. We
+first briefly review this construction: A variable gadget (see Fig. 4 in [22]) is created for each 3SAT variable x
+and a clause gadget (see Fig. 5 in [22]) is created for each 3SAT clause. Positive literals in each clause receive
+flow from the left side of the corresponding variable gadget, whereas negative literals receive flow from the
+right side. Theorem VI.1 in [22] establishes that a 3SAT instance is satisfiable if and only if the constructed
+flow network has a minimum flow decomposition of a certain size. Any flow decomposition achieving this
+size must have a specific structure; in particular, there must be a flow path of weight 4 that either travels
+up the left side of the gadget (setting x to TRUE), or the right side (setting x to FALSE).
+⋯
+4
+4
+4
+4
+4
+4
+⋯
+4
+4
+4
+4
+t(x)
+s(x)
+Fig. 7: The variable gadget from [22], showing only the weight 4 edges (other edges have weights from
+{1, 2}). A key property established in [22] is that if the 3SAT instance is satisfiable then in a minimum flow
+decomposition, a weight 4 flow path must travel up either the left side of the gadget (as shown), or the
+right side. A left flow path indicates the variable should be set to TRUE, while right indicates FALSE. We
+leverage this construction to reduce U3SAT to MFD-SAFETY.
+Theorem 1. There is a polynomial time Turing-reduction from U3SAT to MFD-SAFETY.
+17
+
+Proof. To obtain the desired Turing-reduction algorithm, instead of checking the size of the MFD, we will
+instead sequentially check the MFD-SAFETY of the aforementioned side paths traveling up the left and
+right sides of each variable gadget. Provided each variable gadget has exactly one safe side path we can then
+check the corresponding truth assignment to see if each clause is satisfied. If yes, we accept the instance as
+belonging to U3SAT, otherwise we reject.
+Suppose the instance does belong to U3SAT. In this case there is a satisfying assignment so the MFD
+must have the structure as described above. Furthermore, since there is exactly one satisfying assignment,
+exactly one side path of each variable gadget must be safe and so our algorithm finds it and then verifies that
+the truth assignment satisfies each clause, thus accepting the instance. On the other hand, if the instance does
+not belong to U3SAT it could either be unsatisfiable or have multiple satisfying assignments. If unsatisfiable,
+no matter whether the safety checks pass, the corresponding assignment will not satisfy all clauses, so the
+instance will be rejected. If there are multiple solutions, then any variable that can be both TRUE and
+FALSE will not have a safe side path in the MFD. This means the safety check will fail and the instance
+will again be rejected.
+⊓⊔
+18
+
diff --git a/3NFQT4oBgHgl3EQfGjVY/content/tmp_files/load_file.txt b/3NFQT4oBgHgl3EQfGjVY/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf,len=885
+page_content='A Safety Framework for Flow Decomposition Problems via  Integer Linear Programming  Fernando H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Dias1,⋆[0000−0002−6398−919X], Manuel C´aceres1,⋆[0000−0003−0235−6951], Lucia  Williams2,⋆[0000−0003−3785−0247], Brendan Mumey2,⋆⋆[0000−0001−7151−2124], and  Alexandru I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Tomescu1,⋆⋆[0000−0002−5747−8350]  1 Department of Computer Science, University of Helsinki, Finland  {fernando.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='cunhadias,manuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='caceres,alexandru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='tomescu}@helsinki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='fi  2 School of Computing, Montana State University, Bozeman, MT, USA  {lucia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='williams,brendan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='mumey}@montana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='edu  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Many important problems in Bioinformatics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=', assembly or multi-assembly) admit mul-  tiple solutions, while the final objective is to report only one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' A common approach to deal with this  uncertainty is finding safe partial solutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=', contigs) which are common to all solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Previous  research on safety has focused on polynomially-time solvable problems, whereas many successful and  natural models are NP-hard to solve, leaving a lack of “safety tools” for such problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We propose  the first method for computing all safe solutions for an NP-hard problem, minimum flow decomposi-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We obtain our results by developing a “safety test” for paths based on a general Integer Linear  Programming (ILP) formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Moreover, we provide implementations with practical optimizations  aimed to reduce the total ILP time, the most efficient of these being based on a recursive group-testing  procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Results: Experimental results on the transcriptome datasets of Shao and Kingsford (TCBB, 2017)  show that all safe paths for minimum flow decompositions correctly recover up to 90% of the full RNA  transcripts, which is at least 25% more than previously known safe paths, such as (C´aceres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' TCBB,  2021), (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=', RECOMB 2021), (Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=', RECOMB 2022, ESA 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Moreover, despite the  NP-hardness of the problem, we can report all safe paths for 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='8% of the over 27,000 non-trivial graphs  of this dataset in only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='5 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Our results suggest that, on perfect data, there is less ambiguity than  thought in the notoriously hard RNA assembly problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Availability: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='com/algbio/mfd-safety  Contact: alexandru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='tomescu@helsinki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='fi  Keywords: RNA assembly · Network flow · Flow decomposition · Integer linear programming · Safety  ⋆ Shared first-author contribution  ⋆⋆ Shared last-author contribution  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='13245v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='DS]  30 Jan 2023  1  Introduction  In real-world scenarios where an unknown object needs to be discovered from the input data, we would like  to formulate a computational problem loosely enough so that the unknown object is indeed a solution to  the problem, but also tightly enough so that the problem does not admit many other solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' However,  this goal is difficult in practice, and indeed, various commonly used problem formulations in Bioinformatics  still admit many solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' While a naive approach is to just exhaustively enumerate all these solutions, a  more practical approach is to report only those sub-solutions (or partial solutions) that are common to all  solutions to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  In the graph theory community such sub-solutions have been called persistent [14,21], and in the Bioin-  formatics community reliable [54], or more recently, safe [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' The study of safe sub-solutions started in  Bioinformatics in the 1990’s [54,11,37] with those amino-acid pairs that are common to all optimal and  suboptimal alignments of two protein sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  In the genome assembly community, the notion of contig, namely a string that is guaranteed to appear in  any possible assembly of the reads, is at the core of most genome assemblers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' This approach originated in 1995  with the notion of unitigs [25] (non-branching paths in an assembly graph), which were progressively [42,6]  generalized to paths made up of a prefix of nodes with in-degree one followed by nodes with out-degree  one [35,24,29] (also called extended unitigs, or Y-to-V contigs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Later, [51] formalized all such types of contigs as those safe strings that appear in all solutions to a  genome assembly problem formulation, expressed as a certain type of walk in a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' [10,9] proposed more  efficient and unifying safety algorithms for several types of graph walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' [45] recently studied the safety of  contigs produced by state-of-the-art genome assemblers on real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Analogous studies were recently made also for multi-assembly problems, where several related genomic  sequences need to be assembled from a sample of mixed reads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' [8] studied safe paths that appear in all  constrained path covers of a directed acyclic graph (DAG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Zheng, Ma and Kingsford studied the more  practical setting of a network flow in a DAG by finding those paths that appear in any flow decomposition  of the given network flow, under a probabilistic framework [34], or a combinatorial framework [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='3 [27]  presented a simple characterization of safe paths appearing in any flow decomposition of a given acyclic  network flow, leading to a more efficient algorithm than the one of [58], and further optimized by [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Despite the significant progress in obtaining safe algorithms for a range of different appli-  cations, current safe algorithms are limited to problems where computing a solution itself is achievable in  polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' However, many natural problems are NP-hard, and safe algorithms for such problems are  fully missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Apart from the theoretical interest, usually such NP-hard problems correspond to restrictions  of easier (polynomially-computable) problems, and thus by definition, also have longer safe sub-solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  As such, current safety algorithms miss data that could be reported as correct, just because they do not  constrain the solution space enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' A major reason for this lack of progress is that if a problem is NP-hard,  then its safety version is likely to be hard too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' This phenomenon can be found both in classically studied NP-  hard problems — for example, computing nodes present in all maximum independent sets of an undirected  graph is NP-hard [21] — as well as in NP-hard problems studied for their application to Bioinformatics, as  we discuss further in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  We introduce our results by focusing on the flow decomposition problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' This is a classical model at the  core of multi-assembly software for RNA transcripts [33,31,5,50] and viral quasi-species genomes [3,2,44,12],  and also a standard problem with applications in other fields, such as networking [36,22,13,23] or transporta-  tion [39,38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' In its most basic optimization form, minimum flow decomposition (MFD), we are given a flow  in a graph, and we need to decompose it into a minimum number of paths with associated weights, such  that the superposition of these weighted paths gives the original flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' This is an NP-hard problem, even  when restricted to DAGs [53,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Various approaches have been proposed to tackle the problem, including  fixed-parameter tractable algorithms [30], approximation algorithms [36,7] and Integer Linear Programming  formulations [15,46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  3 The problem AND-Quant from [58] actually handles a more general version of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  1  In Bioinformatics applications, reads or contigs originating from a mixed sample of genomic sequences  with different abundances are aligned to a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' A graph model, such as a splice graph or a variation  graph, is built from these alignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Read abundances assigned to the nodes and edges of this graph  then correspond to a flow in case of perfect data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' If this is not the case, the abundance values can either  be minimally corrected to become a flow, or one can consider variations of the problem where e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=', the  superposition of the weighted paths is closest (or within a certain range) to the edge abundances [50,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Current safety algorithms for flow decompositions such as [58,27,26,28] compute paths appearing in all  possible flow decompositions (of any size), even though decompositions of minimum size are assumed to  better model the RNA assembly problem [30,48,55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Even dropping the minimality constraint, but adding  other simple constraints easily renders the problem NP-hard (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=', [56]), motivating further study of  practical safe algorithms for NP-hard problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Integer Linear Programming (ILP) is a general and flexible method that has been suc-  cessfully applied to solve NP-hard problems, including in Bioinformatics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' In this paper, we consider graph  problems whose solution consists of a set of paths (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=', not repeating nodes) that can be formulated in  ILP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We introduce a technique that, given an ILP formulation of such a graph problem, can enhance it  with additional variables and constraints in order to test the safety of a given set of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' An obvious first  application of this safety test is to use it with a single path in a straightforward avoid-and-test approach,  using a standard two-pointer technique that has been used previously to find safe paths for flow decomposi-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' However, we find that a top-down recursive approach that uses the group-testing capability halves the  number of computationally-intensive ILP calls, resulting in a 3x speedup over the straightforward approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Additionally, we prove that computing all the safe paths for MFDs is an intractable problem, confirming  the above intuitive claim that if a problem is hard, then also its safety version is hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We give this proof  in the appendix by showing that the NP-hardness reduction for MFD by [22] can be modified into a Turing  reduction from the UNIQUE 3SAT problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  On the dataset [47] containing splice graphs from human, zebrafish and mouse transcriptomes, safe  paths for MFDs (SafeMFD) correctly recover up to 90% of the full RNA transcripts while maintaining a  99% precision, outperforming, by a wide margin (25% increase), state-of-the-art safety approaches, such as  extended unitigs [35,24,29], safe paths for constrained path covers of the edges [8], and safe paths for all  flow decompositions [28,27,26,58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' On the harder dataset by [26], SafeMFD also dominates in a significant  proportion of splice graphs (built from t ≤ 15 RNA transcripts), recovering more than 95% of the full  transcripts while maintaining a 98% precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' For larger t, precision drastically drops (91% precision in the  entire dataset), suggesting that in more complex splice graphs smaller solutions are introduced as an artifact  of the combinatorial nature of the splice graph, and the minimality condition [30,48,55] is thus incorrect in  this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  2  Methods  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='1  Preliminaries  ILP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' In this paper we use ILP models as blackboxes, with as few assumptions as possible to further  underline the generality of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Let M(V, C) be an ILP model consisting of a set V of variables  and a set C of constraints on these variables, built from an input graph G = (V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We make only two  assumptions on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' First, that a solution to this model consists of a given number k ≥ 1 of paths P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' , Pk  in G (in this paper, paths do not repeat vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Second, we assume that the k paths are modeled via  binary edge variables xuvi, for all (u, v) ∈ E and for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' More specifically, for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' , k},  we require that the edges (u, v) ∈ E for which the corresponding variable xuvi equals 1 induce a path in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  For example, if G is a DAG, it is a standard fact (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=', [49]) that a path from a given s ∈ V to a given  2  t ∈ V (an s-t path) can be expressed with the following constraints:  �  (u,v)∈E  xuvi −  �  (v,u)∈E  xvui =  �  �  �  �  �  0,  if v ∈ V \\ {s, t},  1,  if v = t,  −1,  if v = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  (1)  If G is not a DAG, there are other types of constraints that can be added to the xuvi variables to ensure  that they induce a path;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' see, for example, the many formulations in [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We will assume that such constraints  are part of the set C of constraints of M(V, C), but their exact formulation is immaterial for our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' In  fact, one could even add additional constraints to C to further restrict the solution space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' For example, some  ILP models from [15,46] handle the case when the input also contains a set of paths (subpath constraints)  that must appear in at least one of the k solution paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Flow decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' In the flow decomposition problem we are given a flow network (V, E, f), where  G = (V, E) is a (directed) graph with unique source s ∈ V and unique sink t ∈ V , and f assigns a positive  integer flow value fuv to every edge (u, v) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Flow conservation must hold for every node different from s  and t, namely, the sum of the flow values entering the node must equal the sum of the flow values exiting the  node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' See Figure 1(a) for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We say that k s-t paths P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' , Pk, with associated positive integer  weights w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' , wk, are a flow decomposition (FD) if their superposition equals the flow f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Formally, for  every (u, v) ∈ E it must hold that  �  i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=',k} s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  (u,v)∈Pi  wi = fuv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  (2)  See Figures 1(b) and 1(c) for two examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' The number k of paths is also called the size of the flow  decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' In the minimum flow decomposition (MFD) problem, we need to find a flow decomposition  of minimum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='4 On DAGs, a flow decomposition into paths always exists [1], but in general graphs, cycles  may be necessary to decompose the flow (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' [16] for different possible formulations of the problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  For concreteness, we now describe the ILP models from [15] for finding a flow decomposition into k  weighted paths in a DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' They consist of (i) modeling the k paths via the xuvi variables (with constraints  (1)), (ii) adding path-weight variables w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' , wk, and (iii) requiring that these weighted paths form a flow  decomposition, via the following (non-linear) constraint:  �  i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=',k}  xuviwi = fuv,  ∀(u, v) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  (3)  This constraint can then be easily linearized by introducing additional variables and constraints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' [15]  for these technical details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' However, as mentioned above, the precise formulation of the ILP model M for a  problem is immaterial for our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Only the two assumptions on M made above matter for obtaining  our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Given a problem on a graph G whose solutions consist of k paths in G, we say that a path P is safe  if for any solution P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' , Pk to the problem, there exists some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' , k} such that P is a subpath of Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  If the problem is given as an ILP model M, we also say that P is safe for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We say that P is a maximal  safe path, if P is a safe path and there is no larger safe path containing P as subpath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' [27] characterized safe  paths for all FDs (not necessarily of minimum size) using the excess flow fP of a path P, defined as the flow  on the first edge of P minus the flow on the edges out-going from the internal nodes of P, and different from  the edges of P (see Figure 1(d) for an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' It holds that P is safe for all FDs if and only if fP > 0 [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  4 In this paper we work only with integer flow values and weights for simplicity and since this is the most studied  version of the problem, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=', [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' However, the problem can also be defined with fractional weights [41], and  in this case the two problems can have different minima on the same input [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' This fractional case can also be  modeled by ILP [15], and all the results from our paper also immediately carry over to this variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  3  8  9  9  3  5  7  s  t  b  a  d  e  7  c  10  10  f  s  t  b  a  d  e  c  f  s  t  b  a  d  e  c  f  5  3  2  7  3  4  1  9  3  3  Figure 1  Figure 2  (a) A flow network with source s and sink t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  8  9  9  3  5  7  s  t  b  a  d  e  7  c  10  10  f  s  t  b  a  d  e  c  f  s  t  b  a  d  e  c  f  5  3  2  7  3  4  1  9  3  3  Figure 1  Figure 2  8  9  9  3  5  7  s  t  b  a  d  e  7  c  10  10  f  3  3  s  t  b  a  d  e  c  f  xsai = 1  xabi = 1  xbci = 1  xcdi = 1  xdfi = 1  Figure 3  xadi = 0  xeti = 0  xdei = 0  xfti = 1  xsbi = 0  xbdi = 0  (b) An MFD into 4 paths of weights 5,3,7,2, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' The green dashed path is a subpath of the  orange path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  8  9  9  3  5  7  s  t  b  a  d  e  7  c  10  10  f  s  t  b  a  d  e  c  f  s  t  b  a  d  e  c  f  5  3  2  7  3  4  1  9  3  3  Figure 1  Figure 2  8  9  9  3  5  7  s  t  b  a  d  e  7  c  10  10  f  3  3  s  t  b  a  d  e  c  f  xsai = 1  xabi = 1  xbci = 1  xcdi = 1  xdfi = 1  Figure 3  xadi = 0  xeti = 0  xdei = 0  xfti = 1  xsbi = 0  xbdi = 0  (c) An MFD into 4 paths of weights 3,4,1,9, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' The green dashed path is a subpath of the pink  path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  8  9  9  3  5  7  s  t  b  a  d  e  7  c  10  10  f  s  t  b  a  d  e  c  f  s  t  b  a  d  e  c  f  5  3  2  7  3  4  1  9  3  3  Figure 1  Figure 2  8  9  9  3  5  7  s  t  b  a  d  e  7  c  10  10  f  3  3  s  t  b  a  d  e  7  c  10  f  3  xsai = 1  xabi = 1  xbci = 1  xcdi = 1  xdfi = 1  (d) The two subpaths (red and blue) of the green  dashed path that are maximal safe paths for all FDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' 1: Flow decompositions and safe paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' The flow network in (a) admits different MFDs, in (b) and in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  The path (s, a, b, c, d) (dashed green) is a maximal safe path for MFDs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=', it is a subpath of some path  of all MFDs and it cannot be extended without losing this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' However, the path (s, a, b, c, d) is not  safe for all FDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Indeed, its two subpaths (s, a, b) (dashed red in (d)) and (b, c, d) (dashed blue in (d)) are  maximal safe paths for all FDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' To see this, note that the excess flow of (s, a, b) is 3, while the excess flow of  (s, a, b, c) (and of (s, a, b, c, d)) is −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  4  The excess flow can be computed in time linear in the length of P (assuming we have pre-computed the flow  outgoing from every node), giving thus a linear-time verification of whether P is safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  A basic property of safe solutions is that any sub-solution of them is also safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Computing safe paths  for MFDs can thus potentially lead to joining several safe paths for FDs, obtaining longer paths from the  unknown sequences we are trying to assemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' See Figure 1 for an example of a maximal safe path for MFDs  and two maximal subpaths of it that are safe for FDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='2  Finding Maximal Safe Paths for MFD via ILP  We now present a method for finding all maximal safe paths for MFD via ILP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' The basic idea is to define an  inner “safety test” which can be repeatedly called as part of an outer algorithm over the entire instance to  find all maximal safe paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Because calls to the ILP solver are expensive, the guiding choice for our overall  approach is to minimize the number of ILP calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' This inspires us to test the safety of a group of paths as the  inner safety test, which we achieve by augmenting our ILP model so that it can give us information about  the safety of the paths in the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We use this to define a recursive algorithm to fully determine the safety  status of each path in a group of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We can then structure the safety test in either a top-down manner  (starting with long unsafe paths and shrinking them until they are safe) or a bottom-up manner (starting  with short safe paths and lengthening them until they become unsafe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Safety test (inner algorithm) Let M(V, C) be an ILP model as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' namely, its k  solution paths are modeled by binary variables xuvi for each (u, v) ∈ E and each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We assume  that M(V, C) is feasible (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=', the problem admits at least one solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We first show how to modify the  ILP model so that, for a given set of paths, it can tell us one of the following: (1) a set of paths that are not  safe (the remaining being of unknown status), or (2) that all paths are safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' The idea is to maximize the  number of paths that can be simultaneously avoided from the given set of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Let P be a set of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' For each path P ∈ P, we create an auxiliary binary variable γP that indicates:  γP ≡  �  1  if P was avoided in the solution,  0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  (4)  Since the model solutions are paths (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=', not repeating nodes), we can encode whether P appears in the  solution by whether all of the ℓ − 1 edges of P appear simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Using this fact, we add a new set of  constraints R(P) that include the γP indicator variables for each path P ∈ P:  R(P) := {xv1v2i + xv2v3i + · · · + xvℓ−1vℓi  ≤ ℓ − 1 − γP : ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' , k}, ∀P ∈ P}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  (5)  Next, as the objective function of the ILP model, we require that it should maximize the number of  avoided paths from P, i,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=', the sum of the γP variables:  max  �  P ∈P  γP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  (6)  All paths P such that γP = 1 are unsafe, since they were avoided in some minimum flow decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Conversely, if the objective value of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' (6) was 0, then γP = 0 for all paths in P, and it must be that all  paths in P are safe (if not, at least one path could be avoided and increase the objective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We encapsulate  this group testing ILP in a function GroupTest(M, P) that returns a set N ⊆ P with the properties that:  (1) if N = ∅, then all paths in P are safe, and (2) if N ̸= ∅, then all paths in N are unsafe (and |N| is  maximized).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  We employ GroupTest(M, P) to construct a recursive procedure GetSafe(M, P) that determines all safe  paths in P, as shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  5  8  9  9  3  5  7  s  t  b  a  d  e  7  c  10  10  f  s  t  b  a  d  e  c  f  s  t  b  a  d  e  c  f  5  3  2  7  3  4  1  9  3  3  Figure 1  Figure 2  8  9  9  3  5  7  s  t  b  a  d  e  7  c  10  10  f  3  3  s  t  b  a  d  e  c  f  xsai = 1  xabi = 1  xbci = 1  xcdi = 1  xdfi = 0  Figure 3  xadi = 0  xeti = 1  xdei = 1  xfti = 0  xsbi = 0  xbdi = 0  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' 2: Illustration of modeling a solution path and a tested path via binary edge variables and safety  verification constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' The ith solution path Pi is shown in orange, and a tested path P is shown in dashed  green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Constraint (5) includes xsai +xabi +xbci +xcdi +xdei ≤ 5−γP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' This simplifies to γP ≤ 0, thus forcing  γP = 0, which indicates P was not avoided in the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Algorithm 1: Testing a set of paths P for safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Input: A feasible ILP model M(V, C), and a set of paths P  Output: Those paths P ∈ P that are safe for M(V, C)  1 Procedure GetSafe(M, P)  2  N = GroupTest(M, P) if N = ∅ then  3  return P  4  else  5  return GetSafe(M, P \\ N)  We note that in the special case that |P| = 1, GetSafe(M, P) makes only a single call to the ILP via  GroupTest(M, P) to determine whether not the given path is safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' With this safety test for a single path, we  can easily adapt a standard two-pointer approach as the outer algorithm to find all maximal safe paths for  MFD by starting with some MFD solution P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' , Pk of M(V, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' This same procedure was used in [26] to  find all maximal safe paths for FD, using an excess flow check as the inner safety algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Find all maximal safe paths (outer algorithm) We give two algorithms for finding all maximal safe  paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Both algorithms use a similar approach, however the first uses a top-down approach starting from the  original full solution paths and reports all safe paths (these again must be maximal safe), and then trims all  the unsafe paths to find new maximal safe paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' The second is bottom-up in that it tries to extend known  safe subpaths until they cannot be further extended (and at this point must be maximal safe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We present  the first algorithm in detail and defer discussion of the second to the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  We say a set of subpaths T = {Pi[li, ri]} is a trimming core provided that for any unreported maximal  safe path P = Pi[l, r], there is a Pi[li, ri] ∈ T , where li ≤ l ≤ r ≤ ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  We will use the original k solution paths {Pi} as our initial trimming core;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' the complete algorithm is  given in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' 3 in the appendix for an illustration of the algorithm’s initial steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' The  algorithm first checks if any of the paths in T are safe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' if so, these are reported as maximal safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' For those  paths that were unsafe, it then considers trimming one vertex from the left and one vertex from the right  to create new subpaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Of these subpaths, some may be contained in a safe path in T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' these subpaths  can be ignored as they are not maximal safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' The algorithm recurses on those subpaths whose safety status  cannot be determined (lines 6–10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' In this way, the algorithm maintains the invariant that no paths in T are  properly contained in a safe path;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' thus paths reported in line 4 must be maximal safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  6  Algorithm 2: An algorithm to compute all maximal safe paths that can be trimmed from a  trimming core set T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Input: An ILP model M and a trimming core set T  Output: All maximal safe paths for M that are trimmed subpaths of T  1 Procedure AllMaxSafe-TopDown(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' T )  2  S = GetSafe(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' T ) for Pi[li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' ri] ∈ S do  3  output Pi[li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' ri]  4  U = T \\ S L = {Pi[li + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' ri] : Pi[li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' ri] ∈ U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' (ri = |Pi| or Pi[li + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' ri + 1] ∈ U)}  R = {Pi[li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' ri − 1] : Pi[li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' ri] ∈ U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' (li = 1 or Pi[li − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' ri − 1] ∈ U)} P = L ∪ R if P ̸= ∅ then  5  AllMaxSafe-TopDown(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' P)  3  Experiments  To test the performance of our methods,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' we computed safe paths using different safety approaches and re-  ported the quality and running time performances as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Additional details on the experimental  setup are given in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Implementation details – SafeMFD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We implemented the previously described algorithms to compute  all maximal safe paths for minimum flow decompositions in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' The implementation, SafeMFD, uses  the package NetworkX [20] for graph processing and the package gurobipy [19] to model and solve the ILPs  and it is openly available5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Our fastest variant (see Table 2 in the appendix for a comparison of running  times) implements Algorithm 2 using the group testing in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We used this variant to compare  against other safety approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' All tested variants of SafeMFD implement the following two optimizations:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Before processing an input flow graph we contract it using Y-to-V contraction [51], which is known [30]  to maintain (M)FD solution paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Moreover, since edges in the contracted graph correspond to extended  unitigs [35,24,29], source-to-sink edges are further removed from the contracted graph and reported as  safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' As such, our algorithms compute all maximal safe paths for funnels [17,26] without using the ILP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Before testing the safety of a path we check if its excess-flow [26] is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' If this is the case, the  path is removed from the corresponding test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Having positive excess flow implies safety for all flow  decomposition and thus also safety for minimum flow decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Safety approaches tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We compare the following state-of-the-art safety approaches:  EUnitigs:  Maximal paths made up of a prefix of nodes with in-degree one followed by nodes with out-  degree one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' also called extended unitigs [51,35,24,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We use the C++ implementation provided by Khan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' [26] (which computes only the extended unitigs contained in FD paths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  SafeFlow:  Maximal safe paths for all flow decompositions [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We use the C++ implementation provided  by Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  SafeMFD:  Maximal safe paths for all minimum flow decompositions, as proposed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Every  flow graph processed is given a time budget of 2 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' If a flow graph consumes its time budget, the  solution of SafeFlow is output instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  SafeEPC:  Maximal safe paths for all constrained path covers of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Previous authors [8,26] have con-  sidered safe path covers of the nodes, but for a more fair comparison, we instead use path covers of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  To this end, we transform the input graphs by splitting every edge by adding a node in the middle and  run the C++ implementation provided by the authors of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Since flow decompositions are path covers  of edges, safe paths for all edge path covers are subpaths of safe paths for MFD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' However, we restrict  the path covers to those of minimum size and minimum size plus one, as recommended by the authors  of [8] to obtain good coverage results while maintaining high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  5 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='com/algbio/mfd-safety  7  All safety approaches require a post processing step for removing duplicates, prefixes and suffixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We  use the C++ implementation provided by [26] for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We use two datasets of flow graphs inspired by RNA transcript assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' The datasets were  created by simulating abundances on a set of transcripts and then perfectly superposing them into a splice  graphs that are guaranteed to respect flow conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' As such, the ground truth corresponds to a flow  decomposition (not necessarily minimum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' To avoid a skewed picture of our results we filtered out trivial  instances with a unique flow decomposition (or funnels, see [17,26]) from the two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='6  Catfish:  Created by [48], it includes 100 simulated human, mouse and zebrafish transcriptomes using Flux-  Simulator [18] as well as 1,000 experiments from the Sequence Read Archive simulating abundances using  Salmon [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We took one experiment per dataset, which corresponds to 27,696 non-trivial flow graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  RefSim:  Created by [8] from the Ensembl [57] annotated transcripts of GRCh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='104 homo sapiens reference  genome, and later augmented by Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' [26] with simulated abundances using the RNASeqRead-  Simulator [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' This dataset has 10,323 non-trivial graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Quality metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We use the same quality metrics employed by previous multi-assembly safety approaches [8,26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  We provide a high-level description of them for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Weighted precision of reported paths:  As opposed to normal precision, the weighted version considers  the length of the reported subpaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' It is computed as the total length of the correctly reported subpaths  divided by the total length of all reported subpaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' A reported subpath is considered correct if and only  if it is a subpath of some path in the ground truth (exact alignment of exons/nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Maximum coverage of a ground truth path P:  The longest segment of P covered by some reported  subpath (exact alignment of exons/nodes), divided by |P|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  We compute the weighted precision of a graph as the average weighted precision over all reported  paths in the graph, and the maximum coverage of a graph as the average maximum coverage over all  ground truth paths in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  F-Score of a graph:  Harmonic mean between weighted precision and maximum coverage of a graph,  which assigns a global score to the corresponding approach on the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  These metrics are computed per flow graph and reported as an average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' In the case of the Catfish dataset  the metrics are computed in terms of exons (nodes), since genomic coordinates of exons are missing, whereas  in the case of the RefSim dataset the metrics are computed in terms of genomic positions, as this information  is present in the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  4  Results and Discussion  In the Catfish dataset, EUnitigs and SafeFlow ran in less than a second, while SafeEPC took approximately  30 seconds to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' On the other hand, solving a harder problem, SafeMFD took approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='5  hours to compute in the rest of the dataset, timing out in only 54 graphs (we use a cutoff of 2 minutes), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=',  only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='2% of the entire dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' This equates to only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='2 seconds on average per solved graph, underlying  the scalability of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Table 1 shows that SafeMFD, on average, covers close to 90% of the ground truth paths, while maintaining  a high precision (99%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' This corresponds to an increase of approximately 25% in coverage against its closest  competitor SafeFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' SafeMFD also dominates in the combined metric of F-Score, being the only safe  approach with F-Score over 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Figure 4 in the appendix shows the metrics on graphs grouped by number  t of ground truth paths, indicating the dominance in coverage and F-Score of SafeMFD across all values of  t, and indicating that the decrease in precision appears for large values of t (t ≥ 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  6 The exact datasets used in our experiments can be found at https://zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='org/record/7182096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  8  Table 1: Summary of quality metrics for both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' For Catfish, the metrics are computed in terms of  nodes/exons and for RefSim in terms of genomic positions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' t is the number of ground truth paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Dataset  Graphs  Algorithm Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Coverage Wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Precision F-Score  Catfish  All  (100%)  EUnitigs  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='60  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='74  SafeEPC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='74  SafeFlow  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='71  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='82  SafeMFD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='93  RefSim  t ≤ 10  (68%)  EUnitigs  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='72  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='83  SafeEPC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='73  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='84  SafeFlow  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='84  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='91  SafeMFD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='97  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='98  t ≤ 15  (84%)  EUnitigs  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='70  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='82  SafeEPC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='71  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='83  SafeFlow  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='83  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='90  SafeMFD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='97  All  (100%)  EUnitigs  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='68  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='80  SafeEPC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='69  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='81  SafeFlow  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='81  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='89  SafeMFD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='93  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='91  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='90  In the harder RefSim dataset, EUnitigs and SafeFlow also ran in less than a second, while SafeEPC  took approximately 2 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' In this case, SafeMFD ran out of time in 1,562 graphs (15% of the entire  dataset);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' however, recall that in these experiments we allow a time budget of only 2 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' In the rest of  the dataset, it took approximately 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='5 hours in total, corresponding to only 3 seconds on average per graph,  again underlying that our method, even though it solves many NP-hard problems for each input graph,  overall scales sufficiently well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Table 1 shows that again SafeMFD dominates in coverage, being the only approach obtaining coverage  over 90%, with is a 15% improvement over SafeFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' This time its precision drops to close to 90%, and  obtaining an F-Score of 90%, very similar to its closest competitor, SafeFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' However, recall that coverage  is computed only from correctly aligned paths, thus the drop in precision comes only from safe paths not  counting in the coverage metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' If we restrict the metrics to graphs with at most 15 ground truth paths,  which is still a significant proportion (84%) of the entire dataset, then SafeMFD has a very high precision  (98%) while improving coverage by 15% with respect to SafeFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Thus, the drop in precision occurs in  graphs with a large number of ground truth paths, which can also be corroborated by Figure 5 in the  appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  These drops in precision (both in RefSim and Catfish) for large t can be explained by the fact that a  larger number of ground truth paths produces more complex splice graphs and introduces more artificial  solutions of potentially smaller size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' As such, the larger t, the less likely that the ground truth is a minimum  flow decomposition of the graph, and thus the more likely that SafeMFD reports incorrect solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' This  motivates future work on safety not only on minimum flow decompositions but also in flow decompositions  of at most a certain size, analogously to how it is done for SafeEPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' This is still easily achievable with  our framework by just changing the ILP blackbox, and keeping everything else unchanged (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=', the inner  and outer algorithms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Namely, instead of formulating the ILP model M(V, C) to admit solutions of exactly  optimal k paths, it can be changed to allow solutions of at most some k′ paths, with k′ greater than the  optimal k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' If k′ is also greater than the number of ground truth paths in these complex graphs, then safe  paths are fully correct, meaning that we overall increase precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  9  5  Conclusion  RNA assembly is a difficult problem in practice, with even the top tools reporting low precision values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' While  there are still many issues that can introduce uncertainty in practice, we can now provide a major source  of additional information during the process: which RNA fragments must be included in any parsimonious  explanation of the data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Though others have considered RNA assembly in the safety framework [58,26], we  are the first to show that safety can be practically used even when we look for optimal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=', minimum) size  solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Our experimental results show that safe paths for MFD clearly outperform other safe approaches  for the Catfish dataset, commonly used in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' On a significant proportion of the second dataset, safe  paths for MFD still significantly outperforms other safe methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  More generally, this is the first work to show that the safety framework can be practically applied to  NP-hard problems, where the inner algorithm is an efficient test of safety of a group of paths, and the outer  algorithm guides the applications of this test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Because our method was very successful on our test data set,  there is strong motivation to try the approach to on other NP-hard graph problems whose solutions are  sets of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' For example, we could study other variations on MFD, such as finding flow decompositions  minimizing the longest path (NP-hard when flow values are integer [4,43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' The approach given in this paper  can also be directly extended to find decompositions into both cycles and paths [16], though not trails and  walks, because they repeat edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We could also formulate a safety test for classic NP-hard graph problems  like Hamiltonian path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Acknowledgements This work was partially funded by the European Research Council (ERC) under  the European Union’s Horizon 2020 research and innovation programme (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' 851093,  SAFEBIO), partially by the Academy of Finland (grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
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+page_content=' Elsa Bernard, Laurent Jacob, Julien Mairal, and Jean-Philippe Vert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Efficient RNA isoform identification and  quantification from RNA-Seq data with network flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Bioinformatics, 30(17):2447–2455, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
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+page_content=' Franco, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Manca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Spectral concepts in genome informational analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Theoretical Computer  Science, 894:23–30, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Building Bridges – Honoring Nataˇsa Jonoska on the Occasion of Her 60th Birthday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Manuel C´aceres, Massimo Cairo, Andreas Grigorjew, Shahbaz Khan, Brendan Mumey, Romeo Rizzi, Alexan-  dru I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Tomescu, and Lucia Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Width helps and hinders splitting flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  In Shiri Chechik, Gonzalo  Navarro, Eva Rotenberg, and Grzegorz Herman, editors, 30th Annual European Symposium on Algorithms, ESA  2022, September 5-9, 2022, Berlin/Potsdam, Germany, volume 244 of LIPIcs, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' 31:1–31:14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Schloss Dagstuhl  Leibniz-Zentrum f¨ur Informatik, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Manuel C´aceres, Brendan Mumey, Edin Husi´c, Romeo Rizzi, Massimo Cairo, Kristoffer Sahlin, and Alexandru I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Tomescu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Safety in multi-assembly via paths appearing in all path covers of a dag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' IEEE/ACM Transactions on  Computational Biology and Bioinformatics, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Massimo Cairo, Shahbaz Khan, Romeo Rizzi, Sebastian S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Schmidt, Alexandru I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Tomescu, and Elia C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Ziron-  delli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' The hydrostructure: a universal framework for safe and complete algorithms for genome assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' arXiv,  abs/2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
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+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Massimo Cairo, Paul Medvedev, Nidia Obscura Acosta, Romeo Rizzi, and Alexandru I Tomescu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' An optimal o  (nm) algorithm for enumerating all walks common to all closed edge-covering walks of a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' ACM Transactions  on Algorithms (TALG), 15(4):1–17, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  10  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Kun-Mao Chao, Ross C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Hardison, and Webb Miller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Locating well-conserved regions within a pairwise alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Bioinformatics, 9(4):387–396, 08 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
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+page_content=' Jiao Chen, Yingchao Zhao, and Yanni Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' De novo haplotype reconstruction in viral quasispecies using paired-  end read guided path finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
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+page_content=' Rami Cohen, Liane Lewin-Eytan, Joseph Seffi Naor, and Danny Raz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' On the effect of forwarding table size on SDN  network utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' In IEEE INFOCOM 2014-IEEE conference on computer communications, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
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+page_content=' Marie Costa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Persistency in maximum cardinality bipartite matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
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+page_content=' Tomescu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Fast, Flexible, and Exact  Minimum Flow Decompositions via ILP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' In RECOMB 2022 - 26th Annual International Conference on Research  in Computational Molecular Biology, volume 13278 of Lecture Notes in Computer Science, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
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+page_content=' Springer,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
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+page_content=' Minimum flow decomposition  in graphs with cycles using integer linear programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
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+page_content=' Marcelo Garlet Millani, Hendrik Molter, Rolf Niedermeier, and Manuel Sorge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Efficient algorithms for measuring  the funnel-likeness of dags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
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+page_content=' Modelling and simulating generic rna-seq experiments with the flux simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
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+page_content=' Krzysztof Pie´nkosz and Kamil Ko�lty´s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Integral flow decomposition with minimum longest path length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
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+page_content=' Susana Posada-C´espedes, David Seifert, Ivan Topolsky, Kim Philipp Jablonski, Karin J Metzner, and Niko Beeren-  winkel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' V-pipe: a computational pipeline for assessing viral genetic diversity from high-throughput data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Bioin-  formatics, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Amatur Rahman and Paul Medvedev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Assembler artifacts include misassembly because of unsafe unitigs and  under-assembly because of bidirected graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Genome Research, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
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+page_content=' Palash Sashittal, Chuanyi Zhang, Jian Peng, and Mohammed El-Kebir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Jumper enables discontinuous transcript  assembly in coronaviruses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Nature Communications, 12(1):6728, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Mingfu Shao and Carl Kingsford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Accurate assembly of transcripts through phase-preserving graph decomposi-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Nature biotechnology, 35(12):1167–1169, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Mingfu Shao and Carl Kingsford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Theory and a heuristic for the minimum path flow decomposition problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  IEEE/ACM transactions on computational biology and bioinformatics, 16(2):658–670, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Leonardo Taccari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Integer programming formulations for the elementary shortest path problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' European Journal  of Operational Research, 252(1):122–130, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Alexandru I Tomescu, Anna Kuosmanen, Romeo Rizzi, and Veli M¨akinen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' A novel min-cost flow method for  estimating transcript expression with RNA-Seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' In BMC bioinformatics, volume 14, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' S15:1–S15:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Springer,  2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Alexandru I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Tomescu and Paul Medvedev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Safe and complete contig assembly through omnitigs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Journal of  Computational Biology, 24(6):590–602, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Preliminary version appeared in RECOMB 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Valiant and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Vazirani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' NP is as easy as detecting unique solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Theoretical Computer Science,  47:85–93, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Vatinlen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Chauvet, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Chr´etienne, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Mahey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Simple bounds and greedy algorithms for decomposing a  flow into a minimal set of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' European Journal of Operational Research, 185(3):1390–1401, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Martin Vingron and Patrick Argos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Determination of reliable regions in protein sequence alignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Protein  Engineering, Design and Selection, 3(7):565–569, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Lucia Williams, Alexandru Tomescu, Brendan Marshall Mumey, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Flow decomposition with subpath con-  straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  In 21st International Workshop on Algorithms in Bioinformatics (WABI 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Schloss Dagstuhl-  Leibniz-Zentrum f¨ur Informatik, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Lucia Williams, Alexandru I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Tomescu, and Brendan Mumey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Flow decomposition with subpath constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  IEEE/ACM Transactions on Computational Biology and Bioinformatics, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Andrew D Yates, Premanand Achuthan, Wasiu Akanni, James Allen, Jamie Allen, Jorge Alvarez-Jarreta, M Rid-  wan Amode, Irina M Armean, Andrey G Azov, Ruth Bennett, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Ensembl 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Nucleic acids research,  48(D1):D682–D688, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Hongyu Zheng, Cong Ma, and Carl Kingsford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Deriving ranges of optimal estimated transcript expression due  to nonidentifiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Journal of Computational Biology, 29(2):121–139, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Presented at RECOMB 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  12  A  Additional Figures  P1  P2  P3  P4  P1  P2  P3  P4  a) Group Testing - Initial Iteration  c) Group Testing -  Second Iteration  b) Identifying Safe Paths  P2  P2  P4  P4  (a) First group test  P1  P2  P3  P4  P1  P2  P3  P4  a) Group Testing - Initial Iteration  c) Group Testing -  Second Iteration  b) Identifying Safe Paths  P2  P2  P4  P4  (b) Result: {P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' P3} are safe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' {P2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' P4}  are unsafe  P1  P2  P3  P4  P1  P2  P3  P4  a) Group Testing - Initial Iteration  c) Group Testing -  Second Iteration  b) Identifying Safe Paths  P2  P2  P4  P4  (c) Second group test  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' 3: Illustration of the initial group tests performed by Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' 3(a) shows the first group test  (using Algorithm 1) on MFD solution paths {P1, P2, P3, P4};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' suppose {P1, P3} were safe (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' 3(b));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' these  are then reported as maximal safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' In this case we trim {P2, P4} on both the left and right and make the  next group test shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  13  (a) Weighted Precision  (b) Maximum Coverage  (c) F-Score  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' 4: Quality metrics on graphs distributed by number of paths in the ground truth for the Catfish dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  The metrics are computed in terms of exons/nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  (a) Weighted Precision  (b) Maximum Coverage  (c) F-Score  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' 5: Quality metrics on graphs distributed by number of paths in the ground truth for the RefSim dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  The metrics are computed in terms of genomic positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  B  Additional Algorithms and Experimental Results  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='1  The bottom-up algorithm  Algorithm 3, detailed below, uses a bottom-up group-testing strategy to find all maximal safe paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We say a set of subpaths E = {Pi[li, ri]} is an extending core provided all paths in E are safe  and for any unreported maximal safe path P = Pi[l, r], there is a Pi[li, ri] ∈ E, where l ≤ li ≤ ri ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Note that maximal FD-safe subpaths provide an extending core (as well just the set of all single-edge  subpaths in each path).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Algorithm 3 provides an algorithm to find all maximal safe paths based on group  testing, starting from an extending core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' The idea is to try both left-extending (by one) and right-extending  (by one) each subpath in the core;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' if neither of these extensions are safe, then we know that that core subpath  must be maximal safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Testing all extensions can done quickly using Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We then recurse on a new  core set consisting of those extensions that were found to be safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  14  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='2  EUnitigs  SafeEPC  SafeFlow  SafeMFD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='0  3  4  5  6  7  8  9  10 11 12  13  14  15  # Ground truth paths1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='2  EUnitigs  SafeEPC  SafeFlow  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='0  SafeMFD  3  4  5  6  7  8  9  1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='12  13  14  15  # Ground truth paths1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
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+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='2  EUnitigs  SafeEPC  SafeFlow  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='0  SafeMFD  3  4  5  6  7  8  9  1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='12  13  14  15  # Ground truth paths1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='2  EUnitigs  SafeEPC  SafeFlow  SafeMFD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='0  3  4  5  6  7  8  9  10 11 12  13  14  15  # Ground truth paths1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='2  EUnitigs  SafeEPC  SafeFlow  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='0  SafeMFD  3  4  5  6  7  8  9  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='12  13  14  15  # Ground truth paths1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='2  EUnitigs  SafeEPC  SafeFlow  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='0  SafeMFD  3  4  5  6  7  8  9  1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='12  13  14  15  # Ground truth pathsAlgorithm 3: An algorithm to output all maximal safe subpaths that can be extended from an  extending core set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Input: An ILP model M and an extending core set E  Output: All maximal safe paths for M that extend some path from E  1 Procedure AllMaxSafe-BottomUp(M, E)  2  L = {Pi[li − 1, ri] : Pi[li, ri] ∈ E, li > 1} R = {Pi[li, ri + 1] : Pi[li, ri] ∈ E, ri < |Pi|} P = L ∪ R S  = GetSafe(M, P) for Pi[li, ri] ∈ E do  3  if Pi[li − 1, ri] /∈ S and Pi[li, ri + 1] /∈ S then  4  output Pi[li, ri]  5  if S ̸= ∅ then  6  AllMaxSafe-BottomUp(M, S)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='2  The two-pointer algorithm  As we observed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='2, we can test whether a single path P is safe using one ILP call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We will  assume that this test is encapsulated as a procedure IsSafe(M, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Once we can test whether a single path  is safe for M(V, C), we can adopt a standard approach to compute all maximal safe paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Namely, we start  by computing one solution of M(V, C), P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' , Pk and then compute maximal safe paths by a two-pointer  technique that for each path Pi, finds all maximal safe paths by just a linear number of calls to the procedure  IsSafe [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  This works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We use two pointers, a left pointer L, and a right pointer R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Initially, L points to  the first node of path Pi and R to the second node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' As long as the subpath of Pi between L and R is safe,  we move the right pointer to the next node on Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' When this subpath is not safe, we output the subpath  between L and the previous location of R as a maximal safe path, and we start moving the left pointer to  the next node on Pi, until the subpath between L and R is safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We stop the procedure once we reach the  end of Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We summarize this procedure as Algorithm 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' see also Figure 6 for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  1  8  3  2  5  6  4  7  s  t  b  a  d  e  c  f  s  t  b  a  d  e  c  f  s  t  b  a  d  e  c  f  xsai + xabi + xbci + xcdi + xdfi ≤ 4  xsai + xabi + xbci + xcdi ≤ 3  xabi + xbci + xcdi + xdfi ≤ 3  R  L  R  ∀i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' k}  ∀i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' k}  ∀i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' k}  L  L  R  (a) Current iteration  1  8  3  2  5  6  4  7  s  t  b  a  d  e  c  f  s  t  b  a  d  e  c  f  s  t  b  a  d  e  c  f  xsai + xabi + xbci + xcdi + xdfi ≤ 4  xsai + xabi + xbci + xcdi ≤ 3  xabi + xbci + xcdi + xdfi ≤ 3  R  L  R  ∀i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' k}  ∀i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' k}  ∀i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' k}  L  L  R  (b) Right pointer movement  1  8  3  2  5  6  4  7  s  t  b  a  d  e  c  f  s  t  b  a  d  e  c  f  s  t  b  a  d  e  c  f  xsai + xabi + xbci + xcdi + xdfi ≤ 4  xsai + xabi + xbci + xcdi ≤ 3  xabi + xbci + xcdi + xdfi ≤ 3  R  L  R  ∀i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' k}  ∀i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' k}  ∀i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='…,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' k}  L  L  R  (c) Left pointer movement  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' 6: Illustration of the two-pointer algorithm applied on a flow decomposition path Pi (in orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' In each  sub-figure, the subpath P (dashed green) between the nodes pointed by the left pointer L and the right  pointer R is tested for safety, by adding constraints S(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' In (a), IsSafe(M, P) returns True, and the right  pointer advances on Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' In (b), IsSafe(M, P) returns False, and the previous subpath from (a) is output as  a maximal safe path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' In (c), the left pointer has advanced, and the new path P is tested for safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  15  Algorithm 4: The two-pointer algorithm applied to compute all maximal subpaths of a given  solution path Pi  Input: An ILP model M and one of its k solution paths, Pi = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' , vt), t ≥ 2  Output: All maximal safe subpaths of Pi for M  1 Procedure AllMaxSafe-TwoPointer(M, Pi)  2  L ← 1, R ← 2 while True do  3  while IsSafe(M, Pi[L, R]) and R ≤ t do  4  R ← R + 1  5  output Pi[L, R − 1] if R > t then return;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  6  while not IsSafe(M, Pi[L, R]) do  7  L ← L + 1  Dataset (# Graphs)  Variant  Time (hh:mm:ss) # ILP calls  Catfish  (27,613)  TopDown  01:13:27  124,676  BottomUp  03:22:13  212,774  TwoPointer  04:21:44  226,365  TwoPointerBin  03:31:57  216,540  RefSim  (5,808)  TopDown  04:38:41  55,450  BottomUp  11:55:20  76,837  TwoPointer  13:48:00  127,352  TwoPointerBin  11:34:02  119,218  Table 2: Running times and number of ILP calls in four different variants of SafeMFD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='3  Running time experiments among different variants proposed  We conducted the experiments on an isolated Linux server with AMD Ryzen Threadripper PRO 3975WX  CPU with 32 cores (64 virtual) and 504GB of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Time and peak memory usage of each program were  measured with the GNU time command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' SafeMFD was allowed to run Gurobi with 12 threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' All C++  implementations were compiled with optimization level 3 (-O3 flag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Running time and peak memory is  computed and reported per dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  SafeMFD includes the following four variants computing maximal safe paths:  TopDown : Implements Algorithm 2 using the group testing in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  BottomUp : Implements Algorithm 3 (Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='1) using the group testing in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  TwoPointer : Implements Algorithm 4 (Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='2), the traditional two-pointer algorithm [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  TwoPointerBin : Same as previous variant, but it additionally replaces the linear scan employed to extend  and reduce the currently processed safe path by a binary search7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  To compare between our four different variants we first run them all on every dataset, and then filter  out those graphs that ran out of time in some variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' This way we ensure that no variant consumes its  time budget and thus our running time measurements are not skewed by the unsuccessful inputs’ timeouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Applying this filter we removed 83 graphs from the Catfish dataset (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='3%) and 4,515 graphs from the RefSim  dataset (43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='74%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Table 2 shows the running times and number of ILP calls of the different variants on both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  TopDown clearly outperforms the rest, being at least twice as fast, and performing (roughly) half many ILP  calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' While BottomUp is analogous to TopDown, the superiority of the latter can be explained by the length  maximal safe paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Since maximal safe paths are long it is faster to obtain them by reducing unsafe paths  7 The binary search is only applied if the search space is larger than a constant threshold set experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  16  (TopDown) than by extending safe paths (BottomUp and both TwoPointer variants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' On the other hand,  TwoPointer is the slowest variant and BottomUp and TwoPointerBin obtain similar improvements (over  TwoPointer) by following different strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' While BottomUp reduces the number of ILP calls more than  TwoPointerBin (better appreciated in the RefSim dataset), the ILP calls of BottomUp take longer (since  BottomUp tests several paths at the same time and TwoPointerBin only one), and thus the total running  times of both is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' This motivates future work on combining both approaches, while processing the  paths starting from unsafe (as in TopDown) for better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  C  Hardness of Testing MFD Safety  In this section we give a Turing-reduction from the UNIQUE 3SAT problem (U3SAT) to the problem of  determining if a given path P in a flow network G is safe for minimum flow decomposition (call this problem  MFD-SAFETY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' A 3SAT instance belongs to U3SAT if and only if it has exactly one satisfying assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  U3SAT has been shown to be NP-hard under randomized reductions [52], but it is open as to whether it is  NP-hard in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  The reduction leverages the construction in [22] that reduces 3SAT to minimum flow decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We  first briefly review this construction: A variable gadget (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' 4 in [22]) is created for each 3SAT variable x  and a clause gadget (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' 5 in [22]) is created for each 3SAT clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Positive literals in each clause receive  flow from the left side of the corresponding variable gadget, whereas negative literals receive flow from the  right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Theorem VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='1 in [22] establishes that a 3SAT instance is satisfiable if and only if the constructed  flow network has a minimum flow decomposition of a certain size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Any flow decomposition achieving this  size must have a specific structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' in particular, there must be a flow path of weight 4 that either travels  up the left side of the gadget (setting x to TRUE), or the right side (setting x to FALSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  ⋯  4  4  4  4  4  4  ⋯  4  4  4  4  t(x)  s(x)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' 7: The variable gadget from [22], showing only the weight 4 edges (other edges have weights from  {1, 2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' A key property established in [22] is that if the 3SAT instance is satisfiable then in a minimum flow  decomposition, a weight 4 flow path must travel up either the left side of the gadget (as shown), or the  right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' A left flow path indicates the variable should be set to TRUE, while right indicates FALSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' We  leverage this construction to reduce U3SAT to MFD-SAFETY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' There is a polynomial time Turing-reduction from U3SAT to MFD-SAFETY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  17  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' To obtain the desired Turing-reduction algorithm, instead of checking the size of the MFD, we will  instead sequentially check the MFD-SAFETY of the aforementioned side paths traveling up the left and  right sides of each variable gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Provided each variable gadget has exactly one safe side path we can then  check the corresponding truth assignment to see if each clause is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' If yes, we accept the instance as  belonging to U3SAT, otherwise we reject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  Suppose the instance does belong to U3SAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' In this case there is a satisfying assignment so the MFD  must have the structure as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' Furthermore, since there is exactly one satisfying assignment,  exactly one side path of each variable gadget must be safe and so our algorithm finds it and then verifies that  the truth assignment satisfies each clause, thus accepting the instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' On the other hand, if the instance does  not belong to U3SAT it could either be unsatisfiable or have multiple satisfying assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' If unsatisfiable,  no matter whether the safety checks pass, the corresponding assignment will not satisfy all clauses, so the  instance will be rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' If there are multiple solutions, then any variable that can be both TRUE and  FALSE will not have a safe side path in the MFD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content=' This means the safety check will fail and the instance  will again be rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
+page_content='  ⊓⊔  18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQfGjVY/content/2301.13245v1.pdf'}
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+arXiv:2301.00278v1  [math.CO]  31 Dec 2022
+Isometric path antichain covers: beyond hyperbolic graphs∗
+Dibyayan Chakraborty†
+Florent Foucaud‡
+January 3, 2023
+Abstract
+The isometric path antichain cover number of a graph G, denoted by ipacc (G), is a graph pa-
+rameter that was recently introduced to provide a constant factor approximation algorithm for Iso-
+metric Path Cover, whose objective is to cover all vertices of a graph with a minimum number
+of isometric paths (i.e. shortest paths between their end-vertices). This parameter was previously
+shown to be bounded for chordal graphs and, more generally, for graphs of bounded chordality and
+bounded treelength. In this paper, we show that the isometric path antichain cover number remains
+bounded for graphs in three seemingly unrelated graph classes, namely, hyperbolic graphs, (theta,
+prism, pyramid)-free graphs, and outerstring graphs. Hyperbolic graphs are extensively studied in
+Metric Graph Theory. The class of (theta, prism, pyramid)-free graphs are extensively studied in
+Structural Graph Theory, e.g. in the context of the Strong Perfect Graph Theorem. The class of
+outerstring graphs is studied in Geometric Graph Theory and Computational Geometry. Our results
+imply a constant factor approximation algorithm for Isometric Path Cover on all the above graph
+classes. Our results also show that the distance functions of these (structurally) different graph classes
+are more similar than previously thought.
+1
+Introduction
+A path is isometric if it is a shortest path between its endpoints. An isometric path cover of a graph G
+is a set of isometric paths such that each vertex of G belongs to at least one of the paths. The isometric
+path number of G is the smallest size of an isometric path cover of G. Given a graph G and an integer k,
+the objective of the algorithmic problem Isometric Path Cover is to decide if there exists an isometric
+path cover of cardinality at most k. Isometric Path Cover has been introduced and studied in the
+context of pursuit-evasion games [1, 2] and used in the context of Product Structure Theorems [15].
+The goal of this paper is to continue the study of approximation algorithms for Isometric Path
+Cover on several graph classes.
+We do so by continuing the study of a recently introduced graph
+parameter which seems interesting in its own right, as it encapsulates several previously unrelated graph
+classes.
+Isometric Path Cover has also been studied from a structural point of view: the cardinalities
+of the optimal solution have been determined for square grids [17], hypercubes [18], complete r-partite
+graphs [24] and Cartesian products of complete graphs [24], and it was recently proved that the pathwidth
+of a graph is always upper-bounded by the size of its smallest isometric path cover [16]. However, until
+recently the algorithmic aspects of Isometric Path Cover remained unexplored. The problem is easy
+to solve on trees and more generally, on block graphs [23] but remains hard on chordal graph, i.e. graphs
+without any induced cycle of length at least 4 [7]. It can be approximated in polynomial time within a
+factor of log(d) for graphs of diameter d by a greedy algorithm [27] and solved in polynomial time for every
+∗This
+research
+was
+partially
+financed
+by
+the
+IFCAM
+project
+“Applications
+of
+graph
+homomorphisms”
+(MA/IFCAM/18/39), the ANR project GRALMECO (ANR-21-CE48-0004) and the French government IDEX-ISITE ini-
+tiative 16-IDEX-0001 (CAP 20-25).
+†Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
+‡Université Clermont-Auvergne, CNRS, Mines de Saint-Étienne, Clermont-Auvergne-INP, LIMOS, 63000 Clermont-
+Ferrand, France
+1
+
+fixed value of k by an XP algorithm [16]. In a quest to find constant factor approximation algorithms
+for Isometric Path Cover, Chakraborty et al. [7] introduced a parameter called the isometric path
+antichain cover number of graphs, denoted by ipacc (G) (see Section 2 for a definition) and proved a
+result directly implying the following (see [7, Proposition 10]).
+Proposition 1 ([7]). For a graph G, if ipacc (G) ≤ c, then Isometric Path Cover admits a polynomial-
+time c-approximation algorithm on G.
+Proposition 1 is proved by a simple approximation algorithm described as follows. For each vertex r
+of the graph, perform a Breadth-First Search at this vertex. Remove edges joining any vertices at the
+same distance from r, and orient all edges towards r. The resulting directed acyclic graph can be seen as
+the Hasse diagram of a poset. Compute a chain covering of that poset using classic methods related to
+Dilworth’s theorem. The chains are the isometric paths of the solution. Keep the smallest of all solutions
+over all choices of r.
+Using Proposition 1, the above algorithm was shown to be a constant factor approximation algo-
+rithm for many graph classes, including interval graphs, chordal graphs, and more generally, graphs with
+bounded treelength. Indeed, on all these graph classes, the isometric path antichain cover number is
+shown to be bounded by a constant (note that one does not need to compute this parameter for the
+algorithm to function: it serves only in the analysis of the approximation ratio of the algorithm). As
+noted in [7], this parameter may be unbounded on general graphs, for example for the class of hypercubes
+or square grids.
+In this paper, we continue to study the boundedness of the isometric path antichain cover number
+of various graph classes. Specifically, we consider three structurally unrelated graph classes, namely,
+hyperbolic graphs, (theta, prism, pyramid)-free graphs, and outerstring graphs, which extends the above
+work to strictly larger graph classes.
+Hyperbolic graphs: A graph G is said to be δ-hyperbolic [19] if for any four vertices u, v, x, y, the two
+larger of the three distance sums d (u, v)+d (x, y), d (u, x)+d (v, y) and d (u, y)+d (v, x) differ by at most
+2δ. A graph class G is hyperbolic if there exists a constant δ such that every graph G ∈ G is δ-hyperbolic.
+This parameter was first introduced by Gromov in the context of automatic groups [19] in relation with
+their Cayley graphs. The hyperbolicity of a tree is 0, and in general, hyperbolicity seems to measure
+how much the distance function of a graph deviates from a tree metric. Many structurally defined graph
+classes like chordal graphs, cocomparability graphs, asteroidal-triple free graphs, graphs with bounded
+chordality or treelength are hyperbolic graphs [8, 21]. Moreover, hyperbolicity has been found to capture
+important properties of several large practical graphs such as the Internet [26] or database relations [31].
+Due to its importance in discrete mathematics, algorithms, metric graph theory, researchers have studied
+various algorithmic aspects of hyperbolic graphs [8, 12, 9, 13]. Note that graphs with diameter 2 are
+hyperbolic, which may contain any graph as an induced subgraph.
+(theta, prism, pyramid)-free graphs: A theta is a graph made of three vertex-disjoint induced paths
+P1 = a . . . b, P2 = a . . . b, P3 = a . . . b of lengths at least 2, and such that no edges exist between the paths
+except the three edges incident to a and the three edges incident to b. See Figure 2 for an illustration. A
+pyramid is a graph made of three induced paths P1 = a . . . b1, P2 = a . . . b2, P3 = a . . . b3, two of which
+have lengths at least 2, vertex-disjoint except at a, and such that b1b2b3 is a triangle and no edges exist
+between the paths except those of the triangle and the three edges incident to a. A prism is a graph
+made of three vertex-disjoint induced paths P1 = a1 . . . b1, P2 = a2 . . . b2, P3 = a3 . . . b3 of lengths at
+least 1, such that a1a2a3 and b1b2b3 are triangles and no edges exist between the paths except those of
+the two triangles. A graph G is (theta, pyramid, prism)-free if G does not contain any induced subgraph
+isomorphic to a theta, pyramid or prism. A graph is a 3-path configuration if it is a theta, pyramid or
+prism. The study of 3-path configurations dates back to the works of Watkins and Meisner [32] in 1967
+and plays “special roles” in the proof of the celebrated Strong Perfect Graph Theorem [10, 14, 28, 30].
+Important graph classes like chordal graphs, circular arc graphs, universally-signable graphs [11] exclude
+all 3-path configurations. Popular graph classes like perfect graphs, even hole-free graphs exclude some
+2
+
+Bounded isometric path
+antichain cover number
+bounded hyperbolicity *
+(t-theta, t-prism, t-pyramid)-
+free *
+Outerstring *
+circle *
+(theta,prism,pyramid)-
+free *
+Universally signable *
+bounded tree-length
+bounded chordality
+bounded diameter
+chordal
+AT-free
+Interval
+circular arc *
+Permutation
+Figure 1: Inclusion diagram for graph classes discussed here (and related ones). If a class A has an
+upward path to class B, then A is included in B. For graphs in the gray classes, the complexity of
+Isometric Path Cover is open; for all other graph classes, it is NP-complete. For all shown graph
+classes, Isometric Path Cover is constant-factor approximable in polynomial time. Constant factor
+approximation algorithms for Isometric Path Cover on graph classes marked with * are contributions
+of this paper.
+b
+a
+b1
+b2
+b3
+a
+b1
+b2
+b3
+a1
+a2
+a3
+(a)
+(b)
+(c)
+(d)
+Figure 2: (a) Theta, (b) Pyramid, (c) Prism, (d) Outerstrings. The figure shows that the graph K2,3,
+which is also a theta, is an outerstring graph.
+of the 3-path configurations. Note that, (theta, prism, pyramid)-free graphs are not hyperbolic. To see
+this, consider a cycle C of order n. Clearly, C excludes all 3-path configurations and has hyperbolicity
+Ω(n).
+Outerstring graphs: A set S of simple curves on the plane is grounded if there exists a horizontal line
+containing one endpoint of each of the curves in S. A graph G is an outerstring graph if there is a collection
+C of grounded simple curves and a bijection between V (G) and C such that two curves in S if and only
+if the corresponding vertices are adjacent in G. See Figure 2(d) for an illustration. The term “outerstring
+graph” was first used in the early 90’s [22] in the context of studying intersection graphs of simple curves
+on the plane. Many well-known graph classes like chordal graphs, circular arc graphs, circle graphs
+(intersection graphs of chords of a circle), or cocomparability graphs are also outerstring graphs and
+thus, motivated researchers from the geometric graph theory and computational geometry communities
+to study algorithmic and structural aspects of outerstring graphs and its subclasses [4, 5, 6, 20, 25]. Note
+that, in general, outerstring graphs may contain a prism, pyramid or theta as an induced subgraph.
+Moreover, cycles of arbitrary order are outerstring graphs, implying that outerstring graphs are not
+hyperbolic.
+It is clear from the above discussion that the classes of hyperbolic graphs, (theta, prism, pyramid)-free
+3
+
+graphs, and outerstring graphs are pairwise incomparable (with respect to the containment relationship).
+1.1
+Our contributions
+The main contribution of this paper is to show that the isometric path antichain cover number (see
+Section 2 for a definition) remains bounded on hyperbolic graphs, (theta, pyramid, prism)-free graphs,
+and outerstring graphs. Specifically, we prove the following theorems.
+Theorem 2. Let G be a graph with hyperbolicity δ. Then, ipacc (G) ≤ 12δ + 6.
+Theorem 3. Let G be a (theta, pyramid, prism)-free graph. Then, ipacc (G) ≤ 71.
+Theorem 4. Let G be an outerstring graph. Then, ipacc (G) ≤ 95.
+To the best of our knowledge, the isometric path antichain cover number being bounded (by con-
+stant(s)) is the only known non-trivial property shared by any two or all three of these graph classes.
+To provide a unified proof of Theorem 3 and 4, we study a more general graph class called (t-theta,
+t-pyramid, t-prism)-free graphs [29] (see Section 4 for definition). When t = 1, (t-theta, t-pyramid, t-
+prism)-free graphs are exactly (theta, prism, pyramid)-free graphs. Moreover, we show that all outerstring
+graphs are (4-theta, 4-pyramid, 4-prism)-free graphs (Lemma 16). We prove the following.
+Theorem 5. For t ≥ 1, let G be a (t-theta, t-pyramid, t-prism)-free graph. Then ipacc (G) ≤ 8t + 63.
+Due to Proposition 1 and the above theorems, we also have the following corollary.
+Corollary 6. There is an approximation algorithm for Isometric Path Cover with approximation
+ratio
+(a) 12δ + 6 on δ-hyperbolic graphs,
+(b) 73 on (theta, prism, pyramid)-free graphs, and
+(c) 95 on outerstring graphs.
+(d) 8t + 63 on (t-theta, t-pyramid, t-prism)-free graphs.
+Organisation: In Section 2, we introduce the recall some definitions and some results. In Section 3 we
+prove Theorem 2. In Section 4, we prove Theorems 3 and 5. In Section 5, we prove Theorem 4. We
+conclude in Section 6.
+2
+Definitions and preliminary observations
+In this section, we formally recall the definition of isometric path antichain cover number of graphs from
+[7] and some related observations. A sequence of distinct vertices forms a path P if any two consecutive
+vertices are adjacent. Whenever we fix a path P of G, we shall refer to the subgraph formed by the
+edges between the consecutive vertices of P. The length of a path P, denoted by |P|, is the number of
+its vertices minus one. A path is induced if there are no graph edges joining non-consecutive vertices.
+In a directed graph, a directed path is a path in which all arcs are oriented in the same direction. For a
+path P of a graph G between two vertices u and v, the vertices V (P) \ {u, v} are internal vertices of P.
+A path between two vertices u and v is called a (u, v)-path. Similarly, we have the notions of isometric
+(u, v)-path and induced (u, v)-path. For a vertex r of G and a set S of vertices of G, the distance of S from
+r, denoted as d (r, S), is the minimum of the distance between any vertex of S and r. For a subgraph
+H of G, the distance of H w.r.t. r is d (r, V (H)). Formally, we have d (r, S) = min{d (r, v) : v ∈ S} and
+d (r, H) = d (r, V (H)).
+4
+
+For a graph G and a vertex r ∈ V (G), consider the following operations on G. First, remove all
+edges xy from G such that d (r, x) = d (r, y).
+Let G′
+r be the resulting graph.
+Then, for each edge
+e = xy ∈ E(G′
+r) with d (r, x) = d (r, y) − 1, orient e from y to x. Let −→
+Gr be the directed acyclic graph
+formed after applying the above operation on G′. Note that this digraph can easily be computed in linear
+time using a Breadth-First Search (BFS) traversal with starting vertex r.
+The following definition is inspired by the terminology of posets (as the graph −→
+Gr can be seen as the
+Hasse diagram of a poset).
+Definition 7. For a graph G and a vertex r ∈ V (G), two vertices x, y ∈ V (G) are antichain vertices if
+there are no directed paths from x to y or from y to x in −→
+Gr. A set X of vertices of G is an antichain
+set if any two vertices in X are antichain vertices. The cardinality of the largest antichain set in −→
+Gr will
+be denoted by β
+�−→
+Gr
+�
+. The cardinality of the largest antichain set of G, is defined as
+β (G) = min
+�
+β
+�−→
+Gr
+�
+: r ∈ V (G)
+�
+Definition 8 ([7]). Let r be a vertex of a graph G. For a subgraph H, Ar (H) shall denote the maximum
+antichain set of H in −→
+Gr. The isometric path antichain cover number of −→
+Gr, denoted by ipacc
+�−→
+Gr
+�
+, is
+defined as follows:
+ipacc
+�−→
+Gr
+�
+= max {|Ar (P) |: P is an isometric path}
+The isometric path antichain cover number of graph G, denoted as ipacc (G), is defined as the minimum
+over all possible antichain covers of its associated directed acyclic graphs:
+ipacc (G) = min
+�
+ipacc
+�−→
+Gr
+�
+: r ∈ V (G)
+�
+We recall the proof of the following proposition from [7] which will be used heavily in this paper.
+Proposition 9 ([7]). Let G be a graph and r, an arbitrary vertex of G. Consider the directed acyclic
+graph −→
+Gr, and let P be an isometric path between two vertices x and y in G. Then |P| ≥ |d (r, x) −
+d (r, y) | + |Ar (P) | − 1.
+Proof. Orient the edges of P from y to x in G. First, observe that P must contain a set E1 of oriented
+edges such that |E1| = |d (r, y) − d (r, x) | and for any −→
+ab ∈ E1, d (r, a) = d (r, b) + 1. Let the vertices of
+the largest antichain set of P in −→
+Gr, i.e., Ar (P), be ordered as a1, a2, . . . , at according to their occurrence
+while traversing P from y to x. For i ∈ [2, t], let Pi be the subpath of P between ai−1 and ai. Observe that
+for any i ∈ [2, t], since ai and ai−1 are antichain vertices, there must exist an oriented edge −→
+bici ∈ E(Pi)
+such that either d (r, bi) = d (r, ci) or d (r, bi) = d (r, ci) − 1.
+Let E2 = {bici}i∈[2,t].
+Observe that
+E1 ∩ E2 = ∅ and therefore |P| ≥ |E1| + |E2| = |d (r, y) − d (r, x) | + |Ar (P) | − 1.
+3
+Proof of Theorem 2
+In this section, we shall show that isometric path antichain cover number of graphs with hyperbolicity
+at most δ is at most 12δ + 6. To achieve our goal we need to recall a few definitions from the literature.
+For three vertices x, y, z of a graph G, a geodesic triangle [3], denoted as ∆(x, y, z) is the union P(x, y) ∪
+P(y, z)∪P(x, z) of three isometric paths connecting these vertices. A geodesic triangle ∆(x, y, z) is called
+ρ-slim if for any vertex u ∈ P(x, y) the distance d (u, P(y, z) ∪ P(x, z)) is at most ρ. The smallest value
+of ρ for which every geodesic triangle of G is ρ-slim is called the slimness of G and is denoted by sl (G).
+In the following lemma, we shall show that if the isometric path antichain cover number of a graph is
+large then so is the slimness of the graph.
+Lemma 10. For any graph G, ipacc (G) ≤ 4sl (G) + 2.
+5
+
+u
+v
+c
+c′
+Figure 3: An example of a 4-fat turtle. Let C be the cycle induced by the black vertices, P be the path
+induced by the white vertices. Then the tuple (4, C, P, c, c′) defines a 4-fat turtle.
+Proof. Let ρ = sl (G).
+Aiming for a contradiction, let r be a vertex of G such that there exists an
+isometric path P such that |Ar (P) | ≥ 4ρ + 3. Let the vertices of Ar (P) be named and ordered as
+a1, a2, . . . , a2ρ+2, . . . , a4ρ+3 as they are encountered while traversing P from one end-vertex to the other.
+Let x = a1, y = a4ρ+3. Let −→
+Px be an oriented path from x to r in −→
+Gr. Observe that Px, the path of
+G obtained by removing the orientation of −→
+Px, is an (x, r)-isometric path. Let −→
+Py be an oriented path
+from y to r in −→
+Gr. Similarly, Py, the path of G obtained by removing the orientation of −→
+Py, is an (y, r)-
+isometric path. Observe that P, Px, Py form a geodesic triangle with x, r, y as end-vertices. Consider the
+vertex z = a2ρ+2 on the path P. Since ρ = sl (G), there exists a vertex w ∈ V (Px) ∪ V (Py) such that
+d (w, z) ≤ ρ. Without loss of generality, assume w ∈ V (Px). Then, d (x, z) ≤ d (x, w) + d (w, z). By
+using that d (r, z) ≤ d (r, w) + d (w, z) ≤ d (r, w) + ρ, we get d (x, z) ≤ |d (r, x) − d (r, z) | + 2ρ. But this
+contradicts Proposition 9, due to which we have d (x, z) ≥ |d (r, x) − d (r, z) | + 2ρ + 1.
+Now we shall use the following result.
+Proposition 11 ([3]). For any graph G, sl (G) ≤ 3hb (G).
+Proposition 11 and Lemma 10, imply the theorem.
+4
+Proofs of Theorem 3 and 5
+In this section, we shall prove Theorems 3 and 5. First we shall define the notions of t-theta, t-prism,
+and t-pyramid [29].
+For an integer t ≥ 1, a t-prism is a graph made of three vertex-disjoint induced paths P1 = a1 . . . b1,
+P2 = a2 . . . b2, P3 = a3 . . . b3 of lengths at least t, such that a1a2a3 and b1b2b3 are triangles and no edges
+exist between the paths except those of the two triangles. For an integer t ≥ 1, a t-pyramid is a graph
+made of three induced paths P1 = a . . . b1, P2 = a . . . b2, P3 = a . . . b3 of lengths at least t, two of which
+have lengths at least t + 1, they are pairwise vertex-disjoint except at a, such that b1b2b3 is a triangle
+and no edges exist between the paths except those of the triangle and the three edges incident to a. For
+an integer t ≥ 1, a t-theta is a graph made of three internally vertex-disjoint induced paths P1 = a . . . b,
+P2 = a . . . b, P3 = a . . . b of lengths at least t+1, and such that no edges exist between the paths except the
+three edges incident to a and the three edges incident to b. A graph G is (t-theta, t-pyramid, t-prism)-free
+if G does not contain any induced subgraph isomorphic to a t-theta, t-pyramid or t-prism. When t = 1,
+(t-theta, t-pyramid, t-prism)-free graphs are exactly (theta, prism, pyramid)-free graphs.
+Now we shall show that the isometric path antichain cover number of (t-theta, t-pyramid, t-prism)-
+free graphs are bounded above by a linear function on t. We shall show that, when the isometric path
+antichain cover number of a graph is large, the existence of a structure called “t-fat turtle” (defined
+below) as an induced subgraph is forced, which, cannot be present in a ((t − 1)-theta, (t − 1)-pyramid,
+(t − 1)-prism)-free graph.
+Definition 12. For an integer t ≥ 1, a “t-fat turtle” consists of a cycle C and an induced (u, v)-path P
+of length at least t such that all of the following hold.
+6
+
+(a) V (P) ∩ V (C) = ∅,
+(b) For any vertex w ∈ (V (P) \ {u, v}), N(w) ∩ V (C) = ∅ and both u and v have at least one neighbour
+in C.
+(c) For any vertex w ∈ N(u) ∩ V (C) and w′ ∈ N(v) ∩ V (C), the distance between w and w′ in C is at
+least t,
+(d) There exist two vertices {c, c′} ⊂ V (C) and two distinct components Cu, Cv of C − {c, c′} such that
+N(u) ∩ V (C) ⊆ V (Cu) and N(v) ∩ V (C) ⊆ V (Cv).
+The tuple (t, C, P, c, c′) defines the t-fat turtle. See Figure 3 for an example.
+In the following observation, we show that any (t-theta, t-pyramid,t-prism)-free graph cannot contain
+a (t + 1)-fat turtle as an induced subgraph.
+Lemma 13. For some integer t ≥ 1, let G be a graph containing a (t + 1)-fat turtle as an induced
+subgraph. Then G is not (t-theta, t-pyramid, t-prism)-free.
+Proof. Let (t+1, C, P, c, c′) be a (t+1)-fat turtle in G. Let the vertices of C be named c = a0, a1, . . . , ak =
+c′, ak+1, . . . , a|V (C)| as they are encountered while traversing C starting from c in a counter-clockwise
+manner. Denote by u, v the end-vertices of P. By definition, there exist two distinct components Cu, Cv of
+C−{c, c′} such that N(u)∩V (C) ⊆ V (Cu) and N(v)∩V (C) ⊆ V (Cv). Without loss of generality, assume
+V (Cu) = {a1, a2, . . . , ak−1} and V (Cv) = {ak+1, ak+2, . . . , a|V (C)|}. Let i− and i+ be the minimum and
+maximum indices such that ai− and ai+ are adjacent to u. Let j− and j+ be the minimum and maximum
+indices such that aj− and aj+ are adjacent to v. By definition, i− ≤ i+ < j− ≤ j+. Let P1 be the
+(ai−, aj+)-subpath of C containing c. Let P2 be the (ai+, aj−)-subpath of C that contains c′. Observe
+that P1 and P2 have length at least t (by definition). Now we show that P, P1, P2 together form one of
+theta, pyramid or prism. If ai− = ai+ and aj− = aj+, then P, P1, P2 form a t-theta. If i− ≤ i+ − 2 and
+j− ≤ j+ − 2, then also P, P1, P2 form a t-theta. If j− = j+ − 1 and i− = i+ − 1, then P, P1, P2 form a
+t-prism. In any other case, P, P1, P2 form a t-pyramid.
+In the remainder of this section, we shall prove that there exists a linear function f(t) such that if
+the isometric path antichain cover number of a graph is more than f(t), then G is forced to contain a
+(t + 1)-fat turtle as an induced subgraph, and therefore is not (t-theta, t-pyramid,t-prism)-free. We shall
+use the following observation.
+Observation 14. Let G be a graph, r be an arbitrary vertex, P be an isometric (u, v)-path in G and Q
+be a subpath of an isometric (v, r)-path in G such that one endpoint of Q is v. Let P ′ be the maximum
+(u, w)-subpath of P such that no internal vertex of P ′ is a neighbour of some vertex of Q. We have that
+|Ar (P ′) | ≥ |Ar (P) | − 3.
+Proof. Suppose |Ar (P ′) | ≤ |Ar (P) | − 4 and consider the (w, v)-subpath, say P ′′, of P. Observe that
+|Ar (P ′′) | ≥ 4. Now let w′ be a vertex of Q which is a neighbour of w. Observe that |d (r, w)−d (r, w′) | ≤ 1
+and therefore d (w, v) = |E(P ′′)| ≤ |d (r, w)−d (r, v) |+2. But this contradicts Proposition 9, which implies
+that the length of P ′′ is at least |d (r, w) − d (r, v) | + 3.
+Lemma 15. For an integer t ≥ 1, let G be a graph with ipacc (G) ≥ 8t + 64. Then G has a (t + 1)-fat
+turtle as an induced subgraph.
+Proof. Let r be a vertex of G such that ipacc
+�−→
+Gr
+�
+is at least 8t+64. Then there exists an isometric path
+P such that |Ar (P) | ≥ 8t+ 64. Let the two endpoints of P be a and b. (See Figure 4.) Let u be a vertex
+of P such that d (r, u) = d (r, P). Let Pau be the (a, u)-subpath of P and Pbu be the (b, u)-subpath of P.
+Both Pau and Pbu are isometric paths and observe that either |Ar (Pau) | ≥ 4t+32 or |Ar (Pbu) | ≥ 4t+32.
+Without loss of generality, assume that |Ar (Pbu) | ≥ 4t + 32. Let Qr
+b be an isometric (b, r)-path in G.
+7
+
+r
+z2
+w2
+u
+z
+z1
+w
+b
+w1
+c
+(= a2t+13)
+x
+c1
+a
+c2
+T (c1, c2)
+≥ t
+≥ t
+≥ t
+Qr
+b
+Qr
+u
+Figure 4: Illustration of the notations used in the proof of Lemma 15.
+Let Ruw be the maximum (u, w)-subpath, of Pbu such that no internal vertex of Ruw is a neighbour
+of Qr
+b. Note that Ruw is an isometric path and w has a neighbour in Qr
+b. Applying Observation 14, we
+have the following:
+Claim 15.1. |Ar (Ruw) | ≥ 4t + 29.
+Let Qr
+u be any isometric (u, r)-path of G and let Rzw be the maximum (z, w)-subpath of Ruw such
+that no internal vertex of Rzw has a neighbour in Qr
+u. Observe that Rzw is an isometric path, and z has
+a neighbour in Qr
+u. Again applying Observation 14, we have the following:
+Claim 15.2. |Ar (Rzw) | ≥ 4t + 26.
+Let a1, a2, . . . , ak be the vertices of Ar (Rzw) ordered according to their appearance while traversing
+Rzw from z to w. Due to Claim 15.2, we have that k ≥ 4t + 26. Let c = a2t+13 and Qr
+c denote an
+isometric (c, r)-path. Let T (r, c1) be the maximum subpath of Qr
+c such that no internal vertex of T (r, c1)
+is adjacent to any vertex of Rzw.
+Claim 15.3. Let x be a neighbor of c1 in Rzw, X be the (x, b)-subpath of Pub and Y be the (x, u)-subpath
+of Pub. Then |Ar (X) | ≥ 2t + 11 and |Ar (Y ) | ≥ 2t + 11.
+Proof. Let Rcw denote the (c, w)-subpath of Rzw. Observe that |Ar (Rcw) | ≥ 2t + 14. First, consider
+the case when x lies in the (z, c)-subpath of Rzw. In this case, Rcw is a subpath of X and therefore
+|Ar (X) | ≥ 2t + 14. Now consider the case when x lies in Rcw. In this case, applying Observation 14,
+we have that |Ar (X) | ≥ |Ar (Rcw) | − 3 ≥ 2t + 11. Using a similar argument, we have that |Ar (Y ) | ≥
+2t + 11.
+Let T (c1, c2) be the maximum (c1, c2)-subpath of T (c1, r) such that no internal vertex of T (c1, c2) is
+adjacent to a vertex of Qr
+b or Qr
+u. We have the following claim.
+Claim 15.4. The length of T (c1, c2) is at least t + 3.
+Proof. Assume that the length of T (c1, c2) is at most t + 2 and x be a neighbour of c1 in Rzw. Observe
+that all vertices of Rzw are at distance at least d (r, u) i.e. d (r, Rzw) ≥ d (r, u), since d (r, u) = d (r, P).
+Hence,
+(+) d (r, x) ≥ d (r, u) and d (r, c1) ≥ d (r, u) − 1.
+8
+
+Now, suppose c2 has a neighbor c3 in Qr
+u. Hence d (c3, x) ≤ d (c3, c2) + d (c2, c1) + d (c1, x) ≤ t + 4.
+Now, using (+) and the fact that c3 lies on an isometric (r, u)-path (Qr
+u), we have that d (c3, u) ≤ t + 4.
+Therefore, d (u, x) ≤ d (c3, u) + d (c3, x) ≤ 2t + 8. But this contradicts Proposition 9 and Claim 15.3, as
+they together imply that d (u, x) is at least d (r, x) − d (r, u) + 2t + 10≥ 2t + 10.
+Hence, c2 must have a neighbour c3 in Qr
+b. First, assume that d (r, x) ≥ d (r, b). Then, as d (c3, x) ≤
+d (c3, c2) + d (c2, c1) + d (c1, x) ≤ t + 4 and c3 lies on an isometric (r, b)-path (Qr
+b), we have that d (x, b) ≤
+2t + 8. But again this contradicts Proposition 9 and Claim 15.3, as they together imply that the length
+of d (x, b) is at least d (r, x) − d (r, u) + 2t + 10. Now, assume that d (r, x) < d (r, b). Let b′ be a vertex of
+Qr
+b such that d (r, b′) = d (r, x). Using a similar argumentation as before, we have that d (x, b′) ≤ 2t + 8.
+Hence, d (x, b) ≤ d (x, b′) + d (b′, b) ≤ d (r, b) − d (r, x) + 2t + 8. But this contradicts Proposition 9 which,
+due to Claim 15.3, implies that d (x, b) ≥ d (r, b) − d (r, x) + 2t + 10.
+The path T (c1, c2) forms the first ingredient to extract a (t + 1)-fat turtle. Let z1 be the neighbor of
+z in Qr
+u and w1 be the neighbour of w in Qr
+b. We have the following claim.
+Claim 15.5. The vertices w1 and z1 are non adjacent.
+Proof. Recall that z1 lies in Qr
+u and d (r, z) ≥ d (r, u). Hence z1 must be a neighbor of u. If w1 and z1 are
+adjacent, then observe that d (u, b) ≤ d (r, b) − d (r, w1) + 2 ≤. This implies d (u, b) ≤ d (r, b) − d (r, u)+ 3.
+But this shall again contradict Proposition 9.
+Now we shall construct a (w1, z1)-path as follows: Consider the maximum (w1, w2)-subpath, say
+T (w1, w2), of Qr
+b such that no internal vertex of T (w1, w2) has a neighbour in Qr
+u. Similarly, consider the
+maximum (z1, z2)-subpath, say T (z1, z2), of Qr
+b such that no internal vertex of T (z1, z2) is a neighbor of
+w2. Let T be the path obtained by taking the union of T (w1, w2) and T (z1, z2). Observe that z2 must
+be a neighbour of w2 and T is an induced (w1, z1)-path. The definitions of T and Rzw imply that their
+union induces a cycle Z. Here we have the second and final ingredient to extract the (t + 1)-fat turtle.
+Suppose that c2 has a neighbour in T . Let T ′ be the maximum subpath of T (c1, c2) which is vertex-
+disjoint from Z. Due to Claim 15.4, the length of T ′ is at least t + 1. Let e1 and e2 be the end-vertices
+of T ′. Observe the following.
+• Each of e1 and e2 has at least one neighbor in Z.
+• Z −{z, w} contains two distinct components C1, C2 such that for i ∈ {1, 2}, N(ei)∩V (Z) ⊆ V (Ci).
+• For a vertex e′
+1 ∈ N(e1) ∩ V (Z) and e′
+2 ∈ N(e2) ∩ V (Z), the distance between e′
+1 and e′
+2 is at least
+t + 1. This statement follows from Claim 15.3.
+Hence, we have that the tuple (t + 1, Z, T ′, z, w) defines a (t + 1)-fat turtle. Now consider the case
+when c2 does not have a neighbor in T . By definition, c2 has at least one neighbor in Qr
+u or Qr
+b. Without
+loss of generality, assume that c2 has a neighbor c3 in Qr
+u such that the (z2, c3)-subpath, say, T ′′ of Qr
+u
+has no neighbor of c2 other than c3. Observe that the path T ∗ = (T ′ ∪ (T ′′ − {z2})) is vertex-disjoint
+from Z and has length at least t + 1. Let e1, e2 be the two end-vertices of T ∗. Observe the following.
+• Each of e1 and e2 has at least one neighbor in Z.
+• Z −{z, w} contains two distinct components C1, C2 such that for i ∈ {1, 2}, N(ei)∩V (Z) ⊆ V (Ci).
+• For a vertex e′
+1 ∈ N(e1) ∩ V (Z) and e′
+2 ∈ N(e2) ∩ V (Z), the distance between e′
+1 and e′
+2 is at least
+t + 1. This statement follows from Claim 15.3.
+Hence, (t + 1, Z, T ∗, z, w) is a (t + 1)-fat turtle
+Proof of Theorem 5 and 3: Lemma 13 and 15 together imply the theorems.
+9
+
+5
+Proof of Theorem 4
+Next we shall show that outerstring graphs are (4-theta, 4-prism, 4-pyramid)-free.
+Lemma 16. Let G be an outerstring graph. Then, G is (4-theta, 4-prism, 4-pyramid)-free.
+Proof. To prove the lemma, we shall need to recall a few definitions and results from the literature. A
+graph G is a string graph if there is a collection S of simple curves on the plane and a bijection between
+V (G) and S such that two curves in S intersect if and only if the corresponding vertices are adjacent in
+G. Let G be a graph with an edge e. The graph G \ e is obtained by contracting the edge e into a single
+vertex. Observe that string graphs are closed under edge contraction [22]. We shall use the following
+result.
+Proposition 17 ([22]). Let G be an outerstring graph with an edge e. Then G\e is an outerstring graph.
+A full subdivision of a graph means replacing each edge of G with a new path of length at least two.
+We shall use the following result implied from Theorem 1 of [22].
+Proposition 18 ([22]). Let G be a string graph. Then G does not contain a full subdivision of K3,3 as
+an induced subgraph.
+For a graph G, the graph G+ is constructed by introducing a new apex vertex a and connecting a
+with all vertices of G by new copies of paths of length at least 2. We shall use the following result of
+Biedl et al. [4].
+Proposition 19 (Lemma 1, [4]). A graph G is an outerstring graph if and only if G+ is a string graph.
+Now we are ready to prove the lemma.
+Let G be an outerstring graph. Assume for the sake of
+contradiction that G contains an induced subgraph H which is a 4-theta, 4-pyramid, or a 4-prism. Since
+every induced subgraph of an outerstring graph is also an outerstring graph, we have that H is an
+outerstring graph. Let E be the set of edges of H whose both endpoints are part of some triangle. Now
+consider the graph H1 = H \ E which is obtained by contracting all edges in E. By Proposition 17, H1
+is an outerstring graph and it is easy to check that H1 is a 3-theta. Let u and v be the vertices of H1
+with degree 3 and w1, w2, w3 be the set of mutually non-adjacent vertices such that for each i ∈ {1, 2, 3}
+d (u, wi) = 2 and d (v, wi) ≥ 2. Since H1 is a 3-theta, w1, w2, w3 exist. Now consider the graph H+
+1 and
+a be the new apex vertex. Due to Proposition 19, we have that H+
+1 is a string graph. But notice that,
+for each pair of vertices in {x, y} ⊂ {w1, w2, w3, u, v, a}, there exists a unique path of length at least 2
+connecting x, y. This implies that H+
+1 (which is a string graph) contains a full subdivision of K3,3, which
+contradicts Proposition 18.
+Proof of Theorem 4: Lemma 16 and Theorem 5 together imply the theorem.
+6
+Conclusion
+In this paper, we derived upper bounds on the isometric path antichain cover number of three seemingly
+(structurally) different classes of graphs, namely hyperbolic graphs, (theta,pyramid,prism)-free graphs
+and outerstring graphs. We have not made any efforts in reducing the constants in our bounds. In
+particular, we believe that a careful analysis of the structure of outerstring graphs would help in reducing
+its isometric path antichain cover number. (Note that outerstring graphs may contain a theta, 2-pyramid
+or a 2-prism). We note that the isometric path antichain cover number of a (n × n)-grid is Ω(n), which
+implies that the isometric path antichain cover number of planar graphs (which are also string graphs)
+is not bounded. Similarly, we note that the isometric path antichain cover number of G1, G2 and G3
+are unbounded where G1 denotes the class of (theta, prism)-free graphs, G2 denotes the class of (prism,
+pyramid)-free graphs and G3 denotes the class of (theta, pyramid)-free graphs. An interesting direction
+10
+
+of research is to generalise the properties of hyperbolic graphs to graphs with bounded isometric path
+antichain cover number.
+We also note that recognizing graphs with a given value of isometric path antichain cover number
+might be computationally hard. This problem does not seem to be in NP: to certify that a graph has
+isometric path antichain cover number at most k, (intuitively) one would need to check, for all possible
+isometric paths, that it does not contain any antichain of size k + 1 (with respect to all possible roots r).
+On the contrary, it is in coNP: to certify that its isometric path antichain cover number is not at most k,
+one may exhibit, for every possible root r, one isometric path and one antichain of size k + 1 contained
+in the path. Checking the validity of this certificate can be done in polynomial time. We do not know if
+the problem is coNP-hard. Nevertheless, this parameter seems interesting from a structural graph theory
+point of view, since it encapsulates several seemingly unrelated graph classes with, as a consequence,
+common algorithmic behaviours of these classes (recall that the value of the parameter does not need to
+be computed for the approximation algorithm to work). Using our framework, perhaps other common
+properties of these classes could be exhibited?
+Our results imply a constant factor approximation algorithm for Isometric Path Cover on hyper-
+bolic graphs, (theta, pyramid, prism)-free graphs and outerstring graphs. However, the existence of a
+constant factor approximation algorithm for Isometric Path Cover on general graphs is not known
+(it was observed that the algorithm from [7] also used here, can have non-constant approximation ratios,
+for example on hypercube graphs, whose isometric path antichain cover numbers are unbounded).
+Polynomial-time solvability of Isometric Path Cover on restricted graph classes like split graphs,
+interval graphs, planar graphs etc. also remains unknown, see [7].
+Acknowledgement: We thank Nicolas Trotignon for suggesting us to study the class of (t-theta, t-
+pyramid, t-prism)-free graphs.
+References
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+13
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf,len=697
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='00278v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='CO]  31 Dec 2022  Isometric path antichain covers: beyond hyperbolic graphs∗  Dibyayan Chakraborty†  Florent Foucaud‡  January 3, 2023  Abstract  The isometric path antichain cover number of a graph G, denoted by ipacc (G), is a graph pa-  rameter that was recently introduced to provide a constant factor approximation algorithm for Iso-  metric Path Cover, whose objective is to cover all vertices of a graph with a minimum number  of isometric paths (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' shortest paths between their end-vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' This parameter was previously  shown to be bounded for chordal graphs and, more generally, for graphs of bounded chordality and  bounded treelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' In this paper, we show that the isometric path antichain cover number remains  bounded for graphs in three seemingly unrelated graph classes, namely, hyperbolic graphs, (theta,  prism, pyramid)-free graphs, and outerstring graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Hyperbolic graphs are extensively studied in  Metric Graph Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' The class of (theta, prism, pyramid)-free graphs are extensively studied in  Structural Graph Theory, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' in the context of the Strong Perfect Graph Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' The class of  outerstring graphs is studied in Geometric Graph Theory and Computational Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Our results  imply a constant factor approximation algorithm for Isometric Path Cover on all the above graph  classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Our results also show that the distance functions of these (structurally) different graph classes  are more similar than previously thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  1  Introduction  A path is isometric if it is a shortest path between its endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' An isometric path cover of a graph G  is a set of isometric paths such that each vertex of G belongs to at least one of the paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' The isometric  path number of G is the smallest size of an isometric path cover of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Given a graph G and an integer k,  the objective of the algorithmic problem Isometric Path Cover is to decide if there exists an isometric  path cover of cardinality at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Isometric Path Cover has been introduced and studied in the  context of pursuit-evasion games [1, 2] and used in the context of Product Structure Theorems [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  The goal of this paper is to continue the study of approximation algorithms for Isometric Path  Cover on several graph classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  We do so by continuing the study of a recently introduced graph  parameter which seems interesting in its own right, as it encapsulates several previously unrelated graph  classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Isometric Path Cover has also been studied from a structural point of view: the cardinalities  of the optimal solution have been determined for square grids [17], hypercubes [18], complete r-partite  graphs [24] and Cartesian products of complete graphs [24], and it was recently proved that the pathwidth  of a graph is always upper-bounded by the size of its smallest isometric path cover [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' However, until  recently the algorithmic aspects of Isometric Path Cover remained unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' The problem is easy  to solve on trees and more generally, on block graphs [23] but remains hard on chordal graph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' graphs  without any induced cycle of length at least 4 [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' It can be approximated in polynomial time within a  factor of log(d) for graphs of diameter d by a greedy algorithm [27] and solved in polynomial time for every  ∗This  research  was  partially  financed  by  the  IFCAM  project  “Applications  of  graph  homomorphisms”  (MA/IFCAM/18/39), the ANR project GRALMECO (ANR-21-CE48-0004) and the French government IDEX-ISITE ini-  tiative 16-IDEX-0001 (CAP 20-25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  †Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France  ‡Université Clermont-Auvergne, CNRS, Mines de Saint-Étienne, Clermont-Auvergne-INP, LIMOS, 63000 Clermont-  Ferrand, France  1  fixed value of k by an XP algorithm [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' In a quest to find constant factor approximation algorithms  for Isometric Path Cover, Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' [7] introduced a parameter called the isometric path  antichain cover number of graphs, denoted by ipacc (G) (see Section 2 for a definition) and proved a  result directly implying the following (see [7, Proposition 10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Proposition 1 ([7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' For a graph G, if ipacc (G) ≤ c, then Isometric Path Cover admits a polynomial-  time c-approximation algorithm on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Proposition 1 is proved by a simple approximation algorithm described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' For each vertex r  of the graph, perform a Breadth-First Search at this vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Remove edges joining any vertices at the  same distance from r, and orient all edges towards r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' The resulting directed acyclic graph can be seen as  the Hasse diagram of a poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Compute a chain covering of that poset using classic methods related to  Dilworth’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' The chains are the isometric paths of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Keep the smallest of all solutions  over all choices of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Using Proposition 1, the above algorithm was shown to be a constant factor approximation algo-  rithm for many graph classes, including interval graphs, chordal graphs, and more generally, graphs with  bounded treelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Indeed, on all these graph classes, the isometric path antichain cover number is  shown to be bounded by a constant (note that one does not need to compute this parameter for the  algorithm to function: it serves only in the analysis of the approximation ratio of the algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' As  noted in [7], this parameter may be unbounded on general graphs, for example for the class of hypercubes  or square grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  In this paper, we continue to study the boundedness of the isometric path antichain cover number  of various graph classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Specifically, we consider three structurally unrelated graph classes, namely,  hyperbolic graphs, (theta, prism, pyramid)-free graphs, and outerstring graphs, which extends the above  work to strictly larger graph classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Hyperbolic graphs: A graph G is said to be δ-hyperbolic [19] if for any four vertices u, v, x, y, the two  larger of the three distance sums d (u, v)+d (x, y), d (u, x)+d (v, y) and d (u, y)+d (v, x) differ by at most  2δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' A graph class G is hyperbolic if there exists a constant δ such that every graph G ∈ G is δ-hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  This parameter was first introduced by Gromov in the context of automatic groups [19] in relation with  their Cayley graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' The hyperbolicity of a tree is 0, and in general, hyperbolicity seems to measure  how much the distance function of a graph deviates from a tree metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Many structurally defined graph  classes like chordal graphs, cocomparability graphs, asteroidal-triple free graphs, graphs with bounded  chordality or treelength are hyperbolic graphs [8, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Moreover, hyperbolicity has been found to capture  important properties of several large practical graphs such as the Internet [26] or database relations [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Due to its importance in discrete mathematics, algorithms, metric graph theory, researchers have studied  various algorithmic aspects of hyperbolic graphs [8, 12, 9, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Note that graphs with diameter 2 are  hyperbolic, which may contain any graph as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  (theta, prism, pyramid)-free graphs: A theta is a graph made of three vertex-disjoint induced paths  P1 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' b, P2 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' b, P3 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' b of lengths at least 2, and such that no edges exist between the paths  except the three edges incident to a and the three edges incident to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' See Figure 2 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' A  pyramid is a graph made of three induced paths P1 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' b1, P2 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' b2, P3 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' b3, two of which  have lengths at least 2, vertex-disjoint except at a, and such that b1b2b3 is a triangle and no edges exist  between the paths except those of the triangle and the three edges incident to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' A prism is a graph  made of three vertex-disjoint induced paths P1 = a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' b1, P2 = a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' b2, P3 = a3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' b3 of lengths at  least 1, such that a1a2a3 and b1b2b3 are triangles and no edges exist between the paths except those of  the two triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' A graph G is (theta, pyramid, prism)-free if G does not contain any induced subgraph  isomorphic to a theta, pyramid or prism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' A graph is a 3-path configuration if it is a theta, pyramid or  prism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' The study of 3-path configurations dates back to the works of Watkins and Meisner [32] in 1967  and plays “special roles” in the proof of the celebrated Strong Perfect Graph Theorem [10, 14, 28, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Important graph classes like chordal graphs, circular arc graphs, universally-signable graphs [11] exclude  all 3-path configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Popular graph classes like perfect graphs, even hole-free graphs exclude some  2  Bounded isometric path  antichain cover number  bounded hyperbolicity *  (t-theta, t-prism, t-pyramid)-  free *  Outerstring *  circle *  (theta,prism,pyramid)-  free *  Universally signable *  bounded tree-length  bounded chordality  bounded diameter  chordal  AT-free  Interval  circular arc *  Permutation  Figure 1: Inclusion diagram for graph classes discussed here (and related ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' If a class A has an  upward path to class B, then A is included in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' For graphs in the gray classes, the complexity of  Isometric Path Cover is open;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' for all other graph classes, it is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' For all shown graph  classes, Isometric Path Cover is constant-factor approximable in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Constant factor  approximation algorithms for Isometric Path Cover on graph classes marked with * are contributions  of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  b  a  b1  b2  b3  a  b1  b2  b3  a1  a2  a3  (a)  (b)  (c)  (d)  Figure 2: (a) Theta, (b) Pyramid, (c) Prism, (d) Outerstrings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' The figure shows that the graph K2,3,  which is also a theta, is an outerstring graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  of the 3-path configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Note that, (theta, prism, pyramid)-free graphs are not hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' To see  this, consider a cycle C of order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Clearly, C excludes all 3-path configurations and has hyperbolicity  Ω(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Outerstring graphs: A set S of simple curves on the plane is grounded if there exists a horizontal line  containing one endpoint of each of the curves in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' A graph G is an outerstring graph if there is a collection  C of grounded simple curves and a bijection between V (G) and C such that two curves in S if and only  if the corresponding vertices are adjacent in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' See Figure 2(d) for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' The term “outerstring  graph” was first used in the early 90’s [22] in the context of studying intersection graphs of simple curves  on the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Many well-known graph classes like chordal graphs, circular arc graphs, circle graphs  (intersection graphs of chords of a circle), or cocomparability graphs are also outerstring graphs and  thus, motivated researchers from the geometric graph theory and computational geometry communities  to study algorithmic and structural aspects of outerstring graphs and its subclasses [4, 5, 6, 20, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Note  that, in general, outerstring graphs may contain a prism, pyramid or theta as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Moreover, cycles of arbitrary order are outerstring graphs, implying that outerstring graphs are not  hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  It is clear from the above discussion that the classes of hyperbolic graphs, (theta, prism, pyramid)-free  3  graphs, and outerstring graphs are pairwise incomparable (with respect to the containment relationship).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='1  Our contributions  The main contribution of this paper is to show that the isometric path antichain cover number (see  Section 2 for a definition) remains bounded on hyperbolic graphs, (theta, pyramid, prism)-free graphs,  and outerstring graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Specifically, we prove the following theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let G be a graph with hyperbolicity δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Then, ipacc (G) ≤ 12δ + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let G be a (theta, pyramid, prism)-free graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Then, ipacc (G) ≤ 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let G be an outerstring graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Then, ipacc (G) ≤ 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  To the best of our knowledge, the isometric path antichain cover number being bounded (by con-  stant(s)) is the only known non-trivial property shared by any two or all three of these graph classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  To provide a unified proof of Theorem 3 and 4, we study a more general graph class called (t-theta,  t-pyramid, t-prism)-free graphs [29] (see Section 4 for definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' When t = 1, (t-theta, t-pyramid, t-  prism)-free graphs are exactly (theta, prism, pyramid)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Moreover, we show that all outerstring  graphs are (4-theta, 4-pyramid, 4-prism)-free graphs (Lemma 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' We prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' For t ≥ 1, let G be a (t-theta, t-pyramid, t-prism)-free graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Then ipacc (G) ≤ 8t + 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Due to Proposition 1 and the above theorems, we also have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' There is an approximation algorithm for Isometric Path Cover with approximation  ratio  (a) 12δ + 6 on δ-hyperbolic graphs,  (b) 73 on (theta, prism, pyramid)-free graphs, and  (c) 95 on outerstring graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  (d) 8t + 63 on (t-theta, t-pyramid, t-prism)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Organisation: In Section 2, we introduce the recall some definitions and some results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' In Section 3 we  prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' In Section 4, we prove Theorems 3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' In Section 5, we prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' We  conclude in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  2  Definitions and preliminary observations  In this section, we formally recall the definition of isometric path antichain cover number of graphs from  [7] and some related observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' A sequence of distinct vertices forms a path P if any two consecutive  vertices are adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Whenever we fix a path P of G, we shall refer to the subgraph formed by the  edges between the consecutive vertices of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' The length of a path P, denoted by |P|, is the number of  its vertices minus one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' A path is induced if there are no graph edges joining non-consecutive vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  In a directed graph, a directed path is a path in which all arcs are oriented in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' For a  path P of a graph G between two vertices u and v, the vertices V (P) \\ {u, v} are internal vertices of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  A path between two vertices u and v is called a (u, v)-path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Similarly, we have the notions of isometric  (u, v)-path and induced (u, v)-path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' For a vertex r of G and a set S of vertices of G, the distance of S from  r, denoted as d (r, S), is the minimum of the distance between any vertex of S and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' For a subgraph  H of G, the distance of H w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' r is d (r, V (H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Formally, we have d (r, S) = min{d (r, v) : v ∈ S} and  d (r, H) = d (r, V (H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  4  For a graph G and a vertex r ∈ V (G), consider the following operations on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' First, remove all  edges xy from G such that d (r, x) = d (r, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Let G′  r be the resulting graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Then, for each edge  e = xy ∈ E(G′  r) with d (r, x) = d (r, y) − 1, orient e from y to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let −→  Gr be the directed acyclic graph  formed after applying the above operation on G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Note that this digraph can easily be computed in linear  time using a Breadth-First Search (BFS) traversal with starting vertex r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  The following definition is inspired by the terminology of posets (as the graph −→  Gr can be seen as the  Hasse diagram of a poset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' For a graph G and a vertex r ∈ V (G), two vertices x, y ∈ V (G) are antichain vertices if  there are no directed paths from x to y or from y to x in −→  Gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' A set X of vertices of G is an antichain  set if any two vertices in X are antichain vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' The cardinality of the largest antichain set in −→  Gr will  be denoted by β  �−→  Gr  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' The cardinality of the largest antichain set of G, is defined as  β (G) = min  �  β  �−→  Gr  �  : r ∈ V (G)  �  Definition 8 ([7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let r be a vertex of a graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' For a subgraph H, Ar (H) shall denote the maximum  antichain set of H in −→  Gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' The isometric path antichain cover number of −→  Gr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' denoted by ipacc  �−→  Gr  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' is  defined as follows:  ipacc  �−→  Gr  �  = max {|Ar (P) |: P is an isometric path}  The isometric path antichain cover number of graph G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' denoted as ipacc (G),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' is defined as the minimum  over all possible antichain covers of its associated directed acyclic graphs:  ipacc (G) = min  �  ipacc  �−→  Gr  �  : r ∈ V (G)  �  We recall the proof of the following proposition from [7] which will be used heavily in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Proposition 9 ([7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let G be a graph and r, an arbitrary vertex of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Consider the directed acyclic  graph −→  Gr, and let P be an isometric path between two vertices x and y in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Then |P| ≥ |d (r, x) −  d (r, y) | + |Ar (P) | − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Orient the edges of P from y to x in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' First, observe that P must contain a set E1 of oriented  edges such that |E1| = |d (r, y) − d (r, x) | and for any −→  ab ∈ E1, d (r, a) = d (r, b) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let the vertices of  the largest antichain set of P in −→  Gr, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=', Ar (P), be ordered as a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' , at according to their occurrence  while traversing P from y to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' For i ∈ [2, t], let Pi be the subpath of P between ai−1 and ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Observe that  for any i ∈ [2, t], since ai and ai−1 are antichain vertices, there must exist an oriented edge −→  bici ∈ E(Pi)  such that either d (r, bi) = d (r, ci) or d (r, bi) = d (r, ci) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Let E2 = {bici}i∈[2,t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Observe that  E1 ∩ E2 = ∅ and therefore |P| ≥ |E1| + |E2| = |d (r, y) − d (r, x) | + |Ar (P) | − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  3  Proof of Theorem 2  In this section, we shall show that isometric path antichain cover number of graphs with hyperbolicity  at most δ is at most 12δ + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' To achieve our goal we need to recall a few definitions from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  For three vertices x, y, z of a graph G, a geodesic triangle [3], denoted as ∆(x, y, z) is the union P(x, y) ∪  P(y, z)∪P(x, z) of three isometric paths connecting these vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' A geodesic triangle ∆(x, y, z) is called  ρ-slim if for any vertex u ∈ P(x, y) the distance d (u, P(y, z) ∪ P(x, z)) is at most ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' The smallest value  of ρ for which every geodesic triangle of G is ρ-slim is called the slimness of G and is denoted by sl (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  In the following lemma, we shall show that if the isometric path antichain cover number of a graph is  large then so is the slimness of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' For any graph G, ipacc (G) ≤ 4sl (G) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  5  u  v  c  c′  Figure 3: An example of a 4-fat turtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let C be the cycle induced by the black vertices, P be the path  induced by the white vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Then the tuple (4, C, P, c, c′) defines a 4-fat turtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let ρ = sl (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Aiming for a contradiction, let r be a vertex of G such that there exists an  isometric path P such that |Ar (P) | ≥ 4ρ + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let the vertices of Ar (P) be named and ordered as  a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' , a2ρ+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' , a4ρ+3 as they are encountered while traversing P from one end-vertex to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Let x = a1, y = a4ρ+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let −→  Px be an oriented path from x to r in −→  Gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Observe that Px, the path of  G obtained by removing the orientation of −→  Px, is an (x, r)-isometric path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let −→  Py be an oriented path  from y to r in −→  Gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Similarly, Py, the path of G obtained by removing the orientation of −→  Py, is an (y, r)-  isometric path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Observe that P, Px, Py form a geodesic triangle with x, r, y as end-vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Consider the  vertex z = a2ρ+2 on the path P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Since ρ = sl (G), there exists a vertex w ∈ V (Px) ∪ V (Py) such that  d (w, z) ≤ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Without loss of generality, assume w ∈ V (Px).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Then, d (x, z) ≤ d (x, w) + d (w, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' By  using that d (r, z) ≤ d (r, w) + d (w, z) ≤ d (r, w) + ρ, we get d (x, z) ≤ |d (r, x) − d (r, z) | + 2ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' But this  contradicts Proposition 9, due to which we have d (x, z) ≥ |d (r, x) − d (r, z) | + 2ρ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Now we shall use the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Proposition 11 ([3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' For any graph G, sl (G) ≤ 3hb (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Proposition 11 and Lemma 10, imply the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  4  Proofs of Theorem 3 and 5  In this section, we shall prove Theorems 3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' First we shall define the notions of t-theta, t-prism,  and t-pyramid [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  For an integer t ≥ 1, a t-prism is a graph made of three vertex-disjoint induced paths P1 = a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' b1,  P2 = a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' b2, P3 = a3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' b3 of lengths at least t, such that a1a2a3 and b1b2b3 are triangles and no edges  exist between the paths except those of the two triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' For an integer t ≥ 1, a t-pyramid is a graph  made of three induced paths P1 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' b1, P2 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' b2, P3 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' b3 of lengths at least t, two of which  have lengths at least t + 1, they are pairwise vertex-disjoint except at a, such that b1b2b3 is a triangle  and no edges exist between the paths except those of the triangle and the three edges incident to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' For  an integer t ≥ 1, a t-theta is a graph made of three internally vertex-disjoint induced paths P1 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' b,  P2 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' b, P3 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' b of lengths at least t+1, and such that no edges exist between the paths except the  three edges incident to a and the three edges incident to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' A graph G is (t-theta, t-pyramid, t-prism)-free  if G does not contain any induced subgraph isomorphic to a t-theta, t-pyramid or t-prism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' When t = 1,  (t-theta, t-pyramid, t-prism)-free graphs are exactly (theta, prism, pyramid)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Now we shall show that the isometric path antichain cover number of (t-theta, t-pyramid, t-prism)-  free graphs are bounded above by a linear function on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' We shall show that, when the isometric path  antichain cover number of a graph is large, the existence of a structure called “t-fat turtle” (defined  below) as an induced subgraph is forced, which, cannot be present in a ((t − 1)-theta, (t − 1)-pyramid,  (t − 1)-prism)-free graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' For an integer t ≥ 1, a “t-fat turtle” consists of a cycle C and an induced (u, v)-path P  of length at least t such that all of the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  6  (a) V (P) ∩ V (C) = ∅,  (b) For any vertex w ∈ (V (P) \\ {u, v}), N(w) ∩ V (C) = ∅ and both u and v have at least one neighbour  in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  (c) For any vertex w ∈ N(u) ∩ V (C) and w′ ∈ N(v) ∩ V (C), the distance between w and w′ in C is at  least t,  (d) There exist two vertices {c, c′} ⊂ V (C) and two distinct components Cu, Cv of C − {c, c′} such that  N(u) ∩ V (C) ⊆ V (Cu) and N(v) ∩ V (C) ⊆ V (Cv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  The tuple (t, C, P, c, c′) defines the t-fat turtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' See Figure 3 for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  In the following observation, we show that any (t-theta, t-pyramid,t-prism)-free graph cannot contain  a (t + 1)-fat turtle as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' For some integer t ≥ 1, let G be a graph containing a (t + 1)-fat turtle as an induced  subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Then G is not (t-theta, t-pyramid, t-prism)-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let (t+1, C, P, c, c′) be a (t+1)-fat turtle in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let the vertices of C be named c = a0, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' , ak =  c′, ak+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' , a|V (C)| as they are encountered while traversing C starting from c in a counter-clockwise  manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Denote by u, v the end-vertices of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' By definition, there exist two distinct components Cu, Cv of  C−{c, c′} such that N(u)∩V (C) ⊆ V (Cu) and N(v)∩V (C) ⊆ V (Cv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Without loss of generality, assume  V (Cu) = {a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' , ak−1} and V (Cv) = {ak+1, ak+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' , a|V (C)|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let i− and i+ be the minimum and  maximum indices such that ai− and ai+ are adjacent to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let j− and j+ be the minimum and maximum  indices such that aj− and aj+ are adjacent to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' By definition, i− ≤ i+ < j− ≤ j+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let P1 be the  (ai−, aj+)-subpath of C containing c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let P2 be the (ai+, aj−)-subpath of C that contains c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Observe  that P1 and P2 have length at least t (by definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Now we show that P, P1, P2 together form one of  theta, pyramid or prism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' If ai− = ai+ and aj− = aj+, then P, P1, P2 form a t-theta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' If i− ≤ i+ − 2 and  j− ≤ j+ − 2, then also P, P1, P2 form a t-theta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' If j− = j+ − 1 and i− = i+ − 1, then P, P1, P2 form a  t-prism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' In any other case, P, P1, P2 form a t-pyramid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  In the remainder of this section, we shall prove that there exists a linear function f(t) such that if  the isometric path antichain cover number of a graph is more than f(t), then G is forced to contain a  (t + 1)-fat turtle as an induced subgraph, and therefore is not (t-theta, t-pyramid,t-prism)-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' We shall  use the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Observation 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let G be a graph, r be an arbitrary vertex, P be an isometric (u, v)-path in G and Q  be a subpath of an isometric (v, r)-path in G such that one endpoint of Q is v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let P ′ be the maximum  (u, w)-subpath of P such that no internal vertex of P ′ is a neighbour of some vertex of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' We have that  |Ar (P ′) | ≥ |Ar (P) | − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Suppose |Ar (P ′) | ≤ |Ar (P) | − 4 and consider the (w, v)-subpath, say P ′′, of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Observe that  |Ar (P ′′) | ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Now let w′ be a vertex of Q which is a neighbour of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Observe that |d (r, w)−d (r, w′) | ≤ 1  and therefore d (w, v) = |E(P ′′)| ≤ |d (r, w)−d (r, v) |+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' But this contradicts Proposition 9, which implies  that the length of P ′′ is at least |d (r, w) − d (r, v) | + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' For an integer t ≥ 1, let G be a graph with ipacc (G) ≥ 8t + 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Then G has a (t + 1)-fat  turtle as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let r be a vertex of G such that ipacc  �−→  Gr  �  is at least 8t+64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Then there exists an isometric path  P such that |Ar (P) | ≥ 8t+ 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let the two endpoints of P be a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' (See Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=') Let u be a vertex  of P such that d (r, u) = d (r, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let Pau be the (a, u)-subpath of P and Pbu be the (b, u)-subpath of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Both Pau and Pbu are isometric paths and observe that either |Ar (Pau) | ≥ 4t+32 or |Ar (Pbu) | ≥ 4t+32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Without loss of generality, assume that |Ar (Pbu) | ≥ 4t + 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let Qr  b be an isometric (b, r)-path in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  7  r  z2  w2  u  z  z1  w  b  w1  c  (= a2t+13)  x  c1  a  c2  T (c1, c2)  ≥ t  ≥ t  ≥ t  Qr  b  Qr  u  Figure 4: Illustration of the notations used in the proof of Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Let Ruw be the maximum (u, w)-subpath, of Pbu such that no internal vertex of Ruw is a neighbour  of Qr  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Note that Ruw is an isometric path and w has a neighbour in Qr  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Applying Observation 14, we  have the following:  Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' |Ar (Ruw) | ≥ 4t + 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Let Qr  u be any isometric (u, r)-path of G and let Rzw be the maximum (z, w)-subpath of Ruw such  that no internal vertex of Rzw has a neighbour in Qr  u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Observe that Rzw is an isometric path, and z has  a neighbour in Qr  u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Again applying Observation 14, we have the following:  Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' |Ar (Rzw) | ≥ 4t + 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Let a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' , ak be the vertices of Ar (Rzw) ordered according to their appearance while traversing  Rzw from z to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Due to Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='2, we have that k ≥ 4t + 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let c = a2t+13 and Qr  c denote an  isometric (c, r)-path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let T (r, c1) be the maximum subpath of Qr  c such that no internal vertex of T (r, c1)  is adjacent to any vertex of Rzw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let x be a neighbor of c1 in Rzw, X be the (x, b)-subpath of Pub and Y be the (x, u)-subpath  of Pub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Then |Ar (X) | ≥ 2t + 11 and |Ar (Y ) | ≥ 2t + 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let Rcw denote the (c, w)-subpath of Rzw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Observe that |Ar (Rcw) | ≥ 2t + 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' First, consider  the case when x lies in the (z, c)-subpath of Rzw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' In this case, Rcw is a subpath of X and therefore  |Ar (X) | ≥ 2t + 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Now consider the case when x lies in Rcw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' In this case, applying Observation 14,  we have that |Ar (X) | ≥ |Ar (Rcw) | − 3 ≥ 2t + 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Using a similar argument, we have that |Ar (Y ) | ≥  2t + 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Let T (c1, c2) be the maximum (c1, c2)-subpath of T (c1, r) such that no internal vertex of T (c1, c2) is  adjacent to a vertex of Qr  b or Qr  u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' We have the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' The length of T (c1, c2) is at least t + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Assume that the length of T (c1, c2) is at most t + 2 and x be a neighbour of c1 in Rzw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Observe  that all vertices of Rzw are at distance at least d (r, u) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' d (r, Rzw) ≥ d (r, u), since d (r, u) = d (r, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Hence,  (+) d (r, x) ≥ d (r, u) and d (r, c1) ≥ d (r, u) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  8  Now, suppose c2 has a neighbor c3 in Qr  u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Hence d (c3, x) ≤ d (c3, c2) + d (c2, c1) + d (c1, x) ≤ t + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Now, using (+) and the fact that c3 lies on an isometric (r, u)-path (Qr  u), we have that d (c3, u) ≤ t + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Therefore, d (u, x) ≤ d (c3, u) + d (c3, x) ≤ 2t + 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' But this contradicts Proposition 9 and Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='3, as  they together imply that d (u, x) is at least d (r, x) − d (r, u) + 2t + 10≥ 2t + 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Hence, c2 must have a neighbour c3 in Qr  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' First, assume that d (r, x) ≥ d (r, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Then, as d (c3, x) ≤  d (c3, c2) + d (c2, c1) + d (c1, x) ≤ t + 4 and c3 lies on an isometric (r, b)-path (Qr  b), we have that d (x, b) ≤  2t + 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' But again this contradicts Proposition 9 and Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='3, as they together imply that the length  of d (x, b) is at least d (r, x) − d (r, u) + 2t + 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Now, assume that d (r, x) < d (r, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let b′ be a vertex of  Qr  b such that d (r, b′) = d (r, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Using a similar argumentation as before, we have that d (x, b′) ≤ 2t + 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Hence, d (x, b) ≤ d (x, b′) + d (b′, b) ≤ d (r, b) − d (r, x) + 2t + 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' But this contradicts Proposition 9 which,  due to Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='3, implies that d (x, b) ≥ d (r, b) − d (r, x) + 2t + 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  The path T (c1, c2) forms the first ingredient to extract a (t + 1)-fat turtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let z1 be the neighbor of  z in Qr  u and w1 be the neighbour of w in Qr  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' We have the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' The vertices w1 and z1 are non adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Recall that z1 lies in Qr  u and d (r, z) ≥ d (r, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Hence z1 must be a neighbor of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' If w1 and z1 are  adjacent, then observe that d (u, b) ≤ d (r, b) − d (r, w1) + 2 ≤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' This implies d (u, b) ≤ d (r, b) − d (r, u)+ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  But this shall again contradict Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Now we shall construct a (w1, z1)-path as follows: Consider the maximum (w1, w2)-subpath, say  T (w1, w2), of Qr  b such that no internal vertex of T (w1, w2) has a neighbour in Qr  u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Similarly, consider the  maximum (z1, z2)-subpath, say T (z1, z2), of Qr  b such that no internal vertex of T (z1, z2) is a neighbor of  w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let T be the path obtained by taking the union of T (w1, w2) and T (z1, z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Observe that z2 must  be a neighbour of w2 and T is an induced (w1, z1)-path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' The definitions of T and Rzw imply that their  union induces a cycle Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Here we have the second and final ingredient to extract the (t + 1)-fat turtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Suppose that c2 has a neighbour in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let T ′ be the maximum subpath of T (c1, c2) which is vertex-  disjoint from Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Due to Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='4, the length of T ′ is at least t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let e1 and e2 be the end-vertices  of T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Observe the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Each of e1 and e2 has at least one neighbor in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Z −{z, w} contains two distinct components C1, C2 such that for i ∈ {1, 2}, N(ei)∩V (Z) ⊆ V (Ci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  For a vertex e′  1 ∈ N(e1) ∩ V (Z) and e′  2 ∈ N(e2) ∩ V (Z), the distance between e′  1 and e′  2 is at least  t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' This statement follows from Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Hence, we have that the tuple (t + 1, Z, T ′, z, w) defines a (t + 1)-fat turtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Now consider the case  when c2 does not have a neighbor in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' By definition, c2 has at least one neighbor in Qr  u or Qr  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Without  loss of generality, assume that c2 has a neighbor c3 in Qr  u such that the (z2, c3)-subpath, say, T ′′ of Qr  u  has no neighbor of c2 other than c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Observe that the path T ∗ = (T ′ ∪ (T ′′ − {z2})) is vertex-disjoint  from Z and has length at least t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let e1, e2 be the two end-vertices of T ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Observe the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Each of e1 and e2 has at least one neighbor in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Z −{z, w} contains two distinct components C1, C2 such that for i ∈ {1, 2}, N(ei)∩V (Z) ⊆ V (Ci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  For a vertex e′  1 ∈ N(e1) ∩ V (Z) and e′  2 ∈ N(e2) ∩ V (Z), the distance between e′  1 and e′  2 is at least  t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' This statement follows from Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Hence, (t + 1, Z, T ∗, z, w) is a (t + 1)-fat turtle  Proof of Theorem 5 and 3: Lemma 13 and 15 together imply the theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  9  5  Proof of Theorem 4  Next we shall show that outerstring graphs are (4-theta, 4-prism, 4-pyramid)-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let G be an outerstring graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Then, G is (4-theta, 4-prism, 4-pyramid)-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' To prove the lemma, we shall need to recall a few definitions and results from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' A  graph G is a string graph if there is a collection S of simple curves on the plane and a bijection between  V (G) and S such that two curves in S intersect if and only if the corresponding vertices are adjacent in  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let G be a graph with an edge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' The graph G \\ e is obtained by contracting the edge e into a single  vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Observe that string graphs are closed under edge contraction [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' We shall use the following  result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Proposition 17 ([22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let G be an outerstring graph with an edge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Then G\\e is an outerstring graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  A full subdivision of a graph means replacing each edge of G with a new path of length at least two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  We shall use the following result implied from Theorem 1 of [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Proposition 18 ([22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let G be a string graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Then G does not contain a full subdivision of K3,3 as  an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  For a graph G, the graph G+ is constructed by introducing a new apex vertex a and connecting a  with all vertices of G by new copies of paths of length at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' We shall use the following result of  Biedl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Proposition 19 (Lemma 1, [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' A graph G is an outerstring graph if and only if G+ is a string graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Now we are ready to prove the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Let G be an outerstring graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Assume for the sake of  contradiction that G contains an induced subgraph H which is a 4-theta, 4-pyramid, or a 4-prism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Since  every induced subgraph of an outerstring graph is also an outerstring graph, we have that H is an  outerstring graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let E be the set of edges of H whose both endpoints are part of some triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Now  consider the graph H1 = H \\ E which is obtained by contracting all edges in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' By Proposition 17, H1  is an outerstring graph and it is easy to check that H1 is a 3-theta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Let u and v be the vertices of H1  with degree 3 and w1, w2, w3 be the set of mutually non-adjacent vertices such that for each i ∈ {1, 2, 3}  d (u, wi) = 2 and d (v, wi) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Since H1 is a 3-theta, w1, w2, w3 exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Now consider the graph H+  1 and  a be the new apex vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Due to Proposition 19, we have that H+  1 is a string graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' But notice that,  for each pair of vertices in {x, y} ⊂ {w1, w2, w3, u, v, a}, there exists a unique path of length at least 2  connecting x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' This implies that H+  1 (which is a string graph) contains a full subdivision of K3,3, which  contradicts Proposition 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Proof of Theorem 4: Lemma 16 and Theorem 5 together imply the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  6  Conclusion  In this paper, we derived upper bounds on the isometric path antichain cover number of three seemingly  (structurally) different classes of graphs, namely hyperbolic graphs, (theta,pyramid,prism)-free graphs  and outerstring graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' We have not made any efforts in reducing the constants in our bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' In  particular, we believe that a careful analysis of the structure of outerstring graphs would help in reducing  its isometric path antichain cover number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' (Note that outerstring graphs may contain a theta, 2-pyramid  or a 2-prism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' We note that the isometric path antichain cover number of a (n × n)-grid is Ω(n), which  implies that the isometric path antichain cover number of planar graphs (which are also string graphs)  is not bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Similarly, we note that the isometric path antichain cover number of G1, G2 and G3  are unbounded where G1 denotes the class of (theta, prism)-free graphs, G2 denotes the class of (prism,  pyramid)-free graphs and G3 denotes the class of (theta, pyramid)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' An interesting direction  10  of research is to generalise the properties of hyperbolic graphs to graphs with bounded isometric path  antichain cover number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  We also note that recognizing graphs with a given value of isometric path antichain cover number  might be computationally hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' This problem does not seem to be in NP: to certify that a graph has  isometric path antichain cover number at most k, (intuitively) one would need to check, for all possible  isometric paths, that it does not contain any antichain of size k + 1 (with respect to all possible roots r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  On the contrary, it is in coNP: to certify that its isometric path antichain cover number is not at most k,  one may exhibit, for every possible root r, one isometric path and one antichain of size k + 1 contained  in the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Checking the validity of this certificate can be done in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' We do not know if  the problem is coNP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Nevertheless, this parameter seems interesting from a structural graph theory  point of view, since it encapsulates several seemingly unrelated graph classes with, as a consequence,  common algorithmic behaviours of these classes (recall that the value of the parameter does not need to  be computed for the approximation algorithm to work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' Using our framework, perhaps other common  properties of these classes could be exhibited?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Our results imply a constant factor approximation algorithm for Isometric Path Cover on hyper-  bolic graphs, (theta, pyramid, prism)-free graphs and outerstring graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' However, the existence of a  constant factor approximation algorithm for Isometric Path Cover on general graphs is not known  (it was observed that the algorithm from [7] also used here, can have non-constant approximation ratios,  for example on hypercube graphs, whose isometric path antichain cover numbers are unbounded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Polynomial-time solvability of Isometric Path Cover on restricted graph classes like split graphs,  interval graphs, planar graphs etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content=' also remains unknown, see [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
+page_content='  Acknowledgement: We thank Nicolas Trotignon for suggesting us to study the class of (t-theta, t-  pyramid, t-prism)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAyT4oBgHgl3EQfcPdw/content/2301.00278v1.pdf'}
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+Down the Rabbit Hole: Detecting Online Extremism,
+Radicalisation, and Politicised Hate Speech
+JAROD GOVERS, ORKA Lab, Department of Software Engineering, University of Waikato, NZ
+PHILIP FELDMAN, ASRC Federal, US
+AARON DANT, ASRC Federal, US
+PANOS PATROS, ORKA Lab, Department of Software Engineering, University of Waikato, NZ
+Social media is a modern person’s digital voice to project and engage with new ideas and mobilise communities—
+a power shared with extremists. Given the societal risks of unvetted content-moderating algorithms for
+Extremism, Radicalisation, and Hate speech (ERH) detection, responsible software engineering must understand
+the who, what, when, where, and why such models are necessary to protect user safety and free expression.
+Hence, we propose and examine the unique research field of ERH context mining to unify disjoint studies.
+Specifically, we evaluate the start-to-finish design process from socio-technical definition-building and dataset
+collection strategies to technical algorithm design and performance. Our 2015-2021 51-study Systematic
+Literature Review (SLR) provides the first cross-examination of textual, network, and visual approaches
+to detecting extremist affiliation, hateful content, and radicalisation towards groups and movements. We
+identify consensus-driven ERH definitions and propose solutions to existing ideological and geographic
+biases, particularly due to the lack of research in Oceania/Australasia. Our hybridised investigation on Natural
+Language Processing, Community Detection, and visual-text models demonstrates the dominating performance
+of textual transformer-based algorithms. We conclude with vital recommendations for ERH context mining
+researchers and propose an uptake roadmap with guidelines for researchers, industries, and governments to
+enable a safer cyberspace.
+CCS Concepts: • Computing methodologies → Discourse, dialogue and pragmatics; Lexical semantics;
+Information extraction; Machine learning; Natural language processing; Knowledge representa-
+tion and reasoning; Image and video acquisition; • Applied computing → Sociology; • Social and profes-
+sional topics → Hate speech; Technology and censorship; Political speech; Governmental regulations.
+Additional Key Words and Phrases: extremism, radicalisation, machine learning, community detection, natural
+language processing, neural networks, hate speech, sociolinguistics
+ACM Reference Format:
+Jarod Govers, Philip Feldman, Aaron Dant, and Panos Patros. 2021. Down the Rabbit Hole: Detecting Online
+Extremism, Radicalisation, and Politicised Hate Speech. ACM Comput. Surv. 00, 0, Article 000 (December 2021),
+35 pages. https://doi.org/tobereplacedwhenassignedDOI
+Authors’ addresses: Jarod Govers, jg199@students.waikato.ac.nz, ORKA Lab, Department of Software Engineering, University
+of Waikato, Gate 1, Knighton Road, Hamilton, Waikato, NZ, 3216; Philip Feldman, philip.feldman@asrcfederal.com, ASRC
+Federal, Beltsville, Maryland, US; Aaron Dant, aaron.dant@asrcfederal.com, ASRC Federal, Beltsville, Maryland, US; Panos
+Patros, panos.patros@waikato.ac.nz, ORKA Lab, Department of Software Engineering, University of Waikato, Gate 1,
+Knighton Road, Hamilton, Waikato, NZ, 3216.
+Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee
+provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and
+the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored.
+Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires
+prior specific permission and/or a fee. Request permissions from permissions@acm.org.
+© 2021 Association for Computing Machinery.
+0360-0300/2021/12-ART000 $15.00
+https://doi.org/tobereplacedwhenassignedDOI
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+arXiv:2301.11579v1  [cs.SI]  27 Jan 2023
+
+000:2
+Govers et al.
+Contents
+Abstract
+1
+Contents
+2
+1
+Introduction
+3
+1.1
+Motivation and Contributions
+3
+1.2
+Structure
+4
+2
+Social Context to Social Network Analysis
+4
+2.1
+Extremism and Radicalisation Decoupled
+4
+2.2
+Hate Speech Decoupled
+5
+3
+Systematic Literature Review Design and Protocol
+6
+3.1
+Trends and Shortfalls in Prior SLRs
+6
+3.2
+Research Questions
+7
+3.3
+Databases
+7
+3.4
+Search Strings
+7
+3.5
+Inclusion and Exclusion Criteria
+8
+4
+Key Research Question (RQ) Findings
+9
+4.1
+Summary of the Social ERH Definitions Used by Researchers
+9
+4.2
+Summary of the Data Collection, Processing, and Annotation Processes
+9
+4.3
+Summary of the State-of-the-art Computational ERH Detection Models
+10
+4.4
+Summary of ERH Models’ Classification Performance
+10
+4.5
+Geographic Trends—Islamophobia and Exclusion in the Academic Community
+11
+5
+Socio-technical Context in Research—Consensus-driven Definitions
+12
+5.1
+Researchers’ Consensus-driven Definitions for ERH Concepts
+13
+5.2
+Correlation Between Definitions and Algorithmic Approach
+14
+6
+Building ERH Datasets—Collection, Processing, and Annotation
+15
+6.1
+Prominent Platforms, Pulling, and Populations
+15
+6.2
+Feature Extraction Techniques
+17
+6.3
+Data Filtering
+18
+6.4
+Key Takeaways for Dataset Domain, Pre-processing, and Annotation
+19
+7
+Community Detection, Text, and Image ERH Detection Algorithms
+20
+7.1
+Observed Non-deep Machine Learning Algorithms (MLAs)
+20
+7.2
+Deep Learning Algorithms (DLAs)
+22
+8
+Model Performance Evaluation, Validation, and Challenges
+24
+8.1
+Benchmark Dataset Performance (Inter-study Evaluation)
+25
+8.2
+Community Detection Performance
+25
+9
+Future Research Directions
+25
+9.1
+Ideological Isomorphism—a Novel Framework for Radicalisation Detection
+25
+9.2
+Morphological Mapping and Consensus-building—a novel computationally-
+grounded framework for extremism detection
+26
+9.3
+Outwards Dissemination––‘traditional’ hate speech detection updated
+27
+9.4
+Uptake Roadmap for Researchers, Industry, and Government
+28
+10
+Conclusion
+28
+References
+30
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech
+000:3
+1
+INTRODUCTION
+Online social media empowers users to communicate with friends and the wider world, organise
+events and movements, and engage with communities all at the palm of our hands. Social media
+platforms are a frequent aid for modern political exchanges and organisation [115], with extremes
+amplified by algorithmic recommendation systems, such as on Twitter [49], TikTok [87], and
+YouTube [63]. Furthermore, the semantic expression of ideas differs between social media platforms,
+with Twitter’s short 280 character limit resulting in more narcissistic and aggressive content
+compared to Facebook [75], and anonymous platforms such as 4Chan instilling a vitriolic ‘group
+think’ in political threads/‘boards’ [70]. Hateful, emotive, and ‘click-worthy’ content permeates
+virtual discourse, which can radicalise users down an ideological rabbit hole towards real-world
+violent action. The individuals behind the 2014 Isla Vista and 2019 Christchurch shootings appeared
+as individual ‘lone-wolf attacks’ without an allegiance. However, investigations found a deep
+network of perverse and violent communities across social media [40, 106]. Likewise, exploiting
+social media to plan politically motivated attacks towards the civilian population to coerce political
+change (i.e., terrorism) delegitimises democracies, social cohesion, and physical/mental health [28,
+58, 79].
+As a response, social media platforms employ text and visual content-moderation systems to
+detect and remove hate speech, extremism, and radicalising content. This paper offers a state-of-the-
+art Systematic Literature Review (SLR) on the definitions, data collection, annotation, processing,
+and model considerations for Extremism, Radicalisation, and Hate speech (ERH) detection.
+1.1
+Motivation and Contributions
+Existing ERH literature reviews exist as independent microcosms, often focusing on specific types
+of models, typically text-only Natural Language Processing (NLP) models or non-textual network
+analysis via community detection models. Studies seldom cross-examine models and evaluate the
+performance between non-textual network analysis (a ‘who-knows-who’ approach), textual, and/or
+multimedia approaches for ERH detection. While we identified ten prior literature reviews for
+hateful content detection, none consider the similarities and definitional nuance between ERH
+concepts and what Extremism, Hate Speech, or Radicalisation means in practice by researchers [2,
+5, 20, 42, 43, 51, 82, 97, 110, 114]. Evaluating the consensus for ERH definitions, dataset collection
+and extraction techniques, model choice and performance are all essential to create ethical models
+without injurious censorship or blowback.
+Through understanding the groups, beliefs, data, and algorithms behind existing content-
+moderation models—we can reliably critique often overlooked social concepts, such as algorithmic
+bias, and ensure compliance between social definitions and computational practice. Hence, this new
+field of ERH context mining extracts the context to classifications—enabling researchers, industries,
+and governments to assess the state of social discourse.
+Given the rise of state-sponsored disinformation campaigns to undermine democratic institutions
+and social media campaigns, the time is now for ERH research within politicised discussions.
+The three core contributions for this paper are:
+(1) The establishment of consensus-driven working definitions for Extremism, Radicalisation,
+and Hate Speech within the novel field of ERH context mining—and a proposed frame-
+work/roadmap for future researchers, social media platforms, and government advisors.
+(2) The critical examination of existing textual (NLP), network (community detection), and
+hybrid text-image datasets.
+(3) The identification and cross-examination of the state-of-the-art models’ performance on
+benchmark datasets and relevant challenges with the current ERH detection metrics.
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+000:4
+Govers et al.
+1.2
+Structure
+For a high-level summary of this SLR’s findings, refer to Section 4 on Key Research Question Findings,
+and Section 9 for Recommendations for Future Work. These key summaries condense and contextual-
+ise the 51 studies observed between 2015-2021, which we use to build our proposed computational
+ERH definitions and technological roadmap for researchers, industry, and government in Section 9.
+For a holistic understanding, we present a social context to our motivations in Section 2. Related
+work and areas our SLR improves on are outlined in Section 3.1. Section 3 outlines the systematic
+protocol used to collect the 51 studies between 2015-2021. Further to the summaries presented in
+Section 4, we present an in-depth analysis and cross-examination of studies definitions of ERH
+concepts in Section 5, approaches for collecting and processing data in Section 6, algorithmic
+approaches for classification in Section 7, and their performance in Section 8. We conclude with
+recommendations for future SLRs, and studies in Section 9, and conclusions in Section 10.
+2
+SOCIAL CONTEXT TO SOCIAL NETWORK ANALYSIS
+Analysing social media requires the socio-technical considerations of what constitutes hate speech,
+extremism, and radicalisation (ERH). To detect such concepts, computational models can investigate
+multimodal sources—including textual meaning and intent through Natural Language Processing
+(NLP), computer vision for images, and evaluating user relationships through community detection.
+Hence, this section decouples and analyses ERH’s social background and definitions.
+2.1
+Extremism and Radicalisation Decoupled
+Extremism’s definition appears in two main flavours: politically fringe belief systems outside the
+social contract or violence supporting organisational affiliation. The Anti-Defamation League (ADL)
+frames extremism as a concept "used to describe religious, social or political belief systems that exist
+substantially outside of belief systems more broadly accepted in society" [6]. For instance, under the
+ADL’s definition, extremism can be a peaceful positive force for mainstreaming subjugated beliefs,
+such as for civil rights movements. This construct of a socially mainstream belief constitutes the
+Overton window [73]—and is not the target for content moderation.
+Conceptually, extremism typically involves hostility towards an apparent ‘foreign’ group based
+on an opposing characteristic or ideology. Core tenants of extremism can stem from political
+trauma, power vacuums and opportunity, alongside societal detachment and exclusion [58, 78].
+Hence, extremism often relies on defending and congregating people(s) around a central ideology,
+whose followers and devotees are considered ‘in-group’ [116]. Extremists unify through hostility
+and a perceived injustice from an ‘out-group’ of people(s) that do not conform to the extremist
+narrative—typically in a ‘us vs. them’ manner [28, 78, 116]. Hence, extremism detection algorithms
+can use non-textual relationships as an identifying factor via clustering users into communities
+(i.e., community detection) [20, 110]. Thus, extremism can simply reduce to any form of a fringe
+group whose identity represents the vocal antithesis of another group.
+There is no one conceptual factor to make an extremist. Extremism can also emanate from
+political securitisation—whereby state actors transform a specific referent object (such as Buzan’s
+five dimensions of society: societal, military, political, economic, and environmental security [25]; or
+individuals and groups [28]) towards matters of national security, requiring extraordinary political
+measures [25]. As the state normalises policies into matters of existential national security, society
+can adapt and ideate decisions to ones of existential ‘life or death’ nature [25].
+For example, the ‘Great Replacement’ conspiracy theory claims that non-European immigrants
+and children are “colonizers" or "occupiers”, and an “internal enemy”—-with the intent to securitise
+migration, race, religion, and culture into wars with wording to invoke fears of a fifth column or
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech
+000:5
+racial invasion/replacement [22, 106]. The Christchurch Shooter took direct interest in securitising
+migrants as an extreme military threat, as far to name his manifesto after the conspiracy [106].
+Extremism is not a strictly demographic ‘majority vs minority’ concern, as it encapsulates
+movements demanding radical change and earmarked by a sense of rewarding personal and social
+relationships, self-esteem, and belief of a wider purpose against a perceived adversarial force [58].
+Exploiting desires for vengeance and hostility are also key recruitment strategies [20, 58, 70].
+Outside of political, cultural, and socio-economic factors, mental health and media are intrinsically
+inalienable contributing factors [28, 58, 79]. Likewise, repeated media reports of footage and body
+counts can gamify and normalise extremism as a macabre sport for notoriety [20, 79].
+Within industry, Facebook, Twitter, YouTube, and the European Union frame extremism as
+a form of indirect or direct support for civilian-oriented and politically motivated violence for
+coercive change [34]. Facebook expands this industry-wide consensus to include Militarised Social
+Movements and "violence-inducing conspiracy networks such as QAnon" [37].
+Radicalisation focuses on the process of ideological movement towards a different belief, which
+the EU frames as a "phased and complex process in which an individual or a group embraces a
+radical ideology or belief that accepts, uses or condones violence" [35]. Terrorism consists of
+politically motivated violence towards the civilian population to coerce, intimidate, or force specific
+political objectives, as an end-point for violent radicalisation to project extremism [25, 28]. Borum
+delineates the passive ideological movement of radicalisation from active decisions to engage in
+‘action pathways’ consisting of physical terrorism, or hate crimes [20].
+Political radicalisation towards increasingly aggrandising groups can also manifest in Roe’s two
+sides of nationalism: positive socio-cultural and negative ethnic/racial nationalism [98]. These
+balancing forces create a form of societal security dilemma whereby the actions of one society to
+strengthen its identity can cause a reaction in another societal group, weakening security between
+all groups-—a radicalising spiral which can manifest into a polarised ‘culture war’ [70, 98]. However,
+integration over assimilation can inversely undermining culture, self-expression and group cohesion,
+leading to alienation and oppression by the dominant political or normative force [28, 98].
+Nonetheless, ERH detection does not offer a panacea to combating global terrorism, nor does
+surveillance offer a ‘catch-all’ solution. In the case of the livestreamed Christchurch shooter, the
+New Zealand Security Intelligence Service concluded that “the most likely (if not only) way NZSIS
+could have discovered [the individual]’s plans to conduct what is now known of [the individual]’s
+plans to conduct his terrorist attacks would have been via his manifesto.” [106, p. 105]. However,
+the individual did not disseminate this until immediately before the attack, and his 8Chan posts
+did not pass the criteria to garner a search warrant [106, p. 105]. Hence, extremism detection is an
+evolutionary arms race between effective and ethical defences vs. new tactics to evade detection.
+2.2
+Hate Speech Decoupled
+Obtaining viewpoint neutrality to categorise hate speech is challenging due to human biases and
+the risk of hate speech undermining liberties through mainstreaming intolerance—the paradox of
+tolerance where a society tolerant without limit may have their rights seized by those projected
+intolerance [92]. Popper encapsulates this challenge by formulating that "if we are not prepared
+to defend a tolerant society against the onslaught of the intolerant, then the tolerant will be
+destroyed, and tolerance with them" [92]. Defining clear hate speech restrictions are needed to
+protect expression rights and victim groups rights and safety [11, 117].
+The European Union defines hate speech as "all conduct publicly inciting to violence or hatred
+directed against a group of persons or a member of such a group defined by reference to race, colour,
+religion, descent or national or ethnic origin." [34]. Whereas, the U.S. Department of Justice frames
+that: "A hate crime is a traditional offence like murder, arson, or vandalism with an added element
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+000:6
+Govers et al.
+of bias... [consisting of a] criminal offence against a person or property motivated in whole or in
+part by an offender’s bias against a race, religion, disability, sexual orientation, ethnicity, gender, or
+gender identity." [38] Notably, governmental laws may differ from industry content moderation
+policies via the omission of sexual, gender, religious or disability protections, and may include
+threats of violence and non-violent but insulting speech.
+The United Nations outlines the international consensus on hate speech as "any kind of commu-
+nication in speech, writing or behaviour, that attacks or uses pejorative or discriminatory language
+with reference to a person or a group on the basis of who they are, in other words, based on their
+religion, ethnicity, nationality, race, colour, descent, gender or other identity factor." [117, p. 2]
+What all these definitions have in common is that they all involve speech directed at a portion
+of the population based on a protected class.
+3
+SYSTEMATIC LITERATURE REVIEW DESIGN AND PROTOCOL
+This SLR investigates the state-of-the-art approaches, datasets, ethical, socio-legal, and technical
+implementations used for extremism, radicalisation, and politicised hate speech detection. We
+conduct a preliminary review of prior ERH-related SLRs to establish the trends and research gaps.
+For the purposes of our SLR’s design, and to embed Open-Source Intelligence (OSINT) and Social
+Media Intelligence (SOCMINT) principles, we define social media data as any online medium where
+users can interactively communicate, exchange or influence others. We accept external data sources,
+such as manifesto or news sites if interactive—such as via comment sections. Furthermore, we
+propose and use a novel quality assessment criteria to filter irrelevant or ambiguous studies.
+3.1
+Trends and Shortfalls in Prior SLRs
+Searching for Extremism, Radicalisation, Hate speech (ERH) and related terms, resulted in ten
+literature reviews ranging from January 2011 to April 2021 [2, 5, 20, 42, 43, 51, 82, 97, 110, 114].
+Aldera et al. observed only one survey before 2011 (covering 2003-2011) and another in 2013,
+indicating the limited, exclusionary, but developing nature of reviews in this ERH detection area [5].
+Prior SLRs seldom delineated or elaborated on Extremism, Hate Speech and Radicalisation. Neither
+“extremism”, "radicalism" [2, 5, 20, 42, 43, 110, 114] or “hate speech” oriented SLRs [51, 82, 97]
+cross-reference each other despite 26.3% of the data reviewed in the “hate speech” oriented review
+by Adek et al. encompassing hate speech in a political context [114]. This lack of overlap presents
+an industry-wide challenge for social media companies who may oversee developments in ‘hate
+speech’ detection which could transfer to a ‘extremism/radicalisation detection’ model.
+SLRs prior to 2015 found that deep learning approaches (DLAs), such as Convolutional Neural
+Networks (CNN) and Long Short-Term Memory (LSTM), resulted in 5-20% lower F1-scores than
+non-deep approaches (e.g., Naïve Bayes, Support Vector Machines, and Random Forest classifiers) [2,
+5, 42, 51, 97, 114]. DLAs post-2015 indicated a pivotal change towards higher-performing language
+transformers such as Bidirectional Encoder Representations from Transformers (BERT) models [33].
+3.1.1
+Domains and Criteria.
+No review delineated or removed studies that did not use English social media data. This presents
+three areas of concern for researchers when attempting to compare the performance of models:
+(1) Results may not be comparable, if they use culture-specific lexical datasets, or language
+models trained on other languages.
+(2) Linguistic differences and conveyance in language—as what may be culturally appro-
+priate for the majority class may appear offensive to minority groups and vice-versa.
+(3) The choice of language(s) influences the distribution of target groups—with a bias
+towards Islamic extremism given its global reach in both Western (predominantly ISIS) and
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech
+000:7
+Eastern countries (e.g., with studied online movements in the Russian Caucasus Region [77]).
+It is worth investigating whether Gaikwad et al.’s finding that 64% of studies solely target
+‘Jihadism’ corroborates with our study, which targets only English data [42, p. 17].
+Our SLR incorporates the key approaches of dataset evaluation (including their accessibility,
+labels, source and target group(s)), data collection and scraping approaches, Machine Learning and
+Deep Learning algorithms, pre-processing techniques, research biases, and socio-legal contexts.
+Unlike prior SLRs, our SLR conceptualises all elements for ERH context mining—consisting of a
+user’s ideological radicalisation to an extremist position, and then projected via hate speech.
+3.2
+Research Questions
+Our Research Questions (RQ) investigate the full process of ERH Context Mining—incl. data collec-
+tion, annotation, pre-processing, model generation and evaluation. These RQs consist of:
+(1) What are the working definitions for classifying online Extremism, Radicalisation, and Hate
+Speech?
+(2) What are the methodologies for collecting, processing and annotating datasets?
+(3) What are the computational classification methods for ERH detection?
+(4) What are the highest performing models, and what challenges exist in cross-examining them?
+Given the overlap of studies across prior SLRs targeting extremism or radicalisation or hate
+speech, RQ1 addresses the similarities and differences between researchers’ definitions of ERH
+concepts and their computational classification approach. We dissect the ERH component of ERH
+context mining and propose consensus-driven working definitions.
+RQ2 addresses the vital context for ERH models—the data used and features extracted or filtered
+out from it. Furthermore, identifying frequently used benchmark datasets provides the basis for
+critical appraisal of the state-of-the-art algorithmic approaches in the community detection, multi-
+media, and NLP spheres in RQ3/4. Covering algorithmic approaches is not in itself novel. However,
+we consider novel, niche, and overlooked features relevant for an ERH model to make accurate
+classifications—namely, bot/troll detection, transfer learning, the role of bias, and a hybridised eval-
+uation of NLP and non-textual community detection models. We also consider critical challenges,
+choice of metrics, and performance considerations not observed in prior SLRs.
+3.3
+Databases
+Given the cross-disciplinary, global and socio-technical concepts for ERH detection, we queried the
+following range of software engineering, computer science, crime and security studies databases.
+• ProQuest (with the “peer-reviewed” filter, including queries to the below databases)
+• Association for Computing Machinery (ACM) Digital Library
+• SpringerLink
+• ResearchGate
+• Wiley
+• Institute of Electrical and Electronics Engineers (IEEE) Xplore
+• Association for Computational Linguistics Portal
+• Public Library of Science (PLOS) ONE Database
+• Google Scholar—as a last line to capture other journals missed in the above searched databases
+3.4
+Search Strings
+The first round of study collection included automated database search strings. A second round
+included a targeted manual search strategy with dissected keyword combinations to expand
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+000:8
+Govers et al.
+coverage. All results were added to the Title and Abstract Screening list. The following database
+search query also included time filters (2015-2021) and peer-review-only filters:
+(“artificial␣intelligence”␣OR␣“machine␣learning”␣OR␣“data␣mining”␣OR␣“natural
+language␣processing”␣OR␣“multiclass␣classification”␣OR␣“model”␣OR␣“analysis”␣OR
+“intelligence”␣OR␣“modelling”␣OR␣“detection”)
+AND␣(“hate␣speech”␣OR␣“radicalisation”␣OR␣“radicalization”␣OR␣“extremism”)
+AND␣(“social␣media”␣OR␣“forums”␣OR␣“comments”OR␣“virtual␣networks”␣OR␣“virtual
+communities”␣OR␣“online␣communities”␣OR␣“posts”␣OR␣“tweets”␣OR␣“blogs”)
+3.5
+Inclusion and Exclusion Criteria
+After attaining our 251 studies from our search strings, we read the journal metadata, title and
+abstract to screen studies. Our ranked criteria requires that all studies must:
+(1) Originate from a peer-reviewed journal, conference proceeding, reports, or book chapters.
+(2) Be written in English.
+(3) Involve a computational model (network relationship, textual and/or visual machine learning
+model) for identifying and classifying radicalisation, hate speech or extreme affiliation.
+(4) Utilise social media platform(s) for generating their model.
+(5) Computationally identify ERH via binary, multiclass, clustering or score-based algorithms.
+(6) Focus on politicised discourse to exclude cyber-bullying or irrelevant benign discussions.
+(7) Published after the 1st January 2015—until the 1st July 2021.
+(8) Utilise English social media data if evaluating semantics and grammatical structure.
+In addition to those that do not abide to any of the above, we exclude studies that:
+(1) Are duplicates of existing studies.
+(2) Do not specify their target affiliation to exclude broad observational studies.
+We outline our further in-depth critical Quality Assessment (QA) criteria to filter irrelevant or
+ambiguous studies in our supplementary material’s Quality Assessment Criteria subsection.
+After the Title and Abstract Screen, we read the full text of the 57 studies for the screening stages
+displayed in Table 1. With 42 studies passing the Full Text Screen, we then randomly selected studies
+from the bibliographies from this ’snowball sample’ of the 42 studies until 5 studies fail QA.
+Table 1. Studies found and filtered
+Screen Type
+Study Count
+Search Strings
+251
+Title and Abstract Screen
+57
+Full Text Screening
+42
+After Snowball Sampling
+51
+3.5.1
+Threats to Validity.
+While we consider a concerted range of search strings, we recognise that ERH concepts is a wide
+spectrum. To focus on manifestly hateful, politicised, and violent datasets/studies, we excluded
+cyber-bullying or emotion-detection studies. The potential overlap and alternate terms for ERH
+(e.g., sexism as ‘misogyny classification’ [30]) could evade our search strings. Our pilot study,
+subsequent tweaks to our search method, and snowball sampling minimise this lost paper dilemma.
+This study does not involve external funding, and all researchers declare no conflicts of interest.
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech
+000:9
+4
+KEY RESEARCH QUESTION (RQ) FINDINGS
+Across the 51 studies between 2015-2021, ERH research is gaining popularity—with 4 studies from
+2015-2016 increasing to 25 between 2019-2020 (and 4 studies from January to July 2021). We present
+our SLR’s core findings in this section, with in-depth RQ analysis in Sections 5, 6, 7, and 8.
+4.1
+Summary of the Social ERH Definitions Used by Researchers
+RQ1: What are the working definitions for computationally classifying online Extremism, Radicalisation,
+and Hate Speech?
+Across the 51 studies, there are seldom delineations between the researchers choice of Extremism,
+Radicalisation, and Hate Speech as the study’s focus––with the consensus that hate speech is
+equivalent to extremist or radical views. Hence, researchers approach extremism and radicalisation
+as an organisationally affiliated form of hate speech.
+The consensus on hate speech’s working definition is any form of subjective and derogat-
+ory speech towards protected characteristics expressed directly or indirectly in textual form—-
+predominantly via racism or sexism. Benchmark datasets utilise human-annotated labels on single-
+post instances of racially or gender-motivated straw man arguments, stereotyping, or post causing
+offensive towards the majority of annotators (via inter-annotator agreement). Only 20% of studies
+consider explicit rules or legal frameworks for defining hate speech [15, 18, 47, 53, 55, 71, 77, 84,
+123, 128], with others relying on either an implicit ‘consensus’ on hate speech or utilise benchmark
+datasets. Benchmark datasets typically consider categorising their data into explicit categories of
+racism [31, 32, 122, 123], sexism [13, 122, 123], aggression [13, 61], or offensiveness [13, 31, 128];
+including hate categorisation via visual memes and textual captions [3, 57, 99, 111].
+Extremism and radicalisation are equivalent terms in existing academia. Islamic extremism is
+the target group in 77% of US-originating extremism studies. ‘Far-right extremism’ and ‘white
+supremacy’ are used interchangeably, a form of cultural bias given the variety of right-wing politics
+worldwide. Only one study considered radicalisation as an ideological movement over time [12].
+4.2
+Summary of the Data Collection, Processing, and Annotation Processes
+RQ2: What are the methodologies for collecting, processing and annotating datasets?
+Collecting non-hateful and hateful ERH instances varies between supervised and unsupervised
+(clustering) tasks. Supervised learning typically utilises manual human annotation of textual posts
+extracted via tools presented in Figure 1. Semi or unsupervised data collection can include grouping
+ideologies by platform, thread, or relation to a suspended extremist account. Islamic extremism
+studies frequently used manifestos and official Islamic State magazines as a ‘ground truth’ for
+textual similarity-based approaches for extremism detection. We found a direct correlation between
+the availability of open and official research tools, and the platform of choice by researchers. Biases
+extend geographically, with no studies utilising data or groups from Oceanic countries.
+Figure 2 displays the skew for Twitter as the dominant platform for hate speech research. Despite
+the nuance of conversations, 69% of studies classify hate on a single post per Figure 3.
+Data processing often utilises extracting statistically significant ERH features—such as hateful
+lexicons, emotional sentiment, psychological signals, ‘us vs them’ mentality (higher occurrence of
+first and third-person plural words [46]), and references to political entities.
+We categorise and frame the two approaches for dataset annotation: organisational or experience-
+driven annotation. Organisational annotation utilises non-governmental anti-hate organisations [105]
+or ‘expert’ annotator panels—determined via custom tests or by tertiary degree. Organisational an-
+notation relies on crowdsourced annotators, balanced by self-reported political affiliation. Inter-rater
+agreement or Kappa coefficient are the sole metrics for measuring annotator agreement.
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+000:10
+Govers et al.
+Fig. 1. Method to collect data.
+Fig. 2. Number of studies per social media platform studied.
+4.3
+Summary of the State-of-the-art Computational ERH Detection Models
+RQ3: What are the computational classification methods for ERH detection?
+ERH detection includes text-only Natural Language Processing (NLP), network-related community
+detection, and hybrid image-text models. Between 2015-2021, there was a notable shift from
+traditional Machine Learning Algorithms (MLAs) towards contextual Deep Learning Algorithms
+(DLAs) due to higher classification performance—typically measured by macro F1-score.
+Notably, only 3 of the 21 community detection studies utilised Deep Learning Algorithms
+(DLAs) [77, 84, 99]. Instead, community detection researchers tended to opt for graph-based models
+such as heterogeneous graph models converting follower/following, reply/mention, and URL
+networks with numeric representations for logistic regression or decision trees [12, 16, 24, 48, 80, 84].
+Community detection Machine Learning Algorithms (MLAs) performance varied by ~0.3 F1-score
+(mean between studies) dependent on the selection of features. Statistically significant features
+for performant MLA models include gender, topics extracted from a post’s URL(s), location, and
+emotion via separate sentiment algorithms such as ExtremeSentiLex [89] and SentiStrength [112].
+For textual non-deep NLP studies, researchers classified text via converting the input into
+word embeddings via Word2Vec, GloVe, or frequent words via Term Frequency-Inverse Docu-
+ment Frequency (TF-IDF), and parsing it into Support Vector Machines, decision trees, or logistic
+regression models. As these embeddings do not account for word order, context and nuance is
+often lost—leading to higher false positives on controversial political threads. Conversely, DLAs
+utilise positional and contextual word embeddings for context-sensitivity using Long Short Term
+Memory (LSTM) Convolutional Neural Networks and Bidirectional Encoder Representations from
+Transformers (BERT) leading to their higher performance as outlined in RQ4.
+4.4
+Summary of ERH Models’ Classification Performance
+RQ4: What are the highest performing models, and what challenges exist in cross-examining them?
+By 2021, Support Vector Machines on emotional, sentiment, and derogatory lexicon features were
+the last non-deep MLA to attain competitive F1-scores for NLP tasks compared to DLAs such as
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+3%6%
+29%
+32%
+22%
+5%
+3%
+Twitter APi
+Custom web-crawler
+ Pushshift.io APl (Reddit)
+Magazine or Newspaper Archives
+Pre-existing Dataset or Library
+Facebook Graph APl
+Other APlIslamicExtremist
+Sources, 9
+Stormfront, 7
+Other Far-
+Blogs and
+right
+Apolitical
+Extremist
+Sources, 4
+Forums, 4
+News
+Websites, 6
+Far-Left
+Forums,
+Facebook,
+Wikipedia
+2
+3
+Twitter, 36
+Vkont
+akte,
+Religious
+Other, 5
+Reddit, 2
+Texts, 2Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech
+000:11
+Fig. 3. Type of data used for an ERH classification.
+Fig. 4. Distribution of target groups.
+Convolutional Neural Networks (CNNs) and neural language transformers. As of 2021, BERT-base
+attained the highest macro F1-score average across the seven benchmark datasets. However, cross-
+examining models between datasets present various challenges due to varying criteria, social media
+domain, and choice of metrics. Likewise, non-textual community detection and traditional MLA
+studies resulted in lower classification F1-score by ~0.15 and ~0.2 respectively. While BERT, attention-
+layered Long Short Term Memory (BiLSTM), and other ensemble DLAs attain the highest F1-scores,
+no studies consider their performance trade-offs with their high computational complexity. Our
+recommendations propose further research in prompt engineering, distilled models, and hybrid
+multimedia-text studies—as we only identified one hybrid image-text study.
+While textual DLAs outperform community detection models, grouping unknown instances
+enable network models to identify bot networks and emergent terror cells. Hence, there is a growing
+area of research for hybrid semi-supervised NLP and community detection models to identify new
+groups and radical individuals in a domain we frame as meso-level and micro-level classification.
+4.5
+Geographic Trends—Islamophobia and Exclusion in the Academic Community
+To identify ERH hot spots in research, we present the first cross-researcher examination of their
+institution’s location compared to their dataset(s) geolocational origin in Figure 5. For clarity, we
+filter out the 29 indiscriminate global studies.
+Despite the decline of the Islamic State as a conventional state-actor post-2016, western academic
+research remains skewed towards researching Islamic extremist organisations operating from the
+Middle East. 24% of US-originating ERH studies targeted Islamic extremism, compared to 19%
+focusing on violent far-right groups and 19% for left vs right polarised speech (in discussions
+containing hate speech). Despite more Islamic extremist studies from US-oriented research, over
+90% of terrorist attacks and plots in the US were from far-right extremists in 2020 [54].
+European-origin studies have a reduced bias, where 25% target far-right white supremacy and
+29% on Islamic extremism. Islamic extremism’s popularity is a global trend for 20% of all studies,
+shown in Figure 4. Hence, there is a clear Islamophobic trend in academia—given the aversion of
+far-right groups, and the lack of a change in the distribution of targeted groups between 2015-2021.
+Researcher ethics and socio-legal considerations present a critical international research gap, as
+only 13% of US and 28% of European studies included discussions on annotation ethics, expression
+laws, or regulation. This US vs. Europe discrepancy likely emanates from the data collection and
+autonomous decision-making rights guaranteed under the EU’s GDPR [36].
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+4%
+Text. Single post only
+8%
+I Text. Grouped by user
+19%
+No text. Metadata
+69%
+Relations (Community
+Detection), or Images
+Text and Graphing
+(NLP and Community
+Detection).lslamic
+20%
+20%
+IFar-right White
+Supremacy)
+IMainstream Politics
+10%
+Hate Speech
+9%
+41%
+1Other000:12
+Govers et al.
+Fig. 5. Relations between researchers country of origin and their dataset’s country of focus (global/indiscrim-
+inate studies excluded). Created via Flowmap.blue [21] with Mapbox [74] and OpenStreetMap [85]
+4.5.1
+The Case for Oceania and the Global South.
+Despite developments post-2015, such as two New Zealand terrorist attacks [94, 106] and five in
+Australia [8], the rise of racial violence in South Africa [28], and the 2019-ongoing COVID-related
+radicalisation [120]; no studies considered targeting Oceania or English-recognised countries in the
+global south. Likewise, applying these datasets across intersectional ethnic, sexual, and cultural
+domains presents a threat to validity as terms considered mundane or inoffensive to one group may
+be considered inflammatory to another. Datasets are also biased towards racism towards a minority
+group [31, 123], which may bias English hate speech in a white minority country such as South
+Africa. Investigating language trends and model performance on Mela-, Micro- and Polynesian
+groups could also offer insights in the role of religion, values (such as tikanga values in New
+Zealand’s M¯aori population), taboos, lexicons, and social structures unique to indigenous cultures.
+5
+SOCIO-TECHNICAL CONTEXT IN RESEARCH—CONSENSUS-DRIVEN
+DEFINITIONS
+What are the Working Definitions for Classifying Online Extremism, Radicalisation, and Hate Speech?
+Empirically, studies often provide a generalised social definition in their introduction or back-
+ground and utilise technical criteria to annotate instances for (semi)supervised learning tasks.
+Hence, this research question consists of two parts: the socio-legal ERH definitions, and the
+technical implementation and classification thereof outlined in the existing literature.
+We identified an unexpected overlap between the definitions and models between extremism and
+radicalisation studies, whereby researchers frame these concepts as synonymous with hate speech
+with a political/organisational affiliation. Hate speech studies focus on protected groups as binary
+‘hate or not’ [3, 4, 15, 27, 32, 57, 77, 84, 101, 109, 111, 123], or multiclass ‘racism, sexism, offensive,
+or neither’ text [10, 13, 31, 45, 69, 81, 83, 90, 91, 123, 126], with a consensus that ‘Extremism =
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+Iceland
+Sweden
+Finland
+Norya
+Russia
+Denmark
+Belarus:
+Germany-
+Ukraine
+Era
+Kazakhstan
+Romania
+Mongolia
+Atiantic
+ain
+Kyrgyzstan
+Unit
+Qcean
+Turkey
+Turkmenistan
+Afghanistan
+China
+Morocco
+Iran
+‘Nepal-K
+Mexico
+Bangladesh 
+Cuba
+Mauritania
+Oman
+India
+Laos
+Niger
+Senega
+Chad
+Sudan
+Yemen
+Cambodia
+Philipp
+Venezuela
+Ethiopia
+SriLanka
+Colombia
+Maldives
+Malaysia
+Kenya
+Ecuador
+Indonesia
+Tanzania
+Perum
+Brazil
+Angola
+Zambia
+Bolivia
+Mozambique
+Namibia
+Indiar
+Paraguay
+South
+Chile
+Ocear
+Locationtotals
+Atlantic
+SouthAfrica
+Ocean
+outgoing 兰 incoming
+ruguay
+Argoitina
+more outgoing
+more incomingDown the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech
+000:13
+Radicalisation = Hate speech with an affiliation’. Alternatively, we propose a novel computationally
+grounded framework and definitions to seperate and expand ERH in Section 9 to underline the
+holistic stages of extremists temporal radicalisation through disseminating hateful media.
+5.0.1
+Socio-legal Context Provided in Existing Literature.
+The largest discrepancy were between studies that discussed legal or ethical context to ERH, which
+constituted only 20% of studies [15, 18, 47, 53, 55, 71, 77, 84, 123, 128]. The remaining 80% relied on
+an implicit consensus of ‘hate speech’ (often synonymous with toxic, threatening, and targeted
+speech), or ’extremism’ (often UN designated terrorist organisations like ISIS [118]).
+Waseem and Hovy [123] outlined a unique eleven-piece criteria to identify offensive ‘hate
+speech’ including considerations for politicised extremist speech via tweets that “promotes, but
+does not directly use, hate speech or violent crime” and “shows support of problematic hash tags"
+(although "problematic" was not defined). Hate speech as a supervised learning task resulted in two
+categories—sexism and racism. A sexist post requires gender-oriented slurs, stereotypes, promoting
+gender-based violence, or straw man arguments with gender as a focus (defined as a logical fallacy
+aimed at grossly oversimplifying/altering an argument in bad faith [123]). The ambiguity for sexism
+classification by human annotators was responsible for 85% of inter-annotator disagreements [123].
+5.1
+Researchers’ Consensus-driven Definitions for ERH Concepts
+We aggregate the trends in ERH based on the definitions used throughout the 51 studies, and
+observe that ERH concepts reflect their computational approach more than their social definitions.
+Despite radicalisation being a social process of ideological movement, existing work considers the
+term as synonymous to political hate speech/extremism.
+Definition 1: Hate speech (researchers’ consensus)
+Hate speech is the subjective and derogatory speech towards protected characteristics expressed
+directly or indirectly to such groups in textual form.*
+*N.B: there is a significant bias in hate speech categorical classification in practice, whereby no
+studies considered categories outside of sexism (including gender-identity) or racism.
+Definition 2: Extremism (researchers’ consensus)
+Organisational affiliation to an ideology that discriminates against protected inalienable
+characteristics or a violent political organisation. Affiliation does not always include manifestly
+hateful text and may include tacit or explicit organisational support. Extremist studies often
+classify organisational affiliation based on text (NLP) and community networks (follower,
+following, or friend relationships).
+The current academic consensus among researchers demonstrates a considerable overlap between
+‘extremism’ and ‘hate speech’ definitions. In practice, extremism exclusively focused on racism
+detection, or in the specific context of Jihadism [4, 15, 84], white supremacy [53, 86, 99, 109],
+Ukrainian separatists [16], anti-fascism (Antifa) [53], and the sovereign citizen movement [53]. Of
+the 13 studies targeting extremism, only one considered extremism by the ADL’s politically-fringe-
+but-not-violent definition [1]. Tying extremism to the study of mainstream terrorist-affiliated groups
+neglects rising movements, ethical movements using unethical terror-tactics, and non-violent
+fragments of other terrorist groups, such as a reversion to ‘fringe’ activism. Hence, extremism’s
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+000:14
+Govers et al.
+working definition is similar to terrorism when considering group affiliation detection. If investigating
+extremist ideologies, then the definition is synonymous with those in hate speech studies.
+Extremism’s working definition is exceptionally biased towards support for Islamic extremist
+movements (10 out of 13 studies [4, 16, 24, 53, 77, 80, 84, 86, 89, 109]), with far-right ideologies a
+distant second (5 out of 13 studies [16, 53, 86, 99, 109]). These organisational and ideological biases
+is potentially a result of US security discourse and national interests (via the ‘War on Terror’). White
+supremacy and far-right ideologies are separate terms used interchangeably without distinction.
+Definition 3: Radicalisation (researchers’ consensus)
+No discernible difference between extremism’s definition with both terms used interchangeably.
+Radicalisation = extremism = politically affiliated hate invoking or supporting speech.
+Definitions and algorithmic approaches on radicalisation detection relied on political hate speech,
+or extremism via ideological affiliation—with 5 of the 8 radicalisation studies targeting textual or
+network affiliation to the Islamic State (IS) [7, 47, 80, 95, 101], and 2 on white supremacy [46, 103].
+The only other notable deviations from this extremism = radicalisation dilemma is Bartal et
+al.’s [12] focus on radicalisation as a temporal movement with apolitical roles. Their study investig-
+ated the temporal movement from a ‘Novice’ (new poster) classification towards an ‘influencer’
+role based on their network relations and reply/response networks. Chandrasekharan et al. defined
+‘radicalisation’ as the process of an entire subreddit’s patterns up to and including the time of its
+ban to map subreddit-wide radicalisation [27]. Only two studies are the exception to the extremism
+= radicalisation = politicised hate speech consensus per the remainder of the 51 studies [12, 27].
+Fig. 6. ERH Definition Tree—visualising how ERH definitions deviate based on their algorithmic approach.
+5.2
+Correlation Between Definitions and Algorithmic Approach
+Uniquely, 66% of publications in the field of social-science or security studies utilised network-
+driven community detection models, with extremism defined in a law enforcement context by
+emphasising a user’s network-of-interactions between known annotated extremists. Hung et al.
+defined extremism in a semi-supervised OSINT and HUMINT surveillance manner—requiring
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+ERH
+Semantic/Topic-based (Aggregate) Hate
+Radical or Extremist Network Detection
+Speech Detection
+Generally
+Generally semi or fully
+Generally
+unsupervised
+supervised.
+unsupervised
+'Big Data':
+Unless cluster-based
+'Big Data':
+Graphs (non-textual):
+Emotional Sentiment
+Topics and Political Affilation
+Hate Speech
+Crime and Social Science Focused
+E.g. SentiStrength, SIRA, LIWC
+Law Enforcement Pivoted Data:
+E.g. FBI 'Tripwire', Explicit Terror Afillation, TSA's
+Automated Targeting System, etc.
+Multiclass Sentiment
+Binary Affiliation
+Influencer
+Racism
+Terror Network
+Sexism
+Aggression
+Offense
+or ProlAnti/Neutral
+Person-of
+Classification
+Clustering
+Classificatior
+Classification
+Classification
+Classification
+Interest Detection
+Classification
+F
+H
+J
+K
+c
+D
+A
+Multiclass
+Ideological or
+Community
+Radical Scoring
+Organisation or
+Detection
+Topic Classification
+Regression
+B
+E
+GDown the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech
+000:15
+links between an extremist virtual and a physical/offline presence to extremist stimuli through
+incorporating the FBI’s tripwire program [48]. Approaching extremism using relational properties
+via interactions, geographic proximity, profile metadata, and semantic or network similarity raises
+ethical dilemmas vis-à-vis individuals who have/had a solely virtual presence or those interested in
+opposing opinions [24, 29, 76]. The relationship between definitions and algorithmic approaches
+indicate that radicalisation studies skew towards community detection models, extremism towards
+hybrid NLP and community detection models, and hate speech to a text-only NLP endeavour.
+Table 2. Table of references for studies in each category (A, B...K) in the above ERH definition tree diagram.
+Label
+Studies
+Label
+Studies
+Label
+Studies
+A
+[12, 15, 48, 80]
+B
+[15, 16, 80, 89, 99, 108]
+C
+[48, 108]
+D
+[3, 4, 15, 27, 32, 57, 77, 84, 101, 109, 111,
+123]
+E
+[14, 53, 80, 89, 93, 99, 99]
+F
+[1, 4, 7, 13, 52, 55, 86, 89, 90, 95, 100, 107,
+124, 128, 129]
+G
+[15, 18, 46, 47, 103, 104, 113]
+H
+[10, 13, 45, 69, 81, 83, 90, 91, 123, 126]
+I
+[10, 13, 45, 81, 83, 90, 91, 123, 126]
+J
+[13, 56, 83, 90, 126, 128, 129]
+K
+[3, 56, 57, 67, 71, 81, 83, 90, 99, 111, 126,
+128–130]
+-
+-
+5.2.1
+Privacy and Ethics-driven Regulation.
+No studies integrated or mentioned existing AI ethics regulation or standards, such as those
+emerging from the EU [50], or private-sector self-regulation such as the IEEE P700x Series of
+Ethical AI Design [60]. Researchers should consider the application and use cases for their proposed
+models—as autonomous legal decision making, injurious use of data (outside of a reasonable
+purpose), or erasure (a challenge for persistent open-source datasets), may violate regulations such
+as the EU’s General Data Protection Regulation (GDPR)’s Article 22, 4, and 17 respectively [36].
+Prominently, Mozafari et al. evaluated hate speech with a ethno-linguistic context, recognising
+that certain racist slurs were dependent on the culture and demographic using them [81].
+6
+BUILDING ERH DATASETS—COLLECTION, PROCESSING, AND ANNOTATION
+What are the methodologies for collecting, processing and annotating datasets?
+This RQ outlines the dominant platforms of choice for ERH research, the APIs and methods
+for pulling data and its underlying ethical considerations. Geographic mapping demonstrates the
+marginalisation of Oceania and the global south in academia. We critically evaluate the sentimental,
+relationship, and contextual feature extraction and filtering techniques in community detection
+and NLP studies. We conclude with the key recommendations for future data collection research.
+6.1
+Prominent Platforms, Pulling, and Populations
+This subsection outlines the common social media platforms, the method for sampling and extracting
+(‘pulling’) textual and network/relationship data, and the type of data used in ERH datasets.
+40% of studies relied on Twitter tweets for ERH detection, with Twitter being the dominant
+platform for research per Figure 2. Twitter’s mainstream global reach paired with its data-scraping
+Twitter API enabled researchers to target specific hashtags (topics or groups), real-time tweet
+streams and reach of tweets and their community networks. Hence, the Twitter API is also the most
+used method for scraping data, with other methods outlined in Figure 1. Unfortunately, revised
+2021 Twitter Academic API regulations removed access to tweets from suspended accounts [17],
+limiting datasets to those pre-archived. Currently, the Waseem and Hovy datasets are not available
+due to relying on the Twitter API and suspended tweets [122, 123].
+For far-right ERH detection, researchers used custom web-scrapers to pull from the global
+white supremacist forum Stormfront—containing topics ranging from political discussions, radic-
+alising "Strategy and Tactics", and "Ideology and Philosophy" sections, and regional multilingual
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+000:16
+Govers et al.
+chapters [55, 71, 81, 103, 109, 124, 126]. As a supplement or alternate to searching and collecting
+hateful posts en masse, five studies considered extremist ‘ground truth’ instances by comparing tex-
+tual similarity from Tor-based terror-supporting anonymous forums [4] and websites [124], radical
+Islamist magazines and manifestos [7, 84, 95]. Interestingly, no studies considered extracting ground
+truths from far-right manifestos or media. Likewise, no studies considered recent low-moderation
+anonymised forums such as 8Chan (now 8kun) or Kiwifarms, which were extensive hubs for pro-
+paganda dissemination from the Christchurch shooter [106]; Parler, notable for its organisational
+influence during the 2021 Capitol Hill riots [72]; Telegram, TikTok, or Discord, despite reports on
+its use for sharing suicides, mass shootings, and group-lead harassment of minority groups [44, 87]).
+Hence, there is a prevalent and concerning trend towards NLP studies on mainstream platforms,
+which may overlook the role of emergent, pseudo-anonymised or multimedia-oriented platforms.
+6.1.1
+Data Collected.
+62% of all studies evaluate ERH on a single post-by-post basis, with NLP the dominant approach per
+Figure 3. Conversely, grouping posts on a per user basis frequently included annotations from cyber-
+vigilante groups such as the Anonymous affiliated OSINT ‘Controlling Section’ (CtrlSec) group’s
+annotations of ISIS-supporting accounts [47]. However, Twitter claims that CntrlSec annotations
+were "wildly inaccurate and full of academics and journalists" [26]. Hence, researchers should avoid
+unvetted cyber-vigilante data, and consider anonymising datasets to further benefit user privacy,
+researcher ethics, and model performance by reducing false positives (i.e., censorship).
+While NLP text detection is the dominant detection approach, 23 of the 51 studies investigated
+data sources outside of textual posts per Figure 10. Research gaps include the lack of multimedia
+and law enforcement studies, with only three hybrid text-image detection [57, 99, 111] and one
+study utilising FBI anonymous tips, Automated Targeting System and Secure Flight data [48, 119].
+6.1.2
+Data Collection and Annotation Bias.
+Due to the varying fiscal costs, biases, and time trade-offs, there is no consensus for selecting or
+excluding annotators for supervised learning datasets. Hence, we frame that annotator selection
+falls within two varying groups: experience-driven selection and organisation-driven selection.
+For the former, experience-driven selection includes studies that utilised self-proclaimed ‘expert’
+panels as determined by their location and relevant degrees [123], are anti-racism and feminist
+activists [122], or work on behalf of a private anti-discrimination organisation [105]. However,
+assembling annotators by specific characteristics may be time-consuming or costly, such as crowd-
+sourcing tertiary annotators via Amazon Mechanical Turk, or Figure Eight [10, 45, 71, 81, 93].
+Conversely, an organisation-driven selection approach focuses on agreement by a crowdsourced
+consensus. Instead of relying on specific experience, researchers utilised custom-made tests for
+knowledge of hate speech criteria based on the researchers own labelled subset [122, 128]. Likewise,
+organising annotator pools can also include balancing annotators self-reported political affiliation to
+reduce political bias [93]. Researchers use Inter-rater agreement, and Kappa Coefficient to determine
+a post’s ERH classification. For racism, sexism, and neither classifications, annotation Fleiss’ Kappa
+values ranged between 0.607 [32] to 0.83 [128], indicating moderate to strong agreement [125].
+Thirdly, unsupervised clustering enables mass data collection without time-consuming annota-
+tion via Louvain grouping (to automatically group text/networks to identify emergent groups) [15,
+16, 108], or grouping based on a thread’s affiliation (e.g., the now-banned r/altright [46] and v/[n-
+word] [27]). Although not all posts from an extremist platform may be manifestly hateful, as evident
+in the 9507 post ‘non-hate’ class in the Stormfront benchmark dataset from de Gibert et al. [32].
+Research continues to skew towards radical Islamic extremism per Figure 4, while the plurality
+(41%) target generic ‘hate speech’ in ‘hate or not’, or delineations for racism, sexism, and/or offence.
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech
+000:17
+6.1.3
+Benchmark Datasets.
+We define a benchmark dataset as any dataset evaluated by three or more studies. The majority of
+studies used custom web-scrapped datasets or Tweets pulled via the Twitter API per Figure 1.
+Table 3. Datasets used by three or more studies.
+Dataset
+Year
+Categories
+Platform
+of origin
+Collection strategy
+Used By
+Waseem
+and
+Hovy [123]
+2016
+16914 tweets: 3383 Sexist ,
+1972 Racist, 11559 Neutral
+Twitter
+11-point Hate Speech Cri-
+teria
+[10, 52, 55, 56,
+71, 81, 83, 91,
+123, 126]
+FifthTribe
+[39]
+2016
+17350 pro-ISIS accounts
+Twitter
+Annotated pro-ISIS ac-
+counts
+[7, 84, 95, 102]
+de Gibert
+[32]
+2018
+1196 Hate, 9507 Non-hate,
+74 Skip (other) posts
+Stormfront
+3 annotators considering
+prior posts as context
+[32,
+55,
+71,
+126]
+OffenseEval
+(OLID) [128]
+2019
+14100 tweets. (30%) Of-
+fensive or Not; Targeted
+or
+Untargeted
+insult;
+towards an Individual,
+Group, or Other
+Twitter
+Three-level hierarchical
+schema, by 6 annotators
+[55, 67, 128,
+130]
+HatEval [13] 2019
+10000 tweets distributed
+with Hateful or Not, Ag-
+gressive or Not, Individual
+targeted or Generic
+Twitter
+Crowdsourced via Figure
+Eight, with 3 judgements
+per Tweet
+[13, 55, 71, 90,
+126]
+Hatebase-
+Twitter [31]
+2019
+25000 tweets: Hate speech,
+Offensive, Neither
+Twitter
+3 or more CrowdFlower
+annotators per tweet
+[31, 52, 55, 71,
+81, 83, 126]
+TRAC [61]
+2018
+15000 English and Hindi
+posts; Overtly, Covertly,
+or Not Aggressive
+Facebook
+Kumar et al. [62] subset, 3
+annotators per post, com-
+ment or unit of discourse
+[55, 56, 71]
+6.2
+Feature Extraction Techniques
+Figure 7 outlines the three types of feature extraction techniques. Non-contextual lexicon approaches
+relate to word embeddings for entities, slurs, and emotional features. However, non-contextual
+blacklists and Bag of Words (BoW) lexicon approaches cannot identify context, concepts, emergent,
+or dual-use words (see the Supplementary Material’s Algorithm Handbook section for comprehensive
+definitions) [32, 81, 83, 90, 122]. Contextual sentiment embeddings expand on lexicons by embedding
+a form of context via positional tokens to establish an order to sentences.
+We group unsupervised term clustering and dimensionality reduction methods under the
+Probability-Occurrence Models category. The two dominant approaches include weighted ANOVA-
+based BoW approaches and Term Frequency-Inverse Document Frequency (defined in the Supple-
+mentary Material), which weigh the importance of each word in the overall document and class
+corpus. Contextual sentiment embeddings result in higher F1-scoring models (per RQ4) due to their
+context-sensitivity and compatibility as input embeddings for deep learning models [55, 71, 83].
+Community detection features require mapping following, friend, followee, and mention dy-
+namics. Furthermore, other statistically significant metadata includes profile name, geolocation (to
+investigate ERH as a disease), gender, and URLs [16, 81, 84, 123, 126]. URLs can identify rabbit holes
+for misinformation or alternate forums via PageRank [88] and Hyperlink-induced Topic Search
+(HITS) [59]—which extracts keywords, themes and topical relations across the web [77, 88].
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+000:18
+Govers et al.
+Fig. 7. Types of feature extraction techniques.
+6.3
+Data Filtering
+For context-insensitive BoW and non-deep models, stop words (e.g. the, a, etc.), misspellings, and web
+data are often filtered out via regular expressions and parsing libraries [4, 27, 46, 47, 52, 69, 71, 81, 84].
+Compared to semantic or reply networks, community detection models tend to extract metadata
+for separate clustering for entity and concept relationships. All data filtering techniques are thereby
+aggregated and branched in Figure 8.
+No studies considered satire, comedy, or irony to delineate genuine extremism and online culture.
+Researchers’ implicit consensus is to treat all posts as part of the ERH category if it violates their
+criteria, regardless of intent. Conversely, Figure 9 displays the 14% of the studies filtered bots
+by removing but not classifying bot accounts from the ERH datasets [12, 15, 16, 46, 69, 93, 109].
+Strategies include removing duplicate spam text, filtering Reddit bots by name, and setting minimum
+account statistics for verification—such as accounts with that share hashtags to at least five other
+users to combat spam [16]. Likewise, Lozano et al. limited eligible users for their dataset to have at
+least 100 followers, with more than 200 tweets, and at least 20 favourites to other tweets [69]. This
+operates on the assumption that bots are short-lived, experience high in-degree between similar
+(bot) accounts, and seldom have real-world friends or followers—as discovered by Bartal et al. [12]
+Outside of removing suspicious bot accounts via human annotation in dataset generation, com-
+putational means to explicitly categorise bots or trolls remains an area for future research.
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+SenticNet
+LIWC
+('Bag of Concepts/Narratives')
+Hatebase
+Word2Vec
+Doc2Vec
+Blacklists
+Semanticizer
+Non-contextualLexicon
+SentiWordNet
+Contextual Sentiment
+SentiStrength
+Approaches
+Embeddings
+SIMON
+Entity Detection
+Transfer Learning
+TransferLearningDatasets
+Tasks (pre-training)
+(Stanford Sentiment)
+EmoFeat
+GPT
+Dynamic
+Dependency
+FastText
+few shot
+Clustering
+Parse Trees
+learning
+WordPiece
+Theme and Narrative
+Bag of
+Tokenization
+Clustering
+Words
+Entity Clustering
+PCA
+Probability-Occurrence Models
+TF-IDF
+DBSCAN
+ANOVA F-value weighted BoWDown the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech
+000:19
+Fig. 8. Types of data filtering techniques across NLP and community detection studies.
+Fig. 9. Studies with Bot or Troll Filtering.
+Fig. 10. Special Type of Data Used.
+6.4
+Key Takeaways for Dataset Domain, Pre-processing, and Annotation
+Twitter’s accessible API, popularity and potential for relationship modelling via reply and hashtag
+networks, makes it the platform of choice for research (Figure 1). Despite the rise of far-right
+extremism post-2015, Islamic extremism in the US and Europe remains the target group for the
+majority of organisation-based studies, with no studies considering far-right/left manifestos. The
+marginalisation of Oceania and the global south by datasets predominantly containing US white
+hetero males indicates a structural bias in academia. For feature extraction, we recommend using:
+(1) Contextual sentimental embeddings—due to their compatibility with deep learning models
+and highest performance, per Table 4.
+(2) Pre-defined lexicons—assuming they remain up-to-date with online culture.
+(3) Probability-occurrence models—ideal for large-scale clustering of emergent groups [27, 99, 126].
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+RELATIONS BETWEENPOSTS
+6
+SEMANTIC FEATURE DATA AND DICTIONARIES
+5
+NETWORKS
+METADATA (PROFILE DATA, LOCATION)
+SIMILARITY TO CORPUS DATA
+3
+TEMPORAL DATA (TIME ZONE, TOPICAL)
+3
+LAW ENFORCEMENT DATA
+AUDIO-VISUAL DATA
+0
+1
+2
+3
+4
+5
+6
+7Transliteration
+correction or
+removal
+133tsp3akto
+leetspeak
+Spelling
+(alphabetic
+correction
+representation)
+Regular
+Lowercase
+Hashtag
+Spelling
+expressions
+conversion
+expansion
+URL Topic
+("#HelloWorld" to
+Extraction
+"Hello""World")
+Hashtag
+Clustering
+Filtering
+Word Stemming
+Grammar
+Techniques
+Metadata
+Relations
+(Community
+Detection)
+Variant word
+substitution
+(alternate spellings)
+LangDetect
+Location
+Short posts
+(to remove non
+(min word count)
+Removal
+English posts)
+Stop words
+URLS
+Identity pseudo-
+(for context
+anonymisation
+insensitive MLAs
+(usenames, names,
+mentions etc.)6%
+14%
+Yes
+■No
+Unspecified or
+Irrelevant to Study
+80%000:20
+Govers et al.
+We do not recommend pre-defined lexicons on non-English text, new groups or ideologies—as
+these lexicons may not translate to different concepts and slurs. We recommend adaptive semi or
+unsupervised learning via contextual embeddings and entity/concept clustering for edge cases.
+Research currently lacks multidomain datasets, pseudo-anonymous platforms, multimedia (i.e.,
+images, videos, audio and livestreams), and extraction of comedy, satire, or bot features.
+7
+COMMUNITY DETECTION, TEXT, AND IMAGE ERH DETECTION ALGORITHMS
+What are the computational classification methods for ERH detection?
+Between 2015-2021, non-deep Machine Learning Algorithms (MLAs) shifted towards Deep
+Learning Algorithms (DLAs) due to their superior performance and context-sensitivity (Table 4).
+Support Vector Machines (SVM) and a case of a Random Forest (RF) model were the last remaining
+non-deep MLAs post-2018 to outperform DLAs. Studies seldom hybridise relationship network
+modelling and semantic textual analysis. Ongoing areas of research in MLAs consist of identifying
+psychological signals to compete with DLAs such as Bidirectional Encoder Representations from
+Transformers (BERT), Convolutional Neural Networks (CNN) and Bidirectional Long Short-Term
+Memory (BiLSTM) models (defined further in the Supplementary Material’s Algorithm Handbook
+section). DLAs are best oriented for text-only tasks and for hybrid image-caption models [3, 57,
+99, 111]. Future NLP studies should consider higher-performing neural languages models over
+BERT-base—such as RoBERTa [68], Sentence-BERT [96], or multi-billion parameter transformers
+such as GPT-3 [23], GPT-Neo [19], or Jurassic-1 [65].
+7.1
+Observed Non-deep Machine Learning Algorithms (MLAs)
+Studies investigating non-deep MLAs tend to test multiple models, typically Support Vector Ma-
+chines (SVMs), Random Forest (RF), and Logistic Regression. Figure 11 outlines the distribution of
+both deep and non-deep approaches, with SVM again the most popular MLA in 15 of the 51 studies.
+Non-deep MLAs consistently under-performed for multiclass classification, whereby Ahmad et al.
+identified that a prior Naïve Bayes model could not distinguish between ‘Racism’ and ‘Extremism’
+classes due to a low F1-score of 69%; while their LSTM and GRU model could detect such nuance
+with a 84% F1-score [4]. Likewise, application-specific sentimental algorithms paired with MLAs
+resulted in lower F1-scores compared to context-sensitive BERT models—which do not require
+manual feature extraction [55, 71, 83, 126]. Sharma et al. claimed that SentiStrength was "...not
+robust to various sentence structures and hence fails to capture important semantics like sarcasm,
+negation, indirect words, etc. at the phrase level" [107, pg. 5]—a critique shared in six other non-deep
+sentiment scoring studies [47, 69, 84, 89, 103, 130]. Consider the hypothetical case of "I am not happy
+with those people", whereby context-insensitive (orderless) embeddings will not detect the negation
+of happy nor the implicit euphemism for ‘those people’.
+Hence, researchers have three options when designing ERH models:
+(1) Avoid complex textual feature extraction and filtering by prioritising DLA development, or
+(2) Prioritise manual textual and metadata feature extraction, such as psychological signals,
+emotions, sarcasm, irony, temporal data, and/or
+(3) Consider community detection (relationship network or topic modelling) features.
+7.1.1
+Non-deep Machine Learning Algorithms in Community Detection Studies.
+There is a discrepancy in the choice of algorithmic approach compared to NLP-oriented models
+where less than a third of the community detection studies considered Deep Learning (DL) mod-
+els [77, 84, 99], while NLP-only studies were majority DL (15 of 29). A reason for this discrepancy
+would be the limited research in social media network analysis without investigating textual data,
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech
+000:21
+Fig. 11. Number of instances of Machine Learning Algorithms (MLAs) used for ERH detection.
+instead opting to cluster group affiliation via K-means [12, 16, 80], NbClust [12], weighted bipartite
+graphing into Louvain groups [15, 16], and fast greedy clustering algorithms [80].
+We observed that graphing relationship networks result in two types of classification categories:
+(1) Meso-level affiliation—semi or unsupervised affiliation of a user to an extremist group or
+organisation, with a bias towards Islamic extremist groups [15, 16, 80, 84, 99, 101, 108].
+(2) Micro-level affiliation—(semi)supervised person-to-person affiliation to an annotated extremist,
+such as radicalising influencers [12, 24, 77], and legal person-of-interest models [48].
+For organisational affiliation, information for clustering included the use of hashtags shared by
+suspended extremist Twitter users and unknown (test) users [7, 15, 84, 95].
+For identifying a user’s affiliation to other individuals, researchers preferred non-textual graph-
+based algorithms as they reduce memory complexity and avoid the perils in classifying ambiguous
+text [16, 80]. Furthermore, 2016-2019 demonstrated a move from investigative graph search and
+dynamic heterogeneous graphs via queries in SPARQL [48, 80] towards Louvain grouping on
+bipartite graphs as a higher-performing (by F1-score) classification method [15, 16, 108].
+For hybrid NLP-community detection models, researchers mapped text and friend, follower/ing,
+and mention networks via decision trees and kNN [77, 84], or used Principal Component Analysis
+on extracted Wikipedia articles to map the relationships between discussed events and entities [99].
+An emerging field of community detection for extremism consists of knowledge graphs. Know-
+ledge graphs represents a network of real-world entities, such as events, people, ideologies, situ-
+ations, or concepts [125]. Such network representations can be stored within graph databases,
+word-embeddings, or link-state models [48, 125, 127]. Link-state knowledge models consist of
+undirected graphs where nodes represent entities and edges represent links between entities, such
+as linking Wikipedia article titles with related articles based on those referenced in the article, as
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+SUPPORTVECTOR MACHINE (SVM)
+15
+SENTIMENTAL ALGORITHMS
+11
+LOGISTIC REGRESSION
+10
+OTHER DECISION TREES
+OTHER (NOVEL) NEURAL NETWORK
+9
+GRAPHING ALGORITHMS
+7
+NOVEL APPROACHES
+6
+BERT
+6
+CONVOLUTIONAL NEURAL NETWORK (CNN)
+6
+LSTM-ONLY
+6
+RANDOM FOREST
+5
+ENSEMBLE NEURAL NETWORKS
+5
+KNN CLASSIFICATION
+NAIVE BAYES
+GRU NEURAL NETWORK
+PRINCIPAL COMPONENT ANALYSIS (PCA)
+2
+K-MEANS CLUSTERING
+2
+LDA (LINEAR DISCRIMINANT ANALYSIS)
+2
+LINEAR REGRESSION
+1
+LSA (LATENT SEMANTIC ANALYSIS)
+1
+0
+2
+4
+6
+8
+10
+12
+14
+16000:22
+Govers et al.
+used in Wikipedia2Vec [127]. Hung et al. consider a novel hybrid OSINT and law-enforcement
+database graph model-—which unifies textual n-grams from social media to shared relationships
+between other individuals and law enforcement events over time [48].
+Four studies consider model relationships to individual extremist affiliates [12, 24, 48, 77]. In a
+direct comparison between text and relationship detection models, Saif et al. observed that text-only
+semantic analysis outperformed their graph-based network model by a +6% higher F1-score [101].
+7.2
+Deep Learning Algorithms (DLAs)
+DL studies are rising, with less than a third of studies including DLAs pre-2019 [10, 18, 27, 32, 45,
+91, 99]. The percentage of all studies which included a DLA per year was 0% in 2016, 27.3% in
+2017 [10, 27, 45, 99], and 33.3% in 2018 [18, 32, 91], compared to being the majority post-2018 (81.8%
+in 2019 [1, 4, 53, 67, 71, 77, 83, 84, 90, 128–130], 54.5% in 2020 [14, 55, 57, 81, 107, 111] and 80% in
+2021 [3, 52, 126])—with Figure 12 displaying the shift towards DLAs since 2015.
+Fig. 12. Patterns of adoption for ERH detection algorithms over time. Colour change ordered by F1-score
+trend (low to high). Brown = ~0.75 F1-score on benchmark datasets, Red = ~0.9 F1-score, Grey = No Data.
+Between 2017-2018 Convolutional Neural Networks (CNN) using Long-Short Term Memory
+(LSTM), GRU, Recurrent Neural Networks (RNN), or graph-based layers were the sole DLAs [10, 18,
+32, 45, 91, 99]. From 2019-2021, various new approaches such as SenticNet 5 [89], ElMo (Embeddings
+from Language Model) [90], custom neural networks such as an Iterative Opinion Mining using
+Neural Networks (IOM-NN) model [14], and attention-based models such as BiLSTM [81, 83, 90,
+128, 129]. Since 2019, there is an emerging consensus towards BERT [67, 71, 81, 107, 126, 129, 130]
+due to its easy open-source models on the Hugging Face platform and high performance per Table 4.
+7.2.1
+Deep Learning for Community Detection.
+Only 3 of the 21 DL studies considered relationship network models [77, 84, 99]. Whereby, Mashech-
+kin et al. grouped self-proclaimed Jihadist forums and VK users with Jihadist keywords as a
+"suspicious users" category [77]. Uniquely, the researchers implemented a Hyperlink-induced Topic
+Search (HITS) approach to calculate spatial network proximity between annotated extremists and
+unknown instances. HITS identifies hubs, which are influential web pages as they link to numerous
+other information sources/pages known as authorities [59]. The influence of an authority depends
+on the number of hubs that redirect to the authority. An example of HITS in-action would be an ex-
+tremist KavazChat forum (a hub) with numerous links to extremist manifestos (authorities) [59, 77].
+Evaluating influence in these graph networks requires measuring spatial proximity via betweenness
+centrality [41] and depth-first search shortest paths where proximity to a known extremist via
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+Prior SLR trends.
+Algorithmic and Feature Extraction Trends found in our SLR
+Community
+Expansion of Sentimental
+Year-of-Hybrid-Models
+Detection
+algorithms. First
+CNN+LSTM, BiLSTM. Use
+BERT & SVM as
+Blacklist
+with Semantic
+application of CNN
+of language models
+the dominant
+Words.
+Clustering.
+LSTM, & Word2Vec.
+(BERT),
+ approaches.
+1
+2000-2010
+2010-2015
+2015
+2016
+2017
+2018
+2019
+2020
+2021
+1
+Bag of Words
+Generic ML (sentimental
+Year-of-Extremism-
+Context-sensitive models
+Approaches.
+feature extraction in
+detection via SVM, RNN
+& features - phrase,
+MLA) Bipartite Graphing.
+kMeans. Emergence of
+ExtremSentiLex, opinion
+Waseem & Hovy dataset.
+affiliation-based datasets
+& propaganda mining.Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech
+000:23
+following/reposting them constitutes an extremist classification. However, such relations do not
+accommodate for replies to deescalate, deradicalise, or oppose extremist speech.
+7.2.2
+Visual-detection Models for ERH Detection.
+Despite the emergence of multimedia sources for radicalisation and ideological dissemination,
+only three studies considered multimodal image and image-text sources—utilising image memes
+with superimposed text from the Facebook hateful meme dataset [57] and the MultiOFF meme
+dataset [111]. Only one study considered the post’s text (i.e., text not displayed on the image itself)
+as context via the multimedia Stormfront post and image data from Rudinac et al. [99].
+For the Facebook hateful meme and MultiOFF datasets include images with superimposed cap-
+tions [57, 111]. Both Kiela et al. [57] and Aggarwal et al. [3] extract caption text via Optical Character
+Recognition (OCR) models—-a computer vision technique to convert images with printed/visual text
+into machine-encoded text [125]. The three hateful meme studies utilised either both (multimodal)
+or one (unimodal) of the image and its caption [3, 57, 111]. The multimodal Visual BERT-COCO
+model attained the highest accuracy of 69.47%, compared to 62.8% for a caption text-only classifier
+or 52.73% for image only, 64% for the ResNet152 model [3]; and 84.70% for the baseline (human) [57].
+Fig. 13. Deep learning pipeline for visual-text ERH detection based on the hateful meme studies [3, 57, 111].
+The highest performing multimodal model relied on Visual BERT [57]. Visual BERT extends
+textual BERT by merging BERT’s self-attention mechanism to align input text with visual regions
+of an image [64]. VisualBERT models are typically pretrained for object detection, segmentation,
+and captioning using the generic Common Objects in COntext (COCO) dataset [66], such that
+the model can segment and create a textual description of the objects behind an image such as
+“two terrorists posing with a bomb” (Figure 13). Training otherwise acts the same as BERT-—which
+involves masking certain words/tokens from a textual description of the image of what the image
+depicts, and VisualBERT predicting the masked token(s) based on the image regions. We aggregate
+and generalise all visual ERH detection studies architectural pipelines in Figure 13 [3, 57, 99, 111].
+No hateful meme dataset studies consider accompanying text from the original post. This raises
+concerns regarding posts satirising, reporting, or providing counter-speech on hateful memes.
+Only one study investigated a contextual textual post and accompanying images through a
+proposed Graph Convolutional Neural Network (GCNN) model [99]. This GCNN approach extracted
+semantic concepts extracted from Wikipedia, such as identifying that an image was a KKK rally—
+attaining a 0.2421 F1-score for detecting forum thread affiliation across 40 Stormfront threads [99].
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+Key:
+Dotted lines = Optional path. Full line = necessary.
+Red outline = Computer Vision (lmage Embeddings)
+ResNet152
+Extract 1000-
+Computer Vision
+dimension
+Model
+embeddings
+Extract Regions of
+Create Textual Representation
+Entity and Object
+the Images (Visual
+VisualBERT
+("Two individuals with a
+Detection
+BERT tokens)
+bomb")
+Separate contextual
+NLP model (e.g.,
+Top Text
+BERT)
+Textual Optical
+Textual Feature
+Concatenate
+Hate Classifier
+Character
+Extraction
+Embeddings
+Neural Network
+Recognition
+"Top
+"Bottom
+Bottom Text
+Text"
+Text"000:24
+Govers et al.
+8
+MODEL PERFORMANCE EVALUATION, VALIDATION, AND CHALLENGES
+What are the highest performing models, and what challenges exist in cross-examining them?
+Evaluating model performance presents three core challenges for future researchers:
+(1) Dataset domain differences—which may include or exclude relevant features (e.g., gender,
+location, or sentiment) and may involve numerous languages or groups (e.g., Islamic extrem-
+ists vs. white supremacists) who will express themselves with different lexicons [20, 53].
+(2) Criteria differences—different standards for ERH definitions, criteria, filtering, and annota-
+tion threaten cross-dataset analysis between models [24, 122]. Binary classification can result
+in higher accuracy compared to classifying nuanced and non-trivial subsets of hate such as
+racism/sexism [122], overt/covert aggression [61], or hateful group affiliation [53].
+(3) Varying and non-standard choice of metrics—Figure 14 displays the 28 metrics, which
+vary depending on whether the study investigates community detection via closeness, in-
+betweenness, and eigenvectors; or NLP, often via accuracy, precision, and F1-scores.
+Fig. 14. Distribution of metrics used across the 51 studies—demonstrating a lack of standardisation.
+Table 4. Models ranked by macro F1-score for the benchmark datasets across studies (inter-study evaluation).
+Dataset
+1st Highest
+2nd Highest
+3rd Highest
+Waseem
+and
+Hovy [123]
+0.966 (BERT with GPT-2
+fine-tuned dataset [126])
+0.932 (Ensemble RNN [91])
+0.930 (LSTM + Random Em-
+bedding + GBDT [10])
+FifthTribe [39]
+1.0 (RF [84])
+0.991-0.862 (SVM [7])
+0.87 (SVM [95])
+de Gibert [32]
+0.859 (SP-MTL LSTM, CNN
+and GRU Ensemble [55])
+0.82 (BERT [71])
+0.73 (LSTM baseline met-
+ric [32])
+TRAC FB [61]
+0.695 (CNN + GRU [55])
+0.64 (LSTM [61])
+0.548 (FEDA SVM [56])
+Hatebase Twit-
+ter [31]
+0.923 (BiLSTM with Atten-
+tion modeling [83])
+0.92 (BERTbase+CNN / BiL-
+STM [81], 0.86 (with racial/-
+sexual debiasing module)
+0.912
+(Neural
+En-
+semble [71])
+HatEval [13]
+0.7481
+(Neural
+En-
+semble [71])
+0.738
+(LSTM-
+ELMo+BoW) [90]
+0.695 (BERT with GPT-2
+fine-tuned dataset [126])
+OffensEval [128]
+0.924 (SP-MTL CNN [55])
+0.839 (BERT [130])
+0.829 (BERT 3-epochs [67])
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+F1-SCORE
+35
+ACCURACY
+25
+PRECISION
+21
+RECALL
+19
+P-VALUE
+3
+CORRELATION COEFFICIENT
+NETWORKCENTRALITIES
+2
+MEAN ABSOLUTE ERROR
+STANDARDIZED CO-INCIDENCE RATIO
+LOUVAIN (Q) MODULARITY
+PATIAL PROXIMITY
+1
+GINI COEFFICIENT
+1
+INFORMATION GAIN
+PARTITION SENSITIVITY
+MEANLOG ACCURACY
+1
+RACIST SCORE (CUSTOM METRIC)
+1
+WILCOXON SIGNED-RANK TEST
+CHI-SQUARE INDEPENDENCE
+PEARSON CORRELATION COEFFICIENT
+1
+PAGERANK SCORES
+RADICAL SCORE (SENTISTRENGTH)
+1
+HYBRID STRUCTURAL-SEMANTIC SIMILARITY
+1
+POSSIBILISTIC SIMILARITY MEASURE
+ROC AREA
+SIRA RADICAL SCORE
+1
+R^2 VALUE
+0
+10
+15
+20
+25
+30
+5
+35
+40Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech
+000:25
+8.1
+Benchmark Dataset Performance (Inter-study Evaluation)
+We use macro F1-score as the target metric as it balances true and false positives among all classes,
+and is a shared metric across the benchmark datasets [13, 31, 32, 39, 123, 128]. Table 4 outlines that
+the highest F1-scoring models reflect the move towards context-sensitive DLAs like BERT, as also
+displayed in Figure 12. SVMs and a single instance of a Random Forest classifier on sentimental
+features were the last standing non-deep MLAs [7, 56, 84, 95]. Given the variety of MLA and DLAs
+(Figure 11), approaches that frequently underperformed included Word2Vec, non-ensemble neural
+networks such as CNN-only models, and baseline models [31, 46]. These baseline models include
+the HateSonar model by Davidson et al. [31], Waseem and Hovy’s n-grams and gender-based
+approach [123], LSTM model by de Gibert et al. [32], and C-Support Vector Classification by Basile
+et al. [13]. No studies discuss memory or computational complexity, an area worthy of future
+research as expanded in our Supplementary Material’s Section 3.
+8.2
+Community Detection Performance
+While community detection models tend to produce F1-scores ~0.15 lower than DLAs [12, 15, 16, 48,
+80, 108], these comparisons rely on different datasets/metrics. Shi and Macy recommended using
+Standardised Cosine Ratio as the standardised metric for structural similarity in network analysis, as
+it is not biased towards the majority class, unlike Jaccard or cosine similarity [108]. For community
+detection models on the same pro/anti-ISIS dataset [39], F1-scores ranged from 0.74-0.93 [7, 15, 95].
+Only one study cross-examined text and network features [101], with a hybrid dataset consisting
+of annotated anti/pro-ISIS users’ posts and number of followers/ing, hashtags, mentions, and
+location. Text-only semantic analysis outperformed their network-only model (0.923 F1 vs. 0.866
+respectively) [101]. However, topic (hashtag) clustering and lexicon-based sentiment detection
+via SentiStrength underperformed compared to the network-only approach by a 0.07-0.1 lower
+F1 [101]. Thus, unsupervised clustering models are ideal for temporal radicalisation detection and
+identification of emergent or unknown groups or ideologies. There is insufficient evidence to conclude
+whether community detection is superior to NLP due to the lack of shared NLP-network datasets.
+For supervised community detection tasks, researchers [15, 80, 101] used network features via
+Naïve Bayes [80], k-means [12, 16, 80], SVM [84], and decision trees [77, 84]. The highest F1-score
+community detection model was a hybrid NLP and community detection model using network
+features, keywords and metadata (i.e., language, time, location, tweet/retweet status, and whether
+the post contained links or media) with a Naïve Bayes classifier—attaining a 0.89 F1-score [80].
+9
+FUTURE RESEARCH DIRECTIONS
+In this section, we offer an alternate to the radicalisation = extremism = political hate speech consensus
+from RQ1 and models observed in RQ3/4 to present a new framework for delineating and expanding
+ERH for future work. Overall, we propose an uptake roadmap for ERH context mining to expand the
+field into new research domains, deployments for industries, and elicit governance requirements.
+9.1
+Ideological Isomorphism—a Novel Framework for Radicalisation Detection
+Definition 4: Ideological Isomorphism (Computational Definition for Radicalisation)
+The temporal movement of one’s belief space and network of interactions from a point of
+normalcy towards an extremist belief space. It is an approach to detecting radicalisation with
+an emphasis on non-hateful sentiment as ringleaders and/or influencers pull and absorb others
+towards their hateful group’s identity, relationships, and beliefs.
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+000:26
+Govers et al.
+As outlined in our novel tree-diagram dissection of ERH definitions to their computational
+approach in Figure 6, there is considerable overlap in approaches between the otherwise unique
+fields of extremism, radicalisation and politicised hate speech. Radicalisation’s working definition
+suffers from ambiguity in the majority of studies due to its interchangeability towards extremist
+affiliation and no considerations for temporal changes. Radicalisation’s computational definition
+should reflect a behavioural, psychological, and ideological move towards extremism over time.
+While extremist ideologies and outwards discourse towards victim groups may be manifestly
+hateful, radicalisation towards target audiences may involve non-hateful uniting and persuasive
+speech [58]. Hence, we propose that radicalisation detection should not be a single-post classification.
+Rather, models should consider micro (individual), meso (group dynamics), and macro (global
+events and trends) relations. The roots for radicalisation result from an individual’s perceptions of
+injustice, threat, and self-affecting fears on a micro-level. On a meso-level, this can include the rise
+of community clusters based on topics and relationships. Socially, a radicalised user draws on an
+extremist group’s legitimacy, connections and group identity, trends, culture and memes [58, 70, 79].
+Hence mapping ideological isomorphism requires temporal modelling to:
+(1) Detect the role of users or groups polarising or pulling others towards extreme belief spaces
+(i.e., ideological isomorphism), akin to detecting online influencers [12, 46, 80, 121]. Studies
+should also consider the role of alienation as a radicalising factor via farthest-first clustering.
+(2) Further research into the role of friendship and persuasion by adapting sentimental ap-
+proaches to consider positive reinforcement towards hateful ideologies akin to existing
+research in detecting psycho-behavioural signals [84]. Furthermore, there lacks research in
+computationally detecting social factors such as suicidal ideation or mental health.
+(3) Investigate the interactions between groups across social media platforms as radicalisers
+themselves, such as the promotion of extremist content by recommendation algorithms.
+(4) Utilise community detection metrics such as centrality, Jaccard similarity, and semantic
+similarity over time as measurements for classifying radicalisation for meso-level NLP (topic)
+and graph-based (relational) clustering, leaving content moderation as a separate task.
+(5) Consider the role of satire, journalism, and martyrs as areas for radicalisation clustering.
+9.2
+Morphological Mapping and Consensus-building—a novel
+computationally-grounded framework for extremism detection
+Definition 5: Morphological Mapping and Consensus-building (Extremism)
+The congregation of users into collective identities (‘in-groups’) in support of manifestly
+unlawful actions or ideas.
+While ideological isomorphism focuses on micro-level inter-personal relations, morphological
+mapping pertains to clustering meso-level beliefs and community networks to extremist ideologies.
+While we discovered various affiliation-based clustering approaches, no studies identified novel or
+emergent movements. Establishing a ground truth for a novel extremist organisation is challenging
+if such groups are decentralised or volatile. Hence, we recommend using manifestos, particu-
+larly unconsidered far-right sources, and influential offline and online extremists as a benchmark
+for identifying martyrdom networks and new organisations. Areas for future research include
+investigating the role of trolls, physical world attacks, or misinformation in narrative-building.
+Our morphological mapping framework proposes to delineate Extremism by considering the role
+of group identity and ideological themes behind hate speech by considering affiliation across users
+and posts. When targeting extremism, pledging ‘support’ to a terrorist organisation may not violate
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech
+000:27
+context-insensitive BoW hate speech classifiers—-hence it is not appropriate to categorise extremist
+affiliation under the same guise as post-by-post hate speech. Currently, extremism detection
+constitutes a binary ‘pro vs anti group’ classification, which fails to capture the inner trends of
+radicalisation from peaceful, to fringe beliefs, to committing to violent-inducing beliefs online, and
+potentially to offline extremism. Investigating semi or unsupervised clustering (mapping) of groups
+will also aid Facebook’s commitment to moderating militarised social movements, violence-inducing
+conspiracy networks, terrorist organisations, or hate speech inducing communities [37].
+Thus, we propose four prerequisites for studies to fall under the extremism detection category:
+(1) Investigate the interactions and similarities between groups on mainstream and anonymous
+platforms to map group dynamics and extremist networks. For privacy, we recommend
+group-level (non-individualistic) network and semantic clustering.
+(2) Map affiliation and group dynamics. Given the lack of definitions for extreme affiliation, we
+recommend using Facebook’s definition of affiliation as a basis—being the positive praise of
+a designated entity or event, substantive (financial) support, or representation on behalf of a
+group (i.e., membership/pledges) [37].
+(3) Investigate hateful and non-hateful community interactions, memes and trends, that reinforce
+group cohesion.
+(4) Map affiliation as a clustering task, akin to our proposed radicalisation framework but without
+the temporal component.
+9.3
+Outwards Dissemination––‘traditional’ hate speech detection updated
+Definition 6: Outwards Dissemination (Hate Speech)
+Targeted, harassing, or violence-inducing speech towards other members or groups based on
+protected characteristics.
+Hence, the projection and mainstreaming of hateful ideologies through speech, text, images,
+and videos requires an outwards dissemination of views shared by extremists, such as racism. The
+outwards dissemination of hate is a strictly NLP (text) and computer vision (entity and object)
+classification problem. We delineate hate speech with affiliation to violent extremist groups as such
+misappropriation could have devastating effects on one’s image, well-being, and safety [9, 24, 29].
+All researchers should be aware that malicious actors may exploit existing ERH models for
+injurious surveillance and censorship. Future work should also consider the impacts of labels
+on society at large, whereby terms such as ‘far-right’ as an alias for white supremacy is both
+misleading, infers a ‘right vs wrong’ left-to-right spectrum, and ambiguous. We recommended
+decoupling religious contexts in favour of technical terms such as ‘radical Islamic extremism’ or
+‘terror-supporting martyrdom’ to avoid grouping religiosity to a political ideology and terrorism.
+Thus, we propose three key prerequisites for a study to be in the hate speech category:
+(1) Investigates textual or multimedia interactions only, whereby detecting cyber-bullying or
+extremist community networks should be separate tasks.
+(2) Decouple affiliation where possible. For instance, white supremacy instead of far-right (an
+ambiguous term) or organisational affiliation.
+(3) Consider models which include latent information, such as news, entities, or implied hate.
+Datasets should explain each classification with categories for disinformation and fallacies.
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+000:28
+Govers et al.
+Future work in outlining hate speech would be a systematic socio-legal cross-examination of hate
+speech laws from governments and policies from social media platforms—including the emerging
+consensus vis-à-vis the harmonised EU Code of Conduct for countering illegal hate speech [34].
+9.4
+Uptake Roadmap for Researchers, Industry, and Government
+We present a pipeline for researchers, industries, and government analysts to approach ERH
+context mining per Figure 15. In addition to this summary visualisation of our key dataset and
+model recommendations, we expand on our actionable recommendations for immediate next steps
+and long-term software requirements for ERH detection in our supplementary material.
+Future SLRs should consider a mixture of academic studies, grey material, and technical reports
+to further encompass our proposed ERH context mining field’s socio-legal component and explore
+industry approaches. We recommend transforming our ethical recommendations for responsible
+research outlined in our SLR design into formalised interdisciplinary guidelines to protect privacy
+and researcher safety. ERH is never a singular end-goal, post, or unexpected event. Hence, detecting
+erroneous behaviour emanating from mental health crises can both avoid ERH online and offline,
+and present avenues for cooperation with third-parties such as suicide prevention and counselling
+groups. Finally, we recommend searching for multimedia-only studies including for livestreams.
+10
+CONCLUSION
+ERH context mining is a novel and wide field that funnels to one fundamental aim—the pursuit
+to computationally identify hateful content to enact accurate content moderation. In our work,
+we harmonised Extremist affiliation, Radicalisation, and Hate Speech in politicised discussions from
+2015-2021 in a socio-technical context to deconstruct and decouple all three fields of our proposed
+ERH context mining framework. Hence, we propose a novel framework consisting of ideological
+isomorphism (radicalisation), morphological mapping (extremism), and outwards dissemination
+(politicised hate speech) based on our findings in RQ1. While hate speech included racism and
+sexism, other forms of discrimination were seldom considered. Extremism and radicalisation
+frequently targeted Islamic groups, particularly from US and European researchers. Binary post-by-
+post classification remain the dominant approach despite the complexity of online discourse.
+There is a clear and present danger in current academia emanating from the unresolved biases in
+dataset collection, annotation, and algorithmic approaches per RQ2. We observed a recurring lack
+of consideration for satire/comedic posts, misinformation, or multimedia sources. Likewise, data
+lacked nuance without contextual replies or conversational dynamics, and were skewed towards
+the US and Europe—with the global south, indigenous peoples, and Oceania all marginalised.
+Computationally, we identified that deep learning algorithms result in higher F1-scores at the
+expense of algorithmic complexity via RQ3/4. Context-sensitive neural language DLAs and SVM
+with sentimental, semantic, and network-based features outperformed models found in prior
+SLRs. However, state-of-the-art models still lack a contextual understanding of emergent entities,
+conversational dynamics, events, entities and ethno-linguistic differences. To combat injurious
+censorship and vigilantism, we recommended several areas for future work in context-sensitive
+models, researcher ethics, and a novel approach to framing ERH in SLRs and computational studies.
+The poor design and abuse of social media threatens the fabric of society and democracy. Re-
+searchers, industries, and governments must consider the full start-to-finish ecosystem to ERH
+context mining to understand the data, their criteria, and model performance. Without a holistic
+approach to delineating and evaluating Extremism, Radicalisation, and Hate Speech, threat actors
+(extremists, bots, trolls, (non-)state actors) will continue to exploit and undermine content modera-
+tion systems. Hence, informed, accurate and ethical content moderation are core to responsible
+platform governance while averting injurious censorship from biased models.
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech
+000:29
+Fig. 15. ERH Context Mining pipeline—with key identified research gaps.
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+Sociolinguistics
+Red = Computer-
+science oriented.
+Computational
+Blue = social-
+Research
+Mixed =
+science
+Gaps:
+Social Media
+Criteria-building
+misinformation.
+Satire &
+sexism.
+beyond racism &
+Hate speech
+affiliation.
+non-violent
+Extremism as a
+temporal process.
+a friendly
+Radicalisation as
+Privacy & Ethics
+Consider:
+ERH
+-
+(RQ1, RQ2, RQ3)
+Dataset Creation
+Data Selection &
+-
+ Collection
+movements
+Conspiracy
+COVID-19 posts
+extremist
+Non-lslamic 
+anonymisation)
+(data
+Researcher safety
+manifestos.
+New APls 
+datasets
+Multiple platform
+platforms 
+/anonymous
+Controversial
+Consider:
+Data Filtering
+Extraction
+Topic mapping to
+Belief space
+entities/events)
+(Wikipedia/news
+sources 
+informative 
+mapping
+emotion detection
+Context-sensitive
+features
+Suicidal ideation
+signals 
+Psychological
+Consider:
+80
+Deployment (RQ3/4)
+Model Choice &
+ Model Creation &
+ Performance
+Privacy Paradigm.
+Image detection
+Hybrid NLP-
+Meme detection
+text generation.
+conversational
+Synthetic
+learning.
+Few-shot
+(GPT)
+Pretrained Models
+Generative
+Differential
+design.
+Privacy-by-
+Consider:
+Model Governance
+benchmarks
+metrics &
+Industry standard
+Meta-analysis 
+Interdisciplinary
+Self-regulation
+recognition
+context 
+New event
+updating)
+(frequently
+models
+Dynamic 'online'
+Consider:000:30
+Govers et al.
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+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+Supplementary Material for:
+Down the Rabbit Hole: Detecting Online Extremism,
+Radicalisation, and Politicised Hate Speech
+JAROD GOVERS, ORKA Lab, Department of Software Engineering, University of Waikato, NZ
+PHILIP FELDMAN, ASRC Federal, US
+AARON DANT, ASRC Federal, US
+PANOS PATROS, ORKA Lab, Department of Software Engineering, University of Waikato, NZ
+Contents
+Contents
+1
+1
+Definitions—The Algorithm Handbook
+1
+1.1
+Definitions for Traditional (non-deep) Machine Learning Algorithms
+2
+1.2
+Definitions for Deep Learning Approaches
+5
+1.3
+Language Transformer Models
+6
+1.4
+Definitions for Prominent Feature Extraction Techniques
+7
+2
+SLR Design Considerations
+8
+2.1
+Quality Assessment Criteria
+8
+3
+The Case for Performance Engineering when Evaluating Models
+10
+4
+Uptake Roadmap Expanded
+10
+4.1
+Model Recommendations
+10
+4.2
+Dataset Recommendations
+11
+References
+12
+1
+DEFINITIONS—THE ALGORITHM HANDBOOK
+This supplementary material document includes the supplementary material referenced in the
+main Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech
+Systematic Literature Review. This document offers a ‘dictionary/look-up table’ for the core al-
+gorithmic architectures for the non-deep machine learning and deep learning models mentioned
+throughout the SLR, alongside other side findings and design considerations. We contextualise the
+Authors’ addresses: Jarod Govers, jg199@students.waikato.ac.nz, ORKA Lab, Department of Software Engineering, University
+of Waikato, Gate 1, Knighton Road, Hamilton, Waikato, NZ, 3216; Philip Feldman, philip.feldman@asrcfederal.com, ASRC
+Federal, Beltsville, Maryland, US; Aaron Dant, aaron.dant@asrcfederal.com, ASRC Federal, Beltsville, Maryland, US; Panos
+Patros, panos.patros@waikato.ac.nz, ORKA Lab, Department of Software Engineering, University of Waikato, Gate 1,
+Knighton Road, Hamilton, Waikato, NZ, 3216.
+Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee
+provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and
+the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored.
+Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires
+prior specific permission and/or a fee. Request permissions from permissions@acm.org.
+© 2021 Association for Computing Machinery.
+0360-0300/2021/12-ART000 $15.00
+https://doi.org/tobereplacedwhenassignedDOI
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+arXiv:2301.11579v1  [cs.SI]  27 Jan 2023
+
+000:2
+Govers et al.
+relevant strengths and weaknesses of the various algorithmic approaches for text and visual models
+for ERH detection. No new findings are in this handbook/‘look-up table’. Hence, those familiar
+with the models listed in the contents above need not read this section.
+1.1
+Definitions for Traditional (non-deep) Machine Learning Algorithms
+We aggregate common and historic non-deep machine learning algorithms into the ‘traditional’
+MLA category. Hence, this section defines each of the baseline models used for textual or community
+detection models—consisting of:
+(1) Sentimental Bag of Words approaches,
+(2) Naïve Bayes,
+(3) Decision Trees,
+(4) Support Vector Machines,
+(5) Clustering Models.
+1.1.1
+Bag of Words (BoW).
+BoW approaches simplify complex contextual sentences into a multiset (‘bag’) of individual words
+by assigning a value or probability to each word in its relation to a specific document class. For
+instance, a BoW approach would deconstruct the contiguous sentence, “The Eldian people are
+the spawn of the devil” (where Eldian is a fictitious race), into an unordered bag of individual
+words. While ‘are’, ‘the’ are unlikely to have a considerable influence on whether a sentence is hate
+speech or not, the use of ‘devil’ and ‘Eldian [race]’ is more frequently paired in hate speech than
+for non-hateful/off-topic text. The disregard of word order and the relationship of BoW approaches,
+and MLA models at large, constitute context-insensitive models. For instance, a BoW model does
+not know that ‘I love the Eldian people but hate their food’ is paring love -> Eldian, and hate -> food,
+and thus would consider ‘I hate the Eldian people but love their food’ as identical. Likewise, BoW
+approaches do not consider alternate word meanings/uses (e.g., ‘I ran for government’ vs. ‘I ran
+away’). Nonetheless, BoW approaches are core to word-specific ‘blacklists’ in content moderation,
+such as banning users who use slurs in a post. However, for nuanced and often politicised discussions
+on controversial topics, simple blacklists can lead to injurious censorship—due to the context and
+use of such words.
+Sentimental algorithms, such as SentiStrength [42] aggregate individual words into individual
+emotions—whereby ‘love’ indicates a positive sentiment, while ‘hate’ generally appears in vitriolic
+speech. Figure 1 outlines an abstracted representation of the sentiment classification based on the
+average sentiment score of a sentence. However, the context-insensitive BoW models again fails for
+nuanced cases, whereby Sharma et al. [41] identified that SentiStrength cannot detect negations
+(e.g., “I am NOT happy” where happy skews the final sentiment scores).
+1.1.2
+Naïve Bayes.
+Naïve Bayes classifiers represent types of probabilistic classifiers utilising Bayes theorem with
+the assumption that the influence of each variable for classification is independent of each other
+(i.e., naïve) [44]. For document classification, notable features are assigned a probability for their
+occurrence given a specific class. For instance, a hate speech post that has an angry sentiment may
+have a P(0.8) (Probability of 80%) of being hateful, given that a test hate speech dataset may be 80%
+angry speech. Bayes rule represents these chains of (assumed) independent/unrelated probabilities
+to form a final probability for a test instance.
+Notable features for probability models include:
+• Textual features—(e.g., sentimental scores, appearance of certain slurs/terms),
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+Supplementary Material for:
+Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech
+000:3
+Fig. 1. An abstracted example of Bag of Words approach within a Naïve Bayes classifier—demonstrating its
+lack of context sensitivity and the focus on key ‘racist’ words for ERH detection tasks.
+• Network data—(e.g., probability that someone who is friends with a supremacist is also a
+supremacist, retweet relationships),
+• Metadata—(e.g., length of a post, readability via a Flesch Reading Ease score, number of posts).
+In the example of Figure 1, the probability that the tweet is racist depends on the probability that
+the racist tweet is angry, contains racial terms (‘Eldian’), the semantic similarity between known
+hate speech posts, and the appearance of a negative lexicon. Naïve Bayes can be a final classifier
+for aggregating context-sensitive embeddings (e.g., deep learning models) and multiple ‘ensembles’
+of approaches/models—via chaining their probabilities together with this Bayes rule.
+1.1.3
+Decision Trees.
+Fig. 2. An example of a decision tree, with the leaf nodes constituting the classification. In the Eldian
+hate speech example, this would require traversing the left branches recursively for the final ‘Hate Speech’
+classification leaf (shown via the red arrows).
+Chaining the correlations between features and their class likelihood can also span a tree of
+scenarios. If an annotated dataset indicates that a post is 80% likely to be racist if a sentiment-scoring
+algorithm detects anger, then a binary decision emerges—if post contains angry words, then likely
+hate speech; if not, then not hate speech. These rules construct decision trees, where the root
+constitutes the instance (text, network, metadata, or image), and each node is a decision, with the
+leaves (final node) being the expected class value (i.e., the classification) [44]. Hence, decision trees
+are not naïve as they rely on specific values of other features when traversing a tree’s branches for
+a prediction.
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+Example: Naive Bayes (simplified):
+"The Eldian people are the spawn of the devil"
+Sentiment
+Target Entity
+Semantic Similarity to
+Lexicon also affiliated
+Annotated Posts
+with extremists:
+Angry
+“hate'
+Eldian
+0.9
+1 (devil)
+P(Racist I Tweet) =
+“Eldian'
+'devil'
+P(Rac.IAngry) * P(Rac.I Eldian) * P(Rac.I O.9) * P(Rac.I 1 lexicon)
+“scum"
+P(Racist)Post to Classify
+Sentiment = Not
+Sentiment =
+Angry
+Angry
+Contains slurs
+>=0.5 semantic
+<0.5 semantic
+No slurs
+similarity
+similarity
+= Yes (Devil)
+Not Hate
+Not Hate
+Hate Speech
+Hate Speech
+Speech
+Speech000:4
+Govers et al.
+Creating an optimal tree that maximises accuracy and precision is not trivial due to the feature
+explosion of possible rules and tree nodes. Hence, Random Forest classifiers rely on a divide-
+and-conquer algorithm for generalising feature pairings into class classifications with a random
+initialisation [18]. This recursive process requires finding optimal splits to maximise the separation
+of classes for a final leaf, with an example tree presented in Figure 2—where a random forest would
+consists of multiple trees as a forest. Ideally, a leaf node should encapsulate instances of one class.
+Random forests generate multiple decision trees and select the final prediction based on the
+predictions from the majority of decision trees. Utilising multiple trees with a random initial tree
+state increases the range of features and values selected during the training step. Utilising multiple
+trees and testing the models on untrained ‘test’ data minimises the risk of over-fitting to the training
+(i.e., a classifier which performs reliably on the training dataset but not on real-world data).
+Random forests strengths include its ability to tie dependent and complex features while reducing
+over-fitting through pruning (i.e., reducing tree size to generalise the model). Hence, decision trees
+capture related concepts in hate speech where naïve BoW approaches do not—such as the appearance
+of anger/negative sentiment invoking the use of charged terms (e.g., racism as an emotional outlet)
+or frequency of posts and sentiment.
+1.1.4
+Support Vector Machines (SVM).
+Fig. 3. Support Vector Machine where instances beyond the boundaries (support vectors) are automatically
+assigned to the class.
+SVMs are another supervised learning model for classification and regression tasks, seeking to
+map instances in vector spaces to maximise the distance between classes [14], visualised in Figure 3.
+Mapping features to multidimensional vectors can exponentially increase dimensions (an issue
+shared in deep-learning models). Thus, SVMs reduce irrelevant features through specific kernels—
+typically a linear, polynomial, Gaussian or sigmoid function. These kernels reduce the feature
+set to draw boundaries between two classes, similar to logistic regression. These boundaries are
+either hard (i.e., a binary classification) or soft—allowing outliers near the boundary for edge cases,
+like niche controversial and offensive, but not ostensibly targeting protected characteristics. SVM
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+Support Vector
+(with cost hyperparameter)
+Neutral
+Speech.
+Y-Axis
+Hate
+Speech
+Optimal hyperplane
+to maximise
+distance (y=mx+c)
+X-AxisSupplementary Material for:
+Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech
+000:5
+models are computationally faster and reduce memory compared to deep learning models [3, 45],
+while achieving comparative performance outlined in RQ4. Dimensionality reduction techniques
+can also reduce runtime by reducing the complexity of large feature spaces from textual or network
+data, such as via Principle Component Analysis [27].
+SVMs are the consistently highest performing MLAs per RQ4, while lowest complexity, with
+𝑂(𝑚∗𝑛) complexity for a Linear Kernel SVC—where m = feature count, and n = number of instances.
+1.1.5
+Clustering and Nearest Neighbour Classifiers.
+Instead of annotated hate speech datasets, clustering methods group by textual similarity via Natural
+Language Processing (NLP), and network relations via Community detection. Hence, clustering can
+work in cases of fully annotated datasets as supervised learning, semi-annotated datasets as semi-
+supervised learning, or unlabelled raw web scrapped data for unsupervised learning.
+For supervised learning, K-Nearest Neighbour (KNN) classifiers work via evaluating the nearest
+neighbours’ likeliness when projecting the textual, network, or metadata features onto a multidi-
+mensional space [2]. The ‘distance’ between feature spaces typically rely on Euclidean, Manhattan,
+or Minkowski distance—where the latter two are suited for non-linear feature spaces. Non-euclidean
+distances are ideal where dimensions are not comparable, as Manhatten distance reduces noise/er-
+rors from outliers since the gradient has a constant magnitude.
+Clustering examples for hate speech detection includes K-Means, which partitions n observations
+into k clusters [27]. K-Means automatically generates clusters, thus does not require annotated
+datasets. Hence, K-Means can detect novel groups, including emergent extremist organisations,
+or influential individuals [4]. Unsupervised clustering’s strength for ERH detection is how it
+circumvents the definition issues for annotating data and can cluster large movements without
+costly annotation. However, K-Means may not identify manifestly hateful posts, as it does not abide
+by any standard imbued within strict annotation criteria. Evidently, in the cross examination of
+a naïve approach vs their proposed K-Means derived model by Moussaouri et al. [30], the naïve
+approach outperformed the possibilistic clustering by 0.07-0.14 for accuracy 0.04-0.05 for precision.
+1.2
+Definitions for Deep Learning Approaches
+Deep learning represents a family of machine learning algorithms with multiple layers and com-
+plexity, typically via neural network architectures. Neural networks rely on training a network with
+a set of weights at each layer, known as neurons. The first layer of a neural network utilises numeric
+representation of an instance (e.g., hateful text) in numeric ‘tokenised’ form, which is adjusted
+throughout the hidden lower layers towards a final output (typically) classification layer. Each
+downwards training step results in readjusting the weights of the upper layers for the neurons—
+known as backpropagation [38]. Figure 4 displays this architecture for neural networks per our
+example. The benefit of DLAs in ERH detection is the preservation of word order and meaning (e.g.,
+“I ran” vs “I ran for president”), thus displaying context-sensitivity. Given dual-use words such as
+‘queer’, or racially motivated slurs, understanding the surrounding contextual words is essential to
+reduce bias via misclassifications [31]. DLAs dominant the benchmark dataset leader-board in RQ4.
+1.2.1
+Convolutional Neural Networks (CNN).
+Convolutional Neural Networks (CNNs) expand on the neural network model through a convolu-
+tional layer—which acts as a learnable filter for textual or image embeddings [44]. Moreover, CNNs
+include a pooling layer(s) to reduce the spatial complexity of the network’s features. Reducing
+spatial size helps reduce the number of parameters and thus training time and memory footprint,
+while reducing over-fitting by generalising patterns in the training data.
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+000:6
+Govers et al.
+1.2.2
+Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU).
+LSTM and GRU aim to increase contextual awareness to process data sequences with long-term
+gradients to retain information on prior tokens [12, 19]. LSTM and GRU seek to reduce the vanishing
+gradients caused during backpropagation steps, which reduces classification performance as older
+trained instances are effectively ‘forgotten’ due to later weight changes. Similarly, GRU’s are a
+gating approach with fewer parameters and thus higher runtime, enabling larger neural networks
+overall. CNN models with LSTM and GRU connections outperform CNNs on their own for hate
+speech detection [22, 23, 32]. The highest performing BiLSTM model expands LSTM for bidirectional
+input, via two LSTMs—where tokens in the network utilise information from past (backwards)
+tokens/data and future (forwards) data [32]. The ability to uphold the temporal memory of prior
+tokens (attention) constitutes a Recurrent Neural Network (RNN).
+1.3
+Language Transformer Models
+The state-of-the-art transformer architecture relies on self-attention—the memory retention of
+neural networks where each token of a sequence is differentially weighted [8, 16]. Unlike Recurrent
+Neural Networks (i.e., neural networks where nodes follow a temporal sequence), a transformer’s
+attention mechanism utilises context for any position for the token sequence. Hence, transformers
+can handle words out of order to increase understanding. Transformers offer greater classification
+performance (see RQ4) at the expense of memory and computational overhead. A considerable
+ethical threat of transformer models is their capability to predict future tokens (i.e., text generation).
+For instance, a malicious actor could create realistic automated trolls or radicalising synthetic
+agents as bots. Language models also risk data leakage of their trained data through predicting
+tokens found in the original trained dataset, such as names or addresses [8].
+Fig. 4. An abstracted example of a neural network for text. The top text represents its raw syntactic form,
+with its converted numeric embedding representation. These embeddings are responsible for altering the
+weights to increase token prediction or generation (for transformers) via backpropagation. The final output
+layer for this example would offer the probability that the given text is racist, sexist, or benign.
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+topic
+relation
+relation
+entity
+The
+Eldian
+people
+are
+the
+spawn
+of
+the
+devil
+0.125
+0.553
+0.814
+0.125
+0.25
+0.5
+0.19
+0.25
+0.666
+Segment and Positional tokens
+(i.e. computational embeddings for position/order of words).
+KKKKKKKK
+Contextual interconnected layers (e.g. neural network)
+Racism
+Sexism
+Neither
+Probability
+Probability
+ProbabilitySupplementary Material for:
+Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech
+000:7
+1.3.1
+Cross-encoders (e.g., BERT).
+Bidirectional Encoding Representations from Transformers (BERT) is the most common cross-
+encoder observed for ERH detection [16], with the highest performance of all NLP models. Cross-
+encoders offer higher performance for classification tasks, through retaining information over a
+given sequence with a label (i.e., self-attention). BERT’s core strength is its memory retention of all
+tokens in a sentence, thus upholding full context-sensitivity of every word in the post it seeks to
+classify. However, cross-encoders are computationally expensive due to high parameter counts
+(110 million parameters for BERT-base, 365 million for BERT-large), an issue further outlined in
+RQ4. Hence, an area of ongoing research includes model distillation (optimising and reducing
+parameter count to reduce memory requirements and training time), specialised training datasets,
+and alternate layers [26, 37, 46]. BERT is pre-trained on entries from English Wikipedia (2.5 million
+words) and the English BookCorpus (800 million words) [16]. Hence, such pre-trained models are
+then fine-tuned on a smaller dataset (typically 1000+ instances, per RQ2’s benchmark datasets) to
+optimise the BERT weights to detect hate speech with the context of its pre-trained corpus.
+1.3.2
+Generative Pre-trained Models (GPT).
+Similarly, the state-of-the-art GPT transformer architecture expands on the encoder blocks (shared
+with BERT) to include decoder blocks [8]. Hence, GPT works on a single token (i.e., word vector)
+and produces estimates for the sequence’s next token—ideal for tasks such as text generation,
+summarising, question answering, and information retrieval.
+GPT models differ from BERT-based models via masked self-attention—an alternate form of
+context-sensitivity where the model only knows the context of the prior words in the sentence.
+GPT-2/3 [8], GPT-Neo [7], and Jurassic-1 [25], are notable 2019-2021 era multi-billion parameter
+models—where larger datasets and parameter count result in more human-like text generation and
+higher performance in information retrieval tasks [8].
+GPT’s core strength in ERH detection synthetic hate speech generation via a GPT model fine-
+tuned on a hateful corpus—as investigated by Wullah et al. (see RQ3) [45]. However, state-of-the-art
+GPT models utilise up to 178B parameters, whereby memory and computational requirements
+scale linearly. Hence, future GPT work in synthetic text generation should consider inference tasks
+over fine-tuning. Specifically, inference utilises a pre-trained model’s on-demand text generation
+capability through prompts rather than altering each of the billions of weights. Using the auto-
+complete-like inference capabilities for generating realistic synthetic hate speech posts constitutes
+a novel case of prompt engineering in ERH detection and thus is a potential future research project.
+1.4
+Definitions for Prominent Feature Extraction Techniques
+This subsection outlines the three most common feature extraction techniques used for textual ERH
+detection—as outlined in RQ2 in the SLR. These models seek to identify hateful lexicons from text,
+or create numerical representations for word or sentence meaning via embeddings. We deconstruct
+the six most common feature extraction techniques observed in our SLR.
+1.4.1
+Word2Vec.
+Word2Vec is a model to convert words into vector embeddings, which compares synonymous words
+(e.g., ‘hate’ and ‘disgust’) via numerical vectors [29]. Word2Vec compares these word-to-vector
+embeddings via semantic similarity by evaluating their cosine similarity between their vectors
+(e.g., comparing word vectors of an unknown class instance to words from a known ‘hate speech’
+instance(s) to make a ‘hate or not’ classification). On a word-level basis, the vector value for ‘king’
+- value for man + value for woman would result in a vector similar to queen [29]. In our case, a
+‘Islamist extremist’ and ‘ISIS’ are semantically similar akin to ‘White Supremacy’ and ‘Nazism’.
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+000:8
+Govers et al.
+1.4.2
+Doc2Vec.
+Similar to Word2Vec, Doc2Vec aggregates vector embeddings for paragraphs in addition to indi-
+vidual words [24]. Thereby offering memory of the current context and paragraph’s topic—useful
+for understanding a whole post’s sentiment and meaning.
+1.4.3
+N-grams.
+N-grams represent contiguous sequences of n-number of characters for frequency analysis given
+their non-linear distribution in English, as well as when comparing a radical vs non-radical cor-
+pus [44]. This linguistic model is often paired with methods such as TF-IDF or BoW.
+1.4.4
+Term Frequency-Inverse Document Frequency (TF-IDF).
+TF-IDF determines the relevance of a word in a document by comparing its frequency in the
+document compared to its inverse number for the frequency of that word across all documents [44].
+Thereby, assigning each word a weight to signify its semantic importance compared to the wider
+corpus. For instance, radical Islamist dog-whistle terms (i.e., coded or suggestive political messages
+intended to support a group) appeared disproportionately in extremist text compared to a neutral
+religious corpus [36].
+1.4.5
+SenticNet.
+SenticNet embeds pattern matching, parser trees, and LSTM-CNN models for sentiment analysis,
+with the aim to replace a naïve BoW approach within a proclaimed bag of concepts and narrat-
+ives [10]. Specifically, it includes feature extraction methods of concept parsing (i.e., understanding
+linguistic patterns in natural language into conceptual pairs), subjectivity and polarity inference,
+alongside personality and emotion extraction.
+1.4.6
+Global Vectors for Word Representation (GloVe).
+GloVe offers an unsupervised learning algorithm for context-independent word-to-vector embed-
+dings [34]. While similar in creating vectors akin to Word2Vec, GloVe instead establishes word
+co-occurrences using matrix factorization (i.e., co-occurrence matrix of word [row] and context
+[usage of the word in the document]) and dimensionality reduction techniques.
+2
+SLR DESIGN CONSIDERATIONS
+This supplementary material section outlines the additional criteria and considerations for selecting
+papers and ensuring privacy-protections for users, groups and collected data. In essence, this section
+offers a meta-analysis of the ethics and selection process used throughout the SLR.
+2.1
+Quality Assessment Criteria
+The following includes our paper inclusion quality check criteria—with a score of 13 or higher
+required for inclusion in the final paper selection (i.e., final 51 papers included).
+We propose a critical criteria for quality assessment to filter irrelevant or ambiguous studies.
+Specifically, for a study that passed a title and abstract screen, we assess the study’s clarity for
+ERH definitions and annotations (for objective and legible classifications), methodical clarity (i.e.
+outlining each study’s algorithmic model, methods, data collection processes, and statistical analys-
+is/evaluation methods), and socio-technical considerations. We weighted each quality assessment
+section to prioritise their research methodology and clarity in their technical methods over their
+Conceptual Quality for studies encompassing broader socio-technical issues such as ethics, legality,
+or ERH clarity. After a ten paper pilot study, we selected a score threshold of 65% to exclude
+irrelevant or ambiguous studies. Our supplementary material document includes the criteria and
+scoring for our quality assessment rubric.
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+Supplementary Material for:
+Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech
+000:9
+2.1.1
+Computational Quality (0 = None, 1 = Partial, 2 = Full).
+(1) Is the radicalisation/affiliation detection model clearly defined?
+(2) Is the radicalisation/affiliation detection model’s algorithm clearly defined?
+(3) Is the training data reputable?
+(4) Are the models results compared to similar state-of-the-art methods?
+(5) Is the methodology for designing and conducting their experiment clearly defined?
+(6) Are patterns and trends discussed and presented clearly?
+2.1.2
+Epistemological Quality (0 = None, 1 = Partial, 2 = Full).
+(1) Does the source(s) (data or researchers) avoid any conflict of interests or expressed biases?
+(i.e., explicit support/funding from a political think tank or state agency).
+(2) Does the study provide a cited or evidence-based definition for “radicalisation”, “hate speech”
+or "extremist" affiliation?
+(3) Are the dataset annotations vetted by more than one annotator to reduce bias?
+2.1.3
+Conceptual Quality (0 or 0.5 value, as not critical but useful).
+(1) Does the study discuss social or ethical issues in ERH detection (e.g. censorship)?
+(2) Do the authors discuss the legality of their model or definitions?
+(3) Does the study evaluate its model across multiple social media platforms?
+(4) Does the study discuss regulatory frameworks or recommendations for social media platforms
+based on their findings?
+2.1.4
+Researcher Ethics.
+We focus on key terms and compositions of ERH examples to protect the privacy of the individuals
+exposed, as recommended by meta-studies on extremism research ethics [9, 13, 28]. When linking
+ERH detection to real-world groups and events, we solely focus on events and organisations which
+resulted in media attention or criminal convictions. In no part during this SLR did we attempt to
+track users, groups, or correlate online users to any personally identifiable information (name,
+location, username etc.) given the ease of composing online data into a traceable online fingerprint.
+Similar to the social norms in New Zealand in the aftermath of the Christchurch shooting, no
+extremists, terrorists, and/or criminals are referred by name to minimise publicity. We recognise
+the potential for political or cultural bias in this charged field by citing international non-partisan
+Non-governmental Organisations when framing ERH concepts, and avoid searching any party or
+ideology in our search strategy. Moreover, we encourage that our findings and recommendations
+invoke an open debate among social media platforms, governments, and the wider public. However,
+we do not condone the use of ERH detection in social media as a form of autonomous law. We
+recommend human-in-the-loop processes when handling or classifying data via independent
+reviews, privacy protections, and complaint and redress mechanisms for deployed models.
+Our recommendations thereby focus on Open Source Intelligence (OSINT) oriented studies
+that do not consider governmental or private-conversation surveillance (with the exception of
+one hybridised study that appeared in our search [20]). We thereby consider ERH detection as
+a computational method aimed at garnering community-insights, trends, and flagging for social
+media platforms themselves to use. Whether ERH detection policies should encourage deplatform-
+ing, deranking, demonetisation, fact-checking, or targeted counter-speech/prevention programs
+require further research. We encourage open interdisciplinary research in public and private-
+communications—particularly ethical and legal discussions.
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+000:10
+Govers et al.
+3
+THE CASE FOR PERFORMANCE ENGINEERING WHEN EVALUATING MODELS
+While high F1-scores help enforce community guidelines via accurate predictions and reduce
+injurious censorship from false positives, runtime performance trade-offs are seldom discussed.
+DLAs may perform within 1% (F1-score) of their MLA counterparts in NLP studies but require
+significantly higher computational resources. For instance, fine-tuning a BERT-large model for NLP
+tasks requires Graphics or Tensor Processing Units (GPU or TPU), restricting researchers from
+testing large language models [45, 47]. For community detection, uncompressed network models
+can include up to 27.4 million links [6], which significantly increases computational and memory
+requirements for a minimal 1-5% performance gain. Specifically, using a Possibilistic Approach (PA)
+with dimensionality reduction reduced subgraph mining runtime by up to 67% (1500 seconds to
+500 seconds on an 8-core 3.2GHz system), while reducing accuracy by only 4% [30]. Furthermore,
+community-level insights on topics with millions of tweets, relations, and discussions can lead
+to a network explosion with a non-deterministic polynomial runtime [5, 30]. In graph-detection
+approaches, performance engineering and optimisation for mining frequent subgraphs and graph-
+traversal is an active area of research [30]). No NLP studies consider performance engineering for
+DLAs despite developments in model distillation and sentence-level embeddings [37].
+Thus, we recommend that researchers consider performance trade-offs in future work and
+investigate a possible standardised performance-complexity metric (e.g. parameter count vs. F1-
+score ratio) to build scalable, energy-efficient and fiscally-viable models. Moreover, fine-tuning
+or retraining DLAs, or regenerating frequent subgraphs for community detection, should be a
+frequent endeavour to adapt to the rapidly evolving topics, entities, and events throughout online
+discourse. Due to the computational costs of fine-tuning or training multi-billion parameter models,
+we recommend approaches that do not require expensive training, such as few-shot learning (i.e.,
+giving several known instances of ERH and a unseen ‘test’ instance) and prompt engineering [8].
+4
+UPTAKE ROADMAP EXPANDED
+This supplementary section expands on the dataset and model research gaps highlighted in Figure
+16 of the main Down the Rabbit Hole SLR document. We categorise these research recommendations
+into eight core components for our proposed ERH Context Mining research field.
+4.1
+Model Recommendations
+The two predominant recommendations for future work are investigating the role of changes in
+hateful affiliation or speech over time to satisfy the temporal requirement for Radicalisation detection,
+and to train models on multiclass datasets from multiple platforms. We note that only one study
+considered temporal data on both meso and macro (changes within and between groups), and micro
+(individual) levels, although recommended as future work within four other studies [3, 11, 20, 40].
+Moreover, we recommend expanding on DLAs as the target for future research based on their
+leading performance in RQ4. Neural language models offer a macro-level societal understanding
+due to their pre-trained corpus on academic sources, OpenWebText2 Reddit discussions, and
+Wikipedia [8]. Furthermore, transformer models beyond 764 million parameters are untested.
+Bot, troll, meme, entity, dis/misinformation and satire detection remain underdeveloped—-which
+could lead to censorship or undermine democratic institutions. Five studies recommended multi-
+media detection as future work [1, 5, 11, 17, 35].
+To protect user privacy from recreating user content from neural language models, we encourage
+privacy-by-design software engineering through machine learning paradigms such as Differential
+Privacy (DP). DP-paradigm models and datasets reduce the potential for self-identification from
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+Supplementary Material for:
+Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech
+000:11
+trained models (i.e. data leakage, such as names or usernames in open-source datasets), as DP-
+paradigm models use pseudo-anonymised patterns of groups and hate.
+4.2
+Dataset Recommendations
+To investigate the roles of radicalisation, we recommend expanding on the dataset annotation
+approach by de Gibert et al. [15] by creating a conversation-level dataset with public non-hateful
+replies to a post for context. Moreover, future benchmark datasets should consider pulling data across
+platforms to investigate macro-level radicalisation trends between platforms. We note that only
+two studies considered anti-Asian sentiment in COVID-related tweets, targeting a seldom explored
+topic and demographic [21, 39] worthy of expansion given the ongoing COVID-19 pandemic.
+Likewise, future datasets should consider the role of indigenous discussions and potential re-
+searcher biases given the Anglo-dominant field of ERH research. Given the rise of COVID extrem-
+ism [43], far-right movements, and xenophobia in Oceania. Hence, we recommend geotargeted
+datasets to consider the differences for investigating ERH topics, which would demonstrate NZ’s
+commitment to our Christchurch Call to Action Summit. Investigating unexplored and minority
+groups could also provide imperative insights for social scientists regarding the conversational
+dynamics, morphological mapping, and ideological isomorphism from radical minority groups
+towards the majority. Likewise, research on vulnerable communities (youth, gender and sexual
+minorities, religious, racial, and geographically distant peoples) would aid social media platforms
+in both identifying unique radicalising risks, as well as avenues for support and de-escalation. In
+the mental health end, we recommend building on Nouh et al.’s proposed approach of extracting
+Fig. 5. ERH Context Mining (ERH-CM) eight core components for Research, Industry, and Government.
+ACM Comput. Surv., Vol. 00, No. 0, Article 000. Publication date: December 2021.
+
+Collective
+Researcher
+Researcher,
+Tasks
+Industry,
+Govt.
+1. Harm
+Reduction
+Tasks
+Introspection &
+Reflection
+**
+Community
+2.
+Detection
+7. Privacy
+副
+ERH-CM
+3
+Increased
+& Data
+Context
+Sovereignty
+6. Self-regulation
+Diverse
+& Ethics
+Datasets
+5. Quality Assessment
+Industry
+Collective
+& Control
+Tasks
+(human-in-the-loop)
+Researcher,
+Industry
+Tasks000:12
+Govers et al.
+textual, psychological and behavioural features [33], both due to its performance, as well as its
+potential for analysing societal factors and ERH roots such as correlations between mental health
+issues (isolation, depression etc.) and vulnerability to radicalisation towards violent extremism.
+For any counter-extremism or de-radicalisation studies, we recommend work in ethical and legal
+guidelines to protect privacy, avoid backlash or inadvertent algorithmic amplification.
+Investigating posts from periods of political, or social crisis (e.g., COVID health measures, post-
+terror attack discourse etc.) could also help identify cases of ERH on mainstream platforms before
+they are deplatformed/removed. Event-based datasets would provide unique sociological insights
+on the role of societal stress and emergencies on the human psyche and online group dynamics.
+To reduce the cost, variability in inter-annotator agreement, and psychological impact of human
+annotation, we recommend unsupervised clustering-based research and propose using synthetic
+conversational agents to simulate extremist discourse. Simulating online radicalisation in a closed
+environment would present a safe, ethical, and non-invasive method to build benchmark datasets.
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf,len=3153
+page_content='Down the Rabbit Hole: Detecting Online Extremism,  Radicalisation, and Politicised Hate Speech  JAROD GOVERS, ORKA Lab, Department of Software Engineering, University of Waikato, NZ  PHILIP FELDMAN, ASRC Federal, US  AARON DANT, ASRC Federal, US  PANOS PATROS, ORKA Lab, Department of Software Engineering, University of Waikato, NZ  Social media is a modern person’s digital voice to project and engage with new ideas and mobilise communities—  a power shared with extremists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Given the societal risks of unvetted content-moderating algorithms for  Extremism, Radicalisation, and Hate speech (ERH) detection, responsible software engineering must understand  the who, what, when, where, and why such models are necessary to protect user safety and free expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Hence, we propose and examine the unique research field of ERH context mining to unify disjoint studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Specifically, we evaluate the start-to-finish design process from socio-technical definition-building and dataset  collection strategies to technical algorithm design and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Our 2015-2021 51-study Systematic  Literature Review (SLR) provides the first cross-examination of textual, network, and visual approaches  to detecting extremist affiliation, hateful content, and radicalisation towards groups and movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We  identify consensus-driven ERH definitions and propose solutions to existing ideological and geographic  biases, particularly due to the lack of research in Oceania/Australasia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Our hybridised investigation on Natural  Language Processing, Community Detection, and visual-text models demonstrates the dominating performance  of textual transformer-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We conclude with vital recommendations for ERH context mining  researchers and propose an uptake roadmap with guidelines for researchers, industries, and governments to  enable a safer cyberspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  CCS Concepts: • Computing methodologies → Discourse, dialogue and pragmatics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Lexical semantics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Information extraction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Machine learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Natural language processing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Knowledge representa-  tion and reasoning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Image and video acquisition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' • Applied computing → Sociology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' • Social and profes-  sional topics → Hate speech;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Technology and censorship;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Political speech;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Governmental regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Additional Key Words and Phrases: extremism, radicalisation, machine learning, community detection, natural  language processing, neural networks, hate speech, sociolinguistics  ACM Reference Format:  Jarod Govers, Philip Feldman, Aaron Dant, and Panos Patros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Down the Rabbit Hole: Detecting Online  Extremism, Radicalisation, and Politicised Hate Speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, 0, Article 000 (December 2021),  35 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='org/tobereplacedwhenassignedDOI  Authors’ addresses: Jarod Govers, jg199@students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='waikato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='nz, ORKA Lab, Department of Software Engineering, University  of Waikato, Gate 1, Knighton Road, Hamilton, Waikato, NZ, 3216;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Philip Feldman, philip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='feldman@asrcfederal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='com, ASRC  Federal, Beltsville, Maryland, US;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Aaron Dant, aaron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='dant@asrcfederal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='com, ASRC Federal, Beltsville, Maryland, US;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Panos  Patros, panos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='patros@waikato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='nz, ORKA Lab, Department of Software Engineering, University of Waikato, Gate 1,  Knighton Road, Hamilton, Waikato, NZ, 3216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee  provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and  the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Copyrights for components of this work owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  © 2021 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  0360-0300/2021/12-ART000 $15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='00  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='org/tobereplacedwhenassignedDOI  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='11579v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='SI]  27 Jan 2023  000:2  Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Contents  Abstract  1  Contents  2  1  Introduction  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Motivation and Contributions  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Structure  4  2  Social Context to Social Network Analysis  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Extremism and Radicalisation Decoupled  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Hate Speech Decoupled  5  3  Systematic Literature Review Design and Protocol  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Trends and Shortfalls in Prior SLRs  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Research Questions  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='3  Databases  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='4  Search Strings  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='5  Inclusion and Exclusion Criteria  8  4  Key Research Question (RQ) Findings  9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Summary of the Social ERH Definitions Used by Researchers  9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Summary of the Data Collection, Processing, and Annotation Processes  9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='3  Summary of the State-of-the-art Computational ERH Detection Models  10  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='4  Summary of ERH Models’ Classification Performance  10  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='5  Geographic Trends—Islamophobia and Exclusion in the Academic Community  11  5  Socio-technical Context in Research—Consensus-driven Definitions  12  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Researchers’ Consensus-driven Definitions for ERH Concepts  13  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Correlation Between Definitions and Algorithmic Approach  14  6  Building ERH Datasets—Collection, Processing, and Annotation  15  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Prominent Platforms, Pulling, and Populations  15  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Feature Extraction Techniques  17  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='3  Data Filtering  18  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='4  Key Takeaways for Dataset Domain, Pre-processing, and Annotation  19  7  Community Detection, Text, and Image ERH Detection Algorithms  20  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Observed Non-deep Machine Learning Algorithms (MLAs)  20  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Deep Learning Algorithms (DLAs)  22  8  Model Performance Evaluation, Validation, and Challenges  24  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Benchmark Dataset Performance (Inter-study Evaluation)  25  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Community Detection Performance  25  9  Future Research Directions  25  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Ideological Isomorphism—a Novel Framework for Radicalisation Detection  25  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Morphological Mapping and Consensus-building—a novel computationally-  grounded framework for extremism detection  26  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='3  Outwards Dissemination––‘traditional’ hate speech detection updated  27  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='4  Uptake Roadmap for Researchers, Industry, and Government  28  10  Conclusion  28  References  30  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech  000:3  1  INTRODUCTION  Online social media empowers users to communicate with friends and the wider world, organise  events and movements, and engage with communities all at the palm of our hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Social media  platforms are a frequent aid for modern political exchanges and organisation [115], with extremes  amplified by algorithmic recommendation systems, such as on Twitter [49], TikTok [87], and  YouTube [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Furthermore, the semantic expression of ideas differs between social media platforms,  with Twitter’s short 280 character limit resulting in more narcissistic and aggressive content  compared to Facebook [75], and anonymous platforms such as 4Chan instilling a vitriolic ‘group  think’ in political threads/‘boards’ [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hateful, emotive, and ‘click-worthy’ content permeates  virtual discourse, which can radicalise users down an ideological rabbit hole towards real-world  violent action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The individuals behind the 2014 Isla Vista and 2019 Christchurch shootings appeared  as individual ‘lone-wolf attacks’ without an allegiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' However, investigations found a deep  network of perverse and violent communities across social media [40, 106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Likewise, exploiting  social media to plan politically motivated attacks towards the civilian population to coerce political  change (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', terrorism) delegitimises democracies, social cohesion, and physical/mental health [28,  58, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  As a response, social media platforms employ text and visual content-moderation systems to  detect and remove hate speech, extremism, and radicalising content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' This paper offers a state-of-the-  art Systematic Literature Review (SLR) on the definitions, data collection, annotation, processing,  and model considerations for Extremism, Radicalisation, and Hate speech (ERH) detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Motivation and Contributions  Existing ERH literature reviews exist as independent microcosms, often focusing on specific types  of models, typically text-only Natural Language Processing (NLP) models or non-textual network  analysis via community detection models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Studies seldom cross-examine models and evaluate the  performance between non-textual network analysis (a ‘who-knows-who’ approach), textual, and/or  multimedia approaches for ERH detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' While we identified ten prior literature reviews for  hateful content detection, none consider the similarities and definitional nuance between ERH  concepts and what Extremism, Hate Speech, or Radicalisation means in practice by researchers [2,  5, 20, 42, 43, 51, 82, 97, 110, 114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Evaluating the consensus for ERH definitions, dataset collection  and extraction techniques, model choice and performance are all essential to create ethical models  without injurious censorship or blowback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Through understanding the groups, beliefs, data, and algorithms behind existing content-  moderation models—we can reliably critique often overlooked social concepts, such as algorithmic  bias, and ensure compliance between social definitions and computational practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, this new  field of ERH context mining extracts the context to classifications—enabling researchers, industries,  and governments to assess the state of social discourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Given the rise of state-sponsored disinformation campaigns to undermine democratic institutions  and social media campaigns, the time is now for ERH research within politicised discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  The three core contributions for this paper are:  (1) The establishment of consensus-driven working definitions for Extremism, Radicalisation,  and Hate Speech within the novel field of ERH context mining—and a proposed frame-  work/roadmap for future researchers, social media platforms, and government advisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (2) The critical examination of existing textual (NLP), network (community detection), and  hybrid text-image datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (3) The identification and cross-examination of the state-of-the-art models’ performance on  benchmark datasets and relevant challenges with the current ERH detection metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  000:4  Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Structure  For a high-level summary of this SLR’s findings, refer to Section 4 on Key Research Question Findings,  and Section 9 for Recommendations for Future Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' These key summaries condense and contextual-  ise the 51 studies observed between 2015-2021, which we use to build our proposed computational  ERH definitions and technological roadmap for researchers, industry, and government in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  For a holistic understanding, we present a social context to our motivations in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Related  work and areas our SLR improves on are outlined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Section 3 outlines the systematic  protocol used to collect the 51 studies between 2015-2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Further to the summaries presented in  Section 4, we present an in-depth analysis and cross-examination of studies definitions of ERH  concepts in Section 5, approaches for collecting and processing data in Section 6, algorithmic  approaches for classification in Section 7, and their performance in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We conclude with  recommendations for future SLRs, and studies in Section 9, and conclusions in Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  2  SOCIAL CONTEXT TO SOCIAL NETWORK ANALYSIS  Analysing social media requires the socio-technical considerations of what constitutes hate speech,  extremism, and radicalisation (ERH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' To detect such concepts, computational models can investigate  multimodal sources—including textual meaning and intent through Natural Language Processing  (NLP), computer vision for images, and evaluating user relationships through community detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Hence, this section decouples and analyses ERH’s social background and definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Extremism and Radicalisation Decoupled  Extremism’s definition appears in two main flavours: politically fringe belief systems outside the  social contract or violence supporting organisational affiliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The Anti-Defamation League (ADL)  frames extremism as a concept "used to describe religious, social or political belief systems that exist  substantially outside of belief systems more broadly accepted in society" [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' For instance, under the  ADL’s definition, extremism can be a peaceful positive force for mainstreaming subjugated beliefs,  such as for civil rights movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' This construct of a socially mainstream belief constitutes the  Overton window [73]—and is not the target for content moderation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Conceptually, extremism typically involves hostility towards an apparent ‘foreign’ group based  on an opposing characteristic or ideology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Core tenants of extremism can stem from political  trauma, power vacuums and opportunity, alongside societal detachment and exclusion [58, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Hence, extremism often relies on defending and congregating people(s) around a central ideology,  whose followers and devotees are considered ‘in-group’ [116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Extremists unify through hostility  and a perceived injustice from an ‘out-group’ of people(s) that do not conform to the extremist  narrative—typically in a ‘us vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' them’ manner [28, 78, 116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, extremism detection algorithms  can use non-textual relationships as an identifying factor via clustering users into communities  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', community detection) [20, 110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Thus, extremism can simply reduce to any form of a fringe  group whose identity represents the vocal antithesis of another group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  There is no one conceptual factor to make an extremist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Extremism can also emanate from  political securitisation—whereby state actors transform a specific referent object (such as Buzan’s  five dimensions of society: societal, military, political, economic, and environmental security [25];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' or  individuals and groups [28]) towards matters of national security, requiring extraordinary political  measures [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' As the state normalises policies into matters of existential national security, society  can adapt and ideate decisions to ones of existential ‘life or death’ nature [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  For example, the ‘Great Replacement’ conspiracy theory claims that non-European immigrants  and children are “colonizers" or "occupiers”, and an “internal enemy”—-with the intent to securitise  migration, race, religion, and culture into wars with wording to invoke fears of a fifth column or  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech  000:5  racial invasion/replacement [22, 106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The Christchurch Shooter took direct interest in securitising  migrants as an extreme military threat, as far to name his manifesto after the conspiracy [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Extremism is not a strictly demographic ‘majority vs minority’ concern, as it encapsulates  movements demanding radical change and earmarked by a sense of rewarding personal and social  relationships, self-esteem, and belief of a wider purpose against a perceived adversarial force [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Exploiting desires for vengeance and hostility are also key recruitment strategies [20, 58, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Outside of political, cultural, and socio-economic factors, mental health and media are intrinsically  inalienable contributing factors [28, 58, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Likewise, repeated media reports of footage and body  counts can gamify and normalise extremism as a macabre sport for notoriety [20, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Within industry, Facebook, Twitter, YouTube, and the European Union frame extremism as  a form of indirect or direct support for civilian-oriented and politically motivated violence for  coercive change [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Facebook expands this industry-wide consensus to include Militarised Social  Movements and "violence-inducing conspiracy networks such as QAnon" [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Radicalisation focuses on the process of ideological movement towards a different belief, which  the EU frames as a "phased and complex process in which an individual or a group embraces a  radical ideology or belief that accepts, uses or condones violence" [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Terrorism consists of  politically motivated violence towards the civilian population to coerce, intimidate, or force specific  political objectives, as an end-point for violent radicalisation to project extremism [25, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Borum  delineates the passive ideological movement of radicalisation from active decisions to engage in  ‘action pathways’ consisting of physical terrorism, or hate crimes [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Political radicalisation towards increasingly aggrandising groups can also manifest in Roe’s two  sides of nationalism: positive socio-cultural and negative ethnic/racial nationalism [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' These  balancing forces create a form of societal security dilemma whereby the actions of one society to  strengthen its identity can cause a reaction in another societal group, weakening security between  all groups-—a radicalising spiral which can manifest into a polarised ‘culture war’ [70, 98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' However,  integration over assimilation can inversely undermining culture, self-expression and group cohesion,  leading to alienation and oppression by the dominant political or normative force [28, 98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Nonetheless, ERH detection does not offer a panacea to combating global terrorism, nor does  surveillance offer a ‘catch-all’ solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' In the case of the livestreamed Christchurch shooter, the  New Zealand Security Intelligence Service concluded that “the most likely (if not only) way NZSIS  could have discovered [the individual]’s plans to conduct what is now known of [the individual]’s  plans to conduct his terrorist attacks would have been via his manifesto.” [106, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' However,  the individual did not disseminate this until immediately before the attack, and his 8Chan posts  did not pass the criteria to garner a search warrant [106, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, extremism detection is an  evolutionary arms race between effective and ethical defences vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' new tactics to evade detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Hate Speech Decoupled  Obtaining viewpoint neutrality to categorise hate speech is challenging due to human biases and  the risk of hate speech undermining liberties through mainstreaming intolerance—the paradox of  tolerance where a society tolerant without limit may have their rights seized by those projected  intolerance [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Popper encapsulates this challenge by formulating that "if we are not prepared  to defend a tolerant society against the onslaught of the intolerant, then the tolerant will be  destroyed, and tolerance with them" [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Defining clear hate speech restrictions are needed to  protect expression rights and victim groups rights and safety [11, 117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  The European Union defines hate speech as "all conduct publicly inciting to violence or hatred  directed against a group of persons or a member of such a group defined by reference to race, colour,  religion, descent or national or ethnic origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='" [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Whereas, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Department of Justice frames  that: "A hate crime is a traditional offence like murder, arson, or vandalism with an added element  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  000:6  Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  of bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' [consisting of a] criminal offence against a person or property motivated in whole or in  part by an offender’s bias against a race, religion, disability, sexual orientation, ethnicity, gender, or  gender identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='" [38] Notably, governmental laws may differ from industry content moderation  policies via the omission of sexual, gender, religious or disability protections, and may include  threats of violence and non-violent but insulting speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  The United Nations outlines the international consensus on hate speech as "any kind of commu-  nication in speech, writing or behaviour, that attacks or uses pejorative or discriminatory language  with reference to a person or a group on the basis of who they are, in other words, based on their  religion, ethnicity, nationality, race, colour, descent, gender or other identity factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='" [117, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 2]  What all these definitions have in common is that they all involve speech directed at a portion  of the population based on a protected class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  3  SYSTEMATIC LITERATURE REVIEW DESIGN AND PROTOCOL  This SLR investigates the state-of-the-art approaches, datasets, ethical, socio-legal, and technical  implementations used for extremism, radicalisation, and politicised hate speech detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We  conduct a preliminary review of prior ERH-related SLRs to establish the trends and research gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  For the purposes of our SLR’s design, and to embed Open-Source Intelligence (OSINT) and Social  Media Intelligence (SOCMINT) principles, we define social media data as any online medium where  users can interactively communicate, exchange or influence others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We accept external data sources,  such as manifesto or news sites if interactive—such as via comment sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Furthermore, we  propose and use a novel quality assessment criteria to filter irrelevant or ambiguous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Trends and Shortfalls in Prior SLRs  Searching for Extremism, Radicalisation, Hate speech (ERH) and related terms, resulted in ten  literature reviews ranging from January 2011 to April 2021 [2, 5, 20, 42, 43, 51, 82, 97, 110, 114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Aldera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' observed only one survey before 2011 (covering 2003-2011) and another in 2013,  indicating the limited, exclusionary, but developing nature of reviews in this ERH detection area [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Prior SLRs seldom delineated or elaborated on Extremism, Hate Speech and Radicalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Neither  “extremism”, "radicalism" [2, 5, 20, 42, 43, 110, 114] or “hate speech” oriented SLRs [51, 82, 97]  cross-reference each other despite 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='3% of the data reviewed in the “hate speech” oriented review  by Adek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' encompassing hate speech in a political context [114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' This lack of overlap presents  an industry-wide challenge for social media companies who may oversee developments in ‘hate  speech’ detection which could transfer to a ‘extremism/radicalisation detection’ model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  SLRs prior to 2015 found that deep learning approaches (DLAs), such as Convolutional Neural  Networks (CNN) and Long Short-Term Memory (LSTM), resulted in 5-20% lower F1-scores than  non-deep approaches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Naïve Bayes, Support Vector Machines, and Random Forest classifiers) [2,  5, 42, 51, 97, 114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' DLAs post-2015 indicated a pivotal change towards higher-performing language  transformers such as Bidirectional Encoder Representations from Transformers (BERT) models [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Domains and Criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  No review delineated or removed studies that did not use English social media data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' This presents  three areas of concern for researchers when attempting to compare the performance of models:  (1) Results may not be comparable, if they use culture-specific lexical datasets, or language  models trained on other languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (2) Linguistic differences and conveyance in language—as what may be culturally appro-  priate for the majority class may appear offensive to minority groups and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (3) The choice of language(s) influences the distribution of target groups—with a bias  towards Islamic extremism given its global reach in both Western (predominantly ISIS) and  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech  000:7  Eastern countries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', with studied online movements in the Russian Caucasus Region [77]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  It is worth investigating whether Gaikwad et al.’s finding that 64% of studies solely target  ‘Jihadism’ corroborates with our study, which targets only English data [42, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Our SLR incorporates the key approaches of dataset evaluation (including their accessibility,  labels, source and target group(s)), data collection and scraping approaches, Machine Learning and  Deep Learning algorithms, pre-processing techniques, research biases, and socio-legal contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Unlike prior SLRs, our SLR conceptualises all elements for ERH context mining—consisting of a  user’s ideological radicalisation to an extremist position, and then projected via hate speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Research Questions  Our Research Questions (RQ) investigate the full process of ERH Context Mining—incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' data collec-  tion, annotation, pre-processing, model generation and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' These RQs consist of:  (1) What are the working definitions for classifying online Extremism, Radicalisation, and Hate  Speech?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (2) What are the methodologies for collecting, processing and annotating datasets?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (3) What are the computational classification methods for ERH detection?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (4) What are the highest performing models, and what challenges exist in cross-examining them?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Given the overlap of studies across prior SLRs targeting extremism or radicalisation or hate  speech, RQ1 addresses the similarities and differences between researchers’ definitions of ERH  concepts and their computational classification approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We dissect the ERH component of ERH  context mining and propose consensus-driven working definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  RQ2 addresses the vital context for ERH models—the data used and features extracted or filtered  out from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Furthermore, identifying frequently used benchmark datasets provides the basis for  critical appraisal of the state-of-the-art algorithmic approaches in the community detection, multi-  media, and NLP spheres in RQ3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Covering algorithmic approaches is not in itself novel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' However,  we consider novel, niche, and overlooked features relevant for an ERH model to make accurate  classifications—namely, bot/troll detection, transfer learning, the role of bias, and a hybridised eval-  uation of NLP and non-textual community detection models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We also consider critical challenges,  choice of metrics, and performance considerations not observed in prior SLRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='3  Databases  Given the cross-disciplinary, global and socio-technical concepts for ERH detection, we queried the  following range of software engineering, computer science, crime and security studies databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ProQuest (with the “peer-reviewed” filter, including queries to the below databases)  Association for Computing Machinery (ACM) Digital Library  SpringerLink  ResearchGate  Wiley  Institute of Electrical and Electronics Engineers (IEEE) Xplore  Association for Computational Linguistics Portal  Public Library of Science (PLOS) ONE Database  Google Scholar—as a last line to capture other journals missed in the above searched databases  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='4  Search Strings  The first round of study collection included automated database search strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' A second round  included a targeted manual search strategy with dissected keyword combinations to expand  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  000:8  Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' All results were added to the Title and Abstract Screening list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The following database  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='search query also included time filters (2015-2021) and peer-review-only filters:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='(“artificial␣intelligence”␣OR␣“machine␣learning”␣OR␣“data␣mining”␣OR␣“natural  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='language␣processing”␣OR␣“multiclass␣classification”␣OR␣“model”␣OR␣“analysis”␣OR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='“intelligence”␣OR␣“modelling”␣OR␣“detection”)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='AND␣(“hate␣speech”␣OR␣“radicalisation”␣OR␣“radicalization”␣OR␣“extremism”)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='AND␣(“social␣media”␣OR␣“forums”␣OR␣“comments”OR␣“virtual␣networks”␣OR␣“virtual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='communities”␣OR␣“online␣communities”␣OR␣“posts”␣OR␣“tweets”␣OR␣“blogs”)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='5  Inclusion and Exclusion Criteria  After attaining our 251 studies from our search strings, we read the journal metadata, title and  abstract to screen studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Our ranked criteria requires that all studies must:  (1) Originate from a peer-reviewed journal, conference proceeding, reports, or book chapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (2) Be written in English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (3) Involve a computational model (network relationship, textual and/or visual machine learning  model) for identifying and classifying radicalisation, hate speech or extreme affiliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (4) Utilise social media platform(s) for generating their model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (5) Computationally identify ERH via binary, multiclass, clustering or score-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (6) Focus on politicised discourse to exclude cyber-bullying or irrelevant benign discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (7) Published after the 1st January 2015—until the 1st July 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (8) Utilise English social media data if evaluating semantics and grammatical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  In addition to those that do not abide to any of the above, we exclude studies that:  (1) Are duplicates of existing studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (2) Do not specify their target affiliation to exclude broad observational studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  We outline our further in-depth critical Quality Assessment (QA) criteria to filter irrelevant or  ambiguous studies in our supplementary material’s Quality Assessment Criteria subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  After the Title and Abstract Screen, we read the full text of the 57 studies for the screening stages  displayed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' With 42 studies passing the Full Text Screen, we then randomly selected studies  from the bibliographies from this ’snowball sample’ of the 42 studies until 5 studies fail QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Studies found and filtered  Screen Type  Study Count  Search Strings  251  Title and Abstract Screen  57  Full Text Screening  42  After Snowball Sampling  51  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Threats to Validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  While we consider a concerted range of search strings, we recognise that ERH concepts is a wide  spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' To focus on manifestly hateful, politicised, and violent datasets/studies, we excluded  cyber-bullying or emotion-detection studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The potential overlap and alternate terms for ERH  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', sexism as ‘misogyny classification’ [30]) could evade our search strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Our pilot study,  subsequent tweaks to our search method, and snowball sampling minimise this lost paper dilemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  This study does not involve external funding, and all researchers declare no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech  000:9  4  KEY RESEARCH QUESTION (RQ) FINDINGS  Across the 51 studies between 2015-2021, ERH research is gaining popularity—with 4 studies from  2015-2016 increasing to 25 between 2019-2020 (and 4 studies from January to July 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We present  our SLR’s core findings in this section, with in-depth RQ analysis in Sections 5, 6, 7, and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Summary of the Social ERH Definitions Used by Researchers  RQ1: What are the working definitions for computationally classifying online Extremism, Radicalisation,  and Hate Speech?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Across the 51 studies, there are seldom delineations between the researchers choice of Extremism,  Radicalisation, and Hate Speech as the study’s focus––with the consensus that hate speech is  equivalent to extremist or radical views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, researchers approach extremism and radicalisation  as an organisationally affiliated form of hate speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  The consensus on hate speech’s working definition is any form of subjective and derogat-  ory speech towards protected characteristics expressed directly or indirectly in textual form—-  predominantly via racism or sexism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Benchmark datasets utilise human-annotated labels on single-  post instances of racially or gender-motivated straw man arguments, stereotyping, or post causing  offensive towards the majority of annotators (via inter-annotator agreement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Only 20% of studies  consider explicit rules or legal frameworks for defining hate speech [15, 18, 47, 53, 55, 71, 77, 84,  123, 128], with others relying on either an implicit ‘consensus’ on hate speech or utilise benchmark  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Benchmark datasets typically consider categorising their data into explicit categories of  racism [31, 32, 122, 123], sexism [13, 122, 123], aggression [13, 61], or offensiveness [13, 31, 128];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  including hate categorisation via visual memes and textual captions [3, 57, 99, 111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Extremism and radicalisation are equivalent terms in existing academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Islamic extremism is  the target group in 77% of US-originating extremism studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' ‘Far-right extremism’ and ‘white  supremacy’ are used interchangeably, a form of cultural bias given the variety of right-wing politics  worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Only one study considered radicalisation as an ideological movement over time [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Summary of the Data Collection, Processing, and Annotation Processes  RQ2: What are the methodologies for collecting, processing and annotating datasets?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Collecting non-hateful and hateful ERH instances varies between supervised and unsupervised  (clustering) tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Supervised learning typically utilises manual human annotation of textual posts  extracted via tools presented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Semi or unsupervised data collection can include grouping  ideologies by platform, thread, or relation to a suspended extremist account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Islamic extremism  studies frequently used manifestos and official Islamic State magazines as a ‘ground truth’ for  textual similarity-based approaches for extremism detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We found a direct correlation between  the availability of open and official research tools, and the platform of choice by researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Biases  extend geographically, with no studies utilising data or groups from Oceanic countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Figure 2 displays the skew for Twitter as the dominant platform for hate speech research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Despite  the nuance of conversations, 69% of studies classify hate on a single post per Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Data processing often utilises extracting statistically significant ERH features—such as hateful  lexicons, emotional sentiment, psychological signals, ‘us vs them’ mentality (higher occurrence of  first and third-person plural words [46]), and references to political entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  We categorise and frame the two approaches for dataset annotation: organisational or experience-  driven annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Organisational annotation utilises non-governmental anti-hate organisations [105]  or ‘expert’ annotator panels—determined via custom tests or by tertiary degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Organisational an-  notation relies on crowdsourced annotators, balanced by self-reported political affiliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Inter-rater  agreement or Kappa coefficient are the sole metrics for measuring annotator agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  000:10  Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Method to collect data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Number of studies per social media platform studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='3  Summary of the State-of-the-art Computational ERH Detection Models  RQ3: What are the computational classification methods for ERH detection?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ERH detection includes text-only Natural Language Processing (NLP), network-related community  detection, and hybrid image-text models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Between 2015-2021, there was a notable shift from  traditional Machine Learning Algorithms (MLAs) towards contextual Deep Learning Algorithms  (DLAs) due to higher classification performance—typically measured by macro F1-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Notably, only 3 of the 21 community detection studies utilised Deep Learning Algorithms  (DLAs) [77, 84, 99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Instead, community detection researchers tended to opt for graph-based models  such as heterogeneous graph models converting follower/following, reply/mention, and URL  networks with numeric representations for logistic regression or decision trees [12, 16, 24, 48, 80, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Community detection Machine Learning Algorithms (MLAs) performance varied by ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='3 F1-score  (mean between studies) dependent on the selection of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Statistically significant features  for performant MLA models include gender, topics extracted from a post’s URL(s), location, and  emotion via separate sentiment algorithms such as ExtremeSentiLex [89] and SentiStrength [112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  For textual non-deep NLP studies, researchers classified text via converting the input into  word embeddings via Word2Vec, GloVe, or frequent words via Term Frequency-Inverse Docu-  ment Frequency (TF-IDF), and parsing it into Support Vector Machines, decision trees, or logistic  regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' As these embeddings do not account for word order, context and nuance is  often lost—leading to higher false positives on controversial political threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Conversely, DLAs  utilise positional and contextual word embeddings for context-sensitivity using Long Short Term  Memory (LSTM) Convolutional Neural Networks and Bidirectional Encoder Representations from  Transformers (BERT) leading to their higher performance as outlined in RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='4  Summary of ERH Models’ Classification Performance  RQ4: What are the highest performing models, and what challenges exist in cross-examining them?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  By 2021, Support Vector Machines on emotional, sentiment, and derogatory lexicon features were  the last non-deep MLA to attain competitive F1-scores for NLP tasks compared to DLAs such as  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  3%6%  29%  32%  22%  5%  3%  Twitter APi  Custom web-crawler  Pushshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='io APl (Reddit)  Magazine or Newspaper Archives  Pre-existing Dataset or Library  Facebook Graph APl  Other APlIslamicExtremist  Sources, 9  Stormfront, 7  Other Far-  Blogs and  right  Apolitical  Extremist  Sources, 4  Forums, 4  News  Websites, 6  Far-Left  Forums,  Facebook,  Wikipedia  2  3  Twitter, 36  Vkont  akte,  Religious  Other, 5  Reddit, 2  Texts, 2Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech  000:11  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Type of data used for an ERH classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Distribution of target groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Convolutional Neural Networks (CNNs) and neural language transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' As of 2021, BERT-base  attained the highest macro F1-score average across the seven benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' However, cross-  examining models between datasets present various challenges due to varying criteria, social media  domain, and choice of metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Likewise, non-textual community detection and traditional MLA  studies resulted in lower classification F1-score by ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='15 and ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' While BERT, attention-  layered Long Short Term Memory (BiLSTM), and other ensemble DLAs attain the highest F1-scores,  no studies consider their performance trade-offs with their high computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Our  recommendations propose further research in prompt engineering, distilled models, and hybrid  multimedia-text studies—as we only identified one hybrid image-text study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  While textual DLAs outperform community detection models, grouping unknown instances  enable network models to identify bot networks and emergent terror cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, there is a growing  area of research for hybrid semi-supervised NLP and community detection models to identify new  groups and radical individuals in a domain we frame as meso-level and micro-level classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='5  Geographic Trends—Islamophobia and Exclusion in the Academic Community  To identify ERH hot spots in research, we present the first cross-researcher examination of their  institution’s location compared to their dataset(s) geolocational origin in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' For clarity, we  filter out the 29 indiscriminate global studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Despite the decline of the Islamic State as a conventional state-actor post-2016, western academic  research remains skewed towards researching Islamic extremist organisations operating from the  Middle East.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 24% of US-originating ERH studies targeted Islamic extremism, compared to 19%  focusing on violent far-right groups and 19% for left vs right polarised speech (in discussions  containing hate speech).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Despite more Islamic extremist studies from US-oriented research, over  90% of terrorist attacks and plots in the US were from far-right extremists in 2020 [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  European-origin studies have a reduced bias, where 25% target far-right white supremacy and  29% on Islamic extremism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Islamic extremism’s popularity is a global trend for 20% of all studies,  shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, there is a clear Islamophobic trend in academia—given the aversion of  far-right groups, and the lack of a change in the distribution of targeted groups between 2015-2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Researcher ethics and socio-legal considerations present a critical international research gap, as  only 13% of US and 28% of European studies included discussions on annotation ethics, expression  laws, or regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' This US vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Europe discrepancy likely emanates from the data collection and  autonomous decision-making rights guaranteed under the EU’s GDPR [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  4%  Text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Single post only  8%  I Text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Grouped by user  19%  No text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Metadata  69%  Relations (Community  Detection), or Images  Text and Graphing  (NLP and Community  Detection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='lslamic  20%  20%  IFar-right White  Supremacy)  IMainstream Politics  10%  Hate Speech  9%  41%  1Other000:12  Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Relations between researchers country of origin and their dataset’s country of focus (global/indiscrim-  inate studies excluded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Created via Flowmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='blue [21] with Mapbox [74] and OpenStreetMap [85]  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  The Case for Oceania and the Global South.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Despite developments post-2015, such as two New Zealand terrorist attacks [94, 106] and five in  Australia [8], the rise of racial violence in South Africa [28], and the 2019-ongoing COVID-related  radicalisation [120];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' no studies considered targeting Oceania or English-recognised countries in the  global south.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Likewise, applying these datasets across intersectional ethnic, sexual, and cultural  domains presents a threat to validity as terms considered mundane or inoffensive to one group may  be considered inflammatory to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Datasets are also biased towards racism towards a minority  group [31, 123], which may bias English hate speech in a white minority country such as South  Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Investigating language trends and model performance on Mela-, Micro- and Polynesian  groups could also offer insights in the role of religion, values (such as tikanga values in New  Zealand’s M¯aori population), taboos, lexicons, and social structures unique to indigenous cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  5  SOCIO-TECHNICAL CONTEXT IN RESEARCH—CONSENSUS-DRIVEN  DEFINITIONS  What are the Working Definitions for Classifying Online Extremism, Radicalisation, and Hate Speech?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Empirically, studies often provide a generalised social definition in their introduction or back-  ground and utilise technical criteria to annotate instances for (semi)supervised learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Hence, this research question consists of two parts: the socio-legal ERH definitions, and the  technical implementation and classification thereof outlined in the existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  We identified an unexpected overlap between the definitions and models between extremism and  radicalisation studies, whereby researchers frame these concepts as synonymous with hate speech  with a political/organisational affiliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hate speech studies focus on protected groups as binary  ‘hate or not’ [3, 4, 15, 27, 32, 57, 77, 84, 101, 109, 111, 123], or multiclass ‘racism, sexism, offensive,  or neither’ text [10, 13, 31, 45, 69, 81, 83, 90, 91, 123, 126], with a consensus that ‘Extremism =  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Iceland  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Sweden  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Finland  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content='Russia  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Denmark  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Belarus:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Germany-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Ukraine  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Era  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Kazakhstan  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Romania  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content='Unit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content='Afghanistan  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content='SriLanka  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content='Maldives  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content='Bolivia  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content='Chile  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Ocear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Locationtotals  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Atlantic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='SouthAfrica  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Ocean  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='outgoing 兰 incoming  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='ruguay  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Argoitina  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='more outgoing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='more incomingDown the Rabbit Hole: Detecting Online Extremism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Radicalisation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' and Politicised Hate Speech  000:13  Radicalisation = Hate speech with an affiliation’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Alternatively, we propose a novel computationally  grounded framework and definitions to seperate and expand ERH in Section 9 to underline the  holistic stages of extremists temporal radicalisation through disseminating hateful media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Socio-legal Context Provided in Existing Literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  The largest discrepancy were between studies that discussed legal or ethical context to ERH, which  constituted only 20% of studies [15, 18, 47, 53, 55, 71, 77, 84, 123, 128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The remaining 80% relied on  an implicit consensus of ‘hate speech’ (often synonymous with toxic, threatening, and targeted  speech), or ’extremism’ (often UN designated terrorist organisations like ISIS [118]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Waseem and Hovy [123] outlined a unique eleven-piece criteria to identify offensive ‘hate  speech’ including considerations for politicised extremist speech via tweets that “promotes, but  does not directly use, hate speech or violent crime” and “shows support of problematic hash tags"  (although "problematic" was not defined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hate speech as a supervised learning task resulted in two  categories—sexism and racism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' A sexist post requires gender-oriented slurs, stereotypes, promoting  gender-based violence, or straw man arguments with gender as a focus (defined as a logical fallacy  aimed at grossly oversimplifying/altering an argument in bad faith [123]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The ambiguity for sexism  classification by human annotators was responsible for 85% of inter-annotator disagreements [123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Researchers’ Consensus-driven Definitions for ERH Concepts  We aggregate the trends in ERH based on the definitions used throughout the 51 studies, and  observe that ERH concepts reflect their computational approach more than their social definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Despite radicalisation being a social process of ideological movement, existing work considers the  term as synonymous to political hate speech/extremism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Definition 1: Hate speech (researchers’ consensus)  Hate speech is the subjective and derogatory speech towards protected characteristics expressed  directly or indirectly to such groups in textual form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' *  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='B: there is a significant bias in hate speech categorical classification in practice, whereby no  studies considered categories outside of sexism (including gender-identity) or racism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Definition 2: Extremism (researchers’ consensus)  Organisational affiliation to an ideology that discriminates against protected inalienable  characteristics or a violent political organisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Affiliation does not always include manifestly  hateful text and may include tacit or explicit organisational support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Extremist studies often  classify organisational affiliation based on text (NLP) and community networks (follower,  following, or friend relationships).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  The current academic consensus among researchers demonstrates a considerable overlap between  ‘extremism’ and ‘hate speech’ definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' In practice, extremism exclusively focused on racism  detection, or in the specific context of Jihadism [4, 15, 84], white supremacy [53, 86, 99, 109],  Ukrainian separatists [16], anti-fascism (Antifa) [53], and the sovereign citizen movement [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Of  the 13 studies targeting extremism, only one considered extremism by the ADL’s politically-fringe-  but-not-violent definition [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Tying extremism to the study of mainstream terrorist-affiliated groups  neglects rising movements, ethical movements using unethical terror-tactics, and non-violent  fragments of other terrorist groups, such as a reversion to ‘fringe’ activism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, extremism’s  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  000:14  Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  working definition is similar to terrorism when considering group affiliation detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' If investigating  extremist ideologies, then the definition is synonymous with those in hate speech studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Extremism’s working definition is exceptionally biased towards support for Islamic extremist  movements (10 out of 13 studies [4, 16, 24, 53, 77, 80, 84, 86, 89, 109]), with far-right ideologies a  distant second (5 out of 13 studies [16, 53, 86, 99, 109]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' These organisational and ideological biases  is potentially a result of US security discourse and national interests (via the ‘War on Terror’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' White  supremacy and far-right ideologies are separate terms used interchangeably without distinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Definition 3: Radicalisation (researchers’ consensus)  No discernible difference between extremism’s definition with both terms used interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Radicalisation = extremism = politically affiliated hate invoking or supporting speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Definitions and algorithmic approaches on radicalisation detection relied on political hate speech,  or extremism via ideological affiliation—with 5 of the 8 radicalisation studies targeting textual or  network affiliation to the Islamic State (IS) [7, 47, 80, 95, 101], and 2 on white supremacy [46, 103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  The only other notable deviations from this extremism = radicalisation dilemma is Bartal et  al.’s [12] focus on radicalisation as a temporal movement with apolitical roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Their study investig-  ated the temporal movement from a ‘Novice’ (new poster) classification towards an ‘influencer’  role based on their network relations and reply/response networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Chandrasekharan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' defined  ‘radicalisation’ as the process of an entire subreddit’s patterns up to and including the time of its  ban to map subreddit-wide radicalisation [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Only two studies are the exception to the extremism  = radicalisation = politicised hate speech consensus per the remainder of the 51 studies [12, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' ERH Definition Tree—visualising how ERH definitions deviate based on their algorithmic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Correlation Between Definitions and Algorithmic Approach  Uniquely, 66% of publications in the field of social-science or security studies utilised network-  driven community detection models, with extremism defined in a law enforcement context by  emphasising a user’s network-of-interactions between known annotated extremists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  defined extremism in a semi-supervised OSINT and HUMINT surveillance manner—requiring  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ERH  Semantic/Topic-based (Aggregate) Hate  Radical or Extremist Network Detection  Speech Detection  Generally  Generally semi or fully  Generally  unsupervised  supervised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content="  unsupervised  'Big Data':  Unless cluster-based  'Big Data':  Graphs (non-textual):  Emotional Sentiment  Topics and Political Affilation  Hate Speech  Crime and Social Science Focused  E." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' SentiStrength, SIRA, LIWC  Law Enforcement Pivoted Data:  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=" FBI 'Tripwire', Explicit Terror Afillation, TSA's  Automated Targeting System, etc." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Multiclass Sentiment  Binary Affiliation  Influencer  Racism  Terror Network  Sexism  Aggression  Offense  or ProlAnti/Neutral  Person-of  Classification  Clustering  Classificatior  Classification  Classification  Classification  Interest Detection  Classification  F  H  J  K  c  D  A  Multiclass  Ideological or  Community  Radical Scoring  Organisation or  Detection  Topic Classification  Regression  B  E  GDown the Rabbit Hole: Detecting Online Extremism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Radicalisation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' and Politicised Hate Speech  000:15  links between an extremist virtual and a physical/offline presence to extremist stimuli through  incorporating the FBI’s tripwire program [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Approaching extremism using relational properties  via interactions, geographic proximity, profile metadata, and semantic or network similarity raises  ethical dilemmas vis-à-vis individuals who have/had a solely virtual presence or those interested in  opposing opinions [24, 29, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The relationship between definitions and algorithmic approaches  indicate that radicalisation studies skew towards community detection models, extremism towards  hybrid NLP and community detection models, and hate speech to a text-only NLP endeavour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Table of references for studies in each category (A, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='K) in the above ERH definition tree diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Label  Studies  Label  Studies  Label  Studies  A  [12, 15, 48, 80]  B  [15, 16, 80, 89, 99, 108]  C  [48, 108]  D  [3, 4, 15, 27, 32, 57, 77, 84, 101, 109, 111,  123]  E  [14, 53, 80, 89, 93, 99, 99]  F  [1, 4, 7, 13, 52, 55, 86, 89, 90, 95, 100, 107,  124, 128, 129]  G  [15, 18, 46, 47, 103, 104, 113]  H  [10, 13, 45, 69, 81, 83, 90, 91, 123, 126]  I  [10, 13, 45, 81, 83, 90, 91, 123, 126]  J  [13, 56, 83, 90, 126, 128, 129]  K  [3, 56, 57, 67, 71, 81, 83, 90, 99, 111, 126,  128–130] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Privacy and Ethics-driven Regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  No studies integrated or mentioned existing AI ethics regulation or standards, such as those  emerging from the EU [50], or private-sector self-regulation such as the IEEE P700x Series of  Ethical AI Design [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Researchers should consider the application and use cases for their proposed  models—as autonomous legal decision making, injurious use of data (outside of a reasonable  purpose), or erasure (a challenge for persistent open-source datasets), may violate regulations such  as the EU’s General Data Protection Regulation (GDPR)’s Article 22, 4, and 17 respectively [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Prominently, Mozafari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' evaluated hate speech with a ethno-linguistic context, recognising  that certain racist slurs were dependent on the culture and demographic using them [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  6  BUILDING ERH DATASETS—COLLECTION, PROCESSING, AND ANNOTATION  What are the methodologies for collecting, processing and annotating datasets?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  This RQ outlines the dominant platforms of choice for ERH research, the APIs and methods  for pulling data and its underlying ethical considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Geographic mapping demonstrates the  marginalisation of Oceania and the global south in academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We critically evaluate the sentimental,  relationship, and contextual feature extraction and filtering techniques in community detection  and NLP studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We conclude with the key recommendations for future data collection research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Prominent Platforms, Pulling, and Populations  This subsection outlines the common social media platforms, the method for sampling and extracting  (‘pulling’) textual and network/relationship data, and the type of data used in ERH datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  40% of studies relied on Twitter tweets for ERH detection, with Twitter being the dominant  platform for research per Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Twitter’s mainstream global reach paired with its data-scraping  Twitter API enabled researchers to target specific hashtags (topics or groups), real-time tweet  streams and reach of tweets and their community networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, the Twitter API is also the most  used method for scraping data, with other methods outlined in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Unfortunately, revised  2021 Twitter Academic API regulations removed access to tweets from suspended accounts [17],  limiting datasets to those pre-archived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Currently, the Waseem and Hovy datasets are not available  due to relying on the Twitter API and suspended tweets [122, 123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  For far-right ERH detection, researchers used custom web-scrapers to pull from the global  white supremacist forum Stormfront—containing topics ranging from political discussions, radic-  alising "Strategy and Tactics", and "Ideology and Philosophy" sections, and regional multilingual  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  000:16  Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  chapters [55, 71, 81, 103, 109, 124, 126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' As a supplement or alternate to searching and collecting  hateful posts en masse, five studies considered extremist ‘ground truth’ instances by comparing tex-  tual similarity from Tor-based terror-supporting anonymous forums [4] and websites [124], radical  Islamist magazines and manifestos [7, 84, 95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Interestingly, no studies considered extracting ground  truths from far-right manifestos or media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Likewise, no studies considered recent low-moderation  anonymised forums such as 8Chan (now 8kun) or Kiwifarms, which were extensive hubs for pro-  paganda dissemination from the Christchurch shooter [106];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Parler, notable for its organisational  influence during the 2021 Capitol Hill riots [72];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Telegram, TikTok, or Discord, despite reports on  its use for sharing suicides, mass shootings, and group-lead harassment of minority groups [44, 87]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Hence, there is a prevalent and concerning trend towards NLP studies on mainstream platforms,  which may overlook the role of emergent, pseudo-anonymised or multimedia-oriented platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Data Collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  62% of all studies evaluate ERH on a single post-by-post basis, with NLP the dominant approach per  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Conversely, grouping posts on a per user basis frequently included annotations from cyber-  vigilante groups such as the Anonymous affiliated OSINT ‘Controlling Section’ (CtrlSec) group’s  annotations of ISIS-supporting accounts [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' However, Twitter claims that CntrlSec annotations  were "wildly inaccurate and full of academics and journalists" [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, researchers should avoid  unvetted cyber-vigilante data, and consider anonymising datasets to further benefit user privacy,  researcher ethics, and model performance by reducing false positives (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', censorship).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  While NLP text detection is the dominant detection approach, 23 of the 51 studies investigated  data sources outside of textual posts per Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Research gaps include the lack of multimedia  and law enforcement studies, with only three hybrid text-image detection [57, 99, 111] and one  study utilising FBI anonymous tips, Automated Targeting System and Secure Flight data [48, 119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Data Collection and Annotation Bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Due to the varying fiscal costs, biases, and time trade-offs, there is no consensus for selecting or  excluding annotators for supervised learning datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, we frame that annotator selection  falls within two varying groups: experience-driven selection and organisation-driven selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  For the former, experience-driven selection includes studies that utilised self-proclaimed ‘expert’  panels as determined by their location and relevant degrees [123], are anti-racism and feminist  activists [122], or work on behalf of a private anti-discrimination organisation [105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' However,  assembling annotators by specific characteristics may be time-consuming or costly, such as crowd-  sourcing tertiary annotators via Amazon Mechanical Turk, or Figure Eight [10, 45, 71, 81, 93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Conversely, an organisation-driven selection approach focuses on agreement by a crowdsourced  consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Instead of relying on specific experience, researchers utilised custom-made tests for  knowledge of hate speech criteria based on the researchers own labelled subset [122, 128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Likewise,  organising annotator pools can also include balancing annotators self-reported political affiliation to  reduce political bias [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Researchers use Inter-rater agreement, and Kappa Coefficient to determine  a post’s ERH classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' For racism, sexism, and neither classifications, annotation Fleiss’ Kappa  values ranged between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='607 [32] to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='83 [128], indicating moderate to strong agreement [125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Thirdly, unsupervised clustering enables mass data collection without time-consuming annota-  tion via Louvain grouping (to automatically group text/networks to identify emergent groups) [15,  16, 108], or grouping based on a thread’s affiliation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', the now-banned r/altright [46] and v/[n-  word] [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Although not all posts from an extremist platform may be manifestly hateful, as evident  in the 9507 post ‘non-hate’ class in the Stormfront benchmark dataset from de Gibert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Research continues to skew towards radical Islamic extremism per Figure 4, while the plurality  (41%) target generic ‘hate speech’ in ‘hate or not’, or delineations for racism, sexism, and/or offence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech  000:17  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='3  Benchmark Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  We define a benchmark dataset as any dataset evaluated by three or more studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The majority of  studies used custom web-scrapped datasets or Tweets pulled via the Twitter API per Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Datasets used by three or more studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Dataset  Year  Categories  Platform  of origin  Collection strategy  Used By  Waseem  and  Hovy [123]  2016  16914 tweets: 3383 Sexist ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  1972 Racist,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 11559 Neutral  Twitter  11-point Hate Speech Cri-  teria  [10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 52,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 55,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 56,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  71,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 81,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 83,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 91,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  123,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 126]  FifthTribe  [39]  2016  17350 pro-ISIS accounts  Twitter  Annotated pro-ISIS ac-  counts  [7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 84,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 95,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 102]  de Gibert  [32]  2018  1196 Hate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 9507 Non-hate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  74 Skip (other) posts  Stormfront  3 annotators considering  prior posts as context  [32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  55,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  71,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  126]  OffenseEval  (OLID) [128]  2019  14100 tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' (30%) Of-  fensive or Not;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Targeted  or  Untargeted  insult;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  towards an Individual,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' or Other  Twitter  Three-level hierarchical  schema,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' by 6 annotators  [55,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 67,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  130]  HatEval [13] 2019  10000 tweets distributed  with Hateful or Not,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Ag-  gressive or Not,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Individual  targeted or Generic  Twitter  Crowdsourced via Figure  Eight,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' with 3 judgements  per Tweet  [13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 55,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 71,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 90,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  126]  Hatebase-  Twitter [31]  2019  25000 tweets: Hate speech,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Offensive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Neither  Twitter  3 or more CrowdFlower  annotators per tweet  [31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 52,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 55,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 71,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  81,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 83,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 126]  TRAC [61]  2018  15000 English and Hindi  posts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Overtly, Covertly,  or Not Aggressive  Facebook  Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' [62] subset, 3  annotators per post, com-  ment or unit of discourse  [55, 56, 71]  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Feature Extraction Techniques  Figure 7 outlines the three types of feature extraction techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Non-contextual lexicon approaches  relate to word embeddings for entities, slurs, and emotional features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' However, non-contextual  blacklists and Bag of Words (BoW) lexicon approaches cannot identify context, concepts, emergent,  or dual-use words (see the Supplementary Material’s Algorithm Handbook section for comprehensive  definitions) [32, 81, 83, 90, 122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Contextual sentiment embeddings expand on lexicons by embedding  a form of context via positional tokens to establish an order to sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  We group unsupervised term clustering and dimensionality reduction methods under the  Probability-Occurrence Models category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The two dominant approaches include weighted ANOVA-  based BoW approaches and Term Frequency-Inverse Document Frequency (defined in the Supple-  mentary Material), which weigh the importance of each word in the overall document and class  corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Contextual sentiment embeddings result in higher F1-scoring models (per RQ4) due to their  context-sensitivity and compatibility as input embeddings for deep learning models [55, 71, 83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Community detection features require mapping following, friend, followee, and mention dy-  namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Furthermore, other statistically significant metadata includes profile name, geolocation (to  investigate ERH as a disease), gender, and URLs [16, 81, 84, 123, 126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' URLs can identify rabbit holes  for misinformation or alternate forums via PageRank [88] and Hyperlink-induced Topic Search  (HITS) [59]—which extracts keywords, themes and topical relations across the web [77, 88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  000:18  Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Types of feature extraction techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='3  Data Filtering  For context-insensitive BoW and non-deep models, stop words (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' the, a, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' ), misspellings, and web  data are often filtered out via regular expressions and parsing libraries [4, 27, 46, 47, 52, 69, 71, 81, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Compared to semantic or reply networks, community detection models tend to extract metadata  for separate clustering for entity and concept relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' All data filtering techniques are thereby  aggregated and branched in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  No studies considered satire, comedy, or irony to delineate genuine extremism and online culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Researchers’ implicit consensus is to treat all posts as part of the ERH category if it violates their  criteria, regardless of intent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Conversely, Figure 9 displays the 14% of the studies filtered bots  by removing but not classifying bot accounts from the ERH datasets [12, 15, 16, 46, 69, 93, 109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Strategies include removing duplicate spam text, filtering Reddit bots by name, and setting minimum  account statistics for verification—such as accounts with that share hashtags to at least five other  users to combat spam [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Likewise, Lozano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' limited eligible users for their dataset to have at  least 100 followers, with more than 200 tweets, and at least 20 favourites to other tweets [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' This  operates on the assumption that bots are short-lived, experience high in-degree between similar  (bot) accounts, and seldom have real-world friends or followers—as discovered by Bartal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' [12]  Outside of removing suspicious bot accounts via human annotation in dataset generation, com-  putational means to explicitly categorise bots or trolls remains an area for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='SenticNet  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='LIWC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content="('Bag of Concepts/Narratives')  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Hatebase  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Word2Vec  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Doc2Vec  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Blacklists  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Semanticizer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Non-contextualLexicon  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='SentiWordNet  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Contextual Sentiment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='SentiStrength  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Approaches  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Embeddings  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='SIMON  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Entity Detection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Transfer Learning  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='TransferLearningDatasets  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Tasks (pre-training)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='(Stanford Sentiment)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='EmoFeat  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='GPT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Dynamic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Dependency  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='FastText  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='few shot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Clustering  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Parse Trees  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='learning  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='WordPiece  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Theme and Narrative  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Bag of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Tokenization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Clustering  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Words  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Entity Clustering  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='PCA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Probability-Occurrence Models  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='TF-IDF  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='DBSCAN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='ANOVA F-value weighted BoWDown the Rabbit Hole: Detecting Online Extremism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Radicalisation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' and Politicised Hate Speech  000:19  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Types of data filtering techniques across NLP and community detection studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Studies with Bot or Troll Filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Special Type of Data Used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='4  Key Takeaways for Dataset Domain, Pre-processing, and Annotation  Twitter’s accessible API, popularity and potential for relationship modelling via reply and hashtag  networks, makes it the platform of choice for research (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Despite the rise of far-right  extremism post-2015, Islamic extremism in the US and Europe remains the target group for the  majority of organisation-based studies, with no studies considering far-right/left manifestos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The  marginalisation of Oceania and the global south by datasets predominantly containing US white  hetero males indicates a structural bias in academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' For feature extraction, we recommend using:  (1) Contextual sentimental embeddings—due to their compatibility with deep learning models  and highest performance, per Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (2) Pre-defined lexicons—assuming they remain up-to-date with online culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (3) Probability-occurrence models—ideal for large-scale clustering of emergent groups [27, 99, 126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  RELATIONS BETWEENPOSTS  6  SEMANTIC FEATURE DATA AND DICTIONARIES  5  NETWORKS  METADATA (PROFILE DATA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content='Filtering  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Word Stemming  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Grammar  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Techniques  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Metadata  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content='(Community  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content='LangDetect  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content='(to remove non  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content='English posts)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Stop words  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content='Identity pseudo-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='(for context  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='anonymisation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='insensitive MLAs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='(usenames,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' names,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  mentions etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' )6%  14%  Yes  ■No  Unspecified or  Irrelevant to Study  80%000:20  Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  We do not recommend pre-defined lexicons on non-English text, new groups or ideologies—as  these lexicons may not translate to different concepts and slurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We recommend adaptive semi or  unsupervised learning via contextual embeddings and entity/concept clustering for edge cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Research currently lacks multidomain datasets, pseudo-anonymous platforms, multimedia (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=',  images, videos, audio and livestreams), and extraction of comedy, satire, or bot features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  7  COMMUNITY DETECTION, TEXT, AND IMAGE ERH DETECTION ALGORITHMS  What are the computational classification methods for ERH detection?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Between 2015-2021, non-deep Machine Learning Algorithms (MLAs) shifted towards Deep  Learning Algorithms (DLAs) due to their superior performance and context-sensitivity (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Support Vector Machines (SVM) and a case of a Random Forest (RF) model were the last remaining  non-deep MLAs post-2018 to outperform DLAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Studies seldom hybridise relationship network  modelling and semantic textual analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Ongoing areas of research in MLAs consist of identifying  psychological signals to compete with DLAs such as Bidirectional Encoder Representations from  Transformers (BERT), Convolutional Neural Networks (CNN) and Bidirectional Long Short-Term  Memory (BiLSTM) models (defined further in the Supplementary Material’s Algorithm Handbook  section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' DLAs are best oriented for text-only tasks and for hybrid image-caption models [3, 57,  99, 111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Future NLP studies should consider higher-performing neural languages models over  BERT-base—such as RoBERTa [68], Sentence-BERT [96], or multi-billion parameter transformers  such as GPT-3 [23], GPT-Neo [19], or Jurassic-1 [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Observed Non-deep Machine Learning Algorithms (MLAs)  Studies investigating non-deep MLAs tend to test multiple models, typically Support Vector Ma-  chines (SVMs), Random Forest (RF), and Logistic Regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Figure 11 outlines the distribution of  both deep and non-deep approaches, with SVM again the most popular MLA in 15 of the 51 studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Non-deep MLAs consistently under-performed for multiclass classification, whereby Ahmad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  identified that a prior Naïve Bayes model could not distinguish between ‘Racism’ and ‘Extremism’  classes due to a low F1-score of 69%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' while their LSTM and GRU model could detect such nuance  with a 84% F1-score [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Likewise, application-specific sentimental algorithms paired with MLAs  resulted in lower F1-scores compared to context-sensitive BERT models—which do not require  manual feature extraction [55, 71, 83, 126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' claimed that SentiStrength was ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='not  robust to various sentence structures and hence fails to capture important semantics like sarcasm,  negation, indirect words, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' at the phrase level" [107, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 5]—a critique shared in six other non-deep  sentiment scoring studies [47, 69, 84, 89, 103, 130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Consider the hypothetical case of "I am not happy  with those people", whereby context-insensitive (orderless) embeddings will not detect the negation  of happy nor the implicit euphemism for ‘those people’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Hence, researchers have three options when designing ERH models:  (1) Avoid complex textual feature extraction and filtering by prioritising DLA development, or  (2) Prioritise manual textual and metadata feature extraction, such as psychological signals,  emotions, sarcasm, irony, temporal data, and/or  (3) Consider community detection (relationship network or topic modelling) features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Non-deep Machine Learning Algorithms in Community Detection Studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  There is a discrepancy in the choice of algorithmic approach compared to NLP-oriented models  where less than a third of the community detection studies considered Deep Learning (DL) mod-  els [77, 84, 99], while NLP-only studies were majority DL (15 of 29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' A reason for this discrepancy  would be the limited research in social media network analysis without investigating textual data,  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech  000:21  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Number of instances of Machine Learning Algorithms (MLAs) used for ERH detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  instead opting to cluster group affiliation via K-means [12, 16, 80], NbClust [12], weighted bipartite  graphing into Louvain groups [15, 16], and fast greedy clustering algorithms [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  We observed that graphing relationship networks result in two types of classification categories:  (1) Meso-level affiliation—semi or unsupervised affiliation of a user to an extremist group or  organisation, with a bias towards Islamic extremist groups [15, 16, 80, 84, 99, 101, 108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (2) Micro-level affiliation—(semi)supervised person-to-person affiliation to an annotated extremist,  such as radicalising influencers [12, 24, 77], and legal person-of-interest models [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  For organisational affiliation, information for clustering included the use of hashtags shared by  suspended extremist Twitter users and unknown (test) users [7, 15, 84, 95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  For identifying a user’s affiliation to other individuals, researchers preferred non-textual graph-  based algorithms as they reduce memory complexity and avoid the perils in classifying ambiguous  text [16, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Furthermore, 2016-2019 demonstrated a move from investigative graph search and  dynamic heterogeneous graphs via queries in SPARQL [48, 80] towards Louvain grouping on  bipartite graphs as a higher-performing (by F1-score) classification method [15, 16, 108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  For hybrid NLP-community detection models, researchers mapped text and friend, follower/ing,  and mention networks via decision trees and kNN [77, 84], or used Principal Component Analysis  on extracted Wikipedia articles to map the relationships between discussed events and entities [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  An emerging field of community detection for extremism consists of knowledge graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Know-  ledge graphs represents a network of real-world entities, such as events, people, ideologies, situ-  ations, or concepts [125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Such network representations can be stored within graph databases,  word-embeddings, or link-state models [48, 125, 127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Link-state knowledge models consist of  undirected graphs where nodes represent entities and edges represent links between entities, such  as linking Wikipedia article titles with related articles based on those referenced in the article, as  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='SUPPORTVECTOR MACHINE (SVM)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='SENTIMENTAL ALGORITHMS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='LOGISTIC REGRESSION  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='OTHER DECISION TREES  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='OTHER (NOVEL) NEURAL NETWORK  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='GRAPHING ALGORITHMS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='NOVEL APPROACHES  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='BERT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='CONVOLUTIONAL NEURAL NETWORK (CNN)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='LSTM-ONLY  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='RANDOM FOREST  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='ENSEMBLE NEURAL NETWORKS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='KNN CLASSIFICATION  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='NAIVE BAYES  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='GRU NEURAL NETWORK  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='PRINCIPAL COMPONENT ANALYSIS (PCA)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='K-MEANS CLUSTERING  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='LDA (LINEAR DISCRIMINANT ANALYSIS)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='LINEAR REGRESSION  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='LSA (LATENT SEMANTIC ANALYSIS)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='16000:22  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  used in Wikipedia2Vec [127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' consider a novel hybrid OSINT and law-enforcement  database graph model-—which unifies textual n-grams from social media to shared relationships  between other individuals and law enforcement events over time [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Four studies consider model relationships to individual extremist affiliates [12, 24, 48, 77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' In a  direct comparison between text and relationship detection models, Saif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' observed that text-only  semantic analysis outperformed their graph-based network model by a +6% higher F1-score [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Deep Learning Algorithms (DLAs)  DL studies are rising, with less than a third of studies including DLAs pre-2019 [10, 18, 27, 32, 45,  91, 99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The percentage of all studies which included a DLA per year was 0% in 2016, 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='3% in  2017 [10, 27, 45, 99], and 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='3% in 2018 [18, 32, 91], compared to being the majority post-2018 (81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='8%  in 2019 [1, 4, 53, 67, 71, 77, 83, 84, 90, 128–130], 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='5% in 2020 [14, 55, 57, 81, 107, 111] and 80% in  2021 [3, 52, 126])—with Figure 12 displaying the shift towards DLAs since 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Patterns of adoption for ERH detection algorithms over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Colour change ordered by F1-score  trend (low to high).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Brown = ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='75 F1-score on benchmark datasets, Red = ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='9 F1-score, Grey = No Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Between 2017-2018 Convolutional Neural Networks (CNN) using Long-Short Term Memory  (LSTM), GRU, Recurrent Neural Networks (RNN), or graph-based layers were the sole DLAs [10, 18,  32, 45, 91, 99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' From 2019-2021, various new approaches such as SenticNet 5 [89], ElMo (Embeddings  from Language Model) [90], custom neural networks such as an Iterative Opinion Mining using  Neural Networks (IOM-NN) model [14], and attention-based models such as BiLSTM [81, 83, 90,  128, 129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Since 2019, there is an emerging consensus towards BERT [67, 71, 81, 107, 126, 129, 130]  due to its easy open-source models on the Hugging Face platform and high performance per Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Deep Learning for Community Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Only 3 of the 21 DL studies considered relationship network models [77, 84, 99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Whereby, Mashech-  kin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' grouped self-proclaimed Jihadist forums and VK users with Jihadist keywords as a  "suspicious users" category [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Uniquely, the researchers implemented a Hyperlink-induced Topic  Search (HITS) approach to calculate spatial network proximity between annotated extremists and  unknown instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' HITS identifies hubs, which are influential web pages as they link to numerous  other information sources/pages known as authorities [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The influence of an authority depends  on the number of hubs that redirect to the authority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' An example of HITS in-action would be an ex-  tremist KavazChat forum (a hub) with numerous links to extremist manifestos (authorities) [59, 77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Evaluating influence in these graph networks requires measuring spatial proximity via betweenness  centrality [41] and depth-first search shortest paths where proximity to a known extremist via  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Prior SLR trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Algorithmic and Feature Extraction Trends found in our SLR  Community  Expansion of Sentimental  Year-of-Hybrid-Models  Detection  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' First  CNN+LSTM, BiLSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Use  BERT & SVM as  Blacklist  with Semantic  application of CNN  of language models  the dominant  Words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  LSTM, & Word2Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (BERT),  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  1  2000-2010  2010-2015  2015  2016  2017  2018  2019  2020  2021  1  Bag of Words  Generic ML (sentimental  Year-of-Extremism-  Context-sensitive models  Approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  feature extraction in  detection via SVM, RNN  & features - phrase,  MLA) Bipartite Graphing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  kMeans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Emergence of  ExtremSentiLex, opinion  Waseem & Hovy dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  affiliation-based datasets  & propaganda mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech  000:23  following/reposting them constitutes an extremist classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' However, such relations do not  accommodate for replies to deescalate, deradicalise, or oppose extremist speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Visual-detection Models for ERH Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Despite the emergence of multimedia sources for radicalisation and ideological dissemination,  only three studies considered multimodal image and image-text sources—utilising image memes  with superimposed text from the Facebook hateful meme dataset [57] and the MultiOFF meme  dataset [111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Only one study considered the post’s text (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', text not displayed on the image itself)  as context via the multimedia Stormfront post and image data from Rudinac et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  For the Facebook hateful meme and MultiOFF datasets include images with superimposed cap-  tions [57, 111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Both Kiela et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' [57] and Aggarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' [3] extract caption text via Optical Character  Recognition (OCR) models—-a computer vision technique to convert images with printed/visual text  into machine-encoded text [125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The three hateful meme studies utilised either both (multimodal)  or one (unimodal) of the image and its caption [3, 57, 111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The multimodal Visual BERT-COCO  model attained the highest accuracy of 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='47%, compared to 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='8% for a caption text-only classifier  or 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='73% for image only, 64% for the ResNet152 model [3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' and 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='70% for the baseline (human) [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Deep learning pipeline for visual-text ERH detection based on the hateful meme studies [3, 57, 111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  The highest performing multimodal model relied on Visual BERT [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Visual BERT extends  textual BERT by merging BERT’s self-attention mechanism to align input text with visual regions  of an image [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' VisualBERT models are typically pretrained for object detection, segmentation,  and captioning using the generic Common Objects in COntext (COCO) dataset [66], such that  the model can segment and create a textual description of the objects behind an image such as  “two terrorists posing with a bomb” (Figure 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Training otherwise acts the same as BERT-—which  involves masking certain words/tokens from a textual description of the image of what the image  depicts, and VisualBERT predicting the masked token(s) based on the image regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We aggregate  and generalise all visual ERH detection studies architectural pipelines in Figure 13 [3, 57, 99, 111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  No hateful meme dataset studies consider accompanying text from the original post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' This raises  concerns regarding posts satirising, reporting, or providing counter-speech on hateful memes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Only one study investigated a contextual textual post and accompanying images through a  proposed Graph Convolutional Neural Network (GCNN) model [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' This GCNN approach extracted  semantic concepts extracted from Wikipedia, such as identifying that an image was a KKK rally—  attaining a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2421 F1-score for detecting forum thread affiliation across 40 Stormfront threads [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Key:  Dotted lines = Optional path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Full line = necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Red outline = Computer Vision (lmage Embeddings)  ResNet152  Extract 1000-  Computer Vision  dimension  Model  embeddings  Extract Regions of  Create Textual Representation  Entity and Object  the Images (Visual  VisualBERT  ("Two individuals with a  Detection  BERT tokens)  bomb")  Separate contextual  NLP model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=',  Top Text  BERT)  Textual Optical  Textual Feature  Concatenate  Hate Classifier  Character  Extraction  Embeddings  Neural Network  Recognition  "Top  "Bottom  Bottom Text  Text"  Text"000:24  Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  8  MODEL PERFORMANCE EVALUATION, VALIDATION, AND CHALLENGES  What are the highest performing models, and what challenges exist in cross-examining them?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Evaluating model performance presents three core challenges for future researchers:  (1) Dataset domain differences—which may include or exclude relevant features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', gender,  location, or sentiment) and may involve numerous languages or groups (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Islamic extrem-  ists vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' white supremacists) who will express themselves with different lexicons [20, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (2) Criteria differences—different standards for ERH definitions, criteria, filtering, and annota-  tion threaten cross-dataset analysis between models [24, 122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Binary classification can result  in higher accuracy compared to classifying nuanced and non-trivial subsets of hate such as  racism/sexism [122], overt/covert aggression [61], or hateful group affiliation [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (3) Varying and non-standard choice of metrics—Figure 14 displays the 28 metrics, which  vary depending on whether the study investigates community detection via closeness, in-  betweenness, and eigenvectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' or NLP, often via accuracy, precision, and F1-scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Distribution of metrics used across the 51 studies—demonstrating a lack of standardisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Models ranked by macro F1-score for the benchmark datasets across studies (inter-study evaluation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Dataset  1st Highest  2nd Highest  3rd Highest  Waseem  and  Hovy [123]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='966 (BERT with GPT-2  fine-tuned dataset [126])  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='932 (Ensemble RNN [91])  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='930 (LSTM + Random Em-  bedding + GBDT [10])  FifthTribe [39]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='0 (RF [84])  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='991-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='862 (SVM [7])  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='87 (SVM [95])  de Gibert [32]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='859 (SP-MTL LSTM, CNN  and GRU Ensemble [55])  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='82 (BERT [71])  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='73 (LSTM baseline met-  ric [32])  TRAC FB [61]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='695 (CNN + GRU [55])  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='64 (LSTM [61])  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='548 (FEDA SVM [56])  Hatebase Twit-  ter [31]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='923 (BiLSTM with Atten-  tion modeling [83])  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='92 (BERTbase+CNN / BiL-  STM [81], 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='86 (with racial/-  sexual debiasing module)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='912  (Neural  En-  semble [71])  HatEval [13]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='7481  (Neural  En-  semble [71])  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='738  (LSTM-  ELMo+BoW) [90]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='695 (BERT with GPT-2  fine-tuned dataset [126])  OffensEval [128]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='924 (SP-MTL CNN [55])  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='839 (BERT [130])  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='829 (BERT 3-epochs [67])  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='F1-SCORE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='35  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='ACCURACY  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='PRECISION  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='21  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='RECALL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='P-VALUE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='CORRELATION COEFFICIENT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='NETWORKCENTRALITIES  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='MEAN ABSOLUTE ERROR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='STANDARDIZED CO-INCIDENCE RATIO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='LOUVAIN (Q) MODULARITY  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='PATIAL PROXIMITY  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='GINI COEFFICIENT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='INFORMATION GAIN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='PARTITION SENSITIVITY  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='MEANLOG ACCURACY  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='RACIST SCORE (CUSTOM METRIC)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='WILCOXON SIGNED-RANK TEST  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='CHI-SQUARE INDEPENDENCE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='PEARSON CORRELATION COEFFICIENT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='PAGERANK SCORES  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='RADICAL SCORE (SENTISTRENGTH)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='HYBRID STRUCTURAL-SEMANTIC SIMILARITY  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='POSSIBILISTIC SIMILARITY MEASURE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='ROC AREA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='SIRA RADICAL SCORE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='R^2 VALUE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='35  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='40Down the Rabbit Hole: Detecting Online Extremism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Radicalisation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' and Politicised Hate Speech  000:25  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Benchmark Dataset Performance (Inter-study Evaluation)  We use macro F1-score as the target metric as it balances true and false positives among all classes,  and is a shared metric across the benchmark datasets [13, 31, 32, 39, 123, 128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Table 4 outlines that  the highest F1-scoring models reflect the move towards context-sensitive DLAs like BERT, as also  displayed in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' SVMs and a single instance of a Random Forest classifier on sentimental  features were the last standing non-deep MLAs [7, 56, 84, 95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Given the variety of MLA and DLAs  (Figure 11), approaches that frequently underperformed included Word2Vec, non-ensemble neural  networks such as CNN-only models, and baseline models [31, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' These baseline models include  the HateSonar model by Davidson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' [31], Waseem and Hovy’s n-grams and gender-based  approach [123], LSTM model by de Gibert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' [32], and C-Support Vector Classification by Basile  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' No studies discuss memory or computational complexity, an area worthy of future  research as expanded in our Supplementary Material’s Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Community Detection Performance  While community detection models tend to produce F1-scores ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='15 lower than DLAs [12, 15, 16, 48,  80, 108], these comparisons rely on different datasets/metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Shi and Macy recommended using  Standardised Cosine Ratio as the standardised metric for structural similarity in network analysis, as  it is not biased towards the majority class, unlike Jaccard or cosine similarity [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' For community  detection models on the same pro/anti-ISIS dataset [39], F1-scores ranged from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='74-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='93 [7, 15, 95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Only one study cross-examined text and network features [101], with a hybrid dataset consisting  of annotated anti/pro-ISIS users’ posts and number of followers/ing, hashtags, mentions, and  location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Text-only semantic analysis outperformed their network-only model (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='923 F1 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='866  respectively) [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' However, topic (hashtag) clustering and lexicon-based sentiment detection  via SentiStrength underperformed compared to the network-only approach by a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='07-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1 lower  F1 [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Thus, unsupervised clustering models are ideal for temporal radicalisation detection and  identification of emergent or unknown groups or ideologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' There is insufficient evidence to conclude  whether community detection is superior to NLP due to the lack of shared NLP-network datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  For supervised community detection tasks, researchers [15, 80, 101] used network features via  Naïve Bayes [80], k-means [12, 16, 80], SVM [84], and decision trees [77, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The highest F1-score  community detection model was a hybrid NLP and community detection model using network  features, keywords and metadata (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', language, time, location, tweet/retweet status, and whether  the post contained links or media) with a Naïve Bayes classifier—attaining a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='89 F1-score [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  9  FUTURE RESEARCH DIRECTIONS  In this section, we offer an alternate to the radicalisation = extremism = political hate speech consensus  from RQ1 and models observed in RQ3/4 to present a new framework for delineating and expanding  ERH for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Overall, we propose an uptake roadmap for ERH context mining to expand the  field into new research domains, deployments for industries, and elicit governance requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Ideological Isomorphism—a Novel Framework for Radicalisation Detection  Definition 4: Ideological Isomorphism (Computational Definition for Radicalisation)  The temporal movement of one’s belief space and network of interactions from a point of  normalcy towards an extremist belief space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' It is an approach to detecting radicalisation with  an emphasis on non-hateful sentiment as ringleaders and/or influencers pull and absorb others  towards their hateful group’s identity, relationships, and beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  000:26  Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  As outlined in our novel tree-diagram dissection of ERH definitions to their computational  approach in Figure 6, there is considerable overlap in approaches between the otherwise unique  fields of extremism, radicalisation and politicised hate speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Radicalisation’s working definition  suffers from ambiguity in the majority of studies due to its interchangeability towards extremist  affiliation and no considerations for temporal changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Radicalisation’s computational definition  should reflect a behavioural, psychological, and ideological move towards extremism over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  While extremist ideologies and outwards discourse towards victim groups may be manifestly  hateful, radicalisation towards target audiences may involve non-hateful uniting and persuasive  speech [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, we propose that radicalisation detection should not be a single-post classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Rather, models should consider micro (individual), meso (group dynamics), and macro (global  events and trends) relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The roots for radicalisation result from an individual’s perceptions of  injustice, threat, and self-affecting fears on a micro-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' On a meso-level, this can include the rise  of community clusters based on topics and relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Socially, a radicalised user draws on an  extremist group’s legitimacy, connections and group identity, trends, culture and memes [58, 70, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Hence mapping ideological isomorphism requires temporal modelling to:  (1) Detect the role of users or groups polarising or pulling others towards extreme belief spaces  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', ideological isomorphism), akin to detecting online influencers [12, 46, 80, 121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Studies  should also consider the role of alienation as a radicalising factor via farthest-first clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (2) Further research into the role of friendship and persuasion by adapting sentimental ap-  proaches to consider positive reinforcement towards hateful ideologies akin to existing  research in detecting psycho-behavioural signals [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Furthermore, there lacks research in  computationally detecting social factors such as suicidal ideation or mental health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (3) Investigate the interactions between groups across social media platforms as radicalisers  themselves, such as the promotion of extremist content by recommendation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (4) Utilise community detection metrics such as centrality, Jaccard similarity, and semantic  similarity over time as measurements for classifying radicalisation for meso-level NLP (topic)  and graph-based (relational) clustering, leaving content moderation as a separate task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (5) Consider the role of satire, journalism, and martyrs as areas for radicalisation clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Morphological Mapping and Consensus-building—a novel  computationally-grounded framework for extremism detection  Definition 5: Morphological Mapping and Consensus-building (Extremism)  The congregation of users into collective identities (‘in-groups’) in support of manifestly  unlawful actions or ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  While ideological isomorphism focuses on micro-level inter-personal relations, morphological  mapping pertains to clustering meso-level beliefs and community networks to extremist ideologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  While we discovered various affiliation-based clustering approaches, no studies identified novel or  emergent movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Establishing a ground truth for a novel extremist organisation is challenging  if such groups are decentralised or volatile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, we recommend using manifestos, particu-  larly unconsidered far-right sources, and influential offline and online extremists as a benchmark  for identifying martyrdom networks and new organisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Areas for future research include  investigating the role of trolls, physical world attacks, or misinformation in narrative-building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Our morphological mapping framework proposes to delineate Extremism by considering the role  of group identity and ideological themes behind hate speech by considering affiliation across users  and posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' When targeting extremism, pledging ‘support’ to a terrorist organisation may not violate  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech  000:27  context-insensitive BoW hate speech classifiers—-hence it is not appropriate to categorise extremist  affiliation under the same guise as post-by-post hate speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Currently, extremism detection  constitutes a binary ‘pro vs anti group’ classification, which fails to capture the inner trends of  radicalisation from peaceful, to fringe beliefs, to committing to violent-inducing beliefs online, and  potentially to offline extremism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Investigating semi or unsupervised clustering (mapping) of groups  will also aid Facebook’s commitment to moderating militarised social movements, violence-inducing  conspiracy networks, terrorist organisations, or hate speech inducing communities [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Thus, we propose four prerequisites for studies to fall under the extremism detection category:  (1) Investigate the interactions and similarities between groups on mainstream and anonymous  platforms to map group dynamics and extremist networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' For privacy, we recommend  group-level (non-individualistic) network and semantic clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (2) Map affiliation and group dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Given the lack of definitions for extreme affiliation, we  recommend using Facebook’s definition of affiliation as a basis—being the positive praise of  a designated entity or event, substantive (financial) support, or representation on behalf of a  group (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', membership/pledges) [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (3) Investigate hateful and non-hateful community interactions, memes and trends, that reinforce  group cohesion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (4) Map affiliation as a clustering task, akin to our proposed radicalisation framework but without  the temporal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='3  Outwards Dissemination––‘traditional’ hate speech detection updated  Definition 6: Outwards Dissemination (Hate Speech)  Targeted, harassing, or violence-inducing speech towards other members or groups based on  protected characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Hence, the projection and mainstreaming of hateful ideologies through speech, text, images,  and videos requires an outwards dissemination of views shared by extremists, such as racism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The  outwards dissemination of hate is a strictly NLP (text) and computer vision (entity and object)  classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We delineate hate speech with affiliation to violent extremist groups as such  misappropriation could have devastating effects on one’s image, well-being, and safety [9, 24, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  All researchers should be aware that malicious actors may exploit existing ERH models for  injurious surveillance and censorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Future work should also consider the impacts of labels  on society at large, whereby terms such as ‘far-right’ as an alias for white supremacy is both  misleading, infers a ‘right vs wrong’ left-to-right spectrum, and ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We recommended  decoupling religious contexts in favour of technical terms such as ‘radical Islamic extremism’ or  ‘terror-supporting martyrdom’ to avoid grouping religiosity to a political ideology and terrorism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Thus, we propose three key prerequisites for a study to be in the hate speech category:  (1) Investigates textual or multimedia interactions only, whereby detecting cyber-bullying or  extremist community networks should be separate tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (2) Decouple affiliation where possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' For instance, white supremacy instead of far-right (an  ambiguous term) or organisational affiliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (3) Consider models which include latent information, such as news, entities, or implied hate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Datasets should explain each classification with categories for disinformation and fallacies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  000:28  Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Future work in outlining hate speech would be a systematic socio-legal cross-examination of hate  speech laws from governments and policies from social media platforms—including the emerging  consensus vis-à-vis the harmonised EU Code of Conduct for countering illegal hate speech [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='4  Uptake Roadmap for Researchers, Industry, and Government  We present a pipeline for researchers, industries, and government analysts to approach ERH  context mining per Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' In addition to this summary visualisation of our key dataset and  model recommendations, we expand on our actionable recommendations for immediate next steps  and long-term software requirements for ERH detection in our supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Future SLRs should consider a mixture of academic studies, grey material, and technical reports  to further encompass our proposed ERH context mining field’s socio-legal component and explore  industry approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We recommend transforming our ethical recommendations for responsible  research outlined in our SLR design into formalised interdisciplinary guidelines to protect privacy  and researcher safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' ERH is never a singular end-goal, post, or unexpected event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, detecting  erroneous behaviour emanating from mental health crises can both avoid ERH online and offline,  and present avenues for cooperation with third-parties such as suicide prevention and counselling  groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Finally, we recommend searching for multimedia-only studies including for livestreams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  10  CONCLUSION  ERH context mining is a novel and wide field that funnels to one fundamental aim—the pursuit  to computationally identify hateful content to enact accurate content moderation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' In our work,  we harmonised Extremist affiliation, Radicalisation, and Hate Speech in politicised discussions from  2015-2021 in a socio-technical context to deconstruct and decouple all three fields of our proposed  ERH context mining framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, we propose a novel framework consisting of ideological  isomorphism (radicalisation), morphological mapping (extremism), and outwards dissemination  (politicised hate speech) based on our findings in RQ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' While hate speech included racism and  sexism, other forms of discrimination were seldom considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Extremism and radicalisation  frequently targeted Islamic groups, particularly from US and European researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Binary post-by-  post classification remain the dominant approach despite the complexity of online discourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  There is a clear and present danger in current academia emanating from the unresolved biases in  dataset collection, annotation, and algorithmic approaches per RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We observed a recurring lack  of consideration for satire/comedic posts, misinformation, or multimedia sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Likewise, data  lacked nuance without contextual replies or conversational dynamics, and were skewed towards  the US and Europe—with the global south, indigenous peoples, and Oceania all marginalised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Computationally, we identified that deep learning algorithms result in higher F1-scores at the  expense of algorithmic complexity via RQ3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Context-sensitive neural language DLAs and SVM  with sentimental, semantic, and network-based features outperformed models found in prior  SLRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' However, state-of-the-art models still lack a contextual understanding of emergent entities,  conversational dynamics, events, entities and ethno-linguistic differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' To combat injurious  censorship and vigilantism, we recommended several areas for future work in context-sensitive  models, researcher ethics, and a novel approach to framing ERH in SLRs and computational studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  The poor design and abuse of social media threatens the fabric of society and democracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Re-  searchers, industries, and governments must consider the full start-to-finish ecosystem to ERH  context mining to understand the data, their criteria, and model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Without a holistic  approach to delineating and evaluating Extremism, Radicalisation, and Hate Speech, threat actors  (extremists, bots, trolls, (non-)state actors) will continue to exploit and undermine content modera-  tion systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, informed, accurate and ethical content moderation are core to responsible  platform governance while averting injurious censorship from biased models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech  000:29  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' ERH Context Mining pipeline—with key identified research gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Sociolinguistics  Red = Computer-  science oriented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Computational  Blue = social-  Research  Mixed =  science  Gaps:  Social Media  Criteria-building  misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Satire &  sexism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  beyond racism &  Hate speech  affiliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  non-violent  Extremism as a  temporal process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  a friendly  Radicalisation as  Privacy & Ethics  Consider:  ERH (RQ1, RQ2, RQ3)  Dataset Creation  Data Selection & Collection  movements  Conspiracy  COVID-19 posts  extremist  Non-lslamic  anonymisation)  (data  Researcher safety  manifestos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  New APls  datasets  Multiple platform  platforms  /anonymous  Controversial  Consider:  Data Filtering  Extraction  Topic mapping to  Belief space  entities/events)  (Wikipedia/news  sources  informative  mapping  emotion detection  Context-sensitive  features  Suicidal ideation  signals  Psychological  Consider:  80  Deployment (RQ3/4)  Model Choice &  Model Creation &  Performance  Privacy Paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Image detection  Hybrid NLP-  Meme detection  text generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  conversational  Synthetic  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Few-shot  (GPT)  Pretrained Models  Generative  Differential  design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content="  Privacy-by-  Consider:  Model Governance  benchmarks  metrics &  Industry standard  Meta-analysis  Interdisciplinary  Self-regulation  recognition  context  New event  updating)  (frequently  models  Dynamic 'online'  Consider:000:30  Govers et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  REFERENCES  [1] Umar Abubakar, Sulaimon A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Bashir, Muhammad B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content=' Irfan Uddin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content=' Complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 2021 (jan 2021), 7 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content=' Online  Extremism Detection in Textual Content: A Systematic Literature Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Extremism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Retrieved June 29, 2021 from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='adl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='org/resources/glossary-  terms/extremism  [7] Oscar Araque and Carlos A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Iglesias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' An Approach for Radicalization Detection Based on Emotion Signals and  Semantic Similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content='2967219  [8] Australian Security Intelligence Organisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' ASIO Annual Report 2019-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content=' In Proceedings of the 26th International Conference on World Wide Web Companion (Perth,  Australia) (WWW ’17 Companion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content=' Springer Science & Business Media, New York,  NY, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content=' SemEval-2019 Task 5: Multilingual Detection of Hate Speech Against Immig-  rants and Women in Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content=' Association  for Computational Linguistics, Minneapolis, Minnesota, USA, 54–63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Supplementary Material for:  Down the Rabbit Hole: Detecting Online Extremism,  Radicalisation, and Politicised Hate Speech  JAROD GOVERS, ORKA Lab, Department of Software Engineering, University of Waikato, NZ  PHILIP FELDMAN, ASRC Federal, US  AARON DANT, ASRC Federal, US  PANOS PATROS, ORKA Lab, Department of Software Engineering, University of Waikato, NZ  Contents  Contents  1  1  Definitions—The Algorithm Handbook  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Definitions for Traditional (non-deep) Machine Learning Algorithms  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Definitions for Deep Learning Approaches  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='3  Language Transformer Models  6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='4  Definitions for Prominent Feature Extraction Techniques  7  2  SLR Design Considerations  8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Quality Assessment Criteria  8  3  The Case for Performance Engineering when Evaluating Models  10  4  Uptake Roadmap Expanded  10  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Model Recommendations  10  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Dataset Recommendations  11  References  12  1  DEFINITIONS—THE ALGORITHM HANDBOOK  This supplementary material document includes the supplementary material referenced in the  main Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech  Systematic Literature Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' This document offers a ‘dictionary/look-up table’ for the core al-  gorithmic architectures for the non-deep machine learning and deep learning models mentioned  throughout the SLR, alongside other side findings and design considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We contextualise the  Authors’ addresses: Jarod Govers, jg199@students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='waikato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='nz, ORKA Lab, Department of Software Engineering, University  of Waikato, Gate 1, Knighton Road, Hamilton, Waikato, NZ, 3216;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Philip Feldman, philip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='feldman@asrcfederal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='com, ASRC  Federal, Beltsville, Maryland, US;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Aaron Dant, aaron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='dant@asrcfederal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='com, ASRC Federal, Beltsville, Maryland, US;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='nz, ORKA Lab, Department of Software Engineering, University of Waikato, Gate 1,  Knighton Road, Hamilton, Waikato, NZ, 3216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee  provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and  the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='11579v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='SI]  27 Jan 2023  000:2  Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  relevant strengths and weaknesses of the various algorithmic approaches for text and visual models  for ERH detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' No new findings are in this handbook/‘look-up table’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, those familiar  with the models listed in the contents above need not read this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Definitions for Traditional (non-deep) Machine Learning Algorithms  We aggregate common and historic non-deep machine learning algorithms into the ‘traditional’  MLA category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, this section defines each of the baseline models used for textual or community  detection models—consisting of:  (1) Sentimental Bag of Words approaches,  (2) Naïve Bayes,  (3) Decision Trees,  (4) Support Vector Machines,  (5) Clustering Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Bag of Words (BoW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  BoW approaches simplify complex contextual sentences into a multiset (‘bag’) of individual words  by assigning a value or probability to each word in its relation to a specific document class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' For  instance, a BoW approach would deconstruct the contiguous sentence, “The Eldian people are  the spawn of the devil” (where Eldian is a fictitious race), into an unordered bag of individual  words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' While ‘are’, ‘the’ are unlikely to have a considerable influence on whether a sentence is hate  speech or not, the use of ‘devil’ and ‘Eldian [race]’ is more frequently paired in hate speech than  for non-hateful/off-topic text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The disregard of word order and the relationship of BoW approaches,  and MLA models at large, constitute context-insensitive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' For instance, a BoW model does  not know that ‘I love the Eldian people but hate their food’ is paring love -> Eldian, and hate -> food,  and thus would consider ‘I hate the Eldian people but love their food’ as identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Likewise, BoW  approaches do not consider alternate word meanings/uses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', ‘I ran for government’ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' ‘I ran  away’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Nonetheless, BoW approaches are core to word-specific ‘blacklists’ in content moderation,  such as banning users who use slurs in a post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' However, for nuanced and often politicised discussions  on controversial topics, simple blacklists can lead to injurious censorship—due to the context and  use of such words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Sentimental algorithms, such as SentiStrength [42] aggregate individual words into individual  emotions—whereby ‘love’ indicates a positive sentiment, while ‘hate’ generally appears in vitriolic  speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Figure 1 outlines an abstracted representation of the sentiment classification based on the  average sentiment score of a sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' However, the context-insensitive BoW models again fails for  nuanced cases, whereby Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' [41] identified that SentiStrength cannot detect negations  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', “I am NOT happy” where happy skews the final sentiment scores).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Naïve Bayes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Naïve Bayes classifiers represent types of probabilistic classifiers utilising Bayes theorem with  the assumption that the influence of each variable for classification is independent of each other  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', naïve) [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' For document classification, notable features are assigned a probability for their  occurrence given a specific class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' For instance, a hate speech post that has an angry sentiment may  have a P(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='8) (Probability of 80%) of being hateful, given that a test hate speech dataset may be 80%  angry speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Bayes rule represents these chains of (assumed) independent/unrelated probabilities  to form a final probability for a test instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Notable features for probability models include:  Textual features—(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', sentimental scores, appearance of certain slurs/terms),  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Supplementary Material for:  Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech  000:3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' An abstracted example of Bag of Words approach within a Naïve Bayes classifier—demonstrating its  lack of context sensitivity and the focus on key ‘racist’ words for ERH detection tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Network data—(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', probability that someone who is friends with a supremacist is also a  supremacist, retweet relationships),  Metadata—(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', length of a post, readability via a Flesch Reading Ease score, number of posts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  In the example of Figure 1, the probability that the tweet is racist depends on the probability that  the racist tweet is angry, contains racial terms (‘Eldian’), the semantic similarity between known  hate speech posts, and the appearance of a negative lexicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Naïve Bayes can be a final classifier  for aggregating context-sensitive embeddings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', deep learning models) and multiple ‘ensembles’  of approaches/models—via chaining their probabilities together with this Bayes rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='3  Decision Trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' An example of a decision tree, with the leaf nodes constituting the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' In the Eldian  hate speech example, this would require traversing the left branches recursively for the final ‘Hate Speech’  classification leaf (shown via the red arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Chaining the correlations between features and their class likelihood can also span a tree of  scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' If an annotated dataset indicates that a post is 80% likely to be racist if a sentiment-scoring  algorithm detects anger, then a binary decision emerges—if post contains angry words, then likely  hate speech;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' if not, then not hate speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' These rules construct decision trees, where the root  constitutes the instance (text, network, metadata, or image), and each node is a decision, with the  leaves (final node) being the expected class value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', the classification) [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, decision trees  are not naïve as they rely on specific values of other features when traversing a tree’s branches for  a prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Example: Naive Bayes (simplified):  "The Eldian people are the spawn of the devil"  Sentiment  Target Entity  Semantic Similarity to  Lexicon also affiliated  Annotated Posts  with extremists:  Angry  “hate\'  Eldian  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content="9  1 (devil)  P(Racist I Tweet) =  “Eldian'  'devil'  P(Rac." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='IAngry) * P(Rac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='I Eldian) * P(Rac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='I O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='9) * P(Rac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='I 1 lexicon)  “scum"  P(Racist)Post to Classify  Sentiment = Not  Sentiment =  Angry  Angry  Contains slurs  >=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='5 semantic  <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='5 semantic  No slurs  similarity  similarity  = Yes (Devil)  Not Hate  Not Hate  Hate Speech  Hate Speech  Speech  Speech000:4  Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Creating an optimal tree that maximises accuracy and precision is not trivial due to the feature  explosion of possible rules and tree nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, Random Forest classifiers rely on a divide-  and-conquer algorithm for generalising feature pairings into class classifications with a random  initialisation [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' This recursive process requires finding optimal splits to maximise the separation  of classes for a final leaf, with an example tree presented in Figure 2—where a random forest would  consists of multiple trees as a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Ideally, a leaf node should encapsulate instances of one class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Random forests generate multiple decision trees and select the final prediction based on the  predictions from the majority of decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Utilising multiple trees with a random initial tree  state increases the range of features and values selected during the training step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Utilising multiple  trees and testing the models on untrained ‘test’ data minimises the risk of over-fitting to the training  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', a classifier which performs reliably on the training dataset but not on real-world data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Random forests strengths include its ability to tie dependent and complex features while reducing  over-fitting through pruning (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', reducing tree size to generalise the model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, decision trees  capture related concepts in hate speech where naïve BoW approaches do not—such as the appearance  of anger/negative sentiment invoking the use of charged terms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', racism as an emotional outlet)  or frequency of posts and sentiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='4  Support Vector Machines (SVM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Support Vector Machine where instances beyond the boundaries (support vectors) are automatically  assigned to the class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  SVMs are another supervised learning model for classification and regression tasks, seeking to  map instances in vector spaces to maximise the distance between classes [14], visualised in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Mapping features to multidimensional vectors can exponentially increase dimensions (an issue  shared in deep-learning models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Thus, SVMs reduce irrelevant features through specific kernels—  typically a linear, polynomial, Gaussian or sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' These kernels reduce the feature  set to draw boundaries between two classes, similar to logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' These boundaries are  either hard (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', a binary classification) or soft—allowing outliers near the boundary for edge cases,  like niche controversial and offensive, but not ostensibly targeting protected characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' SVM  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Support Vector  (with cost hyperparameter)  Neutral  Speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Y-Axis  Hate  Speech  Optimal hyperplane  to maximise  distance (y=mx+c)  X-AxisSupplementary Material for:  Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech  000:5  models are computationally faster and reduce memory compared to deep learning models [3, 45],  while achieving comparative performance outlined in RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Dimensionality reduction techniques  can also reduce runtime by reducing the complexity of large feature spaces from textual or network  data, such as via Principle Component Analysis [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  SVMs are the consistently highest performing MLAs per RQ4, while lowest complexity, with  𝑂(𝑚∗𝑛) complexity for a Linear Kernel SVC—where m = feature count, and n = number of instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='5  Clustering and Nearest Neighbour Classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Instead of annotated hate speech datasets, clustering methods group by textual similarity via Natural  Language Processing (NLP), and network relations via Community detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, clustering can  work in cases of fully annotated datasets as supervised learning, semi-annotated datasets as semi-  supervised learning, or unlabelled raw web scrapped data for unsupervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  For supervised learning, K-Nearest Neighbour (KNN) classifiers work via evaluating the nearest  neighbours’ likeliness when projecting the textual, network, or metadata features onto a multidi-  mensional space [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The ‘distance’ between feature spaces typically rely on Euclidean, Manhattan,  or Minkowski distance—where the latter two are suited for non-linear feature spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Non-euclidean  distances are ideal where dimensions are not comparable, as Manhatten distance reduces noise/er-  rors from outliers since the gradient has a constant magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Clustering examples for hate speech detection includes K-Means, which partitions n observations  into k clusters [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' K-Means automatically generates clusters, thus does not require annotated  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, K-Means can detect novel groups, including emergent extremist organisations,  or influential individuals [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Unsupervised clustering’s strength for ERH detection is how it  circumvents the definition issues for annotating data and can cluster large movements without  costly annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' However, K-Means may not identify manifestly hateful posts, as it does not abide  by any standard imbued within strict annotation criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Evidently, in the cross examination of  a naïve approach vs their proposed K-Means derived model by Moussaouri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' [30], the naïve  approach outperformed the possibilistic clustering by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='07-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='14 for accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='04-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='05 for precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Definitions for Deep Learning Approaches  Deep learning represents a family of machine learning algorithms with multiple layers and com-  plexity, typically via neural network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Neural networks rely on training a network with  a set of weights at each layer, known as neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The first layer of a neural network utilises numeric  representation of an instance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', hateful text) in numeric ‘tokenised’ form, which is adjusted  throughout the hidden lower layers towards a final output (typically) classification layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Each  downwards training step results in readjusting the weights of the upper layers for the neurons—  known as backpropagation [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Figure 4 displays this architecture for neural networks per our  example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The benefit of DLAs in ERH detection is the preservation of word order and meaning (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=',  “I ran” vs “I ran for president”), thus displaying context-sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Given dual-use words such as  ‘queer’, or racially motivated slurs, understanding the surrounding contextual words is essential to  reduce bias via misclassifications [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' DLAs dominant the benchmark dataset leader-board in RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Convolutional Neural Networks (CNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Convolutional Neural Networks (CNNs) expand on the neural network model through a convolu-  tional layer—which acts as a learnable filter for textual or image embeddings [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Moreover, CNNs  include a pooling layer(s) to reduce the spatial complexity of the network’s features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Reducing  spatial size helps reduce the number of parameters and thus training time and memory footprint,  while reducing over-fitting by generalising patterns in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  000:6  Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  LSTM and GRU aim to increase contextual awareness to process data sequences with long-term  gradients to retain information on prior tokens [12, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' LSTM and GRU seek to reduce the vanishing  gradients caused during backpropagation steps, which reduces classification performance as older  trained instances are effectively ‘forgotten’ due to later weight changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Similarly, GRU’s are a  gating approach with fewer parameters and thus higher runtime, enabling larger neural networks  overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' CNN models with LSTM and GRU connections outperform CNNs on their own for hate  speech detection [22, 23, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The highest performing BiLSTM model expands LSTM for bidirectional  input, via two LSTMs—where tokens in the network utilise information from past (backwards)  tokens/data and future (forwards) data [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The ability to uphold the temporal memory of prior  tokens (attention) constitutes a Recurrent Neural Network (RNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='3  Language Transformer Models  The state-of-the-art transformer architecture relies on self-attention—the memory retention of  neural networks where each token of a sequence is differentially weighted [8, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Unlike Recurrent  Neural Networks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', neural networks where nodes follow a temporal sequence), a transformer’s  attention mechanism utilises context for any position for the token sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, transformers  can handle words out of order to increase understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Transformers offer greater classification  performance (see RQ4) at the expense of memory and computational overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' A considerable  ethical threat of transformer models is their capability to predict future tokens (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', text generation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  For instance, a malicious actor could create realistic automated trolls or radicalising synthetic  agents as bots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Language models also risk data leakage of their trained data through predicting  tokens found in the original trained dataset, such as names or addresses [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' An abstracted example of a neural network for text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The top text represents its raw syntactic form,  with its converted numeric embedding representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' These embeddings are responsible for altering the  weights to increase token prediction or generation (for transformers) via backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' The final output  layer for this example would offer the probability that the given text is racist, sexist, or benign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  topic  relation  relation  entity  The  Eldian  people  are  the  spawn  of  the  devil  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='125  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='553  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='814  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='125  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='666  Segment and Positional tokens  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' computational embeddings for position/order of words).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  KKKKKKKK  Contextual interconnected layers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' neural network)  Racism  Sexism  Neither  Probability  Probability  ProbabilitySupplementary Material for:  Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech  000:7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Cross-encoders (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', BERT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Bidirectional Encoding Representations from Transformers (BERT) is the most common cross-  encoder observed for ERH detection [16], with the highest performance of all NLP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Cross-  encoders offer higher performance for classification tasks, through retaining information over a  given sequence with a label (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', self-attention).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' BERT’s core strength is its memory retention of all  tokens in a sentence, thus upholding full context-sensitivity of every word in the post it seeks to  classify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' However, cross-encoders are computationally expensive due to high parameter counts  (110 million parameters for BERT-base, 365 million for BERT-large), an issue further outlined in  RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, an area of ongoing research includes model distillation (optimising and reducing  parameter count to reduce memory requirements and training time), specialised training datasets,  and alternate layers [26, 37, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' BERT is pre-trained on entries from English Wikipedia (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='5 million  words) and the English BookCorpus (800 million words) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, such pre-trained models are  then fine-tuned on a smaller dataset (typically 1000+ instances, per RQ2’s benchmark datasets) to  optimise the BERT weights to detect hate speech with the context of its pre-trained corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Generative Pre-trained Models (GPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Similarly, the state-of-the-art GPT transformer architecture expands on the encoder blocks (shared  with BERT) to include decoder blocks [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, GPT works on a single token (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', word vector)  and produces estimates for the sequence’s next token—ideal for tasks such as text generation,  summarising, question answering, and information retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  GPT models differ from BERT-based models via masked self-attention—an alternate form of  context-sensitivity where the model only knows the context of the prior words in the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  GPT-2/3 [8], GPT-Neo [7], and Jurassic-1 [25], are notable 2019-2021 era multi-billion parameter  models—where larger datasets and parameter count result in more human-like text generation and  higher performance in information retrieval tasks [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  GPT’s core strength in ERH detection synthetic hate speech generation via a GPT model fine-  tuned on a hateful corpus—as investigated by Wullah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' (see RQ3) [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' However, state-of-the-art  GPT models utilise up to 178B parameters, whereby memory and computational requirements  scale linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, future GPT work in synthetic text generation should consider inference tasks  over fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Specifically, inference utilises a pre-trained model’s on-demand text generation  capability through prompts rather than altering each of the billions of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Using the auto-  complete-like inference capabilities for generating realistic synthetic hate speech posts constitutes  a novel case of prompt engineering in ERH detection and thus is a potential future research project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='4  Definitions for Prominent Feature Extraction Techniques  This subsection outlines the three most common feature extraction techniques used for textual ERH  detection—as outlined in RQ2 in the SLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' These models seek to identify hateful lexicons from text,  or create numerical representations for word or sentence meaning via embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We deconstruct  the six most common feature extraction techniques observed in our SLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Word2Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Word2Vec is a model to convert words into vector embeddings, which compares synonymous words  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', ‘hate’ and ‘disgust’) via numerical vectors [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Word2Vec compares these word-to-vector  embeddings via semantic similarity by evaluating their cosine similarity between their vectors  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', comparing word vectors of an unknown class instance to words from a known ‘hate speech’  instance(s) to make a ‘hate or not’ classification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' On a word-level basis, the vector value for ‘king’  value for man + value for woman would result in a vector similar to queen [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' In our case, a  ‘Islamist extremist’ and ‘ISIS’ are semantically similar akin to ‘White Supremacy’ and ‘Nazism’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  000:8  Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Doc2Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Similar to Word2Vec, Doc2Vec aggregates vector embeddings for paragraphs in addition to indi-  vidual words [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Thereby offering memory of the current context and paragraph’s topic—useful  for understanding a whole post’s sentiment and meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='3  N-grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  N-grams represent contiguous sequences of n-number of characters for frequency analysis given  their non-linear distribution in English, as well as when comparing a radical vs non-radical cor-  pus [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' This linguistic model is often paired with methods such as TF-IDF or BoW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='4  Term Frequency-Inverse Document Frequency (TF-IDF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  TF-IDF determines the relevance of a word in a document by comparing its frequency in the  document compared to its inverse number for the frequency of that word across all documents [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Thereby, assigning each word a weight to signify its semantic importance compared to the wider  corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' For instance, radical Islamist dog-whistle terms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', coded or suggestive political messages  intended to support a group) appeared disproportionately in extremist text compared to a neutral  religious corpus [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='5  SenticNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  SenticNet embeds pattern matching, parser trees, and LSTM-CNN models for sentiment analysis,  with the aim to replace a naïve BoW approach within a proclaimed bag of concepts and narrat-  ives [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Specifically, it includes feature extraction methods of concept parsing (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', understanding  linguistic patterns in natural language into conceptual pairs), subjectivity and polarity inference,  alongside personality and emotion extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='6  Global Vectors for Word Representation (GloVe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  GloVe offers an unsupervised learning algorithm for context-independent word-to-vector embed-  dings [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' While similar in creating vectors akin to Word2Vec, GloVe instead establishes word  co-occurrences using matrix factorization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', co-occurrence matrix of word [row] and context  [usage of the word in the document]) and dimensionality reduction techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  2  SLR DESIGN CONSIDERATIONS  This supplementary material section outlines the additional criteria and considerations for selecting  papers and ensuring privacy-protections for users, groups and collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' In essence, this section  offers a meta-analysis of the ethics and selection process used throughout the SLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Quality Assessment Criteria  The following includes our paper inclusion quality check criteria—with a score of 13 or higher  required for inclusion in the final paper selection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', final 51 papers included).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  We propose a critical criteria for quality assessment to filter irrelevant or ambiguous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Specifically, for a study that passed a title and abstract screen, we assess the study’s clarity for  ERH definitions and annotations (for objective and legible classifications), methodical clarity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  outlining each study’s algorithmic model, methods, data collection processes, and statistical analys-  is/evaluation methods), and socio-technical considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We weighted each quality assessment  section to prioritise their research methodology and clarity in their technical methods over their  Conceptual Quality for studies encompassing broader socio-technical issues such as ethics, legality,  or ERH clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' After a ten paper pilot study, we selected a score threshold of 65% to exclude  irrelevant or ambiguous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Our supplementary material document includes the criteria and  scoring for our quality assessment rubric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Supplementary Material for:  Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech  000:9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Computational Quality (0 = None, 1 = Partial, 2 = Full).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (1) Is the radicalisation/affiliation detection model clearly defined?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (2) Is the radicalisation/affiliation detection model’s algorithm clearly defined?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (3) Is the training data reputable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (4) Are the models results compared to similar state-of-the-art methods?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (5) Is the methodology for designing and conducting their experiment clearly defined?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (6) Are patterns and trends discussed and presented clearly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Epistemological Quality (0 = None, 1 = Partial, 2 = Full).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (1) Does the source(s) (data or researchers) avoid any conflict of interests or expressed biases?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', explicit support/funding from a political think tank or state agency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (2) Does the study provide a cited or evidence-based definition for “radicalisation”, “hate speech”  or "extremist" affiliation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (3) Are the dataset annotations vetted by more than one annotator to reduce bias?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='3  Conceptual Quality (0 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='5 value, as not critical but useful).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (1) Does the study discuss social or ethical issues in ERH detection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' censorship)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (2) Do the authors discuss the legality of their model or definitions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (3) Does the study evaluate its model across multiple social media platforms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  (4) Does the study discuss regulatory frameworks or recommendations for social media platforms  based on their findings?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='4  Researcher Ethics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  We focus on key terms and compositions of ERH examples to protect the privacy of the individuals  exposed, as recommended by meta-studies on extremism research ethics [9, 13, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' When linking  ERH detection to real-world groups and events, we solely focus on events and organisations which  resulted in media attention or criminal convictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' In no part during this SLR did we attempt to  track users, groups, or correlate online users to any personally identifiable information (name,  location, username etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=') given the ease of composing online data into a traceable online fingerprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Similar to the social norms in New Zealand in the aftermath of the Christchurch shooting, no  extremists, terrorists, and/or criminals are referred by name to minimise publicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We recognise  the potential for political or cultural bias in this charged field by citing international non-partisan  Non-governmental Organisations when framing ERH concepts, and avoid searching any party or  ideology in our search strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Moreover, we encourage that our findings and recommendations  invoke an open debate among social media platforms, governments, and the wider public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' However,  we do not condone the use of ERH detection in social media as a form of autonomous law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We  recommend human-in-the-loop processes when handling or classifying data via independent  reviews, privacy protections, and complaint and redress mechanisms for deployed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Our recommendations thereby focus on Open Source Intelligence (OSINT) oriented studies  that do not consider governmental or private-conversation surveillance (with the exception of  one hybridised study that appeared in our search [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We thereby consider ERH detection as  a computational method aimed at garnering community-insights, trends, and flagging for social  media platforms themselves to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Whether ERH detection policies should encourage deplatform-  ing, deranking, demonetisation, fact-checking, or targeted counter-speech/prevention programs  require further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We encourage open interdisciplinary research in public and private-  communications—particularly ethical and legal discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  000:10  Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  3  THE CASE FOR PERFORMANCE ENGINEERING WHEN EVALUATING MODELS  While high F1-scores help enforce community guidelines via accurate predictions and reduce  injurious censorship from false positives, runtime performance trade-offs are seldom discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  DLAs may perform within 1% (F1-score) of their MLA counterparts in NLP studies but require  significantly higher computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' For instance, fine-tuning a BERT-large model for NLP  tasks requires Graphics or Tensor Processing Units (GPU or TPU), restricting researchers from  testing large language models [45, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' For community detection, uncompressed network models  can include up to 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='4 million links [6], which significantly increases computational and memory  requirements for a minimal 1-5% performance gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Specifically, using a Possibilistic Approach (PA)  with dimensionality reduction reduced subgraph mining runtime by up to 67% (1500 seconds to  500 seconds on an 8-core 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2GHz system), while reducing accuracy by only 4% [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Furthermore,  community-level insights on topics with millions of tweets, relations, and discussions can lead  to a network explosion with a non-deterministic polynomial runtime [5, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' In graph-detection  approaches, performance engineering and optimisation for mining frequent subgraphs and graph-  traversal is an active area of research [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' No NLP studies consider performance engineering for  DLAs despite developments in model distillation and sentence-level embeddings [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Thus, we recommend that researchers consider performance trade-offs in future work and  investigate a possible standardised performance-complexity metric (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' parameter count vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' F1-  score ratio) to build scalable, energy-efficient and fiscally-viable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Moreover, fine-tuning  or retraining DLAs, or regenerating frequent subgraphs for community detection, should be a  frequent endeavour to adapt to the rapidly evolving topics, entities, and events throughout online  discourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Due to the computational costs of fine-tuning or training multi-billion parameter models,  we recommend approaches that do not require expensive training, such as few-shot learning (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=',  giving several known instances of ERH and a unseen ‘test’ instance) and prompt engineering [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  4  UPTAKE ROADMAP EXPANDED  This supplementary section expands on the dataset and model research gaps highlighted in Figure  16 of the main Down the Rabbit Hole SLR document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We categorise these research recommendations  into eight core components for our proposed ERH Context Mining research field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='1  Model Recommendations  The two predominant recommendations for future work are investigating the role of changes in  hateful affiliation or speech over time to satisfy the temporal requirement for Radicalisation detection,  and to train models on multiclass datasets from multiple platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We note that only one study  considered temporal data on both meso and macro (changes within and between groups), and micro  (individual) levels, although recommended as future work within four other studies [3, 11, 20, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Moreover, we recommend expanding on DLAs as the target for future research based on their  leading performance in RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Neural language models offer a macro-level societal understanding  due to their pre-trained corpus on academic sources, OpenWebText2 Reddit discussions, and  Wikipedia [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Furthermore, transformer models beyond 764 million parameters are untested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Bot, troll, meme, entity, dis/misinformation and satire detection remain underdeveloped—-which  could lead to censorship or undermine democratic institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Five studies recommended multi-  media detection as future work [1, 5, 11, 17, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  To protect user privacy from recreating user content from neural language models, we encourage  privacy-by-design software engineering through machine learning paradigms such as Differential  Privacy (DP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' DP-paradigm models and datasets reduce the potential for self-identification from  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Supplementary Material for:  Down the Rabbit Hole: Detecting Online Extremism, Radicalisation, and Politicised Hate Speech  000:11  trained models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' data leakage, such as names or usernames in open-source datasets), as DP-  paradigm models use pseudo-anonymised patterns of groups and hate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='2  Dataset Recommendations  To investigate the roles of radicalisation, we recommend expanding on the dataset annotation  approach by de Gibert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' [15] by creating a conversation-level dataset with public non-hateful  replies to a post for context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Moreover, future benchmark datasets should consider pulling data across  platforms to investigate macro-level radicalisation trends between platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' We note that only  two studies considered anti-Asian sentiment in COVID-related tweets, targeting a seldom explored  topic and demographic [21, 39] worthy of expansion given the ongoing COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Likewise, future datasets should consider the role of indigenous discussions and potential re-  searcher biases given the Anglo-dominant field of ERH research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Given the rise of COVID extrem-  ism [43], far-right movements, and xenophobia in Oceania.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Hence, we recommend geotargeted  datasets to consider the differences for investigating ERH topics, which would demonstrate NZ’s  commitment to our Christchurch Call to Action Summit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Investigating unexplored and minority  groups could also provide imperative insights for social scientists regarding the conversational  dynamics, morphological mapping, and ideological isomorphism from radical minority groups  towards the majority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Likewise, research on vulnerable communities (youth, gender and sexual  minorities, religious, racial, and geographically distant peoples) would aid social media platforms  in both identifying unique radicalising risks, as well as avenues for support and de-escalation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' In  the mental health end, we recommend building on Nouh et al.’s proposed approach of extracting  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' ERH Context Mining (ERH-CM) eight core components for Research, Industry, and Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 0, Article 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Publication date: December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Collective  Researcher  Researcher,  Tasks  Industry,  Govt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Harm  Reduction  Tasks  Introspection &  Reflection  **  Community  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Detection  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Privacy  副  ERH-CM  3  Increased  & Data  Context  Sovereignty  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Self-regulation  Diverse  & Ethics  Datasets  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Quality Assessment  Industry  Collective  & Control  Tasks  (human-in-the-loop)  Researcher,  Industry  Tasks000:12  Govers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  textual, psychological and behavioural features [33], both due to its performance, as well as its  potential for analysing societal factors and ERH roots such as correlations between mental health  issues (isolation, depression etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=') and vulnerability to radicalisation towards violent extremism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  For any counter-extremism or de-radicalisation studies, we recommend work in ethical and legal  guidelines to protect privacy, avoid backlash or inadvertent algorithmic amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  Investigating posts from periods of political, or social crisis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=', COVID health measures, post-  terror attack discourse etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=') could also help identify cases of ERH on mainstream platforms before  they are deplatformed/removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Event-based datasets would provide unique sociological insights  on the role of societal stress and emergencies on the human psyche and online group dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content='  To reduce the cost, variability in inter-annotator agreement, and psychological impact of human  annotation, we recommend unsupervised clustering-based research and propose using synthetic  conversational agents to simulate extremist discourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Simulating online radicalisation in a closed  environment would present a safe, ethical, and non-invasive method to build benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content=' Larochelle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content=' Balcan, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Lin (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' ), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content=', Vancouver, Canada, 17283–17297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
+page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFJT4oBgHgl3EQfkiwH/content/2301.11579v1.pdf'}
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@@ -0,0 +1,1418 @@
+STOCHASTIC APPROACHES: MODELING THE PROBABILITY OF
+ENCOUNTERS BETWEEN H2-MOLECULES AND METALLIC
+ATOMIC CLUSTERS IN A CUBIC BOX
+Maximiliano L. Riddick, Leandro Andrini
+Instituto de Investigaciones Fisicoquimicas Teóricas y Aplicadas
+Departamento de Química, Fac. de Ciencias Exactas (INIFTA/ UNLP-CONICET)
+Departamento de Matemática, Fac. de Ciencias Exactas, UNLP
+La Plata, Argentina
+mriddick@mate.unlp.edu.ar
+Enrique E. Álvarez
+Instituto de Cálculo, Fac. Ciencias Exactas y Naturales, Ciudad Universitaria, Pabellón II, UBA (CABA)
+Departamento de Fisicomatemática, Fac. de Ingeniería, UNLP
+Ciudad Autónoma de Buenos Aires and La Plata, Argentina
+Félix G. Requejo
+Instituto de Investigaciones Fisicoquimicas Teóricas y Aplicadas
+Departamento de Química, Fac. de Ciencias Exactas (INIFTA/ UNLP-CONICET)
+Departamento de Física, Fac. de Ciencias Exactas, UNLP
+La Plata, Argentina
+ABSTRACT
+In recent years the advance of chemical synthesis has made it possible to obtain “naked”clusters of
+different transition metals. It is well known that cluster experiments allow studying the fundamental
+reactive behavior of catalytic materials in an environment that avoids the complications present in
+extended solid-phase research. In physicochemical terms, the question that arises is the chemical
+reduction of metallic clusters could be affected by the presence of H2 molecules, that is, by the
+probability of encounter that these small metal atomic agglomerates can have with these reducing
+species. Therefore, we consider the stochastic movement of N molecules of hydrogen in a cubic
+box containing M metallic atomic clusters in a confined region of the box. We use a Wiener process
+to simulate the stochastic process, with σ given by the Maxwell-Boltzmann relationships, which
+enabled us to obtain an analytical expression for the probability density function. This expression is
+an exact expression, obtained under an original proposal outlined in this work, i.e. obtained from
+considerations of mathematical rebounds. On this basis, we obtained the probability of encounter
+for three different volumes, 0.1
+3, 0.2
+3 and 0.4
+3 m
+3, at three different temperatures in each case,
+293, 373 and 473 K, for 10
+1 ≤ N ≤ 10
+10, comparing the results with those obtained considering
+the distribution of the position as a Truncated Normal Distribution. Finally, we observe that the
+probability is significantly affected by the number N of molecules and by the size of the box, not by
+the temperature.
+Keywords Wiener Process · Probability of encounters · Molecular Collisions · Atomic-Clusters · Mathematical
+Rebounds
+arXiv:2301.13797v1  [cond-mat.mtrl-sci]  10 Jan 2023
+
+M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+1
+Introduction
+In the last two decades there has been an important development in clusters chemistry, and consequently new questions
+arise on the basis of these developments [1, 2, 3, 4, 5, 6, 7]. This interest is due to an atomic clusters containing
+up to a few dozen atoms exhibit features that are very different from the corresponding bulk properties and that can
+depend very sensitively on cluster size [8]. In particular, many of these transition metal clusters are used in the field of
+catalysis [1, 9, 10, 11]. One of the basic principles of catalysis is that when the smaller the metal particles, the larger the
+fraction of the metal atoms that are exposed at surfaces, where they are accessible to reactant molecules and available
+for catalysis [1]. It is well known in chemistry that the encounter between two molecules can give rise to a chemical
+reaction, and from the mathematical aspect there are two fundamental ways to represent these types of situations as
+continuous, represented by differential equations whose variables are concentrations, or as discrete, represented by
+stochastic processes whose variables are the number of molecules [12].
+Without loss of generality, it can be considered that the molecular chemisorption is due to the encounter between
+a molecule and a surface (or a cluster in this case) with the energy necessary for the phenomenon of adsorption to
+occur [13]. Besides, the kinetics of hydrogen chemisorption by neutral gas-phase metal clusters exhibits a complex
+dependence on both cluster size and metal type [14]. For different chemical purposes, for example, in the case of copper
+clusters (Cun) is very important to have control of the chemisorption of hydrogen on these clusters, i.e. the formation of
+Cun-H2 species [15].
+From a reductionist point of view, the molecular chemisorption is a problem of encounter between bodies: metal clusters
+and reactant molecules. In our first approximation (mathematical reduction) we will consider the problem as a problem
+of encounter or collisions between bodies. We are interested in proposing this strategy because we are focused to answer
+what is the probability of meeting between N hydrogen molecules (N-H2) and a fixed M metallic clusters (M-Men),
+for a given time t, where the H2 move freely in a bounded volume V of R3-space. Under this assumption, we are going
+to consider H2-molecules and Men-clusters as rigid spheres of radii r1 and r2, respectively. Then, it is considered that
+there will be a collision whenever the center-to-center distance between an H2-molecule and a Men-cluster is equal to
+r12 = r1 + r2 [16]. Also, in this context we propose the H2-molecules follow a Brownian motion, namely: (a) it has
+continuous trajectories (sample paths) and (b) the increments of the paths in disjoint time intervals are independent zero
+mean Gaussian random variables with variance proportional to the duration of the time interval [17].
+The pioneering work of T.D. Gillespie [16, 18, 19] have given rise to a large number of works that are proposed different
+algorithms for the calculation for numerically simulating the time evolution of a well-stirred chemically reacting system,
+although despite recent major improvements in the efficiency of the stochastic simulation algorithm, its drawback
+remains the great amount of computer time that is often required to simulate a desired amount of system time [20]. While
+our method is a simple reduction to collisions of molecules, allows to calculate the probability of encounter (scheduled
+in R) for a large number of molecules (≈ 106) and clusters (≈ 1020) with advantages regarding the cost of calculation,
+and the effects of first approximation can provide statistical support to the design of experiments. This calculation
+is possible using a stochastic model (Wiener process) in the context of considerations from the Maxwell-Boltzmann
+theory.
+2
+A first theoretical approaching
+As we announced in the introduction, we will assume that hydrogen molecules have a random movement, whence
+let H(t) = (X(t), Y (t), Z(t)) the random variable which specify the space point where the H2 hydrogen molecule
+is at time t. Trivially, H(t) depends on an initially point H(0) = (x0, y0, z0). Thus, when the initial starting point is
+undefined, H(t) = H(t, x0, y0, z0). Our interest is in how probably is that the distance between H(t) and a fixed point
+(a, b, c) is smaller than ϵ. The fixed point (a, b, c) are the coordinates for Men.
+Let’s consider the random variable D(t) as the variable that measures the distance between H(t) and the fixed point
+(a, b, c). Following the classical Pythagorean relationship, D(t) =
+�
+(X(t) − a)2 + (Y (t) − b)2 + (Z(t) − c)2, and
+in general D(t) = D(t, x0, y0, z0, a, b, c).
+Now, given a time window [0, τ], let
+Rτ :=
+�
+�
+�
+1
+if min
+t∈[0,τ) D(t) ≤ ϵ,
+0
+otherwise.
+(1)
+So, for a fixed t0 > 0, we define G(t0) = P(D(t0) ≤ ϵ) =
+� ϵ
+0 D(s)ds. Then, P(Rτ = 1) =
+� τ
+0 G(t)dt.
+2
+
+M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+Thus, given τ > 0, Rτ depends only on the initial values (x0, y0, z0, a, b, c). Now, if we have M-Men, the probability
+that the H2 molecule does not meet with any of the clusters is P(Rτ1 = 0, Rτ2 = 0, ..., RτM = 0) = pA, where
+Rτi, i ∈ {1, ..., M}, follows the definition given in the eq. 1.
+If N-H2 molecules are in the environment, let Aj the event “the j-th hydrogen molecule meet with a metallic cluster”.
+Under random starting points, we are interested in P(AC
+1 ∩ AC
+2 ∩ ... ∩ AC
+N) = pN
+A according to the independence
+among the hydrogen molecules.
+2.1
+Adaptation to our context
+Next, we proceed to realize the analysis according to the Brownian Motion Theory [17], in which the movement of
+the particle is independent among different axis, and we are going to assume that it follows a Wiener process [21, 22].
+Then,
+X(t) = x0 + WX(t)
+Y (t) = y0 + WY (t)
+Z(t) = z0 + WZ(t)
+And we will say that WX(t), WY (t) and WZ(t) are following a Wiener processes with σ =
+�
+kbT
+m , where kb is the
+Boltzmann’s constant, T is the absolute temperature in Kelvin (K) and m is the H2’s mass in kg. That is, we are
+imposing a physical behavior that obeys Maxwell-Boltzmann’s considerations. According this:
+X(t) ∼ N(x0, σ2t)
+Y (t) ∼ N(y0, σ2t)
+Z(t) ∼ N(z0, σ2t)
+With density function fX(x, t|x0), fY (y, t|y0) and fZ(z, t|z0), respectively. Under these assumptions:
+fX(x, t|x0) =
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�x − x0
+σ
+√
+t
+�2�
+fY (y, t|y0) =
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�y − y0
+σ
+√
+t
+�2�
+fZ(z, t|z0) =
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�z − z0
+σ
+√
+t
+�2�
+2.1.1
+Unbounded conditions
+Under unbounded conditions, as it is well known, the density of the particle position in the space for a fixed t follows
+the expression:
+fXY Z(x, y, z, t|x0, y0, z0) = fX(x, t|x0) · fY (y, t|y0) · fZ(z, t|z0) =
+=
+1
+(
+�
+2πσ2t)3 exp
+�
+−1
+2
+�(x − x0)2 + (y − y0)2 + (z − z0)2
+σ2t
+��
+This function is continuous in the variables x, y, z, t, then is also integrable in a measurable context. Because of this
+fact, Fubini’s theorem is aplicable. Now, calling ν =
+�
+(x−x0)2+(y−y0)2+(z−z0)2
+2σ2
+, and integrating over the variable t by
+substitution, results:
+3
+
+M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+fXY Z(x, y, z, τ|x0, y0, z0) =
+1
+−ν(
+√
+2πσ2)3
+� τ
+0
+−ν
+(
+√
+t)3 exp
+�
+−
+� ν
+√
+t
+�2�
+dt
+=
+1
+ν(
+√
+2πσ2)3
+� ∞
+√ ν
+τ
+e−u2du
+Remembering that the erfc function [23] is defined by:
+erfc(z) =
+2
+√π
+� ∞
+z
+e−t2dt
+we conclude:
+fXY Z(x, y, z, τ|x0, y0, z0) =
+1
+2πν(
+√
+2σ2)3 erfc
+��ν
+τ
+�
+From the physical-experimental perspective that the problem is lays out, the unbounded system lacks interest, so we
+will proceed to study the case of the bounded system.
+2.1.2
+Bounded conditions
+We assume that the experiment takes place into a cubic recipe centered at the origin. This implies that X(t), Y (t) and
+Z(t) ∈ [−L; L], for a fixed volume V = L3 in R3-space.
+In a similar issue the traditional way of approaching is by “truncation" [24, 25]. A drawback of this approach is the
+fact that the truncation does not represent precisely the reflection on the boundaries. An illustrative and motivational
+argument is given by the following example: suppose a random walk of N = 4 steps, with starting point at the origin.
+Then, the walker moves 1 step at right or left (with equal probability) at each step. Then, after four steps, the resultant
+probabilities of the walker position are:
+0; with probability 3/8,
+−2 or 2; with probability 2/8,
+−4 or 4; with probability 1/16.
+The probability values (under truncation) in the closed interval [−2, 2] for the values (−2, −1, 0, 1, 2) are, respectively:
+(2/7, 0, 3/7, 0, 2/7)
+With fixed boundaries, considering reflections at [−2, 2], we can construct the following Markov transition matrix P:
+P =
+0
+1
+0
+0
+0
+1/2
+0
+1/2
+0
+0
+0
+1/2
+0
+1/2
+0
+0
+0
+1/2
+0
+1/2
+0
+0
+0
+1
+0
+At the fourth step, after some algebra, we obtain the respectively mass point probability for the position of the walker.
+This is provided by the stochastic vector
+(1/4, 0, 1/2, 0, 1/4)
+(given by the third file of P 4, i.e.: with starting point at the origin). At this point, is clearly the difference between
+truncation and “rebounds" (considering reflection on the boundary).
+We must modify the density of the position H(t) according to the particle rebounds (see Fig. 1). It is important to note
+that the rebounds indicated in the figure in gray colour do not correspond to the physical rebounds of the particles in the
+cubic box, but to the contributions of the displaced distribution considering an infinite behavior.
+4
+
+M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+Figure 1: In red colour an arbitrary normal distribution, N(0, σ). We observe in gray colour the folding of the normal
+distribution at the edge of the box. We could see that A + B = 2L.
+Inside the box, the derived density fB according to the variable X(t) ∼ N(x0, σ2t) with density function fX of the
+particle position (for each dimension, see Fig. 1) follows the expression:
+fB(x) = [fX(x) + fB+(x) + fB−(x)] × I[−L;L](x)
+where:
+fB+(x) = f(x + 2A) + f(x + 2A + 2B) + f(x + 2A + 2B + 2A) + ... =
+= f(x + 2(L − x)) + f(x + 2(L − x) + 2(x − (−L)) + ...
+= f(−x + 2L) + f(x + 4L) + f(−x + 6L) + ...
+=
+∞
+�
+k=1
+f((−1)kx + 2kL)
+=
+∞
+�
+k=1
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�((−1)kx + 2kL) − x0
+σ
+√
+t
+�2�
+and
+fB−(x) = f(x − 2B) + f(x − 2B − 2A) + f(x − 2B − 2A − 2B) + ... =
+= f(x − 2(x − (−L))) + f(x − 2(x − (−L)) − 2(L − x)) + ...
+= f(−x − 2L) + f(x − 4L) + f(−x − 6L) + ...
+=
+∞
+�
+k=1
+f((−1)kx − 2kL)
+=
+∞
+�
+k=1
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�((−1)kx − 2kL) − x0
+σ
+√
+t
+�2�
+5
+
+B
+B'
+B
+0
+XM.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+The proof that fB is a density function is straightforward its definition. Trivially, fB > 0, and by construction:
+� ∞
+−∞
+fB(t)dt =
+� L
+−L
+fB(t)dt =
+� ∞
+−∞
+fX(t)dt = 1
+For practical purposes, we now try to find an upper bound to this expression. Looking at in the model proposed, the
+next constraint is straightforward |(−1)kx − x0| ≤ 2L.
+Following these constraints:
+fB+(x) =
+∞
+�
+k=1
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�((−1)kx + 2kL) − x0
+σ
+√
+t
+�2�
+≤
+∞
+�
+k=1
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�−2L + 2kL
+σ
+√
+t
+�2�
+=
+∞
+�
+k=1
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�2(k − 1)L
+σ
+√
+t
+�2�
+=
+∞
+�
+k=0
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+� 2kL
+σ
+√
+t
+�2�
+=
+1
+√
+2πσ2t
+∞
+�
+k=0
+exp
+�
+−1
+2
+�4L2
+σ2t
+��k2
+It is known that
+∞
+�
+k=0
+rk2 = 1
+2 + 1
+2ΘE[3, 0, r] where ΘE is the Jacobi theta elliptic function [23]. So:
+fB+(x) ≤
+1
+√
+2πσ2t
+∞
+�
+k=0
+exp
+�
+−1
+2
+�4L2
+σ2t
+��k2
+=
+1
+√
+2πσ2t
+�1
+2 + 1
+2ΘE
+�
+3, 0, exp
+�
+−1
+2
+�4L2
+σ2t
+����
+and
+fB−(x) =
+∞
+�
+k=1
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�((−1)kx − 2kL) − x0
+σ
+√
+t
+�2�
+≤
+∞
+�
+k=1
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�−2L − 2kL
+σ
+√
+t
+�2�
+=
+∞
+�
+k=1
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�−2(k + 1)L
+σ
+√
+t
+�2�
+=
+∞
+�
+k=2
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�−2kL
+σ
+√
+t
+�2�
+=
+1
+√
+2πσ2t
+� ∞
+�
+k=0
+exp
+�
+−1
+2
+�4L2
+σ2t
+��k2
+− 1 − exp
+�
+−1
+2
+�4L2
+σ2t
+���
+=
+1
+√
+2πσ2t
+�1
+2 + 1
+2ΘE
+�
+3, 0, exp
+�
+−1
+2
+�4L2
+σ2t
+���
+− 1 − exp
+�
+−1
+2
+�4L2
+σ2t
+���
+6
+
+M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+Then,
+fB+(x) + fB−(x) ≤
+1
+√
+2πσ2t
+�
+ΘE
+�
+3, 0, exp
+�
+−1
+2
+�4L2
+σ2t
+���
+− exp
+�
+−1
+2
+�4L2
+σ2t
+���
+= CB
+For each x ∈ [−L, L], fB(x) ≤ fX(x) + CB. Besides CB does not depends on x. Consequently, we have a maximum
+for the density fB which is equal to f(x0) + CB.
+Calling PB = (f(x0) + CB).2ϵ, we can conclude that:
+P(X(t) ∈ (a0 − ϵ, a0 + ϵ)) ≤ PB, for any a0.
+Analogous, CB is the same for the variables Y (t) and Z(t), and we know that f(x0) = f(y0) = f(z0). Then, the
+same result is available for the variables Y (t) and Z(t). According the bounded CB, it is straightforward the uniform
+convergence of the series fB+ and fB− (by the M Weierstrass criteria). An important fact to remark is that PB is not
+even a probability, but in the case in we are interested, we know that is a real number bigger than the probability desired,
+and then, under certain conditions, we can work with it.
+For practical purposes, the error through the CB implementation can be minimized, since the first S terms are available,
+and the tail can be compared with
+S−1
+�
+k=0
+rk2 ≤
+∞
+�
+k=0
+rk2 =
+S−1
+�
+k=0
+rk2 +
+∞
+�
+k=S
+rk2
+And,
+∞
+�
+k=S
+rk2 =
+∞
+�
+k=0
+r(k+S)2 =
+∞
+�
+k=0
+rk2+2kS+S2 = rS2
+∞
+�
+k=0
+rk2r2kS ≤ rS2
+∞
+�
+k=0
+rk2
+Then,
+∞
+�
+k=S
+rk2 ≤ rS2 �1
+2 + 1
+2ΘE[3, 0, r]
+�
+Controlling the value of S controls the value of the error made by truncating the sum. As we said, CB does not depend
+on x, thus, the desired probability can be estimated with any degree of accuracy, according the computational cost
+necessary to this development.
+Taking into consideration the Brownian Motion Theory, in the time lapse of 1 second, the particle position under
+unbounded conditions follows a N(x0, σ2) distribution. To discretize the problem, if we partitioned the time axis of τ
+seconds in τ intervals of 1 second each one, then:
+P(H(t) ∈ Bϵ(a, b, c)) ≤ P(H(t) ∈ Qϵ(a, b, c))
+where Qϵ(a, b, c) denotes the cube centered in (a, b, c) with side size 2 × ϵ. And, considering the independence
+between X(t), Y (t) and Z(t), with X(t) ∈ (a − ϵ, a + ϵ), Y (t) ∈ (b − ϵ, b + ϵ) and Z(t) ∈ (c − ϵ, c + ϵ),
+P(H(t) ∈ Qϵ(a, b, c)) = P(H ∈ Qϵ) is
+P(H ∈ Qϵ) = P(X(t)) × P(Y (t)) × P(Z(t)) ≤ PB × PB × PB = P 3
+B
+For each second τj for τj ∈ {1 : τ}, P(H(t) ∈ Qϵ(a, b, c)) ≤ P 3
+B. Then, under the Wiener process formulation,
+H(τj) ⊥ H(τk|τj) if j ̸= k, j ≤ k.
+7
+
+M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+P(H(τj) ∈ Qϵ(a, b, c)) ≤ P 3
+B, ∀τj ∈ {1 : τ}. Calling F :=“# of τj ∈ {1 : T} in which H(τj) ∈ Qϵ(a, b, c)", we are
+interesting in the event F = 0.
+According its nature, F is a Binomial random variable B(τ, P 3
+B). Consequently, the non-collision probability is
+pNC = P(F = 0) ≤ (1 − P 3
+B)τ. At this point, we only can conclude that the probability of the encounter between
+a hydrogen molecule and a Men cluster in a time τ is less than p. We proceed to analyze what happens when the
+number of hydrogen molecules and metallic clusters increase. We emphasize that the H2 molecules have a random
+movement while the clusters are confined in a fixed region of space. Since p is the probability that a random hydrogen
+molecule meets in the cube Qϵ in which a Men cluster is, the most unfavorable case with M clusters is when there is no
+intersection among the cubes that contain it. In this case:
+pA = P(Rτ1 = 0, ..., RτM = 0)
+= 1 −
+M
+�
+i=1
+P(Rτi = 1)
+≥ 1 −
+� M
+�
+i=1
+P(Rτi = 1)
+�
+= 1 − M × p
+In view of this analysis, we can conclude that the non-collision probability is higher than pNC.
+In regular conditions, when this approach is used, the values of pNC and N outcomes into a several numerical instability.
+In this case, the small value of pNC and the large value of N place us in conditions to use the Poisson approach to
+the Binomial distribution (with parameter λ = N × p). Then, P(X = 0) ≈ exp(−λ). Even in the cases when the
+probability is still unavailable, the expected number of collisions is presented according a time window, and then we can
+estimate the probability of collisions in a time window T using the relationship between the Poisson and Exponential
+distributions[26].
+Next, we present the results of the analysis whit different box dimensions (in meters) and number of hydrogen molecules
+(N), according to M = 1.9 × 10
+20 Cu20-clusters [27], where the Cu20-clusters have been considered as spheres.
+3
+Results and analysis
+3.1
+Obtaining non-collision probability values
+The situation we consider is approximately a “realistic”situation, with M = 1.9 × 10
+20 Cu20-clusters in a cubic box
+according to the standard dimensions of reaction chambers (0.1
+3, 0.2
+3 and 0.4
+3 m
+3), and a variable N-H2-molecules
+“contamination”(10
+1 ≤ N ≤ 10
+10). It worked with three temperatures, T, 293, 373 and 473 K. The choice of T is
+arbitrary, conditioned by the possible reaction temperatures [28].
+In Fig. 2 we observe the results obtained for the simulations, considering the maximum sum. That is, take S = 10
+6,
+perform the sum, and add the maximum level for the error. Clearly, a greater probability of non-collision, pNC, is
+observed depending on the increase in volume.
+For a detailed study, we proceed as follows: we model the data obtained through a non-linear graphic fitting considering
+a Boltzmann decrease function, g(x) = A2 +
+A1−A2
+1+exp
+� x−x0
+dx
+� (see Fig. 3). In the Appendix A.2 we show the statistical
+results for each parameter in each data fitting.
+Under these considerations, we can calculate the critical value (criticality)[29, 30] of hydrogen molecules, that is “what
+is the value of N for which the non-collision probability is greater than 1
+2”, i.e. the value of the exponent for which
+1
+2 < pNC.
+It should be clarified that, in the strict physical sense, there is no abrupt phase transition to consider “criticality”. As
+we assumed in the introduction, we consider that there is a chemical reaction if there is an encounter between two
+molecules, and under this assumption we are considering as critical the level of presence of hydrogen for a chemical
+reaction to occur. In any case, it can be demonstrated that there is an “abrupt”transition behavior, for a well defined
+interval in the number of molecules. In Fig. 3 we can observe this behavior.
+8
+
+M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+Figure 2: Results for the non-collision probability, pNC, vs. ln(N) for L = 0.05m (V1), L = 0.1m (V2) and L = 0.2m
+(V3), at T = 293 K (blue square), 373 K (black star) and 493 K (red triangle).
+Figure 3: Data (blue square) modeling using a non-linear Boltzmann decrease function (green line).
+9
+
+V1
+V2
+1.0 -
+4
+GD
+△
+0.8
+0
+★
+口
+293 K
+0.6
+V
+373 K
+0
+473 K
+文
+0.4 -
+★
+0.2 -
+0.0-
+支立文支安
+123456789101234567891012345678910
+Ln(N)293 K
+1.0 -
+Boltzmann Fit 293 K
+0.8
+0.6
+0.4 
+0.2
+0.0 
+0
+3
+4
+5
+6
+7
+8
+10
+Ln(N)M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+Table 1: Critical values obtained from the decrease model for each box and each temperature, for mathematical
+robounds.
+L [m]
+293 K
+373 K
+473 K
+0.05
+3.25
+3.16
+3.09
+0.10
+4.97
+4.86
+4.72
+0.20
+5.96
+5.96
+5.96
+Figure 4: Results for the non-collision probability, pNC, vs. ln(N) for L = 0.05m (V1), at T = 293 K, 373 K and 493
+K. Comparison between models:“xxx Trunc”correspond to the truncated normal model and “xxx K”to the mathematical
+rebound model.
+In Table 1 we can see the critical values obtained from the decrease model for each box and each temperature. For the
+smallest volumes, V1 and V2, it is observed that the critical value of N depends more strongly on the temperature than
+in the case of the larger volume (V3). Although it is remarkable the fact of dependence with the size of the box, it can
+be seen directly from Fig. 2. In this way, and under these simplified assumptions, we can obtain control of contaminant
+molecules in relation to the volume and temperature parameters. Linear behavior is evident from the values obtained
+(Table 1, N vs. temperature). Moreover, as the volume increases the slope increases from negative values to null value.
+4
+Conclusion
+By way of conclusion, it can be indicated that considering a Wiener stochastic process, for thermodynamic-statistical
+movements of a gas confined in a box, and considering mathematical rebounds bounded by the physical-geometric
+contour of the problem, the analytical expression could be obtained for the probability density function of encounters
+between two differentiated species of molecules (one of the species fixed in the box -solid or liquid- and the other
+species is a gas whose molecules move stochastically). In addition, the function obtained can be calculated numerically
+or can be bounded. The bounded process allows to reduce the computational cost, and to limit the error from cutting the
+Table 2: Critical values obtained from the decrease model for each box and each temperature, for truncated normal
+model.
+L [m]
+293 K
+373 K
+473 K
+0.05
+3.27
+3.18
+3.12
+0.10
+5.01
+4.91
+4.76
+0.20
+6.00
+5.98
+5.96
+10
+
+293 Trunc
+☆373 Trunc
+473 Trunc
+口
+293 K
+373 K
+473 K
+1.0 0
+0
+★
+Non-collision probability
+0.8
+8
+0.6 -
+0.4
+0.2 +
+0.0+
+口口OO口
+1 2 3 4 5 6 7 8 9101 2 3 4 5 6 7 8 9101 2 3 4 5 6 7 8 910
+Ln(N)M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+Figure 5: Results for the non-collision probability, pNC, vs. ln(N) for L = 0.1m (V1), at T = 293 K, 373 K and 493 K.
+Comparison between models:“xxx Trunc”correspond to the truncated normal model and “xxx K”to the mathematical
+rebound model.
+Figure 6: Results for the non-collision probability, pNC, vs. ln(N) for L = 0.2m (V1), at T = 293 K, 373 K and 493 K.
+Comparison between models:“xxx Trunc”correspond to the truncated normal model and “xxx K”to the mathematical
+rebound model.
+11
+
+293 Trunc
+☆373 Trunc
+→ 473 Trunc
+口
+293
+★373
+473
+★★★
+口
+Non-collision probability
+0.8.
+0.6 -
+0.4 
+0.2 +
+0.0 -
+OOOOG
+1 2 3 4 5 6 7 8 9101 2 3 4 5 6 7 8 9101 2 3 4 5 6 7 8 910
+Ln(N)293 Trunc
+☆373 Trunc
+→ 473 Trunc
+口
+293 K
+373 K
+473K
+1.0-
+Non-collision probability
+0.8
+0.6
+0.4
+0.2 
+0.0-
+123456789101234567891012345678910
+Ln(N)M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+sum in a finite number. In particular, there is an error control that can be made, and it is possible to refine the process
+according to the precision required.
+From the physical-chemical point of view, it is observed that both the number of gas molecules and the dimensions of
+the box affect the probability of encounter. For this model, temperature is a parameter that has a lower incidence on the
+values of the probability of encounter. At this point some considerations have to be made. The first is that in a strict
+sense a chemical reaction is more than the encounter of two chemical entities. The second is the exceptional chemical
+nature of metal clusters, which make them highly reactive. Despite the simplicity of the model we are proposing, this
+model can account in an experiment design about the collision probability between two chemical entities (and this
+collision can lead to a chemical reaction).
+From the point of view of computation, it is a system that requires less computational cost (time + memory) than the
+algorithmic systems developed for this type of problems, so it contributes as a test method in the design of experiments.
+The comparison with an established method (truncated normal model) was optimal. In the method of mathematical
+rebounds the number of molecules needed for a reaction is less than the number obtained by the truncated normal
+model. This is an advantage when strict contamination control is needed.
+On the other hand, in terms of obtaining the density function, mathematical results can be generalized for volumes of
+rectangular prisms of uneven sides. In addition, it remains to calculate the first and second order moments of the density
+function obtained, work that exceeded the purposes of present communication.
+Acknowledgments
+This was was supported in part by PICT-2019-0784, PICT-2017-3944, PICT-2017-1220, PICT-2017-3150 (PICT,
+Agencia Nacional de Promoción de la Investigación, el Desarrollo Tecnológico y la Innovación) and PPID-I231 (PPID,
+Universiad Nacional de La Plata).
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+1992.
+Appendix
+A.1
+Errors in the Boltzmann model for the probability calculated according to mathematical rebounds.
+Program used: Origin 9.1
+In all cases, number of points is 10, and degrees of freedon is 6.
+13
+
+M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+L=0.05 m, T = 293 K
+Parameter
+Value
+Standard Error
+A1
+0.991
+0.009
+A2
+-0.0060
+0.0008
+x0
+4.98
+0.02
+dx
+0.30
+0.03
+Reduced Chi-Sqr 2.66387 × 10
+−4
+Residual Sum of Squares: 0.0016
+Adj. R-Square: 0.99888
+L=0.05 m, T = 373 K
+Parameter
+Value
+Standard Error
+A1
+0.981
+0.006
+A2
+-0.005
+0.003
+x0
+3.16
+0.01
+dx
+0.22
+0.02
+Reduced Chi-Sqr 7.93743 × 10
+−5
+Residual Sum of Squares: 4.76246 × 10
+−4
+Adj. R-Square: 0.99957
+L=0.05 m, T = 473 K
+Parameter
+Value
+Standard Error
+A1
+0.976
+0.008
+A2
+-0.0011
+0.0009
+x0
+3.10
+0.02
+dx
+0.21
+0.03
+Reduced Chi-Sqr 1.30175 × 10
+−4
+Residual Sum of Squares: 7.8105 × 10
+−4
+Adj. R-Square: 0.99927
+L=0.1 m, T = 293 K
+Parameter
+Value
+Standard Error
+A1
+0.991
+0.009
+A2
+-0.0060
+0.0011
+x0
+4.98
+0.02
+dx
+0.30
+0.03
+Reduced Chi-Sqr 2.66387 × 10
+−4
+Residual Sum of Squares: 0.0016
+Adj. R-Square: 0.99888
+L=0.1 m, T = 373 K
+Parameter
+Value
+Standard Error
+A1
+0.994
+0.008
+A2
+-0.006
+0.002
+x0
+4.88
+0.02
+dx
+0.33
+0.02
+Reduced Chi-Sqr 1.88888 × 10
+−4
+Residual Sum of Squares: 0.00113
+Adj. R-Square: 0.9992
+14
+
+M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+L=0.1 m, T = 473 K
+Parameter
+Value
+Standard Error
+A1
+0.996
+0.005
+A2
+-0.0043
+0.0019
+x0
+4.73
+0.01
+dx
+0.33
+0.01
+Reduced Chi-Sqr 7.76599 × 10
+−5
+Residual Sum of Squares: 4.65959 × 10
+−4
+Adj. R-Square: 0.99967
+L=0.2 m, T = 293 K
+Parameter
+Value
+Standard Error
+A1
+0.993
+0.003
+A2
+-0.0082
+0.0025
+x0
+5.97
+0.02
+dx
+0.31
+0.03
+Reduced Chi-Sqr 2.64593 × 10
+−4
+Residual Sum of Squares: 0.00159
+Adj. R-Square: 0.9989
+L=0.2 m, T = 373 K
+Parameter
+Value
+Standard Error
+A1
+0.993
+0.008
+A2
+-0.0082
+0.0025
+x0
+5.97
+0.02
+dx
+0.31
+0.03
+Reduced Chi-Sqr 2.64587 × 10
+−4
+Residual Sum of Squares: 0.00159
+Adj. R-Square: 0.9989
+L=0.2 m, T = 473 K
+Parameter
+Value
+Standard Error
+A1
+0.993
+0.008
+A2
+-0.0082
+0.0025
+x0
+5.97
+0.02
+dx
+0.31
+0.03
+Reduced Chi-Sqr 2.64587 × 10
+−4
+Residual Sum of Squares: 0.00159
+Adj. R-Square: 0.9989
+A.2
+Errors in the Boltzmann model for the probability calculated according to the truncated normal model.
+L=0.05 m, T = 293 K
+Parameter
+Value
+Standard Error
+A1
+0.982
+0.006
+A2
+-0.0003
+0.0001
+x0
+3.29
+0.01
+dx
+0.24
+0.01
+Reduced Chi-Sqr 6.6611 × 10
+−5
+Residual Sum of Squares: 0.000399
+15
+
+M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+Adj. R-Square: 0.99965
+L=0.05 m, T = 373 K
+Parameter
+Value
+Standard Error
+A1
+0.982
+0.006
+A2
+-0.004
+0.003
+x0
+3.19
+0.02
+dx
+0.22
+0.02
+Reduced Chi-Sqr 6.69947 × 10
+−5
+Residual Sum of Squares: 4.0196 × 10
+−4
+Adj. R-Square: 0.99960
+L=0.05 m, T = 473 K
+Parameter
+Value
+Standard Error
+A1
+0.982
+0.005
+A2
+-0.0004
+0.0003
+x0
+3.19
+0.01
+dx
+0.22
+0.02
+Reduced Chi-Sqr 6.69508 × 10
+−4
+Residual Sum of Squares: 4.01705 × 10
+−4
+Adj. R-Square: 0.99964
+L=0.1 m, T = 293 K
+Parameter
+Value
+Standard Error
+A1
+0.991
+0.003
+A2
+-0.0022
+0.0009
+x0
+5.02
+0.07
+dx
+0.21
+0.03
+Reduced Chi-Sqr 5.5392 × 10
+−5
+Residual Sum of Squares: 0.00033
+Adj. R-Square: 0.99977
+L=0.1 m, T = 373 K
+Parameter
+Value
+Standard Error
+A1
+0.992
+0.006
+A2
+-0.0047
+0.0025
+x0
+4.92
+0.01
+dx
+0.29
+0.02
+Reduced Chi-Sqr 1.20178 × 10
+−4
+Residual Sum of Squares: 0.000721
+Adj. R-Square: 0.9995
+L=0.1 m, T = 473 K
+Parameter
+Value
+Standard Error
+A1
+0.995
+0.005
+A2
+-0.0045
+0.0025
+x0
+4.77
+0.05
+dx
+0.33
+0.01
+Reduced Chi-Sqr 8.70051 × 10
+−5
+Residual Sum of Squares: 5.2203 × 10
+−4
+16
+
+M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+Adj. R-Square: 0.99963
+L=0.2 m, T = 293 K
+Parameter
+Value
+Standard Error
+A1
+0.992
+0.003
+A2
+-0.0078
+0.0065
+x0
+6.01
+0.02
+dx
+0.29
+0.03
+Reduced Chi-Sqr 2.75184 × 10
+−4
+Residual Sum of Squares: 0.00165
+Adj. R-Square: 0.99885
+L=0.2 m, T = 373 K
+Parameter
+Value
+Standard Error
+A1
+0.990
+0.007
+A2
+-0.0068
+0.0075
+x0
+5.99
+0.02
+dx
+0.28
+0.03
+Reduced Chi-Sqr 2.07471 × 10
+−4
+Residual Sum of Squares: 0.00124
+Adj. R-Square: 0.99913
+L=0.2 m, T = 473 K
+Parameter
+Value
+Standard Error
+A1
+0.993
+0.008
+A2
+-0.0082
+0.0025
+x0
+5.97
+0.02
+dx
+0.31
+0.03
+Reduced Chi-Sqr 2.64587 × 10
+−4
+Residual Sum of Squares: 0.00159
+Adj. R-Square: 0.9989
+17
+
diff --git a/9tAzT4oBgHgl3EQfg_wg/content/tmp_files/2301.01476v1.pdf.txt b/9tAzT4oBgHgl3EQfg_wg/content/tmp_files/2301.01476v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..c1f934e400af53a1c4757094c76c8bd889c34594
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@@ -0,0 +1,1696 @@
+Lessons Learned Applying Deep Learning Approaches to 
+Forecasting Complex Seasonal Behavior 
+ 
+Andrew T. Karl1, James Wisnowski1, Lambros Petropoulos2 
+ 
+1Adsurgo LLC, Pensacola, FL 
+2USAA, San Antonio, TX 
+ 
+ 
+ 
+Abstract 
+Deep learning methods have gained popularity in recent years through the media and the 
+relative ease of implementation through open source packages such as Keras. We 
+investigate the applicability of popular recurrent neural networks in forecasting call 
+center volumes at a large financial services company. These series are highly complex 
+with seasonal patterns - between hours of the day, day of the week, and time of the year - 
+in addition to autocorrelation between individual observations. Though we investigate the 
+financial services industry, the recommendations for modeling cyclical nonlinear 
+behavior generalize across all sectors. We explore the optimization of parameter settings 
+and convergence criteria for Elman (simple), Long Short-Term Memory (LTSM), and 
+Gated Recurrent Unit (GRU) RNNs from a practical point of view. A designed 
+experiment using actual call center data across many different “skills” (income call 
+streams) compares performance measured by validation error rates of the best observed 
+RNN configurations against other modern and classical forecasting techniques. We 
+summarize the utility of and considerations required for using deep learning methods in 
+forecasting. 
+Key Words: ARIMA, Time Series 
+ 
+ 
+1. Introduction 
+ 
+Member contact call centers receive fluctuating call volumes depending on the day of the 
+week, the time of day, holidays, business conditions, and other factors. It is important for 
+call center managers to have accurate predictions of future call volumes in order to manage 
+staffing levels efficiently. The call center arrival process has been well documented and 
+explored in the literature (Gans, Koole, & Mandelbaum, 2003). In the application presented 
+here, there are several different “skills” (or “splits”) to which an incoming call may be 
+routed – depending on the capabilities of the call center agents – and an arrival volume 
+forecast is required for each skill in the short term for day-ahead or week-ahead predictions. 
+ 
+The weekly seasonality found in call arrivals can be modeled effectively through a variety 
+of methods to include Winter’s Seasonal Smoothing (Winters, 1960) or Autoregressive 
+Integrated Moving Average (Box & Jenkins, 1970). Some accessible references for many 
+of these concepts aimed at the practitioner are well documented in the literature (e.g. 
+Bisgaard & Kulachi, (2007, 2008)) while recommended texts are Bisgaard & Kulachi 
+(2011) and Montgomery, Jennings & Kulachi (2015). 
+ 
+Aiming to improve on these classic methods, “doubly stochastic” linear mixed models 
+(Aldor-Noiman, Feigin, & Mandelbaum, 2009) have effectively modeled additional 
+
+complexities as outlined in a recent review paper from Ibrahim, Ye, L'Ecuyer, & Shen 
+(2016). Similarly, Recurrent Neural Networks (RNNs) have been recommended as deep 
+learning approaches to forecast call volume for a wireless network (Bianchi et al., 2017) in 
+addition to numerous other applications including ride volumes with Uber (Zhu & Laptev, 
+2017). While the doubly stochastic and RNN approaches to predicting call volumes offer 
+greater flexibility in modeling complex arrival behavior by incorporating exogenous 
+variables, this flexibility comes at the cost of greater computational and programming 
+complexity (as well as greater prediction variance). This paper explores practical aspects 
+of managing that complexity for these models, applies the models to actual call volumes 
+recorded by a large financial services company, and compares the prediction capability to 
+that of the more traditional Winters smoothing and ARIMA models.  
+ 
+First, we modify computational aspects of the doubly stochastic approach proposed by 
+Aldor-Noiman, Feigin, & Mandelbaum (2009) to improve call center forecasting 
+performance. Doubly stochastic implies a two-level randomization where not only are call 
+arrivals random variables, but also the call arrival mean parameter. Forecasts are produced 
+by taking advantage of the unique correlation structure for each split while accounting for 
+trend, seasonality, cyclical behavior, and serial dependence. The doubly stochastic model 
+is more complex than ordinary regression as it accounts for both inter- and intra-day 
+correlation. We suggest modifications to the originally proposed approach that lead to more 
+stable convergence and more flexible behavior when many splits need to be fit. 
+ 
+Secondly, we consider how RNNs may be used to model incoming call volume. Whereas 
+“traditional” densely connected, feedforward neural networks process each data point 
+independently, RNNs process sequences according to temporal ordering and retain 
+information from previous points in the sequence. As it processes points within sequences, 
+the RNN maintains states that contain information about what it has seen previously in the 
+sequence (Chollet & Allaire, 2018). This intra-sequence memory is useful in time series 
+applications to autocorrelated data. In the context of call center volumes, these sequences 
+could be constructed to correspond to individual days of observations over a fixed number 
+of (e.g. 30 minute) periods. Bianchi et al. (2017) consider three different RNN architectures 
+to model incoming call volume over a mobile phone network: Elman Recurrent Neural 
+Networks (ERNN) (Elman, 1990), Gated Recurrent Units (GRU) (Cho, et al., 2014), and 
+Long Short Term Memory (LSTM) (Hochrieter & Schmidhuber, 1997), listed in order of 
+increasing complexity.  
+These three RNNs, along with the dense neural network, are now available via the R Keras 
+package (Allaire & Chollet, 2018). Once code has been written for one of the RNNs, the 
+user can switch between the other two by toggling a single option (and, after data 
+reformatting, switch to a dense network). This offers the potential – via a designed 
+experiment – to produce a pragmatic answer to the question of which type of (R)NN 
+provides the best fit to the process at hand. Whereas Bianchi, Maiorino, Kampffmeyer, 
+Rizzi, & Jenssen (2017) created their experimental design by randomly generating points 
+within the design space and then selected the design that lead to the minimum error rate, 
+we create a full factorial design (treating all factors as categorical to allow arbitrary shape 
+in the otherise (discrete) continuous factor of number of nodes) and then explore the 
+behavior of the error rates across the design space with a profiler for the resulitng linear 
+model for the error rate as a function of the NN settings. Unlike ARIMA or regression 
+(including doubly stochastic) modeling approaches for time series, there is a stochastic 
+behavior in the predictions made by neural networks due to the use of randomly initialized 
+weights. Unless the seed for the software’s random number generator is fixed, repeated 
+fitting of the same neural network will lead to different predictions. The amount of 
+
+variation in the resulting predictions depends on the complexity of the network and on the 
+steps that have been taken to avoid overfitting, including early stopping of the optimizer. 
+When selecting a model configuration, we will not only want to minimize the expected 
+error rate, but also minimize the variability in the error rates. To this end, we seek to 
+minimize the upper 95% prediction interval on the testing error rate. The NN study 
+proceeds in two phases where a screening experiment first identifies the most useful 
+(R)NN, followed by a more comprehensive performance study against common 
+forecasting approaches across many more skills.  
+ 
+Section 2 describes the doubly stochastic model for call volumes and how modifications to 
+the originally proposed computational approach can lead to improved convergence. 
+Section 3 details how a full factorial design is used to characterize the performance of RNN 
+options as a function of five factors (and their interactions) on the resulting short-term 
+forecast error rate. Additionally, Section 3 describes the selection of the model 
+configuration that leads to the minimum upper bound on the 95% prediction interval for 
+the testing error rate. Due to the number of different model configurations that must be run 
+along with the computational complexity of RNNs, the first phase discussed in Section 3 
+considers only a limited number of skills and validation days. In Section 4, the best 
+performing RNNs are run over a larger validation set and over all call center skills to 
+compare the performance to the doubly stochastic mixed model approach, and to ARIMA 
+as well as Winters smoothing. 
+ 
+2. Stable Settings for Fitting the Mixed Model 
+ 
+There are two distinct influences on call volumes that induce a correlation between the 
+observed call counts, violating the independence assumption made by ordinary least 
+squares regression models that might be used to model the volumes (Ibrahim & L'Ecuyer, 
+2013). Within a given day, some event may lead to more/fewer calls than expected. For 
+example, unexpected behavior in the stock market in the morning may lead to an increased 
+number of calls for the rest of the day at a financial services contact center. This is intra-
+day correlation. Likewise, there are systemic processes responsible for inter-day 
+correlation. Heuristically, if we noticed that the residuals are very large and positive 
+throughout the day today caused by a weather event for example, we might also expect a 
+larger-than-average call load tomorrow. Ignoring correlation between subsequent 
+observations leads to inaccurate standard errors and prediction intervals. In addition, 
+although the estimates from a linear regression may be unbiased in the presence of 
+correlated residuals, they will not be efficient (Demidenko, 2013). 
+ 
+It is typical for call center regression models to include a day-of-week by period-of-day 
+interaction (Ibrahim, Ye, L'Ecuyer, & Shen, 2016). In a call center open five days per 
+week with 32 half-hour periods per day, this interaction involves 160 fixed-effect 
+parameters. In addition, a call center may require forecasting for holidays. Aldor-
+Noiman, Feigin, & Mandelbaum (2009) exclude holidays when training their model; 
+however, we cannot ignore these days because some splits operate on holidays and may 
+exhibit different behavior on those days. In order to capture this behavior, we include a 
+holiday indicator (holiday_ind) by period-of-day interaction effect in the model. 
+However, some training data sets may include only a single holiday, leading to high 
+variance in the parameter estimates for this effect (each period observation from that one 
+day becomes the new estimate for that period during holidays). To reduce the variability 
+of these estimates, we combine groups of 3 periods together on holidays. That is, periods 
+{1, 2, 3} are assigned p_group = 1, periods {4, 5, 6} are assigned p_group = 2, etc. The 
+
+p_group*holiday_ind interaction is included in the fixed effect structure as an additive 
+effect. 
+ 
+Following Aldor-Noiman et al. (2009), we fit a linear mixed model with correlated errors 
+to the transformed call counts 
+𝑌 = 𝑿𝛽 + 𝒁𝑏 + 𝜀 
+where 
+• 
+𝑌 is the vector transformed call counts, 𝑌 = √𝑐𝑜𝑢𝑛𝑡 + 0.25 
+• 
+𝑿 is a matrix containing the levels of the fixed effects for each observation 
+• 
+𝛽 is the vector of fixed effects parameters containing a day-of-week*period-of-day 
+interaction and a p_group*holiday-indicator interaction 
+• 
+𝒁 is a binary coefficient matrix for the random day-to-day effects in the model. 
+There is one column for each day in the data.  
+• 
+𝑏~𝑁(0, 𝑮 ) is the vector of random day-to-day effects. Each unique day in the data 
+set is represented by one random effect in b. G follows a first-order autoregressive 
+structure, AR(1).  
+• 
+𝜀~𝑁(0, 𝑹 ) is the vector of error terms (residuals), allowing 𝜀 to potentially follow 
+an AR(1) process within days. Thus, R is a block-diagonal matrix, with one AR(1) 
+block for each day in the data set. This accounts for the potential correlation in 
+residuals from proximal periods within days. 
+The full model allows for complex correlation structures. However, for some splits (within 
+particular training data sets), there may be only sporadic and sparse occurrences of call 
+arrivals. This can lead to slow or failed model convergence in some cases. Aldor-Noiman 
+et al. (2009) address this by estimating the doubly stochastic model in two steps: first, the 
+inter-day correlation (G) is estimated using the aggregated total call counts from each day. 
+These parameters are then held constant in a second call to SAS PROC MIXED while 𝛽 
+and 𝑹 are estimated.  
+ 
+Indeed, PROC MIXED can experience convergence problems when the solutions lie on 
+the boundary of the parameter space, such as when variance components are zero (Karl, 
+Yang, & Lohr, 2013).  However, after making modifications to the default PROC MIXED 
+settings, we were reliably able to achieve convergence of the full model with the joint 
+optimization of (𝛽, 𝑮, 𝑹) in a single call to PROC MIXED. In this regard, our approach 
+differs from that of Aldor-Noiman et al. (2009): we fit all of the model parameters jointly 
+(with a single call to PROC MIXED). This will lead to reduced bias in the estimates for 
+the models that do converge.  
+ 
+We improved convergence rates by changing the convergence criterion used by SAS 
+PROC MIXED. By default, SAS ensures that the sum of squared parameter gradients 
+(weighted by the current Hessian of the parameter estimates) is sufficiently small. 
+However, in the presence of strong correlations in the doubly stochastic model, the 
+parameter estimates may lie near the boundary of the parameter space, meaning the 
+gradients may not approach 0 with convergence (Demidenko, 2013). As an alternative, we 
+declare convergence when the relative change in the loglikelihood between iterations is 
+sufficiently small. Additionally, we employ Fisher scoring during the estimation process. 
+Fisher scoring is more stable for models with complex covariance structures and can lead 
+to better estimates of the asymptotic covariance (Demidenko, 2013). Finally, since our 
+
+application only uses the call volume point estimates and not the associated standard errors 
+or tests of significance, we specify ddfm=residual to avoid spending substantial time 
+calculating appropriate degrees of freedom for the approximate F-tests. If confidence or 
+prediction intervals are needed, this value should be set to ddfm=kenwardrodger2 in order 
+to calculate Satterthwaite approximations for the degrees of freedom and to apply the 
+Kenward-Rodger correction (Kenward & Roger, 2009) to the standard errors. The code for 
+our modified approach appears in Figure 1. 
+ 
+Figure 1 Modified SAS code for the Doubly Stochastic Model 
+The square root transformation is applied to reduce the right skew in the observed call 
+volumes, and to stabilize the variance of the observations since quantities such as call 
+volumes tend to follow a Poisson distribution. The approach in Figure 1 employs a normal 
+approximation of this process. We experimented with fitting a mixed Poisson regression to 
+the untransformed call volumes (via PROC GLIMMIX), but found that the run times 
+became unfeasibly long (even when using the default pseudolikelihood approach and 
+avoiding integral approximation) with no noticeable improvement in error rates. 
+ 
+3. Choosing Recurrent Neural Network Configurations with a Designed Experiment 
+ 
+Generally, neural networks consist of layers of weights and nonlinear activation functions 
+that are used to relate inputs (predictors) to outputs (targets). Outputs from each layer are 
+passed sequentially to the next layer as an input vector. The complexity of each layer is 
+determined by the length of the output vector (number of nodes) it produces. A loss 
+function is used to compare the output of the final layer of the neural network to the 
+provided targets (e.g. call volumes), and an optimizer function provides updated values of 
+the weights each node that will decrease the resulting loss. The “depth” of the model is 
+controlled by the number of layers that are used. This “depth” is the source of the phrase 
+“deep learning”. For example, in image processing applications with convolutional neural 
+networks, the different layers can be shown to represent different levels of granularity of 
+detail in an image (Chollet & Allaire, 2018). Besides the number of layers and the 
+number of nodes per layer, there are a number of choices that must be made regarding the 
+properties of the optimizer, the distribution of the random initialization of the parameter 
+weights, and the shape of the activation function(s). 
+ 
+In a traditional, densely connected network, the individual observations are assumed to be 
+independent. A simple example using output from JMP Pro 14.1 helps to illustrate. 
+Suppose we want to fit a densely connected neural network to predict the standardized 
+call count using only the previous day’s standardized call count at the same period (the 
+lag-32 of the call count, since there are 32 periods per day in the example) as a predictor 
+with one node in one layer, using a hyperbolic tangent activation function. This network 
+shown in Figure 2 with resulting weights shown in Figure 3. 
+
+proc mixed data=training_data scoring=50 maxiter=150 maxfunc=10000 convf=1E-6;
+class day_of_week period day_num split p_group;
+by split;
+/* The fixed effects */
+model transf_call_count=day_of_week*period p_group*holiday_ind/
+noint ddfm=residual outp=pred_call_count_output notest;
+/* The day-level random effects */
+/* Note: day num copy is not included in the clAss statment and is numeric *
+random day_num / type=sp(pow)(day_num_copy);
+/* The period-level correlated residuals */
+run; 
+Figure 2 Densely connected neural network with one node in one layer. 
+ 
+Figure 3 Fitted weights from the network with one node in one layer. 
+Suppose that the lag-32 standardized call count is equal to 1 at a given period, t. Then the 
+neural network predicts a standardized call volume of 
+−0.6354 + 3.3923 ∗ TanH(0.5 ∗ (0.4046 + 0.5323 ∗ 1)) = 0.847 
+for the current period, t, where TanH is the hyperbolic tangent function and the 0.5 
+parameter is a fixed value. The nonlinear activation function provides the network with 
+the flexibility to model nonlinear relationships between the inputs and the response, as 
+well as interactions between the inputs. 
+ 
+We next consider a slightly more complex network with two layers using 2 nodes in the 
+first layer and 1 node in the second layer (Figure 4) with parameter estimates shown in 
+Figure 5. 
+ 
+Figure 4 A densely connected neural network with 2 layers using two nodes in the first 
+layer and one node in the second layer. 
+
+Lag[Standardize[cnt_ call], 32]
+Standardize[cnt_call]Estimates
+Parameter
+Estimate
+H1_ 1:Lag[Standardize[cnt_ call], 32]
+0.5323
+H1_1:Intercept
+0.4046
+Standardize[cnt call]_ 1:H1_ 1
+3.3923
+Standardize[cnt call]_2:Intercept
+-0.6354Lag[Standardize[cnt_ call], 32]
+Standardize[cnt_call 
+Figure 5 Fitted weights from the network with two layers. 
+Again assuming that the lag-32 standardized call count is 1, the predicted value is 
+−0.82 + 4.9517
+∗ 𝑇𝑎𝑛𝐻 (0.5
+∗ (0.2416 − 0.7382 ∗ 𝑇𝑎𝑛𝐻(0.5 ∗ (−0.3633 − 0.8785 ∗ 1)) + 0.3057
+∗ 𝑇𝑎𝑛𝐻(0.5 ∗ (−0.0824 + 0.4124 ∗ 1)))) = 0.843 
+In the densely connected network, each observation is processed independently and there 
+is no “memory” of what happened in the previously processed observation. In time series 
+applications, however, there is a temporal ordering that the data are recorded in, and there 
+may be correlation between nearby observations. For example, a spike or drop in call 
+volume might persist over several periods. To address this potential, recurrent neural 
+networks record information when fitting each observation that is then provided as a 
+model input when fitting later observations. 
+ 
+In a simple (Elman) RNN layer, the output from each node (the output of the TanH 
+functions, referred to as the “state”) is recorded and stored, and used as an input for the 
+same node when processing the next observation. Note that there is one state recorded for 
+each node in the layer. For the single layer network example (Figure 3), the state was 
+calculated as 𝑠𝑡 = TanH(0.5 ∗ (0.4046 + 0.5323 ∗ 1)) = 0.44 when the lagged 
+standardized call volume was equal to 1. A simple RNN learns an extra parameter (say, 
+u) to act as a coefficient for the stored state, and the activation function TanH(0.5 ∗
+(𝑤0 + 𝑤1 ∗ 𝑋𝑡)) that is used by the dense network would be replaced by  𝑠𝑡 =
+TanH(0.5 ∗ (𝑤0 + 𝑤1 ∗ 𝑋𝑡 + 𝑢 ∗ 𝑠𝑡−1)) in order to fit a simple RNN. The LSTM and 
+GRU RNNs also use the recorded state when making predictions for the current time 
+period, along with products of additional activation functions that are designed to carry 
+state information further in time. Details of these additional structures are explained in 
+Section 3 of Bianchi et al. (2018). For our purpose it is sufficient to note that the GRU 
+network is extremely similar to the LSTM, albeit less complex due to the ommison of a 
+group of paramters. Chollet & Allaire (2018) remark that Google Translate currently runs 
+using an LSTM with seven large layers. 
+It is not clear a priori which of these four neural networks is most appropriate for a 
+particular call center. Furthermore, it is possible that each of these networks may have a 
+different optimal depth and structure when applied to the call center data. A designed 
+experiment is run to identify the optimal model type and structure. 
+ 
+
+Estimates
+Parameter
+Estimate
+H2_1:Lag[Standardize[cnt _ call], 32]
+-0.8785
+H2_1:lntercept
+-0.3633
+H2_2:Lag[Standardize[cnt_call], 32]
+0.4124
+H2_2:Intercept
+-0.0824
+H1_1:H2_1
+-0.7382
+H1_1:H2 2
+0.3057
+H1_1:lntercept
+0.2416
+Standardize[cnt_ call]_1:H1_1
+4.9517
+Standardize[cnt_ call]_2:Intercept
+-0.82003.1 Data for the Experiment 
+We analyze call volumes aggregated into 30 minute periods from 3 different large-volume 
+skills in an operational call center for the months of March-June 2018. All of the models 
+under consideration are used to forecast next-day call volumes using 5 weeks of training 
+data. Each day contains 32 30-minute periods during which the call center is operating. 
+This application considers the Monday through Friday behavior of the call skills. Due to 
+the use of the one-week lagged observations as a predictor in the neural networks, the first 
+week of training data is not included in the predictor matrix (since the prior week’s call 
+volumes are unknown), meaning each training data set consists of 4*5*32=640 
+observations. The designed experiment in this section will evaluate the methods using a 
+holdout period of the five one-day ahead predictions during last week in June using the 
+three largest skills from the call center. Section 4 will then fit a reduced set of models over 
+all skills for 60 day-ahead predictions. 
+ 
+3.2 Neural Network Input Factors 
+Each network makes use of the one-hot encoding (via a binary indicator matrix) of the day 
+of week and the one-hot encoding of period of day. Furthermore, to capture the day- and 
+week-long correlations, the networks are also fed the call volumes for the same period in 
+the previous day when modeling the current day, as well as the call volumes for the same 
+period on the same day of last week. One-period lagged call volumes are not included, as 
+this is the purpose of the within-sequence memory of the RNNs. Other inputs include a 
+binary indicator for whether the current day is a holiday, a binary indicator for whether 
+yesterday was a holiday, a binary indicator for whether or not last week’s observation was 
+recorded on a holiday, and day number. The day number is a continuous counter for the 
+number of the given day in the data set, which would potentially allow the neural network 
+to detect trends across time. All told, these account for 42 vectors of input for the neural 
+networks. There is no need to create indicator columns for interactions between day of 
+week and period (or any other factors) as the neural network will automatically detect them. 
+ 
+Bianchi et al. (2017) application of hourly call volumes displays strong lag-24 correlation, 
+representing a period-of-day effect. They remove this seasonality by differencing the data 
+at lag-24. By contrast, we do not difference the call volumes, but instead include period-
+of-day (along with day-of-week) as exogenous variables and allow the neural network to 
+detect this seasonality. This approach allows the network to detect the expected interactions 
+between period-of-day and day-of-week, as well as any other input factors. 
+ 
+While these input vectors are included for all models, there are three final input vectors 
+whose (joint) inclusion is treated as an experimental factor: the same-period predictions 
+from the mixed model approach (Aldor-Noiman, Feigin, & Mandelbaum, Workload 
+forecasting for a call center: Methodology and a case study, 2009), from a Winters 
+smoothing model, and from a seasonal ARIMA(1, 0, 1)(0, 1, 1)160 model. The inclusion 
+of the predictions from these models as an input to the neural network is an original 
+approach that gives the network the opportunity to form predictions that may be thought 
+of as corrections to those from these traditional models, based on potential interactions 
+with other included factors. For brevity, we refer to this as the mixed.cheat option, since it 
+allows the neural networks to “cheat” by looking at the predictions generated by these other 
+three models when forming its own predictions for the same time periods. If none of the 
+other 42 input vectors were included with these three, this would represent a supervised 
+learning approach to forming a dynamically weighted average of these three model 
+predictions in order to create a single “bagged” prediction. 
+ 
+
+3.3 Network Configuration Aspects Treated as Experimental Factors 
+The designed experiment considered five different factors of structural settings of the 
+neural networks: model.type {dense, simple (Elman) RNN, GRU, LSTM}, nlayers {1, 
+2}, nnodes (per layer) {25, 50, 75, 100}, kernel.L2.reg {0, 0.0001} and mixed.cheat 
+{FALSE, TRUE}. The L2 regularizer adds kernel.L2.reg times each weight coefficient to 
+the total loss function for the network. Similar to Lasso regression, this helps bound the 
+magnitude of the model coefficients and could potentially help prevent overfitting the 
+training data.  
+ 
+We fit a full factorial design (requiring 128 runs) for these five factors, which allows us 
+to test for the presence of up to five-way interactions between the five factors. Figure 6 
+shows the first 6 runs of the design. The design is replicated over the 3 largest splits and 
+across the 5 subsequent one-day ahead forecasts. This produces a total of 1920 runs for 
+the entire experiment. The replication allows for behavior to be averaged over different 
+days and for a more detailed exploration of how the variance of the error rates depends on 
+each factor. 
+ 
+Figure 6 First 6 runs of the designed experiment 
+3.4 Static Considerations for Neural Network Configuration 
+The input for the classical dense neural network is a 640x42 matrix (640x45 if 
+mixed.cheat=TRUE). The dense network does not consider the temporal ordering of the 
+640 observations (outside of the explicit inclusion of the day- and week-lagged 
+observations as inputs): the observations are shuffled after each epoch and then processed 
+in batches (we used a batch size of 32). 
+ 
+By contrast, the RNNs consider the ordering of the observations. The input for the RNNs 
+is a 20x32x42 array. This indicates to the RNN that there are 20 batches (days) of 32 
+timesteps (periods) with 42 predictors per time step. By default, the batches are treated 
+independently and the timesteps within each batch are potentially correlated (via the 
+persistence of the states in the RNN). If the batches themselves are presented in a 
+temporal order (as is the case in our application), then this can be indicated to the Keras 
+model via the STATEFUL=TRUE option and by disabling the shuffling of batches 
+during training. This retains the model weights from batch-to-batch (day-to-day) to allow 
+for possible long-term behavior. However, we found that using the STATEFUL option 
+led to a failure to converge in some skill*day combinations and the resulting validation 
+error rates were not significantly different from those generated without the STATEFUL 
+option. This seems to indicate that the dependence on prior days’ behavior is already 
+captured by the inclusion of the one-day and one-week lagged observations. Due to the 
+occasional convergence issues, the results in the sequel are generated with 
+STATEFUL=FALSE (and batch shuffling enabled during training). It would have also 
+been possible to fit the model with week- or month-long batches by training the RNN on 
+a 4x160x42 or a 1x640x42 array. We did not consider the week-long batches, but the 
+
+model.type
+nlayers mixed.cheat nnodes
+kernel.L2.reg
+layer_simple_rnn
+2 
+FALSE
+50
+0.0001
+layer_ gru
+FALSE
+25
+0
+layer_Istm
+1
+TRUE
+100
+0
+layer_gru
+TRUE
+75
+0.0001
+layer_gru
+TRUE
+75
+0
+layer_simple_rnn
+FALSE
+100
+0month-long batches tended to produce inferior predictions to the day-long batches. This 
+could possibly be due to the lack of long term correlations in the data and the fact that the 
+smaller batches allow for the shuffling of the ordering that the days are fed through the 
+gradient-optimization routine of the neural network, which can improve the model fit by 
+preventing the model from overweighting the first observations that are provided to the 
+network in each epoch (Chollet & Allaire, 2018). 
+ 
+Note that the model weights are reset and the model is retrained for each skill. This is in 
+contrast to the approach taken by Zhu & Laptev (2017) in which a single network is fit to 
+accommodate disparate behavior from different cities. As discussed in Section 5, a single 
+multi-output network could potentially be built to model all of the skills at once. 
+ 
+While they were not included as experimental factors in this application, we also noticed 
+a significant relationship between the quality of the predictions and the optimization 
+routine employed. Extensive pilot experimentation led to our use of the AMSGrad variant 
+(Reddi, Kale, & Kumar, 2018) of the Adam optimizer (Kingma & Ba, 2014), with a 
+learning rate decay of 0.0001. We would recommend including both the optimizer and 
+the optional learning rate decay as factors in the designed experiment for parameter 
+tuning in future problems. 
+ 
+We found it was important to tune the number of epochs for each model fit using 
+validation data (the last week in the training set) by first fitting 500 epochs for each 
+model, taking a moving average (with a window size of 10 epochs) of the resulting 
+WAPEs on the validation data, and then refitting the model (on both the training and the 
+validation data in order to predict an additional day which was held out as a test set) with 
+the number of epochs that produced the minimum validation WAPE. The moving 
+average is important due to the volatile and non-monotonic behavior we observed in the 
+individual recorded WAPEs and helps to find a relatively stable region. Consistent with 
+previous findings (Bianchi, Maiorino, Kampffmeyer, Rizzi, & Jenssen, 2017), we noticed 
+that the RNNs take many more epochs to converge than the dense neural network. Other 
+researchers (such as Bianchi et al.) have used more than 500 epochs when fitting RNNs, 
+so this upper bound should also be considered as an important factor when building an 
+RNN.  
+ 
+While we also experimented with recurrent dropout (Gal & Ghahramani, 2015) to 
+prevent overfitting, it led to degraded performance in the early iterations of our 
+experiment and we removed it from consideration. However, this could simply be due to 
+features of this particular application, such as the relatively short training period of five 
+weeks: Chollet & Allaire  (2018) strongly advocate the use of dropout and recurrent 
+dropout. 
+ 
+The models experienced improved errror rates after switching the kernel initializer for the 
+random weights to the He normal initializer (He, Zhang, Ren, & Sun, 2015). We used the 
+relu activation function exclusively, although this choice could also impact the quality of 
+the resulting model fit. And while our application did not detect any two-factor 
+interactions among the five experimental factors, it is possible that some of these 
+additional factors could depend on the type of (R)NN being used, meaning there would 
+be an interaction between these factors and model.type.  
+ 
+A final contributing factor is the batch size. This determines how many observations are 
+processed before the gradients are updated: when fitting Keras models on a GPU, amount 
+
+of memory available on the GPU can be a limiting factor on the batch size. This choice is 
+more constrained in the RNNs, where each batch is a single day/week/month (determined 
+by the number of timesteps specified in the input array to the RNN). By contrast, the 
+batch size for a dense network can be set between 1 and the number of observations in 
+the training data. We noticed significant differences in the error rates from the dense 
+model depending on what batch size was used. 
+ 
+3.5 Experimental Response 
+While mean squared error (MSE) is frequently used to evaluate and compare predictive 
+models, this is a poor metric for the call center application as it will give undue focus to 
+the low call volume periods at the beginning and end of each day. Instead, the weighted 
+absolute percentage error (WAPE) is recommended for call volume modeling (Ibrahim, 
+Ye, L'Ecuyer, & Shen, 2016). This weights the absolute percentage error in each period 
+by the number of calls received in that period and is defined by 
+𝑊𝐴𝑃𝐸 =
+∑
+|𝑌𝑖 − 𝑌̂𝑖|
+𝑛
+𝑖=1
+∑
+𝑌𝑖
+𝑛
+𝑖=1
+ 
+where 𝑌𝑖 and 𝑌̂𝑖 are the observed and predicted volumes, respectively, for each 30 minute 
+period 𝑖 = 1, … , 𝑛. Because WAPE is our metric of interest, the models are compiled to 
+use a mean absolute error loss-function, which minimizes the numerator of the WAPE 
+(the denominator is static).  
+ 
+For each row of the experimental table (Figure 6), the specified neural network is fit 
+(independently) to model five subsequent days of each split. That is, day 1 is predicted 
+using the previous 5 weeks leading up to day 1, then the model is reset and day 2 is 
+predicted using the 5 weeks leading up to day 2, etc.   The vector of five one-day ahead 
+predictions is compared with the observed call volumes (that were not visible to the 
+model during training), and the resulting WAPE is recorded. 
+ 
+3.6 Analysis of Experimental Results 
+Figure 7 gives a typical output of the prediction error across the different formulations of 
+the neural networks. GRU often has the lowest forecast error or at least is consistently 
+close to the lowest. The other procedures tend to have much more unstable performance 
+based on the choice of nodes and layers as well as across the days and splits. 
+ 
+Figure 7 Forecast error by model type and settings for forecast day 5 split 3 
+Regression analysis is used to determine the statistically significant factors and 
+interactions. In order to control a heavy right-skew in the recorded WAPEs, an inverse 
+
+model.type
+NN Classic
+RNN GRU
+ RNN LSTM RNN Simple
+WAPE
+WAPE
+WAPE
+WAPE
+nlayers
+nnodes
+Mean
+Mean
+Mean
+Mean
+1
+25
+6.7%
+6.2%
+8.5%
+6.0%
+50
+7.1%
+6.2%
+6.5%
+6.7%
+75
+7.1%
+5.8%
+6.8%
+5.9%
+100
+7.0%
+6.4%
+6.9%
+6.8%
+2
+25
+7.5%
+6.6%
+8.1%
+6.9%
+50
+7.3%
+5.8%
+6.9%
+7.0%
+75
+7.1%
+6.0%
+6.6%
+6.5%
+100
+8.0%
+5.9%
+7.5%
+6.0%transformation is applied to use as the response in the regression models. Figure 12 
+displays the original skewness and the transformation to normality after taking the 
+inverse of the WAPEs.  
+ 
+3.7 Loglinear Variance Regression Model 
+The mean structure of 1/WAPE is modeled by including the main effects of the five 
+experimental factors (model.type, nlayers, mixed.cheat, nnodes, kernel.L2.reg) and all of 
+the interactions. In addition, split, file (which represents the validation day), and split*file 
+are included as blocking factors.   
+ 
+While an ordinary regression model for the full factorial design would assume the same 
+error variance for all responses across the design space, the loglinear variance model 
+allows for the error variance itself to be modelled as a function of the input factors. This 
+is appealing for this application, where parsimonious networks may be expected to 
+demonstrate less variability during repeated fittings.  
+ 
+For each factor setting, the loglinear variance model produces a predicted mean of 
+1/WAPE and a predicted standard deviation of 1/WAPE, representing the variability due 
+to the random initial weighting of the NN. 
+We include model.type, nlayers, mixed.cheat, nnodes, kernel.L2.reg, file, and split as 
+factors in the loglinear variance model (using JMP Pro 14.1), which models the log of the 
+error variance as a linear combination of these factors (which are all treated as 
+categorical). With the exception of kernel.L2.reg, all of these effects are found to be 
+significantly associated with the error variance (Figure 8). 
+ 
+Figure 8 Factors with a significant impact on WAPE variability 
+For the mean model, model.type, nlayers, mixed.cheat, nnodes, day, split, and file*split 
+were found to be significantly associated with 1/WAPE (Figure 9). Neither kernel.L2.reg 
+nor any of the interaction terms involving the NN architecture were found to be 
+significant.  
+ 
+ 
+
+Variance Effect Likelihood Ratio Tests
+Source
+Test Type
+DF
+ ChiSquare
+Prob>ChiSq
+model.type
+Likelihood
+3
+114.126
+<.0001*
+nlayers
+Likelihood
+1
+4.6333
+0.0314*
+mixed.cheat
+Likelihood
+1
+53.6884
+<.0001*
+nnodes
+Likelihood
+3
+16.5624
+0.0009*
+kernel.L2.reg 
+Likelihood
+1
+0.0042
+0.9482
+file
+Likelihood
+4
+36.8213
+<.0001*
+split
+Likelihood
+2
+56.3123
+<.0001* 
+Figure 9 Factors that are significantly associated with the mean WAPE 
+Removing the insignificant interaction terms (but allowing kernel.L2.reg to remain in 
+both the mean and variance models) produces the profiler shown in Figure 10. Notice the 
+large standard deviation associated with the LSTM model. The dependence on file and 
+split is not shown, since there are no interactions modeled between these and the other 
+experimental factors. While we want to maximize 1/WAPE, we also want to minimize 
+the standard deviation of 1/WAPE. That is, we would like to maximize the lower 95% 
+prediction interval of 1/WAPE (or minimize the upper 95% PI of WAPE) in order to find 
+the model configuration that will give a combination of relatively good results on average 
+while being protected against large errors. 
+
+Fixed Effect Tests
+Source
+Nparm
+DF
+DFDen
+F Ratio
+Prob > F
+model.type
+m
+m
+796.5
+44.6120
+<.0001*
+nlayers
+1
+1
+1258
+5.5213
+0.0189*
+model.type*nlayers
+3
+m
+796.5
+1.0691
+0.3614
+mixed.cheat
+1
+1
+1258
+17.0652
+<.0001*
+model.type*mixed.cheat
+3
+3
+796.5
+1.3181
+0.2672
+nlayers*mixed.cheat
+1
+1
+1258
+2.3413
+0.1262
+model.type*nlayers*mixed.cheat
+3
+3
+796.5
+1.9528
+0.1196
+nnodes
+m
+786.2
+4.5600
+0.0036*
+model.type*nnodes
+9
+5
+839.4
+0.7864
+0.6290
+nlayers*nnodes
+3
+3
+786.2
+0.7371
+0.5301
+model.type*nlayers*nnodes
+9
+6
+8394
+0.6668
+0.7395
+mixed.cheat*nnodes
+3
+3
+786.2
+0.6624
+0.5753
+model.type*mixed.cheat*nnodes
+9
+5
+839.4
+0.2224
+0.9913
+nlayers*mixed.cheat*nnodes
+m
+786.2
+0.9291
+0.4261
+model.type*nlayers*mixed.cheat*nnodes
+9
+9
+839.4
+0.4312
+0.9186
+kernel.L2.reg
+1
+1
+1258
+0.0026
+0.9590
+model.type*kernel.L2.reg
+m
+796.5
+0.1263
+0.9445
+nlayers'kernel.L2.reg
+1
+1
+1258
+0.2313
+0.6306
+model.type*nlayers*kernel.L2.reg
+3
+m
+796.5
+0.7812
+0.5046
+mixed.cheat*kernel.L2.reg
+1
+1
+1258
+0.0295
+0.8636
+model.type*mixed.cheat*kernel.L2.reg
+3
+m
+796.5
+0.1395
+0.9364
+nlayers*mixed.cheat*kernel.L2.reg
+1
+1
+1258
+0.1498
+0.6988
+model.type*nlayers*mixed.cheat*kernel.L2.reg
+3
+3
+796.5
+0.3776
+0.7692
+nnodes*kernel.L2.reg
+m
+m
+786.2
+0.9118
+0.4347
+model.type*nnodes*kemel.L2.reg
+9
+6
+839.4
+0.8905
+0.5331
+nlayers*nnodes*kernel.L2.reg
+3
+m
+786.2
+0.7442
+0.5259
+model.type*nlayers*nnodes*kernel.L2.reg
+9
+839.4
+0.6535
+0.7513
+mixed.cheat*nnodes*kernel.L2.reg
+3
+3
+786.2
+0.0812
+0.9703
+model.type*mixed.cheat*nnodes*kernel.L2.reg
+9
+8394
+0.2691
+0.9827
+nlayers*mixed.cheat*nnodes*kernel.L2.reg
+3
+m
+786.2
+0.8510
+0.4662
+model.type*nlayers*mixed.cheat*nnodes*kernel.L2.reg
+9
+6
+8394
+0.6614
+0.7442
+file
+4
+4
+668.1
+1258.316
+<.0001*
+ds
+2
+2
+857.3
+1867.911
+<.0001*
+file*split
+8
+8
+746.8
+244.5502
+<.0001* 
+Figure 10 Mean and Standard Deviation of 1/WAPE as a function of the experimental 
+factors. Goal is to maximize the top (1/mean forecast error) and minimize the bottom 
+(standard deviation) 
+Figure 11 plots the upper 95% prediction interval for WAPE against the experimental 
+factors. The model configuration that minimizes the upper 95% PI of WAPE is a GRU 
+model that is allowed to use the MIXED, ARIMA, and Winters predictions as inputs, 
+with 2 layers and 50 nodes per layer and an L2 penalty of 0.0001 on the kernel weights.  
+However, the L2 penalty was not found to be significant in either the mean or the 
+variance models and the contribution of nlayers is relatively flat.  
+ 
+Figure 11 Upper 95% Prediction Interval for WAPE 
+To summarize, the designed experiment provided insight to effectively select the 
+appropriate levels across several neural network models and parameters. The goal is to 
+balance model performance in minimizing forecast error (that is, maximizing 1/forecast 
+error) and minimizing the variance of this forecast error. Figure 10 and Figure 11 are 
+interpretable across all factors and settings as displayed due to the absence of significant 
+interactions between the factors. The top half of Figure 10 displays the reciprocal forecast 
+error (the larger the value the better) which indicates preferred settings of GRU or Elman 
+(simple) for the model, 1 layer, using the mixed model forecasts, and 25 nodes while the 
+selection of L2.kernel makes little difference. The lower half of Figure 10 displays the 
+variance (lower is better) where the traditional neural net would be preferred along with 2 
+
+Prediction Profiler
+30 -
+28.87292
+29
+[27.9033,
+28
+29.8426]
+27
+26
+Dev
+2.5
+ [1.66636,
+1.5
+dense
+layer_gru
+layer_Istm
+2
+FALSE-
+TRUE-
+5
+0.0001
+layer_gru
+2
+TRUE
+50
+0.0001
+model.type
+nlayers
+mixed.cheat
+nnodes
+kemel.L2.regPrediction Profiler
+0.044
+0.0435
+0.043
+0.040571
+0.0425
+0.042
+0.0415
+0.041
+0.0405
+dense
+layer_gru
+layer_Istm
+layer_simple_rnn
+2
+FALSE-
+TRUE-
+5
+9
+5
+100-
+0.0001
+layer_gru
+TRUE
+50
+0.0001
+model.type
+nlayers
+mixed.cheat
+nnodes
+kenel.L2.reglayers, using mixed model forecasts, with 50 nodes while robust to the regularization 
+parameter value. Note that the traditional dense neural network does have consistently 
+worse forecast error across all scenarios and could not be recommended despite having 
+the lowest variance. The prediction interval on forecast error is an alternative view that is 
+preferred by many practitioners where the goal is to minimize the width. Figure 11 (lower 
+is better) clearly shows GRU is the preferred solution using the mixed model forecasts 
+with 50 nodes.  
+ 
+ 
+Figure 12 Representative distribution of WAPE and 1/WAPE resulting from the designed 
+experiment 
+ 
+4. Comprehensive Performance Study 
+ 
+Based on the results of the designed experiment for NN error rates, we perform a further 
+study with actual call center data across 36 skills rather than only 3. We consider a 
+consistent 5-week training period advancing across multiple months of data to produce 60 
+one-day ahead predictions. These predictions do not use the actual data for the forecasted 
+day during model training and can be viewed as a validation set for the trained models. 
+This allows us to compare the relative performance of the mixed model and the neural 
+networks. We also include the error rates from the ARIMA and Winters seasonal 
+smoothing models for comparison. 
+ 
+Though we could have simply chosen the GRU RNN with 50 nodes on each of 2 layers 
+with kernel.L2.reg complexity parameter set to 0.0001 as the only representative RNN 
+based on the designed experiment, we decided to include all 4 neural network methods 
+
+Distributions
+WAPE
+0 0.1
+0.3
+0.5
+0.7
+0.9 1 1.1
+1.3
+1.5
+1/WAPE
+0
+5
+10
+15
+20
+25 
+30screened by using these same parameter settings. It is possible with the added skills – 
+with many having much lower call volumes – that another RNN model could work better 
+than GRU.  
+ 
+4.1 Results: One-day ahead Predictions Over 60 Separate Validation Days 
+Figure 13 provides the forecast errors averaged across all 60 days for the 12 splits with 
+the highest call volume sorted in descending call volume order. Note that results in this 
+section are generated by the (R)NNs with the mixed.cheat option disabled. The Doubly 
+Stochastic Mixed Model has the lowest average forecast error for all but Split 5540 and is 
+often significantly lower than the competitors. Winters Seasonal Exponential Smoothing 
+also performs quite well given its relative simplicity. The GRU performance confirms the 
+results from the designed experiment as usually the best RNN and always competitive 
+with the best. The highly complex LSTM recurrent neural network has very large forecast 
+errors for many of these splits and cannot be recommended. Note also that forecast error 
+generally increases for all methods as call volume decreases.   
+ 
+Figure 13 Average WAPE forecast errors across 60 separate validation days for high 
+call-volume splits 
+Figure 14 shows the similar trend of increasing error rates with decreasing call volumes 
+for the medium call volume splits. The GRU recurrent neural network is usually 
+outperforming the other neural network methods and is closer to the error rates of the 
+Doubly Stochastic Mixed Model. 
+ 
+Figure 14 Average WAPE forecast errors across 60 separate validation days for medium 
+call-volume splits 
+For the low call volume splits displayed in Figure 15, the GRU and Simple recurrent 
+neural networks perform similarly and slightly better than Doubly Stochastic and Winters 
+for most of the splits. The very low call volumes (last 3 splits) do seem to benefit from 
+the recurrent neural network formulation. 
+
+Split
+Sum Call Vol
+ARIMA
+Doubly Stoch
+NN_Classic
+RNN_GRU
+RNN_LSTM
+RNN_Simple
+Winters
+5000&5240
+929754
+9.0%
+8.4%
+10.2%
+11.2%
+38.0%
+13.9%
+8.3%
+5400&5570
+461256
+7.4%
+6.6%
+8.8%
+8.4%
+26.1%
+14.3%
+7.1%
+5620
+162996
+8.6%
+7.7%
+9.6%
+9.1%
+28.7%
+9.6%
+8.4%
+5660
+137759
+10.2%
+10.1%
+13.2%
+12.0%
+17.1%
+14.0%
+10.5%
+5020
+71930
+11.5%
+11.2%
+14.5%
+12.3%
+27.3%
+14.2%
+11.7%
+5630
+65079
+15.4%
+14.9%
+16.7%
+15.5%
+26.4%
+15.5%
+14.1%
+5840
+63984
+13.1%
+12.1%
+14.6%
+13.4%
+25.3%
+12.9%
+13.0%
+5200
+48728
+13.2%
+13.4%
+15.6%
+13.5%
+52.4%
+13.5%
+13.6%
+5540
+38236
+15.9%
+15.8%
+16.8%
+16.3%
+28.5%
+16.6%
+16.0%
+5670
+34793
+14.0%
+13.2%
+15.6%
+14.2%
+27.3%
+13.9%
+14.0%
+6500
+30534
+16.8%
+16.4%
+18.5%
+17.3%
+23.0%
+18.0%
+17.1%
+5260
+23849
+21.9%
+20.9%
+22.7%
+20.3%
+32.5%
+20.2%
+21.2%Split
+Sum Call Vol
+ARIMA
+Doubly Stoch
+NNClassic
+RNN_GRU
+RNN_LSTM
+RNN_Simple
+Winters
+5460
+20922
+18.2%
+18.1%
+19.5%
+18.1%
+26.0%
+17.7%
+18.2%
+5410
+19461
+19.8%
+19.2%
+21.4%
+20.0%
+59.1%
+20.8%
+19.7%
+6350
+16874
+30.8%
+26.8%
+29.3%
+28.7%
+31.1%
+29.0%
+29.4%
+5060
+16765
+19.3%
+19.1%
+20.7%
+19.6%
+24.4%
+19.6%
+19.6%
+5650
+9911
+23.8%
+23.4%
+24.6%
+24.0%
+27.3%
+24.3%
+24.0%
+5030
+8102
+40.0%
+26.0%
+28.5%
+27.8%
+33.2%
+27.7%
+26.9%
+5680
+7525
+27.3%
+27.1%
+28.2%
+26.7%
+31.9%
+27.8%
+27.7%
+5440
+7402
+28.5%
+28.1%
+30.3%
+28.3%
+33.8%
+29.7%
+29.2%
+5070
+6446
+34.6%
+34.5%
+34.6%
+33.5%
+43.6%
+35.0%
+34.7%
+5420
+5247
+35.0%
+34.6%
+36.2%
+33.9%
+38.1%
+34.7%
+35.0%
+5899
+4844
+36.4%
+35.8%
+37.0%
+35.2%
+64.8%
+36.6%
+36.7%
+5100
+4019
+34.7%
+34.0%
+36.6%
+34.3%
+40.1%
+35.5%
+34.8% 
+Figure 15 Average WAPE forecast errors across 60 separate validation days for low 
+call-volume splits 
+Overall, the best performing procedures were the Doubly Stochastic Mixed Model and 
+the GRU Recurrent Neural Network. Figure 16 shows forecast error by split sorted by 
+call volume. Generally, the mixed model does better for large and medium volume splits 
+(lower is better on the graph) while the RNN 
+model is more effective for the small volume—particularly the very small volume splits. 
+  
+ 
+Figure 16 Forecast error rates ordered by call volume by split for Doubly Stochastic 
+(blue) and GRU (red) 
+Figure 17 presents a different view of this same pattern. For each of the 60 one-day ahead 
+forecasts within each split, the GRU and Doubly Stochastic WAPEs are recorded, along 
+with the number of calls recorded for that split over the training and validation data 
+(sum_all_calls). For each split, Figure 17 plots the percent of the 60 day-ahead forecasts 
+for which GRU “won” (GRU WAPE < Doubly Stochastic WAPE) against the log of the 
+median call volume recorded by that split over the 60 different pairs of training and 
+validation data. There appears to be a linear decrease of the relative performance of GRU 
+against Doubly Stochastic in the log of the call volume. 
+
+Split
+Sum Call Vol
+ARIMA
+Doubly Stoch
+NN_Classic
+RNN_GRU
+RNN_LSTM
+RNN_Simple
+Winters
+5710
+3742
+39.6%
+38.8%
+40.7%
+39.1%
+45.7%
+40.2%
+38.9%
+5820
+2949
+44.1%
+43.1%
+46.1%
+43.2%
+50.4%
+44.0%
+43.4%
+5690
+2238
+56.1%
+51.1%
+51.6%
+51.2%
+58.9%
+51.8%
+52.5%
+5220
+2089
+49.7%
+49.5%
+50.5%
+49.3%
+54.0%
+50.1%
+49.7%
+5470
+1975
+49.2%
+48.8%
+52.0%
+50.5%
+57.0%
+50.7%
+50.2%
+6330
+1556
+60.5%
+60.1%
+61.0%
+60.2%
+65.1%
+61.6%
+60.7%
+5040
+1398
+79.0%
+69.0%
+69.8%
+73.4%
+72.2%
+71.1%
+80.7%
+6310
+922
+98.5%
+92.0%
+92.2%
+88.1%
+91.0%
+89.5%
+92.8%
+5720
+918
+80.2%
+77.8%
+77.5%
+77.6%
+85.0%
+78.4%
+80.1%
+6370
+485
+95.3%
+93.6%
+96.5%
+91.3%
+111.3%
+94.0%
+95.2%
+6360
+171
+150.4%
+136.8%
+126.5%
+107.7%
+118.7%
+111.7%
+142.5%
+6340
+14
+189.7%
+161.4%
+188.2%
+102.7%
+108.9%
+107.8%
+187.7%Forecast ErrorbySplit AverageOver 6oDays
+1.6 
+IDoubly Stoch
+1.5 
+RNN GRU
+1.4
+1.3 
+1.2
+1.1 -
+Errot
+aber
+1.0 
+0.9
+0.8
+0.7
+pa
+ubiay
+0.6
+0.5 
+0.4 
+0.3 
+0.2 
+0.1
+HighCallVolum
+Medium CallVolume
+Low Call Volume 
+Figure 17 Percent of the 60 day-ahead forecasts for which the GRU WAPE was less than 
+the Doubly Stochastic WAPE for each of the 36 splits plotted against the log of the 
+median sum of all calls recorded for each split over each pair of training and validation 
+data. 
+For the large call-volume splits, the extra flexibility of the GRU model does not lead to 
+improvements over the predictions generated by the mixed model approach. Because the 
+mixed model computations are faster and the implementation is less complex, there does 
+not appear to be any benefit to running the neural networks for the high-volume splits for 
+this short-term application. This is consistent with the findings of the Uber traffic volume 
+study when short-term predictions were considered (Zhu & Laptev, 2017), and would 
+possibly change if a longer training period (several months or multiple years) were used 
+for the call center data.  
+ 
+4.2 Improving GRU RNN by Using Doubly Stochastic Forecasts as a Covariate 
+Based on pilot studies during the designed experiment, the forecasting performance of the 
+GRU recurrent neural network often improved by integrating the forecasted value for the 
+validation days from the doubly stochastic, ARIMA, and Winters models. This 
+“cheating” by using other models’ forecasts (shown as mixed.cheat) proved to be a 
+significant benefit for the GRU model over these 60 predictions for each skill. The 
+WAPE for the held-out validation data was again the primary measure of performance. 
+ 
+Figure 18 displays the forecast errors for the high call volume skills averaged over the 60 
+validation days for the each of the neural networks when they use the mixed forecasts. 
+Note the side-by-side comparison of the RNN_GRU (no cheat) and GRU_cheat columns 
+where in most cases the “cheating” does result in improved forecasts and in those cases 
+where it is not better, it has only marginally declined. Additionally, the Simple_cheat 
+error rates compare favorably with the GRU_cheat while the LSTM_cheat suffers from 
+significantly poorer performance and instability issues. The doubly stochastic forecast is 
+still quite good and often the best choice. These results are also consistent when looking 
+at the medium and low call volume splits. Therefore, we recommend using the forecasts 
+from a doubly stochastic or Winters model as inputs to recurrent neural networks.  
+
+Percent won by GRU vs. Log[Median(sum all calls)]
+100%
+ Percent won by GRU
+%06
+%08
+70%
+Percent won by GRU
+%09
+50%
+40%
+30%
+20%
+10%
+0%
+4
+6
+8
+10
+12
+14
+Log[Median(sum all_ calls)] 
+Figure 18 Forecast errors for high call volume splits averaged over 60 validation 
+forecast days allowing NN to “cheat” to improve predictions 
+The median of the GRU WAPE (mixed.cheat=FALSE) minus the GRU WAPE 
+(mixed.cheat=TRUE) (within each split/day combination, for a sample size of 
+36*60=2160) is 0.002 with a p-value of 0.0005 from the Wilcoxon signed rank test with a 
+null hypothesis that the differences were drawn from a population with median equal to 
+0, indicating that mixed.cheat does tend to improve the performance of the GRU model. 
+ 
+A similar comparison of paired differences of error rates (all with mixed.cheat=FALSE) 
+confirms that GRU outperforms the other (R)NNs: LSTM - GRU produces a median of 
+0.0247 with a p-value of 1e-129, NN Classic - GRU produces a median of 0.0690 with a 
+p-value of <1e-185, and Simple RNN - GRU produces a median of 0.0021 with a p-value 
+of 1e-04. 
+ 
+ 
+References 
+Aldor-Noiman, S., Feigin, P. D., & Mandelbaum, A. (2009). Workload forecasting for a 
+call center: Methodology and a case study. Annals of Applied Statistics, 3(4), 
+1403-1447. 
+Aldor-Noiman, S., Feigin, P. D., & Mandelbaum, A. (2009). Workload forecasting for a 
+call center: Methodology and a case study. Annals of Applied Statistics(4), 1403–
+1447. 
+Allaire, J., & Chollet, F. (2018). keras: R Interface to 'Keras'. Retrieved from CRAN: 
+https://CRAN.R-project.org/package=keras 
+Bianchi, F. M., Maiorino, E., Kampffmeyer, M. C., Rizzi, A. R., & Jenssen, R. (2017). 
+Recurrent Neural Networks for Short-Term Load Forecasting: An Overview and 
+Comparative Analysis. Cham: Springer. 
+Bianchi, F. M., Maiorino, E., Kampffmeyer, M. C., Rizzi, A. R., & Jenssen, R. (2018). 
+An overview and comparative analysis of Recurrent Neural Networks for Short 
+Term Load Forecasting. arXiv. Retrieved from https://arxiv.org/abs/1705.04378 
+Box, G., & Jenkins, G. (1970). Time series analysis: Forecasting and control. San 
+Francisco, CA: Holden-Day. 
+
+Split
+Sum Call Vol
+Doubly Stoch
+RNN_GRU
+GRU_cheat
+LSTM_cheat
+Simple_cheat
+5000&5240
+926493
+6.7%
+10.2%
+7.8%
+14.4%
+7.8%
+5400&5570
+459961
+6.3%
+8.1%
+7.4%
+9.4%
+6.9%
+5620
+164138
+7.9%
+8.9%
+8.8%
+15.2%
+8.7%
+5660
+138389
+9.2%
+12.0%
+9.9%
+18.1%
+9.8%
+5020
+71900
+11.2%
+12.4%
+12.4%
+27.0%
+12.3%
+5630
+65279
+11.5%
+13.5%
+13.7%
+421.5%
+13.2%
+5840
+63687
+12.0%
+13.2%
+13.1%
+26.1%
+12.7%
+5200
+48624
+13.0%
+14.2%
+14.4%
+28.1%
+14.0%
+5540
+38318
+17.1%
+16.3%
+16.9%
+20.8%
+17.5%
+5670
+34741
+13.4%
+14.7%
+14.4%
+24.5%
+14.2%
+6500
+30352
+16.3%
+17.5%
+16.8%
+35.2%
+17.2%
+5260
+23674
+20.7%
+21.3%
+20.5%
+28.0%
+20.8%Cho, K., Merrienboer, B. V., Gulcehre, C., Bahdanau, D., Bougares, F., Schwenk, H., & 
+Bengio, Y. (2014). Learning phrase representations using RNN encoder-decoder 
+for statistical machine translation. arXive, arXiv:1406.1078. 
+Chollet, F., & Allaire, J. (2018). Deep Learning with R. Shelter Island: Manning 
+Publications Co. 
+Demidenko, E. (2013). Mixed Models Theory and Applications with R. Hoboken, NJ: 
+John Wiley & Sons, Inc. 
+Elman, J. L. (1990). Finding Structure in Time. Cognitive Science, 14, 179-211. 
+Gal, Y., & Ghahramani, Z. (2015). A Theoretically Grounded Application of Dropout in 
+Recurrent Neural Networks. arXiv. Retrieved from 
+https://arxiv.org/abs/1512.05287 
+Gans, N., Koole, G., & Mandelbaum, A. (2003). Telephone call centers: tutorial, review, 
+and research prospects. Manufacturing and Service Operations Management, 
+5:79-141. 
+Gans, N., Koole, G., & Mandelbaum, A. (2003). Telephone Call Centers: Tutorial, 
+Review, and Research Prospects. Manufacturing & Service Operations 
+Management, 79-141. 
+Harvey, A. (1990). Forecasting, Structural Time Series Models and the Kalman Filter. 
+London: Cambridge University Press. 
+He, K., Zhang, X., Ren, S., & Sun, J. (2015). Delving Deep into Rectifiers: Surpassing 
+Human-Level Performance on ImageNet Classification. arXiv. Retrieved from 
+https://arxiv.org/abs/1502.01852 
+Hochrieter, S., & Schmidhuber, J. (1997). Long short-term memory. Neural 
+Computation, 9(8), 1735-1780. 
+Ibrahim, R., & L'Ecuyer, P. (2013). Forecasting call center arrivals: Fixed-effects, mixed-
+effects, and bivariate models. Manufacturing & Service Operations Management, 
+72-85. 
+Ibrahim, R., Ye, H., L'Ecuyer, P., & Shen, H. (2016). Modeling and forecasting call 
+center arrrivals: A literature survey and a case study. International Journal of 
+Forecasting, 865-874. 
+Karl, A. T., Yang, Y., & Lohr, S. L. (2013). Efficient maximum likelihood estimation of 
+multiple membership linear mixed models, with an application to educational 
+value-added assessments. Computational Statistics & Data Analysis, 59, 13-27. 
+Kenward, M. G., & Roger, J. H. (2009). An Improved Approximation to the Precision of 
+Fixed Effects from Restricted Maximum Likelihood. Computational Statistics 
+and Data Analysis(53), 2583–2595. 
+Kingma, D., & Ba, J. (2014). Adam: A Method for Stochastic Optimization. arXiv. 
+Retrieved from https://arxiv.org/abs/1412.6980v8 
+
+Laptev, N., Yosinski, J., Li, L. E., & Smyl, S. (2017). Time-series Extreme Event 
+Forecasting with Neural Networks at Uber. ICML 2017 Time Series Workshop. 
+Sydney. Retrieved from http://roseyu.com/time-series-
+workshop/submissions/TSW2017_paper_3.pdf 
+Reddi, S., Kale, S., & Kumar, S. (2018). On the Convergence of Adam and Beyond. 
+International Conference on Learning Representations. Retrieved from 
+https://openreview.net/forum?id=ryQu7f-RZ 
+Rushing, H., Karl, A., & Wisnowski, J. (2014). Design and Analysis of Experiments by 
+Douglas Montgomery: A Supplement for Using JMP. Cary: SAS Institute. 
+Winters, P. (1960). Forecasting sales by exponentially weighted moving averages. 
+Management Science, 6(1): 127-137. 
+Zhu, L., & Laptev, N. (2017). Deep and Confident Prediction for Time Series at Uber. 
+arXiv. Retrieved from https://arxiv.org/abs/1709.01907 
+ 
+ 
+
diff --git a/9tAzT4oBgHgl3EQfg_wg/content/tmp_files/load_file.txt b/9tAzT4oBgHgl3EQfg_wg/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf,len=1155
+page_content='Lessons Learned Applying Deep Learning Approaches to  Forecasting Complex Seasonal Behavior  Andrew T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Karl1, James Wisnowski1, Lambros Petropoulos2  1Adsurgo LLC, Pensacola, FL  2USAA, San Antonio, TX  Abstract  Deep learning methods have gained popularity in recent years through the media and the  relative ease of implementation through open source packages such as Keras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' We  investigate the applicability of popular recurrent neural networks in forecasting call  center volumes at a large financial services company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' These series are highly complex  with seasonal patterns - between hours of the day, day of the week, and time of the year -  in addition to autocorrelation between individual observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Though we investigate the  financial services industry, the recommendations for modeling cyclical nonlinear  behavior generalize across all sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' We explore the optimization of parameter settings  and convergence criteria for Elman (simple), Long Short-Term Memory (LTSM), and  Gated Recurrent Unit (GRU) RNNs from a practical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' A designed  experiment using actual call center data across many different “skills” (income call  streams) compares performance measured by validation error rates of the best observed  RNN configurations against other modern and classical forecasting techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' We  summarize the utility of and considerations required for using deep learning methods in  forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Key Words: ARIMA, Time Series  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Introduction  Member contact call centers receive fluctuating call volumes depending on the day of the  week, the time of day, holidays, business conditions, and other factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' It is important for  call center managers to have accurate predictions of future call volumes in order to manage  staffing levels efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The call center arrival process has been well documented and  explored in the literature (Gans, Koole, & Mandelbaum, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' In the application presented  here, there are several different “skills” (or “splits”) to which an incoming call may be  routed – depending on the capabilities of the call center agents – and an arrival volume  forecast is required for each skill in the short term for day-ahead or week-ahead predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  The weekly seasonality found in call arrivals can be modeled effectively through a variety  of methods to include Winter’s Seasonal Smoothing (Winters, 1960) or Autoregressive  Integrated Moving Average (Box & Jenkins, 1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Some accessible references for many  of these concepts aimed at the practitioner are well documented in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Bisgaard & Kulachi, (2007, 2008)) while recommended texts are Bisgaard & Kulachi  (2011) and Montgomery, Jennings & Kulachi (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content="  Aiming to improve on these classic methods, “doubly stochastic” linear mixed models  (Aldor-Noiman, Feigin, & Mandelbaum, 2009) have effectively modeled additional  complexities as outlined in a recent review paper from Ibrahim, Ye, L'Ecuyer, & Shen  (2016)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Similarly, Recurrent Neural Networks (RNNs) have been recommended as deep  learning approaches to forecast call volume for a wireless network (Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=', 2017) in  addition to numerous other applications including ride volumes with Uber (Zhu & Laptev,  2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' While the doubly stochastic and RNN approaches to predicting call volumes offer  greater flexibility in modeling complex arrival behavior by incorporating exogenous  variables, this flexibility comes at the cost of greater computational and programming  complexity (as well as greater prediction variance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' This paper explores practical aspects  of managing that complexity for these models, applies the models to actual call volumes  recorded by a large financial services company, and compares the prediction capability to  that of the more traditional Winters smoothing and ARIMA models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  First, we modify computational aspects of the doubly stochastic approach proposed by  Aldor-Noiman, Feigin, & Mandelbaum (2009) to improve call center forecasting  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Doubly stochastic implies a two-level randomization where not only are call  arrivals random variables, but also the call arrival mean parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Forecasts are produced  by taking advantage of the unique correlation structure for each split while accounting for  trend, seasonality, cyclical behavior, and serial dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The doubly stochastic model  is more complex than ordinary regression as it accounts for both inter- and intra-day  correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' We suggest modifications to the originally proposed approach that lead to more  stable convergence and more flexible behavior when many splits need to be fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Secondly, we consider how RNNs may be used to model incoming call volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Whereas  “traditional” densely connected, feedforward neural networks process each data point  independently, RNNs process sequences according to temporal ordering and retain  information from previous points in the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' As it processes points within sequences,  the RNN maintains states that contain information about what it has seen previously in the  sequence (Chollet & Allaire, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' This intra-sequence memory is useful in time series  applications to autocorrelated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' In the context of call center volumes, these sequences  could be constructed to correspond to individual days of observations over a fixed number  of (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' 30 minute) periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' (2017) consider three different RNN architectures  to model incoming call volume over a mobile phone network: Elman Recurrent Neural  Networks (ERNN) (Elman, 1990), Gated Recurrent Units (GRU) (Cho, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=', 2014), and  Long Short Term Memory (LSTM) (Hochrieter & Schmidhuber, 1997), listed in order of  increasing complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  These three RNNs, along with the dense neural network, are now available via the R Keras  package (Allaire & Chollet, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Once code has been written for one of the RNNs, the  user can switch between the other two by toggling a single option (and, after data  reformatting, switch to a dense network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' This offers the potential – via a designed  experiment – to produce a pragmatic answer to the question of which type of (R)NN  provides the best fit to the process at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Whereas Bianchi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Maiorino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Kampffmeyer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Rizzi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' & Jenssen (2017) created their experimental design by randomly generating points  within the design space and then selected the design that lead to the minimum error rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  we create a full factorial design (treating all factors as categorical to allow arbitrary shape  in the otherise (discrete) continuous factor of number of nodes) and then explore the  behavior of the error rates across the design space with a profiler for the resulitng linear  model for the error rate as a function of the NN settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Unlike ARIMA or regression  (including doubly stochastic) modeling approaches for time series, there is a stochastic  behavior in the predictions made by neural networks due to the use of randomly initialized  weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Unless the seed for the software’s random number generator is fixed, repeated  fitting of the same neural network will lead to different predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The amount of  variation in the resulting predictions depends on the complexity of the network and on the  steps that have been taken to avoid overfitting, including early stopping of the optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  When selecting a model configuration, we will not only want to minimize the expected  error rate, but also minimize the variability in the error rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' To this end, we seek to  minimize the upper 95% prediction interval on the testing error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The NN study  proceeds in two phases where a screening experiment first identifies the most useful  (R)NN, followed by a more comprehensive performance study against common  forecasting approaches across many more skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Section 2 describes the doubly stochastic model for call volumes and how modifications to  the originally proposed computational approach can lead to improved convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Section 3 details how a full factorial design is used to characterize the performance of RNN  options as a function of five factors (and their interactions) on the resulting short-term  forecast error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Additionally, Section 3 describes the selection of the model  configuration that leads to the minimum upper bound on the 95% prediction interval for  the testing error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Due to the number of different model configurations that must be run  along with the computational complexity of RNNs, the first phase discussed in Section 3  considers only a limited number of skills and validation days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' In Section 4, the best  performing RNNs are run over a larger validation set and over all call center skills to  compare the performance to the doubly stochastic mixed model approach, and to ARIMA  as well as Winters smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=" Stable Settings for Fitting the Mixed Model  There are two distinct influences on call volumes that induce a correlation between the  observed call counts, violating the independence assumption made by ordinary least  squares regression models that might be used to model the volumes (Ibrahim & L'Ecuyer,  2013)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Within a given day, some event may lead to more/fewer calls than expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' For  example, unexpected behavior in the stock market in the morning may lead to an increased  number of calls for the rest of the day at a financial services contact center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' This is intra-  day correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Likewise, there are systemic processes responsible for inter-day  correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Heuristically, if we noticed that the residuals are very large and positive  throughout the day today caused by a weather event for example, we might also expect a  larger-than-average call load tomorrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Ignoring correlation between subsequent  observations leads to inaccurate standard errors and prediction intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' In addition,  although the estimates from a linear regression may be unbiased in the presence of  correlated residuals, they will not be efficient (Demidenko, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content="  It is typical for call center regression models to include a day-of-week by period-of-day  interaction (Ibrahim, Ye, L'Ecuyer, & Shen, 2016)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' In a call center open five days per  week with 32 half-hour periods per day, this interaction involves 160 fixed-effect  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' In addition, a call center may require forecasting for holidays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Aldor-  Noiman, Feigin, & Mandelbaum (2009) exclude holidays when training their model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  however, we cannot ignore these days because some splits operate on holidays and may  exhibit different behavior on those days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' In order to capture this behavior, we include a  holiday indicator (holiday_ind) by period-of-day interaction effect in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  However, some training data sets may include only a single holiday, leading to high  variance in the parameter estimates for this effect (each period observation from that one  day becomes the new estimate for that period during holidays).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' To reduce the variability  of these estimates, we combine groups of 3 periods together on holidays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' That is, periods  {1, 2, 3} are assigned p_group = 1, periods {4, 5, 6} are assigned p_group = 2, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The  p_group*holiday_ind interaction is included in the fixed effect structure as an additive  effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Following Aldor-Noiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' (2009), we fit a linear mixed model with correlated errors  to the transformed call counts  𝑌 = 𝑿𝛽 + 𝒁𝑏 + 𝜀  where 𝑌 is the vector transformed call counts, 𝑌 = √𝑐𝑜𝑢𝑛𝑡 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='25 𝑿 is a matrix containing the levels of the fixed effects for each observation 𝛽 is the vector of fixed effects parameters containing a day-of-week*period-of-day  interaction and a p_group*holiday-indicator interaction 𝒁 is a binary coefficient matrix for the random day-to-day effects in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  There is one column for each day in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' 𝑏~𝑁(0, 𝑮 ) is the vector of random day-to-day effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Each unique day in the data  set is represented by one random effect in b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' G follows a first-order autoregressive  structure, AR(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' 𝜀~𝑁(0, 𝑹 ) is the vector of error terms (residuals), allowing 𝜀 to potentially follow  an AR(1) process within days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Thus, R is a block-diagonal matrix, with one AR(1)  block for each day in the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' This accounts for the potential correlation in  residuals from proximal periods within days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  The full model allows for complex correlation structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' However, for some splits (within  particular training data sets), there may be only sporadic and sparse occurrences of call  arrivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' This can lead to slow or failed model convergence in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Aldor-Noiman  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' (2009) address this by estimating the doubly stochastic model in two steps: first, the  inter-day correlation (G) is estimated using the aggregated total call counts from each day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  These parameters are then held constant in a second call to SAS PROC MIXED while 𝛽  and 𝑹 are estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Indeed, PROC MIXED can experience convergence problems when the solutions lie on  the boundary of the parameter space, such as when variance components are zero (Karl,  Yang, & Lohr, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' However, after making modifications to the default PROC MIXED  settings, we were reliably able to achieve convergence of the full model with the joint  optimization of (𝛽, 𝑮, 𝑹) in a single call to PROC MIXED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' In this regard, our approach  differs from that of Aldor-Noiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' (2009): we fit all of the model parameters jointly  (with a single call to PROC MIXED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' This will lead to reduced bias in the estimates for  the models that do converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  We improved convergence rates by changing the convergence criterion used by SAS  PROC MIXED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' By default, SAS ensures that the sum of squared parameter gradients  (weighted by the current Hessian of the parameter estimates) is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  However, in the presence of strong correlations in the doubly stochastic model, the  parameter estimates may lie near the boundary of the parameter space, meaning the  gradients may not approach 0 with convergence (Demidenko, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' As an alternative, we  declare convergence when the relative change in the loglikelihood between iterations is  sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Additionally, we employ Fisher scoring during the estimation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Fisher scoring is more stable for models with complex covariance structures and can lead  to better estimates of the asymptotic covariance (Demidenko, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Finally, since our  application only uses the call volume point estimates and not the associated standard errors  or tests of significance, we specify ddfm=residual to avoid spending substantial time  calculating appropriate degrees of freedom for the approximate F-tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' If confidence or  prediction intervals are needed, this value should be set to ddfm=kenwardrodger2 in order  to calculate Satterthwaite approximations for the degrees of freedom and to apply the  Kenward-Rodger correction (Kenward & Roger, 2009) to the standard errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The code for  our modified approach appears in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Figure 1 Modified SAS code for the Doubly Stochastic Model  The square root transformation is applied to reduce the right skew in the observed call  volumes, and to stabilize the variance of the observations since quantities such as call  volumes tend to follow a Poisson distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The approach in Figure 1 employs a normal  approximation of this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' We experimented with fitting a mixed Poisson regression to  the untransformed call volumes (via PROC GLIMMIX), but found that the run times  became unfeasibly long (even when using the default pseudolikelihood approach and  avoiding integral approximation) with no noticeable improvement in error rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Choosing Recurrent Neural Network Configurations with a Designed Experiment  Generally, neural networks consist of layers of weights and nonlinear activation functions  that are used to relate inputs (predictors) to outputs (targets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Outputs from each layer are  passed sequentially to the next layer as an input vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The complexity of each layer is  determined by the length of the output vector (number of nodes) it produces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' A loss  function is used to compare the output of the final layer of the neural network to the  provided targets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' call volumes), and an optimizer function provides updated values of  the weights each node that will decrease the resulting loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The “depth” of the model is  controlled by the number of layers that are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' This “depth” is the source of the phrase  “deep learning”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' For example, in image processing applications with convolutional neural  networks, the different layers can be shown to represent different levels of granularity of  detail in an image (Chollet & Allaire, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Besides the number of layers and the  number of nodes per layer, there are a number of choices that must be made regarding the  properties of the optimizer, the distribution of the random initialization of the parameter  weights, and the shape of the activation function(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  In a traditional, densely connected network, the individual observations are assumed to be  independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' A simple example using output from JMP Pro 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='1 helps to illustrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Suppose we want to fit a densely connected neural network to predict the standardized  call count using only the previous day’s standardized call count at the same period (the  lag-32 of the call count, since there are 32 periods per day in the example) as a predictor  with one node in one layer, using a hyperbolic tangent activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' This network  shown in Figure 2 with resulting weights shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  proc mixed data=training_data scoring=50 maxiter=150 maxfunc=10000 convf=1E-6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  class day_of_week period day_num split p_group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  by split;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  /* The fixed effects */  model transf_call_count=day_of_week*period p_group*holiday_ind/  noint ddfm=residual outp=pred_call_count_output notest;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  /* The day-level random effects */  /* Note: day num copy is not included in the clAss statment and is numeric *  random day_num / type=sp(pow)(day_num_copy);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  /* The period-level correlated residuals */  run;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Figure 2 Densely connected neural network with one node in one layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Figure 3 Fitted weights from the network with one node in one layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Suppose that the lag-32 standardized call count is equal to 1 at a given period, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Then the  neural network predicts a standardized call volume of  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='6354 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='3923 ∗ TanH(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='5 ∗ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='4046 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='5323 ∗ 1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='847  for the current period, t, where TanH is the hyperbolic tangent function and the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='5  parameter is a fixed value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The nonlinear activation function provides the network with  the flexibility to model nonlinear relationships between the inputs and the response, as  well as interactions between the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  We next consider a slightly more complex network with two layers using 2 nodes in the  first layer and 1 node in the second layer (Figure 4) with parameter estimates shown in  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Figure 4 A densely connected neural network with 2 layers using two nodes in the first  layer and one node in the second layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Lag[Standardize[cnt_ call], 32]  Standardize[cnt_call]Estimates  Parameter  Estimate  H1_ 1:Lag[Standardize[cnt_ call], 32]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='5323  H1_1:Intercept  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='4046  Standardize[cnt call]_ 1:H1_ 1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='3923  Standardize[cnt call]_2:Intercept  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='6354Lag[Standardize[cnt_ call], 32]  Standardize[cnt_call  Figure 5 Fitted weights from the network with two layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Again assuming that the lag-32 standardized call count is 1, the predicted value is  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='82 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='9517  ∗ 𝑇𝑎𝑛𝐻 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='5  ∗ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='2416 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='7382 ∗ 𝑇𝑎𝑛𝐻(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='5 ∗ (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='3633 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='8785 ∗ 1)) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='3057  ∗ 𝑇𝑎𝑛𝐻(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='5 ∗ (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='0824 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='4124 ∗ 1)))) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='843  In the densely connected network, each observation is processed independently and there  is no “memory” of what happened in the previously processed observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' In time series  applications, however, there is a temporal ordering that the data are recorded in, and there  may be correlation between nearby observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' For example, a spike or drop in call  volume might persist over several periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' To address this potential, recurrent neural  networks record information when fitting each observation that is then provided as a  model input when fitting later observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  In a simple (Elman) RNN layer, the output from each node (the output of the TanH  functions, referred to as the “state”) is recorded and stored, and used as an input for the  same node when processing the next observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Note that there is one state recorded for  each node in the layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' For the single layer network example (Figure 3), the state was  calculated as 𝑠𝑡 = TanH(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='5 ∗ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='4046 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='5323 ∗ 1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='44 when the lagged  standardized call volume was equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' A simple RNN learns an extra parameter (say,  u) to act as a coefficient for the stored state, and the activation function TanH(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='5 ∗  (𝑤0 + 𝑤1 ∗ 𝑋𝑡)) that is used by the dense network would be replaced by  𝑠𝑡 =  TanH(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='5 ∗ (𝑤0 + 𝑤1 ∗ 𝑋𝑡 + 𝑢 ∗ 𝑠𝑡−1)) in order to fit a simple RNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The LSTM and  GRU RNNs also use the recorded state when making predictions for the current time  period, along with products of additional activation functions that are designed to carry  state information further in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Details of these additional structures are explained in  Section 3 of Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' For our purpose it is sufficient to note that the GRU  network is extremely similar to the LSTM, albeit less complex due to the ommison of a  group of paramters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Chollet & Allaire (2018) remark that Google Translate currently runs  using an LSTM with seven large layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  It is not clear a priori which of these four neural networks is most appropriate for a  particular call center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Furthermore, it is possible that each of these networks may have a  different optimal depth and structure when applied to the call center data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' A designed  experiment is run to identify the optimal model type and structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Estimates  Parameter  Estimate  H2_1:Lag[Standardize[cnt _ call], 32]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='8785  H2_1:lntercept  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='3633  H2_2:Lag[Standardize[cnt_call], 32]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='4124  H2_2:Intercept  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='0824  H1_1:H2_1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='7382  H1_1:H2 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='3057  H1_1:lntercept  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='2416  Standardize[cnt_ call]_1:H1_1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='9517  Standardize[cnt_ call]_2:Intercept  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='82003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='1 Data for the Experiment  We analyze call volumes aggregated into 30 minute periods from 3 different large-volume  skills in an operational call center for the months of March-June 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' All of the models  under consideration are used to forecast next-day call volumes using 5 weeks of training  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Each day contains 32 30-minute periods during which the call center is operating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  This application considers the Monday through Friday behavior of the call skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Due to  the use of the one-week lagged observations as a predictor in the neural networks, the first  week of training data is not included in the predictor matrix (since the prior week’s call  volumes are unknown), meaning each training data set consists of 4*5*32=640  observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The designed experiment in this section will evaluate the methods using a  holdout period of the five one-day ahead predictions during last week in June using the  three largest skills from the call center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Section 4 will then fit a reduced set of models over  all skills for 60 day-ahead predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='2 Neural Network Input Factors  Each network makes use of the one-hot encoding (via a binary indicator matrix) of the day  of week and the one-hot encoding of period of day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Furthermore, to capture the day- and  week-long correlations, the networks are also fed the call volumes for the same period in  the previous day when modeling the current day, as well as the call volumes for the same  period on the same day of last week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' One-period lagged call volumes are not included, as  this is the purpose of the within-sequence memory of the RNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Other inputs include a  binary indicator for whether the current day is a holiday, a binary indicator for whether  yesterday was a holiday, a binary indicator for whether or not last week’s observation was  recorded on a holiday, and day number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The day number is a continuous counter for the  number of the given day in the data set, which would potentially allow the neural network  to detect trends across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' All told, these account for 42 vectors of input for the neural  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' There is no need to create indicator columns for interactions between day of  week and period (or any other factors) as the neural network will automatically detect them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' (2017) application of hourly call volumes displays strong lag-24 correlation,  representing a period-of-day effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' They remove this seasonality by differencing the data  at lag-24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' By contrast, we do not difference the call volumes, but instead include period-  of-day (along with day-of-week) as exogenous variables and allow the neural network to  detect this seasonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' This approach allows the network to detect the expected interactions  between period-of-day and day-of-week, as well as any other input factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  While these input vectors are included for all models, there are three final input vectors  whose (joint) inclusion is treated as an experimental factor: the same-period predictions  from the mixed model approach (Aldor-Noiman, Feigin, & Mandelbaum, Workload  forecasting for a call center: Methodology and a case study, 2009), from a Winters  smoothing model, and from a seasonal ARIMA(1, 0, 1)(0, 1, 1)160 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The inclusion  of the predictions from these models as an input to the neural network is an original  approach that gives the network the opportunity to form predictions that may be thought  of as corrections to those from these traditional models, based on potential interactions  with other included factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' For brevity, we refer to this as the mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='cheat option, since it  allows the neural networks to “cheat” by looking at the predictions generated by these other  three models when forming its own predictions for the same time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' If none of the  other 42 input vectors were included with these three, this would represent a supervised  learning approach to forming a dynamically weighted average of these three model  predictions in order to create a single “bagged” prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='3 Network Configuration Aspects Treated as Experimental Factors  The designed experiment considered five different factors of structural settings of the  neural networks: model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='type {dense, simple (Elman) RNN, GRU, LSTM}, nlayers {1,  2}, nnodes (per layer) {25, 50, 75, 100}, kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='reg {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='0001} and mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='cheat  {FALSE, TRUE}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The L2 regularizer adds kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='reg times each weight coefficient to  the total loss function for the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Similar to Lasso regression, this helps bound the  magnitude of the model coefficients and could potentially help prevent overfitting the  training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  We fit a full factorial design (requiring 128 runs) for these five factors, which allows us  to test for the presence of up to five-way interactions between the five factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Figure 6  shows the first 6 runs of the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The design is replicated over the 3 largest splits and  across the 5 subsequent one-day ahead forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' This produces a total of 1920 runs for  the entire experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The replication allows for behavior to be averaged over different  days and for a more detailed exploration of how the variance of the error rates depends on  each factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Figure 6 First 6 runs of the designed experiment  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='4 Static Considerations for Neural Network Configuration  The input for the classical dense neural network is a 640x42 matrix (640x45 if  mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='cheat=TRUE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The dense network does not consider the temporal ordering of the  640 observations (outside of the explicit inclusion of the day- and week-lagged  observations as inputs): the observations are shuffled after each epoch and then processed  in batches (we used a batch size of 32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  By contrast, the RNNs consider the ordering of the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The input for the RNNs  is a 20x32x42 array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' This indicates to the RNN that there are 20 batches (days) of 32  timesteps (periods) with 42 predictors per time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' By default, the batches are treated  independently and the timesteps within each batch are potentially correlated (via the  persistence of the states in the RNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' If the batches themselves are presented in a  temporal order (as is the case in our application), then this can be indicated to the Keras  model via the STATEFUL=TRUE option and by disabling the shuffling of batches  during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' This retains the model weights from batch-to-batch (day-to-day) to allow  for possible long-term behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' However, we found that using the STATEFUL option  led to a failure to converge in some skill*day combinations and the resulting validation  error rates were not significantly different from those generated without the STATEFUL  option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' This seems to indicate that the dependence on prior days’ behavior is already  captured by the inclusion of the one-day and one-week lagged observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Due to the  occasional convergence issues, the results in the sequel are generated with  STATEFUL=FALSE (and batch shuffling enabled during training).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' It would have also  been possible to fit the model with week- or month-long batches by training the RNN on  a 4x160x42 or a 1x640x42 array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' We did not consider the week-long batches, but the  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='type  nlayers mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='cheat nnodes  kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='reg  layer_simple_rnn  2  FALSE  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='0001  layer_ gru  FALSE  25  0  layer_Istm  1  TRUE  100  0  layer_gru  TRUE  75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='0001  layer_gru  TRUE  75  0  layer_simple_rnn  FALSE  100  0month-long batches tended to produce inferior predictions to the day-long batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' This  could possibly be due to the lack of long term correlations in the data and the fact that the  smaller batches allow for the shuffling of the ordering that the days are fed through the  gradient-optimization routine of the neural network, which can improve the model fit by  preventing the model from overweighting the first observations that are provided to the  network in each epoch (Chollet & Allaire, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Note that the model weights are reset and the model is retrained for each skill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' This is in  contrast to the approach taken by Zhu & Laptev (2017) in which a single network is fit to  accommodate disparate behavior from different cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' As discussed in Section 5, a single  multi-output network could potentially be built to model all of the skills at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  While they were not included as experimental factors in this application, we also noticed  a significant relationship between the quality of the predictions and the optimization  routine employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Extensive pilot experimentation led to our use of the AMSGrad variant  (Reddi, Kale, & Kumar, 2018) of the Adam optimizer (Kingma & Ba, 2014), with a  learning rate decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' We would recommend including both the optimizer and  the optional learning rate decay as factors in the designed experiment for parameter  tuning in future problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  We found it was important to tune the number of epochs for each model fit using  validation data (the last week in the training set) by first fitting 500 epochs for each  model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' taking a moving average (with a window size of 10 epochs) of the resulting  WAPEs on the validation data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' and then refitting the model (on both the training and the  validation data in order to predict an additional day which was held out as a test set) with  the number of epochs that produced the minimum validation WAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The moving  average is important due to the volatile and non-monotonic behavior we observed in the  individual recorded WAPEs and helps to find a relatively stable region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Consistent with  previous findings (Bianchi, Maiorino, Kampffmeyer, Rizzi, & Jenssen, 2017), we noticed  that the RNNs take many more epochs to converge than the dense neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Other  researchers (such as Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=') have used more than 500 epochs when fitting RNNs,  so this upper bound should also be considered as an important factor when building an  RNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  While we also experimented with recurrent dropout (Gal & Ghahramani, 2015) to  prevent overfitting, it led to degraded performance in the early iterations of our  experiment and we removed it from consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' However, this could simply be due to  features of this particular application, such as the relatively short training period of five  weeks: Chollet & Allaire  (2018) strongly advocate the use of dropout and recurrent  dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  The models experienced improved errror rates after switching the kernel initializer for the  random weights to the He normal initializer (He, Zhang, Ren, & Sun, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' We used the  relu activation function exclusively, although this choice could also impact the quality of  the resulting model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' And while our application did not detect any two-factor  interactions among the five experimental factors, it is possible that some of these  additional factors could depend on the type of (R)NN being used, meaning there would  be an interaction between these factors and model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  A final contributing factor is the batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' This determines how many observations are  processed before the gradients are updated: when fitting Keras models on a GPU, amount  of memory available on the GPU can be a limiting factor on the batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' This choice is  more constrained in the RNNs, where each batch is a single day/week/month (determined  by the number of timesteps specified in the input array to the RNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' By contrast, the  batch size for a dense network can be set between 1 and the number of observations in  the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' We noticed significant differences in the error rates from the dense  model depending on what batch size was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='5 Experimental Response  While mean squared error (MSE) is frequently used to evaluate and compare predictive  models, this is a poor metric for the call center application as it will give undue focus to  the low call volume periods at the beginning and end of each day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=" Instead, the weighted  absolute percentage error (WAPE) is recommended for call volume modeling (Ibrahim,  Ye, L'Ecuyer, & Shen, 2016)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' This weights the absolute percentage error in each period  by the number of calls received in that period and is defined by  𝑊𝐴𝑃𝐸 =  ∑  |𝑌𝑖 − 𝑌̂𝑖|  𝑛  𝑖=1  ∑  𝑌𝑖  𝑛  𝑖=1  where 𝑌𝑖 and 𝑌̂𝑖 are the observed and predicted volumes, respectively, for each 30 minute  period 𝑖 = 1, … , 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Because WAPE is our metric of interest, the models are compiled to  use a mean absolute error loss-function, which minimizes the numerator of the WAPE  (the denominator is static).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  For each row of the experimental table (Figure 6), the specified neural network is fit  (independently) to model five subsequent days of each split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' That is, day 1 is predicted  using the previous 5 weeks leading up to day 1, then the model is reset and day 2 is  predicted using the 5 weeks leading up to day 2, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The vector of five one-day ahead  predictions is compared with the observed call volumes (that were not visible to the  model during training), and the resulting WAPE is recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='6 Analysis of Experimental Results  Figure 7 gives a typical output of the prediction error across the different formulations of  the neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' GRU often has the lowest forecast error or at least is consistently  close to the lowest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The other procedures tend to have much more unstable performance  based on the choice of nodes and layers as well as across the days and splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Figure 7 Forecast error by model type and settings for forecast day 5 split 3  Regression analysis is used to determine the statistically significant factors and  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' In order to control a heavy right-skew in the recorded WAPEs, an inverse  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='type  NN Classic  RNN GRU  RNN LSTM RNN Simple  WAPE  WAPE  WAPE  WAPE  nlayers  nnodes  Mean  Mean  Mean  Mean  1  25  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
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+page_content='0%  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
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+page_content='0%transformation is applied to use as the response in the regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Figure 12  displays the original skewness and the transformation to normality after taking the  inverse of the WAPEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='7 Loglinear Variance Regression Model  The mean structure of 1/WAPE is modeled by including the main effects of the five  experimental factors (model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='type, nlayers, mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='cheat, nnodes, kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='reg) and all of  the interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' In addition, split, file (which represents the validation day), and split*file  are included as blocking factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  While an ordinary regression model for the full factorial design would assume the same  error variance for all responses across the design space, the loglinear variance model  allows for the error variance itself to be modelled as a function of the input factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' This  is appealing for this application, where parsimonious networks may be expected to  demonstrate less variability during repeated fittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  For each factor setting, the loglinear variance model produces a predicted mean of  1/WAPE and a predicted standard deviation of 1/WAPE, representing the variability due  to the random initial weighting of the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  We include model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='type, nlayers, mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
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+page_content='L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='reg, file, and split as  factors in the loglinear variance model (using JMP Pro 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='1), which models the log of the  error variance as a linear combination of these factors (which are all treated as  categorical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' With the exception of kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='reg, all of these effects are found to be  significantly associated with the error variance (Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Figure 8 Factors with a significant impact on WAPE variability  For the mean model, model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='type, nlayers, mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='cheat, nnodes, day, split, and file*split  were found to be significantly associated with 1/WAPE (Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Neither kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
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+page_content='reg  nor any of the interaction terms involving the NN architecture were found to be  significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Variance Effect Likelihood Ratio Tests  Source  Test Type  DF  ChiSquare  Prob>ChiSq  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
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+page_content='0001*  Figure 9 Factors that are significantly associated with the mean WAPE  Removing the insignificant interaction terms (but allowing kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='reg to remain in  both the mean and variance models) produces the profiler shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Notice the  large standard deviation associated with the LSTM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The dependence on file and  split is not shown, since there are no interactions modeled between these and the other  experimental factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' While we want to maximize 1/WAPE, we also want to minimize  the standard deviation of 1/WAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' That is, we would like to maximize the lower 95%  prediction interval of 1/WAPE (or minimize the upper 95% PI of WAPE) in order to find  the model configuration that will give a combination of relatively good results on average  while being protected against large errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Fixed Effect Tests  Source  Nparm  DF  DFDen  F Ratio  Prob > F  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
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+page_content='reglayers, using mixed model forecasts, with 50 nodes while robust to the regularization  parameter value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Note that the traditional dense neural network does have consistently  worse forecast error across all scenarios and could not be recommended despite having  the lowest variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The prediction interval on forecast error is an alternative view that is  preferred by many practitioners where the goal is to minimize the width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Figure 11 (lower  is better) clearly shows GRU is the preferred solution using the mixed model forecasts  with 50 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Figure 12 Representative distribution of WAPE and 1/WAPE resulting from the designed  experiment  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Comprehensive Performance Study  Based on the results of the designed experiment for NN error rates, we perform a further  study with actual call center data across 36 skills rather than only 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' We consider a  consistent 5-week training period advancing across multiple months of data to produce 60  one-day ahead predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' These predictions do not use the actual data for the forecasted  day during model training and can be viewed as a validation set for the trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  This allows us to compare the relative performance of the mixed model and the neural  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' We also include the error rates from the ARIMA and Winters seasonal  smoothing models for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Though we could have simply chosen the GRU RNN with 50 nodes on each of 2 layers  with kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='reg complexity parameter set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='0001 as the only representative RNN  based on the designed experiment, we decided to include all 4 neural network methods  Distributions  WAPE  0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
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+page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='5  1/WAPE  0  5  10  15  20  25  30screened by using these same parameter settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' It is possible with the added skills –  with many having much lower call volumes – that another RNN model could work better  than GRU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='1 Results: One-day ahead Predictions Over 60 Separate Validation Days  Figure 13 provides the forecast errors averaged across all 60 days for the 12 splits with  the highest call volume sorted in descending call volume order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Note that results in this  section are generated by the (R)NNs with the mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='cheat option disabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The Doubly  Stochastic Mixed Model has the lowest average forecast error for all but Split 5540 and is  often significantly lower than the competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Winters Seasonal Exponential Smoothing  also performs quite well given its relative simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The GRU performance confirms the  results from the designed experiment as usually the best RNN and always competitive  with the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The highly complex LSTM recurrent neural network has very large forecast  errors for many of these splits and cannot be recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Note also that forecast error  generally increases for all methods as call volume decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Figure 13 Average WAPE forecast errors across 60 separate validation days for high  call-volume splits  Figure 14 shows the similar trend of increasing error rates with decreasing call volumes  for the medium call volume splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The GRU recurrent neural network is usually  outperforming the other neural network methods and is closer to the error rates of the  Doubly Stochastic Mixed Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Figure 14 Average WAPE forecast errors across 60 separate validation days for medium  call-volume splits  For the low call volume splits displayed in Figure 15, the GRU and Simple recurrent  neural networks perform similarly and slightly better than Doubly Stochastic and Winters  for most of the splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The very low call volumes (last 3 splits) do seem to benefit from  the recurrent neural network formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Split  Sum Call Vol  ARIMA  Doubly Stoch  NN_Classic  RNN_GRU  RNN_LSTM  RNN_Simple  Winters  5000&5240  929754  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
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+page_content='8%  Figure 15 Average WAPE forecast errors across 60 separate validation days for low  call-volume splits  Overall, the best performing procedures were the Doubly Stochastic Mixed Model and  the GRU Recurrent Neural Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Figure 16 shows forecast error by split sorted by  call volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Generally, the mixed model does better for large and medium volume splits  (lower is better on the graph) while the RNN  model is more effective for the small volume—particularly the very small volume splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Figure 16 Forecast error rates ordered by call volume by split for Doubly Stochastic  (blue) and GRU (red)  Figure 17 presents a different view of this same pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' For each of the 60 one-day ahead  forecasts within each split, the GRU and Doubly Stochastic WAPEs are recorded, along  with the number of calls recorded for that split over the training and validation data  (sum_all_calls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' For each split, Figure 17 plots the percent of the 60 day-ahead forecasts  for which GRU “won” (GRU WAPE < Doubly Stochastic WAPE) against the log of the  median call volume recorded by that split over the 60 different pairs of training and  validation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' There appears to be a linear decrease of the relative performance of GRU  against Doubly Stochastic in the log of the call volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
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+page_content='6  IDoubly Stoch  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='5  RNN GRU  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
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+page_content='1 -  Errot  aber  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
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+page_content='7  pa  ubiay  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
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+page_content='1  HighCallVolum  Medium CallVolume  Low Call Volume  Figure 17 Percent of the 60 day-ahead forecasts for which the GRU WAPE was less than  the Doubly Stochastic WAPE for each of the 36 splits plotted against the log of the  median sum of all calls recorded for each split over each pair of training and validation  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  For the large call-volume splits, the extra flexibility of the GRU model does not lead to  improvements over the predictions generated by the mixed model approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Because the  mixed model computations are faster and the implementation is less complex, there does  not appear to be any benefit to running the neural networks for the high-volume splits for  this short-term application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' This is consistent with the findings of the Uber traffic volume  study when short-term predictions were considered (Zhu & Laptev, 2017), and would  possibly change if a longer training period (several months or multiple years) were used  for the call center data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='2 Improving GRU RNN by Using Doubly Stochastic Forecasts as a Covariate  Based on pilot studies during the designed experiment, the forecasting performance of the  GRU recurrent neural network often improved by integrating the forecasted value for the  validation days from the doubly stochastic, ARIMA, and Winters models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' This  “cheating” by using other models’ forecasts (shown as mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='cheat) proved to be a  significant benefit for the GRU model over these 60 predictions for each skill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The  WAPE for the held-out validation data was again the primary measure of performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Figure 18 displays the forecast errors for the high call volume skills averaged over the 60  validation days for the each of the neural networks when they use the mixed forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Note the side-by-side comparison of the RNN_GRU (no cheat) and GRU_cheat columns  where in most cases the “cheating” does result in improved forecasts and in those cases  where it is not better, it has only marginally declined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Additionally, the Simple_cheat  error rates compare favorably with the GRU_cheat while the LSTM_cheat suffers from  significantly poorer performance and instability issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' The doubly stochastic forecast is  still quite good and often the best choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' These results are also consistent when looking  at the medium and low call volume splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Therefore, we recommend using the forecasts  from a doubly stochastic or Winters model as inputs to recurrent neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Percent won by GRU vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Log[Median(sum all calls)]  100%  Percent won by GRU  %06  %08  70%  Percent won by GRU  %09  50%  40%  30%  20%  10%  0%  4  6  8  10  12  14  Log[Median(sum all_ calls)]  Figure 18 Forecast errors for high call volume splits averaged over 60 validation  forecast days allowing NN to “cheat” to improve predictions  The median of the GRU WAPE (mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='cheat=FALSE) minus the GRU WAPE  (mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='cheat=TRUE) (within each split/day combination, for a sample size of  36*60=2160) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='002 with a p-value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='0005 from the Wilcoxon signed rank test with a  null hypothesis that the differences were drawn from a population with median equal to  0, indicating that mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='cheat does tend to improve the performance of the GRU model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  A similar comparison of paired differences of error rates (all with mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='cheat=FALSE)  confirms that GRU outperforms the other (R)NNs: LSTM - GRU produces a median of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='0247 with a p-value of 1e-129, NN Classic - GRU produces a median of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='0690 with a  p-value of <1e-185, and Simple RNN - GRU produces a median of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='0021 with a p-value  of 1e-04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  References  Aldor-Noiman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=', Feigin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=', & Mandelbaum, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Workload forecasting for a  call center: Methodology and a case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Annals of Applied Statistics, 3(4),  1403-1447.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Aldor-Noiman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=', Feigin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=', & Mandelbaum, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
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+page_content=' An Improved Approximation to the Precision of  Fixed Effects from Restricted Maximum Likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
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+page_content=' Time-series Extreme Event  Forecasting with Neural Networks at Uber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' ICML 2017 Time Series Workshop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Sydney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Retrieved from http://roseyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
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+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' On the Convergence of Adam and Beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  International Conference on Learning Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Retrieved from  https://openreview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
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+page_content=' Design and Analysis of Experiments by  Douglas Montgomery: A Supplement for Using JMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Cary: SAS Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Winters, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' (1960).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Forecasting sales by exponentially weighted moving averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  Management Science, 6(1): 127-137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
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+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Deep and Confident Prediction for Time Series at Uber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content='  arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
+page_content=' Retrieved from https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfg_wg/content/2301.01476v1.pdf'}
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+e-print http://www.gm.fh-koeln.de/ciopwebpub/Konen22b.d/TR-Rubiks.pdf
+Towards Learning Rubik’s Cube with
+N-tuple-based Reinforcement Learning
+Wolfgang Konen
+Technical Report,
+Computer Science Institute,
+TH Köln,
+University of Applied Sciences,
+Germany
+wolfgang.konen@th-koeln.de
+Sep 2022,
+last update Jan 2023
+Abstract
+This work describes in detail how to learn and solve the Rubik’s cube game (or
+puzzle) in the General Board Game (GBG) learning and playing framework. We cover
+the cube sizes 2x2x2 and 3x3x3. We describe in detail the cube’s state representation,
+how to transform it with twists, whole-cube rotations and color transformations and
+explain the use of symmetries in Rubik’s cube.
+Next, we discuss different n-tuple
+representations for the cube, how we train the agents by reinforcement learning and
+how we improve the trained agents during evaluation by MCTS wrapping.
+We present results for agents that learn Rubik’s cube from scratch, with and without
+MCTS wrapping, with and without symmetries and show that both, MCTS wrapping
+and symmetries, increase computational costs, but lead at the same time to much
+better results. We can solve the 2x2x2 cube completely, and the 3x3x3 cube in the
+majority of the cases for scrambled cubes up to p = 15 (QTM). We cannot yet reliably
+solve 3x3x3 cubes with more than 15 scrambling twists.
+Although our computational costs are higher with MCTS wrapping and with sym-
+metries than without, they are still considerably lower than in the approaches of McAleer
+et al. (2018, 2019) and Agostinelli et al. (2019) who provide the best Rubik’s cube
+learning agents so far.
+1
+arXiv:2301.12167v1  [cs.LG]  28 Jan 2023
+
+Contents
+1
+Introduction
+4
+1.1
+Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+1.2
+Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+2
+Foundations
+6
+2.1
+Conventions and Symbols
+. . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+2.1.1
+Color arrangement
+. . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+2.1.2
+Twist and Rotation Symbols . . . . . . . . . . . . . . . . . . . . . . .
+6
+2.1.3
+Twist Types
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+2.2
+Facts about Cubes
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+2.2.1
+2x2x2 Cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+2.2.2
+3x3x3 Cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+2.3
+The Cube State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+2.4
+Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+2.4.1
+Twist Transformations . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+2.4.2
+Whole-Cube Rotations (WCR) . . . . . . . . . . . . . . . . . . . . .
+11
+2.4.3
+Color Transformations . . . . . . . . . . . . . . . . . . . . . . . . . .
+13
+2.5
+Symmetries
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+15
+3
+N-Tuple Systems
+16
+4
+N-Tuple Representions for the Cube
+18
+4.1
+CUBESTATE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+18
+4.2
+STICKER
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+18
+4.3
+STICKER2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+20
+4.4
+Adjacency Sets
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+21
+5
+Learning the Cube
+21
+5.1
+McAleer and Agostinelli . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+21
+5.2
+N-Tuple-based TD Learning
+. . . . . . . . . . . . . . . . . . . . . . . . . .
+24
+5.2.1
+Temporal Coherence Learning (TCL) . . . . . . . . . . . . . . . . . .
+25
+5.2.2
+MCTS
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+25
+5.2.3
+Method Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+27
+6
+Results
+27
+6.1
+Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+27
+6.2
+Cube Solving with MCTS Wrapper, without Symmetries
+. . . . . . . . . . .
+28
+6.3
+Number of Symmetric States . . . . . . . . . . . . . . . . . . . . . . . . . .
+28
+6.4
+The Benefit of Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . .
+29
+6.5
+Computational Costs
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+30
+7
+Related Work
+32
+2
+
+8
+Summary and Outlook
+33
+A Calculating sloc from fcol
+37
+A.1 2x2x2 cube
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+37
+A.2 3x3x3 cube
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+38
+B N-Tuple Representations for the 3x3x3 Cube
+38
+B.1
+CUBESTATE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+38
+B.2
+STICKER
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+39
+B.3
+STICKER2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+39
+B.4
+Adjacency Sets
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+40
+C Hyperparameters
+40
+3
+
+1
+Introduction
+1.1
+Motivation
+Game learning and game playing is an interesting test bed for strategic decision making.
+Games usually have large state spaces, and they often require complex pattern recognition
+and strategic planning capabilities to decide which move is the best in a certain situation.
+If algorithms learn a game (or, even better, a variety of different games) just by self-play,
+given no other knowledge than the game rules, it is likely that they perform also well on
+other problems of strategic decision making.
+In recent years, reinforcement learning (RL) and deep neural networks (DNN) achieved
+superhuman capabilities in a number of competitive games (Mnih et al., 2015; Silver et al.,
+2016). This success has been a product of the combination of reinforcement learning,
+deep learning and Monte Carlo Tree Search (MCTS). However, current deep reinforcement
+learning (DRL) methods struggle in environments with a high number of states and a small
+number of reward states.
+(a)
+(b)
+Figure 1: (a) Scrambled 3x3x3 Rubik’s Cube. (b) 2x2x2 cube in the middle of a twist.
+The Rubik’s cube puzzle is an example of such an environment since the classical
+3x3x3 cube has 4.3 · 1019 states and only one state (the solved cube) has a reward. A
+somewhat simpler puzzle is the 2x2x2 cube with 3.6 · 106 state and again only one reward
+state. Both cubes are shown in Fig. 1.
+The difficult task to learn from scratch how to solve arbitrary scrambled cubes (i.e.
+without being taught by expert knowledge, whether from humans or from computerized
+solvers) was not achievable with DRL methods for a long time. Recently, the works of
+McAleer et al. (2018, 2019) and Agostinelli et al. (2019) provided a breakthrough in that
+direction (see Sec. 5.1 and 7 for details): Their approach DAVI (Deep Approximate Value
+Iteration) learned from scratch to solve arbitrary scrambled 3x3x3 cubes.
+This work investigates whether TD-n-tuple learning with much lower computational de-
+mands can solve (or partially solve) Rubik’s cube as well.
+4
+
+1.2
+Overview
+The General Board Game (GBG) learning and playing framework (Konen, 2019; Konen and
+Bagheri, 2020; Konen, 2022) was developed for education and research in AI. GBG allows
+applying the new algorithm easily to a variety of games. GBG is open source and available
+on GitHub1. The main contribution of this paper is to take the TD-n-tuple approach from
+GBG (Scheiermann and Konen, 2022) that was also successful on other games (Othello,
+ConnectFour) and to investigate this algorithm on various cube puzzles. We will show that
+it can solve the 2x2x2 cube perfectly and the 3x3x3 cube partly. At the same time it has
+drastically reduced computational requirements compared to McAleer et al. (2019). We
+will show that wrapping the base agent with an MCTS wrapper, as it was done by McAleer
+et al. (2019) and Scheiermann and Konen (2022), is essential to reach this success.
+This work is at the same time an in-depth tutorial how to represent a cube and its
+transformations within a computer program such that all types of cube operations can be
+computed efficiently. As another important contribution we will show how symmetries
+(Sec. 2.5, 6.3 and 6.4) applied to cube puzzles can greatly increase sample efficiency and
+performance.
+The rest of this paper is organized as follows: Sec. 2 lays the foundation for Rubik’s
+cube, its state representation, its transformations and its symmetries. In Sec. 3 we in-
+troduce n-tuple systems and how they can be used to derive policies for game-playing
+agents. Sec. 4 defines and discusses several n-tuple representations for the cube. Sec. 5
+presents algorithms for learning the cube: first the DAVI algorithm of McAleer et al. (2019);
+Agostinelli et al. (2019) and then our n-tuple-based TD learning (with extensions TCL and
+MCTS). In Sec. 6 we present the results when applying our n-tuple-based TD learning
+method to the 2x2x2 and the 3x3x3 cube. Sec. 7 discusses related work and Sec. 8 con-
+cludes.
+1https://github.com/WolfgangKonen/GBG
+5
+
+2
+Foundations
+2.1
+Conventions and Symbols
+We consider in this paper two well-known cube types, namely the 2x2x2 cube (pocket
+cube) and the 3x3x3 cube (Rubik’s cube).
+2.1.1
+Color arrangement
+Each cube consists of smaller cubies: 8 corner cubies for the 2x2x2 cube and 8 corner,
+12 edge and 6 center cubies for the 3x3x3 cube. A corner cubie has 3 stickers of different
+color on its 3 faces. An edge cubie has two, a center cubie has one sticker.
+We enumerate the 6 cube faces with
+(ULF) = (Up, Left, Front) and
+(DRB) = (Down, Right, Back).
+We number the 6 colors with 0,1,2,3,4,5. My cube has these six colors
+012 = wbo = (white,blue,orange) in the (ULF)-cubie2 and
+345 = ygr = (yellow,green,red) in the opposing (DRB)-cubie.
+The solved cube in default position has colors (012345) for the faces (ULFDRB), i.e. the
+white color is at the Up face, blue at Left, orange as Front and so on. We can cut the cube
+such that up- and bottom-face can be folded away and have a flattened representation as
+shown in Figure 2.
+w
+b
+o
+g
+r
+y
+Figure 2: The face colors of the default cube in flattened representation
+2.1.2
+Twist and Rotation Symbols
+Twists of cube faces are denoted by uppercase letters U, L, F, D, R, B. Each of these twists
+means a 90◦ counterclockwise rotation.3 If U = U1 is a 90◦ rotation, then U2 is a 180◦
+rotation and U3=U−1 is a 270◦ rotation.
+Whole-cube rotations are denoted by lowercase letters u, ℓ, f. (We do not need d, r, b
+here, because d = u−1, r = ℓ−1 and so on.)
+Further symbols like fc[i], sℓ[i] that characterize a cube state will be explained in Sec. 2.3.
+2.1.3
+Twist Types
+Cube puzzles can have different twist types or twist metrics:
+2We run through the faces of a cubie in counter-clockwise orientation.
+3The rotation is counterclockwise when looking at the respective face
+6
+
+• QTM (quarter turn metric): only quarter twists are allowed: e.g. U1 and U−1.
+• HTM (half turn metric): quarter and half turns (twists) are allowed: e.g. U1, U2, U3.
+By allowed we mean what counts as one move. In QTM we can realize U2 via U U as
+well, but it costs us 2 moves. In HTM, U2 counts as one move.
+The twist type influences God’s number and the branching factor of the game, see
+Sec. 2.2.
+2.2
+Facts about Cubes
+2.2.1
+2x2x2 Cube
+The number of distinct states for the 2x2x2 pocket cube is (Wikipedia, 2022a)
+8! · 37
+24
+= 7! · 36 = 3, 674, 160 ≈ 3.6 · 106
+(1)
+Why this formula? — We have 8 cubies which we can place in 8! ways on the 8 cube
+positions. Each but the last cubie has the freedom to appear in 3 orientations, which gives
+the factor 37 (the last cubie is then in a fixed orientation, the other two orientations would
+yield illegal cube states). – Each of these raw states has the (ygr)-cubie in any of the
+24 possible positions. Or, otherwise speaking, each truly different state appears in 24
+whole-cube rotations. To factor out the whole-cube rotations, we count only the states with
+(ygr)-cubie in its default position (DRB) and divide the number of raw states by 24, q.e.d.
+God’s number: What is the minimal number of moves needed to solve any cube posi-
+tion? – For the 2x2x2 pocket cube, it is 11 in HTM (half-turn metric) and 14 in QTM.
+Branching factor: 3 · 3 = 9 in HTM and 3 · 2 = 6 in QTM.
+2.2.2
+3x3x3 Cube
+The number of distinct states for the 3x3x3 Cube is (Wikipedia, 2022b)
+8! · 37 · 12! · 211
+2
+= 43, 252, 003, 274, 489, 856, 000 ≈ 4.3 · 1019
+(2)
+Why this formula? – We have 8 corner cubies which we can place in 8! ways on the 8
+cube positions. Each but the last cubie has the freedom to appear in 3 orientations, which
+gives the factor 37. We have 12 edge cubies which we can place in 12! ways on the edge
+positions. Each but the last cubie has the freedom to appear in 2 orientations, which gives
+the factor 211. The division by 2 stems from the fact, that neither alone two corner cubies
+may be swapped nor alone two edge cubies may be swapped. Instead, the number of such
+swaps must be even (factor 2).
+God’s Number: What is the minimal number of moves needed to solve any cube
+position? — For the 3x3x3 Rubik’s Cube, it is 20 in HTM (half-turn metric) and 26 in QTM.
+This is a result from Rokicki et al. (2014), see also http://www.cube20.org/qtm/.
+Branching factor: 6 · 3 = 18 in HTM and 6 · 2 = 12 in QTM.
+7
+
+3
+2
+0
+1
+5
+4
+8
+11
+18
+17
+23
+22
+6
+7
+9
+10
+19
+16
+20
+21
+14
+13
+15
+12
+Figure 3: Sticker numbering for the 2x2x2 cube
+2.3
+The Cube State
+A cube should be represented by objects in GBG in such a way that
+(a) cube states that are equivalent are represented by identical objects
+(b) if two cube states are equivalent, it should be easy to check this by comparing their
+objects
+(c) cube transformations are easy to carry out on these objects.
+Condition (a) means that if two twist sequences lead to the same cube state (e.g. U−1
+and UUU), this should result also in identical objects. Condition (b) means, that the equality
+should be easy to check, given the objects. That is, a cube should not be represented by
+its twist sequence.
+A cube state is in GBG represented by abstract class CubeState and has two describ-
+ing members
+fc[i]
+=
+fcol[i]
+(3)
+sℓ[i]
+=
+sloc[i]
+(4)
+fc[i] = fcol[i] denotes the face color at sticker location i. The color is one out of
+0,1,2,3,4,5 for the colors w,b,o,y,g,r.
+sℓ[i] = sloc[i] contains the sticker location of the sticker which is in position i for the
+solved cube d.
+Members fc and sℓ are vectors with 24 (2x2x2 cube) or 48 (3x3x3 cube) elements
+where i denotes the ith sticker location.
+The stickers are numbered in a certain way which is detailed in Figures 3 and 4 for the
+flattened representations of the 2x2x2 and 3x3x3 cube, resp.
+In principle, one of the two members fc and sℓ would be sufficient to characterize a
+state, since the fcol-sloc-relation
+fc[sℓ[i]] = d.fc[i]
+(5)
+holds, where d denotes the default cube. This is because sℓ[i] transports the sticker i
+of the default cube d to location sℓ[i], i.e. it has the color d.fc[i]. That is, we can easily
+8
+
+6
+5
+4
+7
+3
+0
+1
+2
+10
+9
+8
+16
+23
+22
+36
+35
+34
+46
+45
+44
+11
+15
+17
+21
+37
+33
+47
+43
+12
+13
+14
+18
+19
+20
+38
+39
+32
+40
+41
+42
+28
+27
+26
+29
+25
+30
+31
+24
+Figure 4: Sticker numbering for the 3x3x3 cube. We do not number the center cubies, they
+stay invariant under twists.
+Table 1: The three relevant twists for the 2x2x2 cube
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10 11
+12 13 14 15
+16 17 18 19
+20 21 22 23
+U twist
+T
+1
+2
+3
+0
+11
+8
+6
+7
+18
+9
+10 17
+12 13 14 15
+16 22 23 19
+20 21
+4
+5
+L twist
+T
+22
+1
+2 21
+5
+6
+7
+4
+3
+0
+10 11
+12 13
+8
+9
+16 17 18 19
+20 14 15 23
+F twist
+T
+7
+4
+2
+3
+14
+5
+6 13
+9
+10 11
+8
+12 18 19 15
+16 17
+0
+1
+20 21 22 23
+U−1
+T −1
+3
+0
+1
+2
+22 23 6
+7
+5
+9
+10
+4
+12 13 14 15
+16 11
+8
+19
+20 21 17 18
+L−1
+T −1
+9
+1
+2
+8
+7
+4
+5
+6
+14 15 10 11
+12 13 21 22
+16 17 18 19
+20
+3
+0
+23
+F−1
+T −1
+18 19 2
+3
+1
+5
+6
+0
+11
+8
+9
+10
+12
+7
+4
+15
+16 17 13 14
+20 21 22 23
+calculate fc given sℓ. With some more effort, it is also possible to calculate sℓ given fc (see
+Appendix A). Although one of these members fc and sℓ would be sufficient, we keep both
+because this allows to better perform assertions or cross checks during transformations.
+Sometime we need the inverse function s−1
+ℓ [i]: Which sticker is at location i? It is easy
+to calculate s−1
+ℓ
+given sℓ with the help of the relation:
+s−1
+ℓ [sℓ[i]] = i
+(6)
+(Note that it is not possible to invert fc, because the face coloring function is not bijective.)
+2.4
+Transformations
+2.4.1
+Twist Transformations
+Each basic twist is a counterclockwise4 rotation of a face by 90◦. Table 1 shows the 2x2x2
+transformation functions for three basic twists. Each twist transformation can be coded in
+two forms:
+1. T[i] (forward transformation): Which is the new location for the sticker being at i
+before the twist?
+4The rotation is counterclockwise when looking at this face.
+9
+
+2
+1
+3
+0
+23
+22
+5
+4
+8
+11
+18
+17
+6
+7
+9
+10
+19
+16
+20
+21
+14
+13
+15
+12
+Figure 5: The default 2x2x2 cube after twist U1
+2. T −1[i] (inverse transformation): Which is the (parent) location of the sticker that lands
+in i after the twist?
+Example (read off from column 0 of Table 1): The L-twist transports sticker at 0 to 22:
+T[0] = 22. The (parent) sticker being at location 9 before the L-twist comes to location 0
+after the twist: T −1[0] = 9. Likewise, for the U-twist we have T[0] = 1 and T −1[0] = 3. We
+show in Fig. 5 the default cube after twist U1.
+How can we apply a twist transformation to a cube state programmatically? – We
+denote with f′
+c and s′
+ℓ the new states for fc and sℓ after transformation. The following
+relations allow to calculate the transformed cube state:
+f′
+c[i]
+=
+fc[T −1[i]]
+(7)
+s′
+ℓ[s−1
+ℓ [i]]
+=
+T[i]
+(8)
+Eq. (7) says: The new color for sticker 0 is the color of the sticker which moves into
+location 0 (fc[9] in the case of an L-twist). To explain Eq. (8), we first note that s−1
+ℓ [i] is the
+sticker being at i before the transformation. Then, Eq. (8) says: „The new location for the
+sticker being at i before the transformation is T[i].“ For example, the L-twist transports the
+current sticker at location 0 to the new location T[0] = 22, i. e. s′
+ℓ[0] = 22.
+For the 2x2x2 cube, these 3 twists U, L, F are sufficient, because D=U−1, R=L−1,
+B=F−1. This is because the 2x2x2 cube has no center cubies. For the 3x3x3 cube, we
+need all 6 twists U, L, F, D, R, B because this cube has center cubies.
+In any case, we will show in Sec. 2.4.2 that only one row in Table 1 or Table 2, say T
+for the U-twist, has to be known or established ’by hand’. All other twists and their inverses
+can be calculated programmatically with the help of Eqs. (9)-(15) that will be derived in
+Sec. 2.4.2.
+Table 2: The U twist for the 3x3x3 cube
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10 11
+12 13 14 15
+16 17 18 19
+20 21 22 23
+U twist T
+2
+3
+4
+5
+6
+7
+0
+1
+22 23 16 11
+12 13 14 15
+36 17 18 19
+20 21 34 35
+24 25 26 27
+28 29 30 31
+32 33 34 35
+36 37 38 39
+40 41 42 43
+44 45 46 47
+U twist T
+24 25 26 27
+28 29 30 31
+32 33 44 45
+46 37 38 39
+40 41 42 43
+8
+9
+10 47
+10
+
+Table 3: Two basic whole-cube rotations for the 2x2x2 cube
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10 11
+12 13 14 15
+16 17 18 19
+20 21 22 23
+u rotation
+T
+1
+2
+3
+0
+11
+8
+9
+10
+18 19 16 17
+15 12 13 14
+21 22 23 20
+6
+7
+4
+5
+f rotation
+T
+7
+4
+5
+6
+14 15 12 13
+9
+10 11
+8
+17 18 19 16
+2
+3
+0
+1
+23 20 21 22
+u−1
+T −1
+3
+0
+1
+2
+22 23 20 21
+5
+6
+7
+4
+13 14 15 12
+10 11
+8
+19
+19 16 17 18
+f−1
+T −1
+18 19 16 17
+1
+2
+3
+0
+11
+8
+9
+10
+6
+7
+4
+5
+15 12 13 14
+21 22 23 20
+Normalizing the 2x2x2 Cube
+As stated above, the 3 twists U, L, F are sufficient
+for the 2x2x2 cube. Therefore, the (DRB)-cubie will never leave its place, whatever the
+twist sequence formed by U, L, F is. The (DRB)-cubie has the stickers (12, 16, 20), and we
+can check in Table 1 that columns (12, 16, 20) are always invariant. If we have an arbitrary
+initial 2x2x2 cube state, we can normalize it by applying a whole-cube rotation such that
+the (ygr)-cubie moves to the (DRB)-location.
+Normalizing the 3x3x3 Cube
+In the case of the 3x3x3 cube, all center cubies
+will be not affected by any twist sequence. Therefore, we normalize a 3x3x3 cube state
+by applying initially a whole-cube rotation such that the center cubies are in their normal
+position (i.e. white up, blue left and so on).
+2.4.2
+Whole-Cube Rotations (WCR)
+Each basic whole-cube rotation (WCR) is a counterclockwise rotation of the whole cube
+around the u, l, f-axis by 90◦. Table 3 shows two of the 2x2x2 transformation functions for
+basic whole-cube rotations. Each rotation can be coded in two forms:
+1. T[i] (forward transformation): Which is the new location for the sticker being at i
+before the twist?
+2. T −1[i] (inverse transformation): Which is the (parent) location of the sticker that lands
+in i after the twist?
+Besides the basic rotation u there is also u2 (180◦) and u3 = u−1 (270◦ = −90◦).
+All whole-cube rotations can be generated from these two forward rotations u and f:
+First, we calculate the inverse transformations via
+T −1[T[i]] = i
+(9)
+where T is a placeholder for u or f. Next, we calculate the missing base rotation ℓ (counter-
+clockwise around the left face) as
+ℓ = fuf−1
+(10)
+We use here the programm-code-oriented notation „first trafo first“: Eq. (10) reads as
+„first f, then u, then f−1“.5
+5In programm code the relation would read cs.fTr(1).uTr().fTr(3). This is „first trafo first“, because
+each transformation is applied to the cube state object to the left and returns the transformed cube state object.
+11
+
+Table 4: All 24 whole-cube rotations (in first-trafo-first notation)
+number
+first rotation
+∗ u0
+∗ u1
+∗ u2
+∗ u3
+00-03
+id (white up)
+id
+u
+u2
+u3
+04-07
+f (green up)
+f
+fu
+fu2
+fu3
+08-11
+f2 (yellow up)
+f2
+f2u
+f2u2
+f2u3
+12-15
+f−1 (blue up)
+f−1
+f−1u
+f−1u2
+f−1u3
+16-19
+ℓ (orange up)
+ℓ
+ℓu
+ℓu2
+ℓu3
+20-23
+ℓ−1 (red up)
+ℓ−1
+ℓ−1u
+ℓ−1u2
+ℓ−1u3
+Table 5: The 24 inverse whole-cube rotations (in first-trafo-first notation)
+number
+first rotation
+∗ u0
+∗ u1
+∗ u2
+∗ u3
+00-03
+id (white up)
+id
+u3
+u2
+u1
+04-07
+f (green up)
+f−1
+ℓu3
+fu2
+ℓ−1u
+08-11
+f2 (yellow up)
+f2
+f2u
+f2u2
+f2u3
+12-15
+f−1 (blue up)
+f
+ℓ−1u3
+f−1u2
+ℓu
+16-19
+ℓ (orange up)
+ℓ−1
+f−1u3
+ℓu2
+fu
+20-23
+ℓ−1 (red up)
+ℓ
+fu−1
+ℓ−1u2
+f−1u
+The other basic whole-cube rotations d, r, b are not needed, because d = u−1, r = ℓ−1
+and b = f−1.
+The basic whole-cube rotations are rotations of the whole cube around just one axis.
+But there are also composite whole-cube rotations which consists of a sequence of basic
+rotations.
+How many different (composite) rotations are there for the cube? – A little thought
+reveals that there are 24 of them: To be specific, we consider the default cube where we
+have 4 rotations with the white face up, 4 with the blue face up, and so on. In total we have
+6 · 4 = 24 rotations since there are 6 faces. Table 4 lists all of them, togehter with the WCR
+numbering convention used in GBG.
+Sometimes we need the inverse whole-cube rotations which are given in Table 5. In
+this table, we read for example from the element with number 5, that the WCR with key 5
+(which is fu according to Table 4) has the inverse WCR ℓu3 such that
+fu ℓu3 = id
+holds.
+For convenience, we list in Table 6 the <Key, InverseKey> relation. For example, the
+trafo with Key=5 (fu) has the inverse trafo with InverseKey=19 (ℓu3). Note that there are
+10 whole-cube rotations which are their own inverse.
+Generating all twists from U twist
+With the help of WCRs we can generate the other
+twists from the U twist only: We simply rotate the face that we want to twist to the up-face,
+12
+
+Table 6: Whole-cube rotations: <Key, InverseKey> relation
+key
+0 1 2 3
+4
+5
+6
+7
+8 9 10 11
+12 13 14 15
+16 17 18 19
+20 21 22 23
+inv key
+0 3 2 1
+12 19 6 21
+8 9 10 11
+4
+23 14 17
+20 15 18 05
+16
+7
+22 13
+apply the U twist and rotate back. This reads in first-trafo-first notation:
+L = f−1Uf
+(11)
+F = ℓUℓ−1
+(12)
+D = f2Uf2
+(13)
+R = fUf−1
+(14)
+B = ℓ−1Uℓ
+(15)
+Thus, given the U twist from Table 1 or Table 2 and the basic WCRs given in Table 3 and
+Eq. (10), we can calculate all other forward transformations with the help of Eqs. (11)–(15).
+Then, all inverse transformations are calculable with the help of Eq. (9).
+2.4.3
+Color Transformations
+Color transformations are special transformations that allow to discover non-trivial symmet-
+ric (equivalent) states.
+One way to describe a color transformation is to select a valid color permutation and to
+paint each sticker with the new color according to this color permutation. This is of course
+nothing one can do with a real cube without destroying or altering it, but it is a theoretical
+concept leading to an equivalent state.
+Another way of looking at it is to record the twist sequence that leads from the default
+cube to a certain scrambled cube state. Then we go back to the default cube, make at
+first a whole-cube rotation (leading to a color-transformed default cube) and then apply the
+recorded twist sequence to the color-transformed default cube.
+In any case, the transformed cube will be usually not in its normal position, so we apply
+finally a normalizing operation to it.
+What are valid color permutations? – These are permutations of the cube colors reach-
+able when applying one of the available 24 WCRs (Table 4) to the default cube. For exam-
+ple, if we apply WCR f (number 04) to the default cube, we get
+g
+w
+o
+y
+r
+b
+Figure 6: The color transformation according to WCR f (number 04)
+that is, g (green) is the new color for each up-sticker that was w (white) before and so
+on. The colors o and r remain untouched under this color permutation. [However, other
+transformations like fu, fu2 and fu3 will change every color.]
+13
+
+2
+1
+3
+0
+23
+22
+5
+4
+8
+11
+18
+17
+6
+7
+9
+10
+19
+16
+20
+21
+14
+13
+15
+12
+Figure 7: The cube of Fig. 5 before color transformation.
+16
+19
+17
+18
+20
+23
+2
+1
+11
+10
+13
+12
+3
+0
+8
+9
+14
+15
+21
+22
+4
+7
+5
+6
+8
+2
+9
+1
+4
+7
+14
+11
+18
+17
+23
+0
+5
+6
+15
+10
+19
+16
+20
+3
+21
+13
+22
+12
+(a)
+(b)
+Figure 8: The cube of Fig. 7 with color transformation from Fig 6: (a) before normalization,
+(b) after normalization.
+How can we apply a color transformation to a cube state programmatically? – We
+denote with f′ and s′
+ℓ the new states for f and sℓ after transformation.
+The following
+relations allow to calculate the transformed cube state:
+f′
+c[i]
+=
+c[fc[i]]
+(16)
+s′
+ℓ[s−1
+ℓ [i]]
+=
+T[i]
+(17)
+where c[] is the 6-element color trafo vector (holding the new colors for current colors 0:w,
+1:b, ..., 5:r) and T is the 24- or 48-element vector of the WCR that produces this color
+transformation. Eq. (16) is simple: If a certain sticker has color 0 (w, white) before the color
+transformation, then it will get the new color c[0], e.g. 4 (g, green), after the transformation.
+Eq. (17) looks complicated, but it has a similar meaning as in the twist trafo: Take i = 0 as
+example: The new place for the sticker being at 0 before the trafo (and coming from s−1
+ℓ [0])
+is T[0]. Therefore, we write the number T[0] into s′
+ℓ[s−1
+ℓ [0]].
+A color transformation example is shown in Figs. 7 and 8. Fig. 7 is just a replication
+of Fig. 5 showing a default cube after U1 twist. The color transformation number 04 applied
+to the cube of Fig. 7 is shown in Fig. 8 (a)-(b) in two steps:
+(a) The stickers are re-painted and re-numbered (white becomes green, blue becomes
+white and so on). The structure of coloring is the same as in Fig. 7. Now the (DRB)-
+cubie is no longer the (ygr)-cubie, it does not carry the numbers (12,16,20).
+14
+
+(b) We apply the proper WCR that brings the (ygr)-cubie back to the (DRB)-location.
+Compared to (a), each 4-sticker cube face is just rotated to another face, but not
+changed internally. We can check that the (DRB)-location now carries again the num-
+bers (12,16,20), as in Fig. 7 and as it should for a normalized cube.
+2.5
+Symmetries
+Symmetries are transformations of the game state (and the attached action, if applicable)
+that lead to equivalent states.
+That is, if s is a certain state with value V (s), then all
+states ssym being symmetric to s have the same value V (ssym) = V (s) because they are
+equivalent. Equivalent means: If s can be solved by a twist sequence of length n, then
+ssym can be solved by an equivalent twist sequence of same length n.
+In the case of Rubik’s cube, all whole-cube rotations (WCRs) are symmetries because
+they do not change the value of a state. But whole-cube rotations are ’trivial’ symmetries
+because they are usually factored out by the normalization of the cube: After 2x2x2 cube
+normalization, which brings the (ygr)-cubie in a certain position, or after 3x3x3 cube nor-
+malization, which brings the center cubies in certain faces, all WCR-symmetric states are
+transformed to the same state.
+Non-trivial symmetries are all color transformations (Sec. 2.4.3): In general, color trans-
+formations transform a state s to a truly different state ssym, even after cube normalization.6
+Since there are 24 color transformations in Rubik’s cube, there are also 24 non-trivial sym-
+metries (including self).
+Symmetries are useful to learn to solve Rubik’s cube for two reasons: (a) to accelerate
+learning and (b) to smooth an otherwise noisy value function.
+(a) Accelerated learning: If a state s (or state-action pair) is observed, not only the
+weights activated by that state are updated, but also the weights of all symmetric states
+ssym, because they have the same V (ssym) = V (s) and thus the same reward. In this
+way, a single observed sample is connected with more weight updates (better sample
+efficiency).
+(b) Smoothed value function: By this we mean that the value function V (s) is replaced
+by
+V (sym)(s) =
+1
+|Fs|
+�
+s′∈Fs
+V (s′)
+(18)
+where Fs is the set of states being symmetric to s. If V (s) were the ideal value function,
+both terms V (s) and V (sym)(s) would be the same.7 But in a real n-tuple network, V (s)
+is non-ideal due to n-tuple-noise (cross-talk from other states that activate the same
+n-tuple LUT entries). If we average over the symmetric states s′ ∈ Fs, the noise will be
+dampened.
+6In rare cases – e.g. for the solved cube – the transformed state may be identical to s or to another
+symmetry state, but this happens seldom for sufficiently scrambled cubes, see Sec. 6.3.
+7because all V (s′) in Eq. (18) are the same for an ideal V
+15
+
+The downside of symmetries is their computational cost: In the case of Rubik’s cube,
+the calculation of color transformations is a costly operation. On the other hand, the number
+of necessary training episodes to reach a certain performance may be reduced. In the
+end, the use of symmetries may pay off, because the total training time may be reduced as
+well. In any case, we will have a better sample efficiency, since we learn more from each
+observed state or state-action pair. Secondly, the smoothing effect introduced with Eq. (18)
+can lead to better overall performance, because the smoothed value function provides a
+better guidance on the path towards the solved cube.
+In order to balance computation time, GBG offers the option to select with nSym the
+number of symmetries actually used. If we specify for example nSym = 8 in GBG’s Rubik’s
+cube implementation, then the state itself and 8 – 1 = 7 random other (non-id) color trans-
+formations will be selected. The resulting set Fs of 8 states is then used for weight update
+and value function computation.
+3
+N-Tuple Systems
+N-tuple systems coupled with TD were first applied to game learning by Lucas (2008), al-
+though n-tuples were already introduced by Bledsoe and Browning (1959) for character
+recognition purposes. The remarkable success of n-tuples in learning to play Othello (Lu-
+cas, 2008) motivated other authors to benefit from this approach for a number of other
+games.
+The main goal of n-tuple systems is to map a highly non-linear function in a low di-
+mensional space to a high dimensional space where it is easier to separate ‘good’ and
+‘bad’ regions. This can be compared to the kernel trick of support-vector machines. An
+n-tuple is defined as a sequence of n cells of the board. Each cell can have m positional
+values representing the possible states of that cell.8 Therefore, every n-tuple will have a
+(possibly large) look-up table indexed in form of an n-digit number in base m. Each entry
+corresponds to a feature and carries a trainable weight. An n-tuple system is a system
+consisting of k n-tuples. As an example we show in Fig. 9 an n-tuple system consisting of
+four 8-tuples.
+Let Θ be the vector of all weights θi of the n-tuple system.9 The length of this vector
+may be large number, e.g. mnk, if all k n-tuples have the same length n and each cell
+has m positional values. Let Φ(s) be a binary vector of the same length representing the
+feature occurences in state s (that is, Φi(s) = 1 if in state s the cell of a specific n-tuple as
+indexed by i has the positional value as indexed by i, Φi(s) = 0 else). The value function
+of the n-tuple network given state s is
+V (s) = σ (Φ(s) · Θ)
+(19)
+with transfer function σ which may be a sigmoidal function or simply the identity function.
+8A typical example is a 2-player board game, where we usually have 3 positional values {0: empty, 1:
+player1, 2: player2 }. But other, user-defined values are possible as well.
+9The index i indexes three qualities: an n-tuple, a cell in this n-tuple and a positional value for this cell.
+16
+
+Figure 9: Example n-tuples: We show 4 random-walk 8-tuples on a 6x7 board. The tuples are
+selected manually to show that not only snake-like shapes are possible, but also bifurcations
+or cross shapes. Tuples may or may not be symmetric.
+An agent using this n-tuple system derives a policy from the value function in Eq. (19)
+as follows: Given state s and the set A(s) of available actions in state s, it applies with a
+forward model f every action a ∈ A(s) to state s, yielding the next state s′ = f(s, a). Then
+it selects the action that maximizes V (s′).
+Each time a new agent is constructed, all n-tuples are either created in fixed, user-
+defined positions and shapes, or they are formed by random walk. In a random walk, all
+cells are placed randomly with the constraint that each cell must be adjacent10 to at least
+one other cell in the n-tuple.
+Agent training proceeds in the TD-n-tuple algorithm as follows: Let s′ be the actual
+state generated by the agent and let s be the previous state generated by this agent. TD(0)
+learning adapts the value function with model parameters Θ through (Sutton and Barto,
+1998)
+Θ ← Θ + αδ∇ΘV (s)
+(20)
+Here, α is the learning rate and V is in our case the n-tuple value function of Eq. (19). δ is
+the usual TD error (Sutton and Barto, 1998) after the agent has acted and generated s′:
+δ = r + γV (s′) − V (s)
+(21)
+where the sum of the first two terms, reward r plus the discounted value γV (s′), is the
+desirable target for V (s).
+10The form of adjacency, e. g. 4- or 8-point neighborhood or any other (might be cell-dependent) form of
+adjacency, is user-defined.
+17
+
+0
+3
+5
+6
+6
+5
+6
+3
+4
+4
+5
+6
+54
+N-Tuple Representions for the Cube
+In order to apply n-tuples to cubes, we have to define a board in one way or the other on
+which we can place the n-tuples. This is not as straightforward as in other board games, but
+we are free to invent abstract boards. Once we have defined a board, we can number the
+board cells k = 0, . . . , K−1 and translate a cube state into a BoardVector: A BoardVector
+b is a vector of K non-negative integer numbers bk ∈ {0, . . . , Nk − 1}. Each k represents
+a board cell and every board cell k has a predefined number Nk of position values.11
+A BoardVector is useful to calculate the feature occurence vector Φ(s) in Eq. (19) for
+a given n-tuple set: If an n-tuple contains board cell k, then look into bk to get the position
+value for this cell k. Set Φi(s) = 1 for that index i that indexes this n-tuple cell and this
+position value.
+In the following we present different options for boards and BoardVectors. We do this
+mainly for the 2x2x2 cube, because it is somewhat simpler to explain. But the same ideas
+apply to the 3x3x3 cube as well, they are just a little bit longer. Therefore, we defer the
+lengthy details of the 3x3x3 cube to Appendix B.
+4.1
+CUBESTATE
+A natural way to translate the cube state into a board is to use the flattened representation
+of Fig. 11 as the board and extract from it the 24-element vector b, according to the given
+numbering. The kth element bk represents a certain cubie face location and gets a number
+from {0, . . . , 5} according to its current face color fc. The solved cube is for example
+represented by b = [0000 1111 2222 . . . 5555].
+This representation CUBESTATE is what the BoardVecType CUBESTATE in our GBG-
+implementation means: Each board vector is a copy of fcol, the face colors of all cubie
+faces. fcol is also the vector that uniquely defines each cube state. An upper bound of
+possible combinations for b is 624 = 4.7 · 1018. If we factor out the (DRB)-cubie, which
+always stays at its home position, we can reduce this to 21 board cells with 6 positional
+values, leading to 621 = 2.1 · 1016 weights. Both numbers are of course way larger than
+the true number of distinct states (Sec. 2.2.1) which is 3.6 · 106. This is because most of
+the combinations are dead weights in the n-tuple LUTs, they will never be activated during
+game play.
+The dead weights occur because many combinations are not realizable, e.g. three
+white faces in one cubie or any of the 63 − 8 · 3 = 192 cubie-face-color combinations that
+are not present in the real cube. The problem is that the dead weights are scattered in a
+complicated way among the active weights and it is thus not easy to factor them out.
+4.2
+STICKER
+McAleer et al. (2019) had the interesting idea for the 3x3x3 cube that 20 stickers (cubie
+faces) are enough. To characterize the full 3x3x3 cube, we need only one (not 2 or 3) sticker
+11In GBG package ntuple2 (base for agent TDNTuple3Agt), all Nk have to be the same.
+In package
+ntuple4( base for agent TDNTuple4Agt), numbers Nk may be different for different k.
+18
+
+(a) Top view
+(b) Bottom view
+Figure 10: The sticker representation used to reduce dimensionality: Stickers that are used
+are shown in white, whereas ignored stickers are dark blue (from McAleer et al. (2019)).
+3
+2
+0
+1
+5
+4
+8
+11
+18
+17
+23
+22
+6
+7
+9
+10
+19
+16
+20
+21
+14
+13
+15
+12
+Figure 11: Tracked stickers for the 2x2x2 cube (white), while ignored stickers are blue.
+for every of the 20 cubies, as shown in Fig. 10. This is because the location of one sticker
+uniquely defines the location and orientation of that cubie. We name this representation
+STICKER in GBG.
+Translated to the 2x2x2 cube, this means that 8 stickers are enough because we have
+only 8 cubies. We may for example track the 4 top stickers 0,1,2,3 plus the 4 bottom
+stickers 12,13,14,15 as shown in Fig. 11 and ignore the 16 other stickers. Since we always
+normalize the cube such that the (DRB)-cubie with sticker 12 stays in place, we can reduce
+this even more to 7 stickers (all but sticker 12).
+How to lay out this representation as a board? – McAleer et al. (2019) create a rect-
+angular one-hot-encoding board with 7 × 21 = 147 cells (7 rows for the stickers and 21
+columns for the locations) carrying only 0’s and 1’s. This is fine for the approach of McAleer
+et al. (2019), where they use this board as input for a DNN, but not so nice for n-tuples.
+Without constraints, such a board amounts to 2147 = 1.7 · 1044 combinations, which is
+unpleasantly large (much larger than in CUBESTATE).12
+STICKER has more dead weights than CUBESTATE, so it seems like a step back. But
+the point is, that the dead weights are better structured: If for example sticker 0 appears at
+column 1 then this column and the two other columns for the same cubie are automatically
+12A possible STICKER BoardVector for the default cube would read b = [1000000 0100000 0010000 . . . ],
+meaning that location 0 has the first sticker, location 1 has the second sticker, and so on. In any STICKER
+BoardVector there are only 7 columns carrying exactly one 1, the other carry only 0’s. Every row carries exactly
+one 1.
+19
+
+Table 7: The correspondence corner location ↔ STICKER2 for the solved cube. The yellow
+colored cells show the location of the 7 (2x2x2) and 8 (3x3x3) corner stickers that we track.
+2x2x2
+location
+0 1 2 3
+4
+5
+6
+7
+8
+9
+10 11
+12 13 14 15
+16 17 18 19
+20 21 22 23
+3x3x3
+location
+0 2 4 6
+8 10 12 14
+16 18 20 22
+24 26 28 30
+32 34 36 38
+40 42 44 46
+STICKER2
+corner
+a b c d
+a
+d
+h
+g
+a
+g
+f
+b
+e
+f
+g
+h
+e
+c
+b
+f
+e
+h
+d
+c
+face ID
+1 1 1 1
+2
+3
+2
+3
+3
+2
+3
+2
+1
+1
+1
+1
+2
+2
+3
+2
+3
+3
+2
+3
+forbidden for all other stickers. Likewise, if sticker 1 is placed in another column, another set
+of 3 columns is forbidden, and so on. We can use this fact to form a much more compact
+representation STICKER2.
+4.3
+STICKER2
+As the analysis in the preceding section has shown, the 21 location columns of STICKER
+cannot carry the tracked stickers in arbitrary combinations. Each cubie (represented by 3
+columns in STICKER) carries only exactly one sticker. We can make this fact explicit by
+choosing another representation for the 21 locations:
+corner location = (corner cubie, face ID).
+That is, each location is represented by a pair: corner cubie a,b,c,d,f,g,h (we number the
+top cubies with letters a,b,c,d and the bottom cubies with letters e,f,g,h and omit e because
+it corresponds to the (DRB)-cubie) and a face ID. To number the faces with a face ID, we
+follow the convention that we start at the top (bottom) face with face ID 1 and then move
+counter-clockwise around the corner cubie to visit the other faces (2,3). Table 7 shows the
+explicit numbering in this new representation.
+To represent a state as board vector we use now a much smaller board shown in
+Table 8: Each cell in the first row has 7 position values (the letters) and each cell in the
+second row has 3 position values (the face IDs). We show in Table 8 the board vector for
+the default cube, b = [abcdfgh 1111111]. Representation STICKER2 allows for 77 · 37 =
+1.8 · 109 combinations in total, which is much smaller than STICKER and CUBESTATE.
+Table 8: STICKER2 board representation for the default 2x2x2 cube. For the BoardVector,
+cells are numbered row-by-row from 0 to 16.
+corner
+a
+b
+c
+d
+f
+g
+h
+7 positions
+face ID
+1
+1
+1
+1
+1
+1
+1
+3 positions
+STICKER2 has some dead weights remaining, because the combinations can carry
+the same letter multiple times, which is not allowed for a real cube state. But this rate of
+dead weights is tolerable.
+It turns out that STICKER2 is in all aspects better than CUBESTATE or STICKER.
+Therefore, we will only report the results for STICKER2 in the following.
+20
+
+4.4
+Adjacency Sets
+To create n-tuples by random walk, we need adjacency sets (sets of neighbors) to be
+defined for every board cell k.
+For CUBESTATE, the board is the flattened representation of the 2x2x2 cube (Fig. 3).
+The adjacency set is defined as the 4-point neighborhood, where two stickers are neigh-
+bors if they share a common edge on the cube, i.e. are neighbors on the cube.
+For STICKER2, the board consists of 16 cells shown in Table 8. Here, the adjacency
+set for cell k contains all other cells different from k.
+Again, the details of ideas similar to Sec. 4.1–4.4, but now for the 3x3x3 cube, are
+shown in Appendix B.1–B.4.
+5
+Learning the Cube
+5.1
+McAleer and Agostinelli
+The works of McAleer et al. (2018, 2019) and Agostinelli et al. (2019) contain up to now
+the most advanced methods for learning to solve the cube from scratch. Agostinelli et al.
+(2019) introduces the cost-to-go function for a general Marko decision process
+J(s) = min
+a∈A(s)
+�
+s′
+P a(s, s′)
+�
+ga(s, s′) + γJ(s′)
+�
+(22)
+where P a(s, s′) is the probability of transitioning from state s to s′ by taking action a and
+ga(s, s′) is the cost for this transition. In the Rubik’s cube case, we have deterministic tran-
+sitions, that is s′ = f(s, a) is deterministically prescribed by a forward model f. Therefore,
+the sum reduces to one term and we specialize to γ = 1. Furthermore, we set ga(s, s′) = 1,
+because only the length of the solution path counts, so that we get the simpler equation
+J(s) = min
+a∈A(s)
+�
+1 + J(s′)
+�
+with
+s′ = f(s, a).
+(23)
+Here, A(s) is the set of available actions in state s. We additionally set J(s∗) = 0 if s∗
+is the solved cube. To better understand Eq. (23) we look at a few examples: If s1 is a state
+one twist away from s∗, Eq. (23) will find this twist and set J(s1) = 1. If s2 is a state two
+twists away from s∗ and all one-twist states have already their correct labels J(s1) = 1,
+then Eq. (23) will find the twist leading to a s1 state and set J(s2) = 1 + 1 = 2. While
+iterations proceed, more and more states (being further away from s∗) will be correctly
+labeled, once their preceding states are correctly labeled. In the end we should ideally
+have
+J(sn) = n.
+However, the number of states for Rubik’s cube is too large to store them all in tabular
+form. Therefore, McAleer et al. (2019) and Agostinelli et al. (2019) approximate J(s) with
+a deep neural network (DNN). To train such a network in the Rubik’s cube case, they
+21
+
+Algorithm 1 DAVI algorithm (from Agostinelli et al. (2019)). Input: B: batch size, K:
+maximum number of twists, M: training iterations, C: how often to check for convergence,
+ϵ: error threshold. Output: Θ, the trained neural network parameters.
+1: function DAVI(B, K, M, C, ϵ)
+2:
+Θ ← INITIALIZENETWORKPARAMETERS
+3:
+ΘC ← Θ
+4:
+for m = 1, . . . , M do
+5:
+X ←GENERATESCRAMBLEDSTATES(B, K)
+▷ B scrambled cubes
+6:
+for xi ∈ X do
+7:
+yi ← mina∈A(s) [1 + jΘC(f(xi, a))]
+▷ cost-to-go function, Eq. (23)
+8:
+(Θ, loss) ← TRAIN(jΘ, X, y)
+▷ loss = MSE(jΘ(xi), yi)
+9:
+if (m mod C = 0 & loss < ϵ) then
+10:
+ΘC ← Θ
+11:
+return Θ
+introduce Deep Approximate Value Iteration (DAVI)13 shown in Algorithm 1. The network
+output jΘ(s) is trained in line 8 to approximate the (unknown) cost-to-go J(s) for every
+state s = xi. The main trick of DAVI is, as Agostinelli et al. (2019) write: „For learning to
+occur, we must train on a state distribution that allows information to propagate from the
+goal state to all the other states seen during training. Our approach for achieving this is
+simple: each training state xi is obtained by randomly scrambling the goal state ki times,
+where ki is uniformly distributed between 1 and K. During training, the cost-to-go function
+first improves for states that are only one move away from the goal state. The cost-to-go
+function then improves for states further away as the reward signal is propagated from the
+goal state to other states through the cost-to-go function.“
+Agostinelli et al. (2019) use in Algorithm 1 two sets of parameters to train the DNN: the
+parameters Θ being trained and the parameters ΘC used to obtain improved estimates
+of the cost-to-go function. If they did not use this two separate sets, performance often
+„saturated after a certain point and sometimes became unstable. Updating ΘC only after
+the error falls below a threshold ϵ yields better, more stable, performance.“ (Agostinelli
+et al., 2019) To train the DNN, they used M = 1 000 000 iterations, each with batch size
+B = 10 000. Thus, the trained DNN has seen ten billion cubes (1010) during training, which
+is still only a small subset of the 4.3 · 1019 possible cube states.
+The heuristic function of the trained DNN alone cannot solve 100% of the cube states.
+Especially for higher twist numbers ki, an additional solver or search algorithm is needed.
+This is in the case of McAleer et al. (2019) a Monte Carlo Tree Search (MCTS), similar to
+AlphaZero (Silver et al., 2017), which uses the DNN as the source for prior probabilities.
+Agostinelli et al. (2019) use instead a variant of A∗-search, which is found to produce
+solutions with a shorter path in a shorter runtime than MCTS.
+13More precisely, McAleer et al. (2019) use Autodidactic Iteration (ADI), a precursor to DAVI, very similar to
+DAVI, just a bit more complicated to explain. Therefore, we describe here only DAVI.
+22
+
+Algorithm 2 TD-n-tuple algorithm for Rubik’s cube. Input: pmax: maximum number of
+twists, M: training iterations, Etrain: maximum episode length during training, c: nega-
+tive cost-to-go, Rpos: positive reward for reaching the solved cube s∗, α: learning rate.
+jΘ(s): n-tuple network value prediction for state s. Output: Θ, the trained n-tuple network
+parameters.
+1: function TDNTUPLE(pmax, M, Etrain, c, Rpos)
+2:
+Θ ← INITIALIZENETWORKPARAMETERS
+3:
+for m = 1, . . . , M do
+4:
+p ∼ U(1, . . . , pmax)
+▷ Draw p uniformly random from {1, 2, . . . , pmax}
+5:
+s ← SCRAMBLESOLVEDCUBE(p)
+▷ start state
+6:
+for k = 1, . . . , Etrain do
+7:
+snew ← arg max
+a∈A(s)
+V (s′)
+with
+s′ = f(s, a)
+and
+8:
+V (s′) = c +
+� Rpos
+if
+s′ = s∗
+jΘ(s′)
+if
+s′ ̸= s∗
+9:
+Train network jΘ with Eq. (20) to bring V (s) closer to target T = V (snew):
+V (s) ← V (s) + α(T − V (s))
+10:
+s ← snew
+11:
+if (s = s∗) then
+12:
+break
+▷ break out of k-loop
+13:
+return Θ
+23
+
+5.2
+N-Tuple-based TD Learning
+To solve the Rubik’s cube in GBG we use an algorithm that is on the one hand inspired by
+DAVI, but on the other hand more similar to traditional reinforcement learning schemes like
+temporal difference (TD) learning. In fact, we want to use in the end the same TD-FARL
+algorithm (Konen and Bagheri, 2021) that we use for all other GBG games.
+We show in Algorithm 2 our method, that we will explain in the following, highlighting
+also the similarities and dissimilarities to DAVI.
+First of all, instead of minimizing the positive cost-to-go as in DAVI, we maximize in
+lines 7-8 a value function V (s′) with a negative cost-to-go. This maximization is functionally
+equivalent, but more similar to the usual TD-learning scheme. The negative cost-to-go, e.g.
+c = −0.1, plays the role of the positive 1 in Eq. (23).
+Secondly, we replace the DNN of DAVI by the simpler-to-train n-tuple network jΘ with
+STICKER2 representation as described in Sec. 3 and 4.
+That is, each time jΘ(s′) is
+requested, we first calculate for state s′ the BoardVector in STICKER2 representation,
+then the occurence vector Φ(s′) and the value function V (s′) according to Eq. (19).
+The central equations for V (s′) in Algorithm 2, lines 7-8, work similar to Eq. (23) in
+DAVI: If s = s1 is a state one twist away from s∗, the local search in arg max V (s′) will find
+this twist and the training step in line 9 moves V (s) closer to c+Rpos.14 Likewise, neighbors
+s2 of s1 will find s1 and thus move V (s2) closer to 2c + Rpos. Similar for s3, s4, . . . under
+the assumption that a ’known’ state is in the neighborhood. We have a clear gradient on
+the path towards the solved cube s∗. If there are no ’known’ states in the neighborhood
+of sn, we get for V (sn) what the net maximally estimates for all those neighbors. We pick
+the neighbor with the highest estimate, wander around randomly until we hit a state with a
+’known’ neighbor or until we reach the limit Etrain of too many steps.
+Note that Algorithm 2 is different from DAVI insofar that it follows the path s → s′ → . . .
+as prescribed by the current V , which may lead to a state sequence ’wandering in the
+unknown’ until Etrain is reached. In contrast to that, DAVI generates many start states s0
+drawn from the distribution of training set states and trains the network just on pairs (s0, T),
+i.e. they do just one step on the path. We instead follow the full path, because we want
+the training method for Rubik’s cube to be as similar as possible to the training method for
+other GBG games.15
+Algorithm 2 is basically the same algorithm as GBG uses for other games. The only
+differences are (i) the cube-specific start state selection borrowed from DAVI (a 1-twist start
+state has the same probability as a 10-twist start state) and (ii) the cube-specific reward in
+line 8 of Algorithm 2 with its negative cost-go-go c which is however a common element of
+many RL rewards.
+Algorithm 2 currently learns with only one parameter vector Θ. However, it could be
+extended as in DAVI to two parameter vectors Θ and ΘC. The weight training step in line
+14It is relevant, that Rpos is a positive number, e.g. 1.0 (and not 0, as it was for DAVI). This is because we
+start with an initial n-tuple network with all weights set to 0, so the initial response of the network to any state
+is 0.0. Thus, if Rpos were 0, a one-twist state would see all its neighbors (including s∗) initially as responding
+0.0 and would not learn the right transition to s∗. With Rpos = 1.0 it will quickly find s∗.
+15We note in passing that we tested the DAVI variant with Etrain = 1 for our TD-n-tuple method as well.
+However, we found that this method gave much worse results, so we stick with our GBG method here.
+24
+
+9 is done with the help of Eq. (20) for Θ using the error signal δ of Eq. (21).
+There are two extra elements, TCL and MCTS, that complete our n-tuple-based TD
+learning. They are described in the next two subsections.
+5.2.1
+Temporal Coherence Learning (TCL)
+The TCL algorithm developed by Beal and Smith Beal and Smith (1999) is an extension
+of TD learning. It replaces the global learning rate α with the weight-individual product
+ααi for every weight θi. Here, the adjustable learning rate αi is a free parameter set by a
+pretty simple procedure: For each weight θi, two counters Ni and Ai accumulate the sum
+of weight changes and the sum of absolute weight changes. If all weight changes have the
+same sign, then αi = |Ni|/Ai = 1, and the learning rate stays at its upper bound. If weight
+changes have alternating signs, then the global learning rate is probably too large. In this
+case, αi = |Ni|/Ai → 0 for t → ∞, and the effective learning rate will be largely reduced
+for this weight.
+In our previous work (Bagheri et al., 2015) we extended TCL to αi = g(|Ni|/Ai) where
+g is a transfer function being either the identity function (standard TCL) or an exponential
+function g(x) = eβ(x−1). It was shown in Bagheri et al. (2015) that TCL with this exponential
+transfer function leads to faster learning and higher win rates for the game ConnectFour.
+5.2.2
+MCTS
+We use Monte Carlo Tree Search (MCTS) (Browne et al., 2012) to augment our trained
+network during testing and evaluation. This is the method also used by McAleer et al.
+(2019) and by AlphaGo Zero (Silver et al., 2017), but they use it also during training.
+MCTS builds iteratively a search tree starting with a tree containing only the start state
+s0 as the root node. Until the iteration budget is exhausted, MCTS does the following: In
+every iteration we start from the root node and select actions following the tree policy until
+we reach a yet unexpanded leaf node sℓ. The tree policy is implemented in our MCTS
+wrapper according to the UCB formula (Silver et al., 2017):
+anew
+=
+arg max
+a∈A(s)
+�W(s, a)
+N(s, a) + U(s, a)
+�
+(24)
+U(s, a)
+=
+cpuctP(s, a)
+�
+ε + �
+b∈A(s) N(s, b)
+1 + N(s, a)
+(25)
+Here, W(s, a) is the accumulator for all backpropagated values that arrive along branch
+a of the node that carries state s. Likewise, N(s, a) is the visit counter and P(s, a) the prior
+probability. A(s) is the set of actions available in state s. ε is a small positive constant for
+the special case �
+b N(s, b) = 0: It guarantees that in this special case the maximum of
+U(s, a) is given by the maximum of P(s, a). The prior probabilities P(s, a) are obtained
+25
+
+Algorithm 3 TD-n-tuple training algorithm. Input: see Algorithm 2. Output: Θ: trained
+n-tuple network parameters.
+1: function TDNTUPLETRAIN(pmax, M, Etrain, c, Rpos)
+2:
+Θ ← INITIALIZENETWORKPARAMETERS
+3:
+INITIALIZETCLPARAMETERS
+▷ Set TCL-accumulators Ni = Ai = 0, αi = 1 ∀i
+4:
+for m = 1, . . . , M do
+5:
+Perform one m-iteration of Algorithm 2 with learning rates ααi instead of α
+6:
+Ni ← Ni + ∆θi and Ai ← Ai + |∆θi|
+▷ Update TCL-accumulators
+7:
+▷ where ∆θi is the last term in Eq. (20)
+8:
+αi ← |Ni|/Ai
+∀i with Ai ̸= 0
+9:
+return Θ
+Algorithm 4 Evaluation algorithm with MCTS solver. Input: trained n-tuple network jΘ,
+p: number of scrambling twists, B: batch size, Eeval: maximum episode length during
+evaluation, I: number of MCTS-iterations, cPUCT : relative weight for U(s, a) in Eq. (24),
+dmax: maximum MCTS tree depth. Output: solved rate.
+1: function TDNTUPLEEVAL(jΘ, p, B, Eeval, I, cPUCT , dmax)
+2:
+X ←GENERATESCRAMBLEDCUBES(B, p)
+▷ B scrambled cubes
+3:
+Csolved ← 0
+4:
+for xi ∈ X do
+5:
+s ← xi
+6:
+for k = 1, . . . , Eeval do
+7:
+T ← PERFORMMCTSSEARCH(s, I, cPUCT , dmax, jΘ)
+8:
+a ← SELECTMOSTVISITEDACTION
+9:
+s ← f(s, a)
+10:
+if (s = s∗) then
+11:
+Csolved ← Csolved + 1
+12:
+break
+▷ break out of k-loop
+13:
+return Csolved/B
+▷ percentage solved
+by sending the trained network’s values of all follow-up states s′ = f(s, a) with a ∈ A(s)
+through a softmax function (see Sec. 3).16
+Once an unexpanded leaf node sℓ is reached, the node is expanded by initializing
+its accumulators: W(s, a) = N(s, a) = 0 and P(s, a) = ps′ where ps′ is the softmax-
+squashed output jΘ(s′) of our n-tuple network for each state s′ = f(s, a). The value of
+the node is the network output of the best state jΘ(sbest) = maxs′ jΘ(s′) and this value is
+backpropagated up the tree.
+More details on our MCTS wrapper can be found in Scheiermann and Konen (2022).
+16Note that the prior probabilities and the MCTS iteration are only needed at test time, so that we – different
+to AlphaZero – do not need MCTS during self-play training.
+26
+
+5.2.3
+Method Summary
+We summarize the different ingredients of our n-tuple-based TD learning method in Algo-
+rithm 3 (training) and Algorithm 4 (evaluation).
+In line 5 of Algorithm 3 we perform one m-iteration of Algorithm 2 which does an update
+step for weight vector Θ, see Eq. (20). All weights of activated n-tuple entries get a weight
+change ∆θi equal to the last term in Eq. (20) where the global α is replaced by ααi.
+Line 2 in Algorithm 4 generates a set X of B scrambled cube states. Line 7 builds for
+each xi ∈ X an MCTS tree (see Sec. 5.2.2) starting from root node xi and line 8 selects
+the most visited action of the root node. If the goal state s∗ is not found during Eeval k-loop
+trials, this xi is considered as not being solved.
+6
+Results
+6.1
+Experimental setup
+We use for all our GBG experiments the same RL method based on n-tuple systems and
+TCL. Only its hyperparameters are tuned to the specific game, as shown below. We refer
+to this method/agent as TCL-base whenever it alone is used for game playing. If we wrap
+such an agent by an MCTS wrapper with a given number of iterations, then we refer to this
+as TCL-wrap.
+We investigate two variants of Rubik’s Cube: 2x2x2 and 3x3x3. We trained all TCL
+agents by presenting them M = 3 000 000 cubes scrambled with p random twists, where
+p is chosen uniformly at random from {1, . . . , pmax}. Here, pmax = 13 [16] for 2x2x2 and
+pmax = 9 [13] for 3x3x3, where the first number is for HTM, while the second number
+in square brackets is for QTM. With these pmax cube twists we cover the complete cube
+space for 2x2x2, where God’s number (Sec. 2.2) is known to be 11 [14]. But we cover only
+a small subset in the 3x3x3 case, where God’s number is known to be 20 [26] (Rokicki
+et al., 2014).17 We train 3 agents for each cube variant { 2x2x2, 3x3x3 } × { HTM, QTM }
+to assess the variability of training.
+The hyperparameters of the agent for each cube variant were found by manual fine-
+tuning. For brevity, we defer the exact explanation and setting of all parameters to Ap-
+pendix C.
+We evaluate the trained agents for each p on 200 scrambled cubes that are created by
+applying the given number p of random scrambling twists to a solved cube. The agent now
+tries to solve each scrambled cube. A cube is said to be unsolved during evaluation if the
+agent cannot reach the solved cube in Eeval = 50 steps.18
+17We limit ourselves to pmax = 9 [13] in the 3x3x3 HTM [QTM ] case, because our network has not enough
+capacity to learn all states of the 3x3x3 Rubik’s cube. Experiments with higher twist numbers during training
+did not improve the solved-rates.
+18During training, we use lower maximum episode lengths Etrain (see Appendix C) than Eeval = 50 in
+order to reduce computation time (in the beginning, many episodes cannot be solved, and 50 would waste a
+lot of computation time). But Etrain is always at least pmax + 3 in order to ensure that the agent has a fair
+chance to solve the cube and collect the reward.
+27
+
+HTM
+0%
+25%
+50%
+75%
+100%
+1
+3
+5
+7
+9
+11
+13
+scrambling twists
+percentage solved
+cubeWidth
+2x2x2
+3x3x3
+iterMWrap
+0
+100
+800
+QTM
+0%
+25%
+50%
+75%
+100%
+1
+3
+5
+7
+9
+11
+13
+15
+scrambling twists
+percentage solved
+cubeWidth
+2x2x2
+3x3x3
+iterMWrap
+0
+100
+800
+Figure 12: Percentage of solved cubes as a function of scrambling twists p for the trained
+TD-N-tuple agent wrapped by MCTS wrapper with different numbers of iterations. The red
+curves are TCL-base without wrapper, the other colors show different forms of TCL-wrap.
+Twist type is HTM (left) and QTM (right). Each point is the average of 3 independently trained
+agents.
+6.2
+Cube Solving with MCTS Wrapper, without Symmetries
+The trained TD-N-tuple agents learn to solve the cubes to some extent, as the red curves
+TCL-base in Fig. 12 show, but they are in many cases (i.e. p > pmax/2) far from being
+perfect. These are the results from training each agent for 3 million episodes, but the
+results would not change considerably, if 10 million training episodes were used.
+Scheiermann and Konen (2022) have shown, that the performance of agents, namely
+TD-N-tuple agents, is largely improved, if the trained agents are wrapped during test, play
+and evaluation by an MCTS wrapper. This holds for Rubik’s cube as well, as Fig. 12 shows:
+For the 2x2x2 cube, the non-wrapped agent TCL-base (red curve) is already quite good,
+but with wrapping it becomes almost perfect. For the 3x3x3 cube, the red curves are not
+satisfactorily: the solved-rates are below 20% for p = 9 [13] in the HTM [QTM ] case. But
+at least MCTS wrapping boosts the solved-rates by a factor of 3 [QTM: from 16% to 48%]
+or 4.5 [HTM: from 10% to 45%].
+All these results are without incorporating symmetries.
+How symmetries affect the
+solved-rates will be investigated in Sec. 6.4. But before this, we look in the next section at
+the number of symmetries that effectively exist in a cube state.
+6.3
+Number of Symmetric States
+Not every cube state has 24 truly different symmetric states (24 = number of color sym-
+metries). For example in the solved cube, all color-symmetric states are the same (after
+normalization). Thus, we have here only one truly different symmetric state.
+However, we show in this section that for the majority of cube states the number of
+truly different symmetric states is close to 24. Two states are truly different if they are
+not the same after the normalizing operation. We generate a cube state by applying p
+random scrambling twists to the default cube. Now we apply all 24 color transformations
+(Sec. 2.4.3) to it and count the truly different states. The results are shown in Fig. 13 for
+28
+
+twistType: HTM
+twistType: QTM
+0
+4
+8
+12
+16
+0
+4
+8
+12
+16
+0
+5
+10
+15
+20
+25
+scrambling twists
+NsymmetricStates
+cubeWidth
+2x2x2
+3x3x3
+Figure 13: Count of truly different symmetric states for cube states generated by p random
+scrambling twists. Each point is an average over 500 such states.
+both cube sizes and both twist types. For the 3x3x3 cube, the number of states quickly (for
+p > 5) approaches the maximum N = 24, while for the 2x2x2 cube it is a bit slower: p > 4
+or p > 8 is needed to surpass N = 20.
+As a consequence, it makes sense to use 16 or even 24 symmetries when training
+and evaluating cube agents. Especially for scrambled states with higher p, the 24 color
+transformations used to construct symmetric states will usually lead to 24 different states.
+6.4
+The Benefit of Symmetries
+In order to investigate the benefits of symmetries, we first train a TCL agent with dif-
+ferent numbers of symmetries. As described in Sec. 2.5, we select in each step nSym
+= 0, 8, 16, 24 symmetric states. Which symmetric states are chosen is selected randomly.
+Symmetries are used (a) to update the weights for each symmetric state and (b) to build
+with Eq. (18) a smoothed value function which is used to decide about the next action dur-
+ing training. For 0, 8, 16, 24 symmetries, we train 3 agents each (3x3x3 cube, STICKER2,
+QTM). The 3 agents differ due to their differently created random-walk n-tuple sets.
+Fig. 14 shows the learning curves for different nSym = 0, 8, 16, 24. It is found that agents
+with nSym > 0 learn faster and achieve a higher asymptotic solved rate.
+Next, we evaluate each of the trained agents by trying to solve for each p ∈ {1, . . . , 15}
+(scrambling twists) 200 different scrambled cubes. During evaluation, we use again the
+same nSym as in training to form a smoothed value function. We compare in Fig. 15 different
+29
+
+50%
+60%
+70%
+80%
+90%
+0e+00
+1e+06
+2e+06
+3e+06
+episodes
+percentage solved
+nSym
+24
+16
+8
+0
+Figure 14: Learning curves for different numbers nSym = 0, 8, 16, 24 of symmetries. Shown is
+the solved rate of (3x3x3, QTM) cubes. The solved rate is the average over all twist numbers
+p = 1, . . . , 13 with 200 testing cubes for each p and over 3 agents with different random-walk
+n-tuple sets.
+symmetry results, both without wrapping (TCL-base, red curves) and with MCTS-wrapped
+agents using 100 (green) or 800 (blue) iterations. It is clearly visible that MCTS wrapping
+has a large effect, as it was also the case in Fig 12. But in addition to that, the use of
+symmetries leads for each agent, wrapped or not, to a substantial increase in solved-rates
+(a surplus of 10-20%). It is remarkable, that even for p=14 or 15 a solved rate above or
+near 50% can be reached19 by the combination (nSym=16, 800 MCTS iterations).
+Surprisingly, it seems that with wrapping it is only important whether we use symme-
+tries, not how many, since the difference between nSym = 8, 16, 24 is only marginal. For
+800 MCTS iterations, the solved rate for nSym = 24 is in most cases even smaller than that
+for nSym = 8, 16. This is surprising because it would have been expected that also with
+wrapping a larger nSym should lead to a smoother value function and thus should in theory
+produce larger solved rates. – Note that this is not a contradiction to Fig. 14, because the
+learning curves were obtained without wrapping and the red TCL-base curves in Fig. 15
+(again without wrapping) show the same positive trend with increasing nSym20. The red
+curves in Fig. 15 show approximately the same average solved rates as the asymptotic
+values in Fig. 14.
+6.5
+Computational Costs
+Table 9 shows the computational costs when training and testing with symmetries. All
+computations were done on a single CPU Intel i7-9850H @ 2.60GHz. If we subtract the
+19p is above pmax=13, the maximum twist number used during training.
+20i.e. nSym= 24 is for every p clearly better than nSym= 16
+30
+
+QTM
+3x3x3
+0%
+25%
+50%
+75%
+100%
+1
+3
+5
+7
+9
+11
+13
+15
+scrambling twists
+percentage solved
+nSym
+0
+8
+16
+24
+iterMWrap
+0
+100
+800
+Figure 15: With symmetries: Percentage of solved cubes (3x3x3, QTM) as a function of
+scrambling twists p for TD-N-tuple agents trained and evaluated with different numbers of
+symmetries nSym and wrapped by MCTS wrappers with different iterations. The red curves
+are TCL-base (without wrapper), the other colors show different forms of TCL-wrap. The
+solved rates are the average over 200 testing cubes for each p and over 3 agents with differ-
+ent random-walk n-tuple sets.
+computational costs for nsym= 0, computation time increases more or less linearly with
+iter and roughly linearly with nSym. Computation times for nSym= 24 are approximately
+10x larger than those for nSym= 0.
+Computation times are dependent on the solved rate: If a cube with p = 13 is solved,
+the episode takes normally 12-15 steps. If the cube is not solved, the episode needs 50
+steps, i.e. a factor of 3-4 more. Thus, the numbers in Table 9 should be taken only as
+rough indication of the trend.
+Bottom line: Training time through symmetries increases by a factor of 13/0.5 = 26
+(nSym= 24) and testing time increases through 800 MCTS iterations by a factor of about
+3130/8 ≈ 400.
+Training with symmetries takes between 5.4h and 13h on a normal CPU, depending
+on the number of symmetries. This is much less than the 44h on a 32-core server with 3
+GPUs that were used by McAleer et al. (2019). But it also does not reach the same quality
+as McAleer et al. (2019).
+31
+
+Table 9: Computation times with symmetries. All numbers are for 3x3x3 cube, STICKER2
+and QTM. Training: 3 million self-play episodes, w/o MCTS in the training loop. Testing: 200
+scrambled cubes with p = 13, agents wrapped by MCTS wrapper with iter iterations.
+nSym
+training
+testing
+[hours]
+[seconds]
+iter
+0
+100
+400
+800
+0
+0.5
+0.5
+48
+196
+390
+8
+5.4
+4.0
+241
+877
+1400
+16
+9.5
+7.3
+464
+1380
+2330
+24
+13.0
+8.0
+550
+1760
+3130
+7
+Related Work
+Ernö Rubik invented Rubik’s cube in 1974. Rubik’s cube has gained worldwide popularity
+with many human-oriented algorithms being developed to solve the cube from arbitrary
+scrambled start states. By ’human-oriented’ we mean algorithms that are simple to mem-
+orize for humans. They usually will find long, suboptimal solutions. For a long time it was
+an unsolved question what is the minimal number of moves (God’s Number) needed to
+solve any given cube state. The early work of Thistlethwaite (1981) put an upper bound on
+this number with his 52-move algorithm. This was one of the first works to systematically
+use group theory as an aid to solve Rubik’s cube. Later, several authors have gradually
+reduced the upper bound 52 (Joyner, 2014), until Rokicki et al. (2014) could prove in 2014
+for the 3x3x3 cube that God’s Number is 20 in HTM and 26 in QTM.
+Computer algorithms to solve Rubik’s cube rely often on hand-engineered features and
+group theory. One popular solver for Rubik’s cube is the two-phase algorithm of Kociemba
+(2015). A variant of A∗ heuristic search was used by Korf (1991), along with a pattern
+database heuristic, to find the shortest possible solutions.
+The problem of letting a computer learn to solve Rubik’s cube turned out to be much
+harder: Irpan (2016) experimented with different neural net baseline architectures (LSTM
+gave for him reportedly best results) and tried to boost them with AdaBoost. However, he
+had only for scrambling twist ≤ 7 solved rates of better than 50% and the baseline turned
+out to be better than the boosted variants. Brunetto and Trunda (2017) found somewhat
+better results with a DNN, they could solve cube states with 18 twists with a rate above
+50%. But they did not learn from scratch because they used an optimal solver based
+on Kociemba (2015) to generate training examples for the DNN. Smith et al. (2016) tried
+to learn Rubik’s cube by genetic programming. However, their learned solver could only
+reliably solve cubes with up to 5 scrambling twists.
+A breakthrough in learning to solve Rubik’s cube are the works of McAleer et al. (2018,
+2019) and Agostinelli et al. (2019): With Autodidactic Iteration (ADI) and Deep Approxi-
+mate Value Iteration (DAVI) they were able to learn from scratch to solve Rubik’s cube in
+QTM for arbitrary scrambling twists. Their method has been explained in detail already
+in Sec. 5.1, so we highlight here only their important findings: McAleer et al. (2019) only
+needs to inspect less than 4000 cubes with its trained network DeepCube when solving
+32
+
+for a particular cube, while the optimal solver of Korf (1991) inspects 122 billion different
+nodes, so Korf’s method is much slower.
+Agostinelli et al. (2019) extended the work of McAleer et al. (2019) by replacing the
+MCTS solver with a batch-weighted A∗ solver which is found to produce shorter solution
+paths and have shorter run times. At the same time, Agostinelli et al. (2019) applied their
+agent DeepCubeA successfully to other puzzles like LightsOut, Sokoban, and the 15-, 24-,
+35- and 48-puzzle21. DeepCubeA could solve all of them.
+The deep network used by McAleer et al. (2019) and Agostinelli et al. (2019) were
+trained without human knowledge or supervised input from computerized solvers. The
+network of McAleer et al. (2019) had over 12 million weights and was trained for 44 hours
+on a 32-core server with 3 GPUs. The network of McAleer et al. (2019) has seen 8 billion
+cubes during training. – Our approach started from scratch as well. It required much less
+computational effort (e.g. 5.4h training time on a single standard CPU for nSym=8, see
+Table 9). It can solve the 2x2x2 cube completely, but the 3x3x3 cube only partly (up to 15
+scrambling twists). Each trained agent for the 3x3x3 cube has seen 48 million scrambled
+cubes22 during training.
+8
+Summary and Outlook
+We have presented new work on how to solve Rubik’s cube with n-tuple systems, reinforce-
+ment learning and an MCTS solver. The main ideas were already presented in Scheier-
+mann and Konen (2022) but only for HTM and up to p = 9 twists. Here we extended
+this work to QTM as well and presented all the details of cube representation and n-tuple
+learning algorithms necessary to reproduce our Rubik’s cube results. As a new aspect,
+we added cube symmetries and studied their effect on solution quality. We found that the
+use of symmetries boosts the solved rates by 10-20%. Based on this, we could increase
+for QTM the number of scrambling twists where at least 45% of the cubes are solved from
+p = 13 without symmetries to p = 15 with symmetries.
+We cannot solve the 3x3x3 cube completely, as McAleer et al. (2019) and Agostinelli
+et al. (2019) do. But our solution is much less computational demanding than their ap-
+proach.
+Further work might be to look into larger or differently structured n-tuple systems, per-
+haps utilizing the staging principle that Ja´skowski (2018) used to produce world-record
+results in the game 2048.
+21a set of 15, 24, ... numbers has to be ordered on a 4 × 4, 5 × 5, ... square with one empty field
+223 · 106 × 16 = training episodes × episode length Etrain. This is an upper bound: some episodes may
+have shorter length, but each unsolved episode has length Etrain.
+33
+
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+35
+
+M. Thistlethwaite.
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+jaapsch.net/puzzles/thistle.htm.
+Reconstructed by Jaap Scherphuis, retrieved
+Sep-01-2022. 32
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+retrieved Aug-17-2022. 7
+Wikipedia.
+Rubik’s Cube, 2022b.
+URL https://en.wikipedia.org/wiki/Rubik’s_
+Cube. retrieved Aug-17-2022. 7
+36
+
+Appendix
+A
+Calculating sloc from fcol
+Given the face colors fc (Eq. (3)) of a transformed cube, how can we calculate the trans-
+formed sticker locations sℓ (Eq. (4))?
+This problem seems ill-posed at first sight, because a certain face color, e.g. white,
+appears multiple times in fc and it is not possible to tell from the appearance of white alone
+to which sticker location sℓ it corresponds. But with a little more effort, i.e. by looking at the
+neighbors of the white sticker, we can solve the problem, as we show in the following.
+A.1
+2x2x2 cube
+All cubies of the 2x2x2 cube are corner cubies. We track for each cubie exactly one sticker.
+This can be for example the set
+B = {0, 1, 2, 3, 12, 13, 14, 15}
+of 8 stickers, which is the same as the set of tracked stickers shown in Fig. 11.
+For each s ∈ B:
+1. Build the cubie that contains s as the first sticker.23
+2. Locate the cubie in fc. That is, find a location in fc with the same color as the 1st
+cubie face. If found, check if the neighbor to the right24 has the color of the 2nd cubie
+face. If yes, check if its neighbor to the right has the color of the 3rd cubie face. If
+yes, we have located the cubie in fc and we return it, i.e. its three sticker locations
+C = [a, b, c].
+3. Having located the cubie, we can infer three elements of sℓ:
+sℓ[s]
+=
+C[0]
+(26)
+sℓ[R[s]]
+=
+C[1]
+(27)
+sℓ[R[R[s]]]
+=
+C[2]
+(28)
+Here R[s] is the right neighbor of sticker s. R[R[s]]] is the left neighbor.
+In total, we have located 8 × 3 = 24 stickers, i.e. the whole transformation for sℓ.25
+23We know for example from looking at the default cube in Fig. 11 that sticker s = 0 is part of the 0-8-4-cubie.
+24By neighbor to the right we mean the next sticker when we march in clockwise orientation around the
+actual cubie.
+25The relevant GBG source code is in CubeState.locate and CubeState2x2.apply_sloc_slow.
+37
+
+A.2
+3x3x3 cube
+The 3x3x3 cube has 8 corner cubies and 12 edge cubies. We track for each cubie exactly
+one sticker. This can be for the corners the set
+B = {0, 2, 4, 6, 24, 26, 28, 30}
+and for the edges the set
+E = {1, 3, 5, 7, 25, 27, 29, 31, 11, 15, 21, 33}.
+We do for the corner set B the same as we did for the 2x2x2 cube.
+For each element s ∈ E of the edge set:
+1. Build the edge cubie cE that contains s as the first sticker.
+2. Locate the cubie in fc. That is, find an edge location in fc with the same color as the
+1st cubie face. If found, check if the other sticker of that cubie has the same color as
+the other sticker of cE. If yes, we have located the edge cubie in fc and we return it,
+i.e. its two stickers C = [a, b].
+3. Having located the cubie, we can infer two elements of sℓ:
+sℓ[s]
+=
+C[0]
+(29)
+sℓ[O[s]]
+=
+C[1]
+(30)
+Here O[s] is the other sticker of the edge cubie that has sticker s as first sticker.
+In total, we have located
+8 × 3 + 12 × 2 = 48
+stickers, i.e. the whole transformation for sℓ.26
+B
+N-Tuple Representations for the 3x3x3 Cube
+In this appendix we describe the n-tuple representations of the cube, analogously to the
+2x2x2 cube Sec. 4, but now for the 3x3x3 cube.
+B.1
+CUBESTATE
+A natural way to translate the cube state into a board is to use the flattened representation
+of Fig. 4 as the board and extract from it the 48-element vector b, according to the given
+numbering. The kth element bk represents a certain cubie face location and gets a number
+from {0, . . . , 5} according to its current face color fc. The solved cube is for example
+represented by b = [00000000 11111111 . . . 55555555].
+This representation CUBESTATE is what the BoardVecType CUBESTATE in our GBG-
+implementation means: Each board vector is a copy of fcol, the face colors of all cubie
+faces. An upper bound of possible combinations for b is 648 = 2.2 · 1032. This is much
+larger than the true number of distinct states (Sec. 2.2.2) which is 4.3 · 1019.
+26The relevant GBG source code is in CubeState.locate, CubeState3x3.locate_edge and CubeState3x3.apply_sloc_slow.
+38
+
+Table 10: The correspondence edge location ↔ STICKER2 for the solved cube. The yellow
+colored cells show the location of the 12 edge stickers that we track.
+3x3x3
+location
+1 3 5
+7
+9 11 13 15
+17 19 21 23
+25 27 29 31
+33 35 37 39
+41 43 45 47
+STICKER2
+edge
+A B C D
+D
+G
+K
+E
+E
+J
+F
+A
+I
+J
+K
+L
+H
+B
+F
+I
+L
+G
+C
+H
+face ID
+1 1 1
+1
+2
+2
+2
+2
+1
+1
+1
+1
+1
+1
+1
+1
+2
+2
+2
+2
+1
+1
+1
+1
+B.2
+STICKER
+McAleer et al. (2019) had the interesting idea for the 3x3x3 cube that 20 stickers (cubie
+faces) are enough. To characterize the 3x3x3 cube, we need according to McAleer et al.
+(2019) only one (not 2 or 3) sticker for every of the 20 cubies, as shown in Fig. 10. This
+is because the location of one sticker uniquely defines the location and orientation of that
+cubie. We name this representation STICKER in GBG.
+We track the 4 top corner stickers 0,2,4,6 plus the 4 bottom corner stickers 24,26,28,30
+plus one sticker for each of the 12 edge stickes as shown in Fig. 10, in total 20 stickers and
+ignore the 28 other stickers.
+How to lay out this representation as a board? – McAleer et al. (2019) create a rect-
+angular one-hot-encoding board with 20 × 24 = 480 cells (20 rows for the stickers and
+24 columns for the locations27) carrying only 0’s and 1’s. This is fine for the approach of
+McAleer et al. (2019), where they use this board as input for a DNN, but not so nice for
+n-tuples. Without constraints, such a board amounts to 2480 ≈ 10145 combinations, which
+is unpleasantly large (much larger than in CUBESTATE).28
+Another possibility to lay out the board: Specify 20 board cells (the stickers) with 24
+position values each. This amounts to 2420 = 4.0 · 1027 combinations.
+B.3
+STICKER2
+Analogously to Sec. 4.3, we represent the 24 corner locations and 24 edge locations as:
+corner location = (corner cubie, face ID),
+edge location = (edge cubie, face ID).
+That is, each corner location is represented by a corner cubie a,b,c,d,e,f,g,h and by a face
+ID 1,2,3. Table 7 shows the explicit numbering in this new representation. Additionally,
+each edge location is represented by an edge cubie A,B,C,D,E,F,G,H,I,J,K,L29 and by a
+face ID 1,2. Convention for face ID numbering of edge cubies: For top- and bottom-layer
+edge cubies, it is 1 for U and D stickers, 2 else. The face ID for middle-layer edge cubies is
+1 for F and B stickers, 2 else. Table 10 shows the explicit numbering in this representation.
+The corresponding board consists of 8 + 8 + 12 +12 = 40 cells shown in Table 11.
+The 8 cell pairs in the first two rows code the locations of the tracked corner stickers
+278 · 3 for the corner stickers and 12 · 2 for the edge stickers
+28McAleer et al. (2019) do not need a weight for every of the 2480 possible states, as the n-tuple network
+would need. Instead they need only 480 · 4096 = 2 · 106 weights to the first hidden layer having 4096 neurons.
+294 U-stickers, 4 D-sticker, 4 middle-layer stickers (2F, 2B)
+39
+
+0,2,4,6,24,26,28,30, see Table 7 in Sec. 4.3. The 12 cell pairs in the last two rows code the
+location of the tracked edge stickers 1,3,5,7,17,21,43,47,25,27,29,31, see Table 10. This
+n-tuple coding requires tuple cells with varying number of position values and leads to
+88 · 38 · 1212 · 212 = 4.0 · 1027
+combinations in representation STICKER2.30
+Table 11: STICKER2 board representation for the default 3x3x3 cube. For the BoardVector,
+cells are numbered row-by-row from 0 to 39.
+corner
+a
+b
+c
+d
+e
+f
+g
+h
+8 positions
+face ID
+1
+1
+1
+1
+1
+1
+1
+1
+3 positions
+edge
+A
+B
+C
+D
+E
+F
+G
+H
+I
+J
+K
+L
+12 positions
+face ID
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+2 positions
+B.4
+Adjacency Sets
+To create n-tuples by random walk, we need to define adjacency sets (sets of neighbors)
+for every board cell k.
+For CUBESTATE, the board is the flattened representation of the 3x3x3 cube (Fig. 4).
+The adjacency set is defined as the 4-point neighborhood, where two stickers are neigh-
+bors if they are neighbors (share a common edge) on the cube.
+For STICKER2, the board consists of 40 cells shown in Table 11. Since it matters for
+the corner stickers mostly where the other corner stickers are and for the edge stickers
+mostly where the other edge stickers are, it is reasonable to form two adjacency subsets
+S1 = {00, . . . , 15} and S2 = {16, . . . , 39} and to define the adjacency set
+Adj(k) = Si \ {k}
+for each k ∈ Si, i = 1, 2.
+C
+Hyperparameters
+In this appendix we list all parameter settings for the GBG agents used in this paper. Pa-
+rameters were manually tuned with two goals in mind: (a) to reach high-quality results and
+(b) to reach stable (robust) performance when conducting multiple training runs with differ-
+ent random seeds. The agents listed further down are the best-so-far agents found (best
+among all agents that learn from scratch by self-play).
+The detailed meaning of RL parameters is explained in Konen and Bagheri (2021):
+30This is, by the way, identical to (8·3)8 ·(12·2)12 = 24(8+12) = 2420 = 4.0·1027, the same number we had
+above in the second mode of STICKER. But STICKER2 has the advantage that the combinations are spread
+over more board cells (40) than in STICKER (20). By having more board cells with fewer position values, the
+n-tuples can better represent the relationships between cube states.
+40
+
+• Algorithms 2, 5 and 7 in Konen and Bagheri (2021) explain parameters α (learning
+rate), γ (discount factor), ϵ (exploration rate) and output sigmoid σ (either identity or
+tanh).
+• Appendix A.3 explains our eligibility method, parameters are: eligibility trace factor λ,
+horizon cut ch, eligibility trace type ET (normal) or RESET (reset on random move).
+If not otherwise stated, we use in this paper λ = 0 (no eligibility traces). For λ = 0,
+horizon cut ch and eligibity trace type are irrelevant. If λ > 0, their defaults ch = 0.1
+and trace type ET apply.
+• Appendix A.5 explains our TCL method (also summarized in Sec. 5.2.1). Parameters
+of TCL are: TC-Init (initialization constant for counters), TC transfer function (TC-id
+or TC-EXP), β (exponential factor in case of TC-EXP), TC accumulation type (delta
+or recommended weight-change).
+Another branch of our algorithm is the MCTS wrapper, which can be used to wrap
+TD-N-tuple agents during evaluation and testing. MCTS wrapping is briefly explained in
+Sec. 5.2.2. The precise algorithm for MCTS wrapping is explained in detail in (Scheiermann
+and Konen, 2022, Sec. II-B).31 Parameters of MCTS are:
+• cPUCT : relative weight for the prior probabilities of the wrapped agent in relation to
+the value that the wrapper estimates
+• dmax: maximum depth of the MCTS tree, if -1: no maximum depth
+• UseSoftMax: boolean, whether to use SoftMax normalization for the priors or not
+• UseLastMCTS: boolean, whether to re-use the MCTS from the previous move within
+an episode or not
+Further parameter explanations:
+• Sec. 4 in this document explains n-tuples, parameters are: number of n-tuples, length
+of n-tuples, and n-tuple creation mode (fixed, random walk, random points).
+• Sec. 2.5 in this document explains symmetries. If parameter nSym = 0, do not use
+symmetries. If nSym > 0, use this number nSym of symmetries. In the Rubik’s cube
+case, nSym is a number between 0 and 24.
+• LearnFromRM: whether to learn from random moves or not. (Does not apply here,
+because we use in Rubiks’s cube always ϵ = 0, i.e. we have no random moves.)
+• ChooseStart-01: whether to start episodes from different 1-ply start states or always
+from the default start state. (Does not apply here, because we start in Rubik’s cube
+never from the default cube, but always from the p-twisted cube.)
+31As (Scheiermann and Konen, 2022, Sec. IV-E) shows, the MCTS wrapper may be used as well during
+training, but due to large computation times needed for this, we do not follow that route in this paper.
+41
+
+• Etrain: maximum episode length during training, if -1: no maximum length.
+• Eeval: maximum episode length during evaluation and play, if -1: no maximum length.
+All agents were trained with no MCTS wrapper inside the training loop. The hyper-
+parameters of the agent for each cube variant were found by manual fine-tuning. See
+also (Konen, 2022).
+In the following, we list the precise settings for all agents used in this paper. If not stated
+otherwise, these common settings apply to all agents: sigmoid σ = id, LearnFromRM =
+false, ChooseStart-01 = false. Wrapper settings during test and evaluation: MCTS wrapper
+with cPUCT = 1.0, dmax = 50, UseSoftMax = true, UseLastMCTS = true.
+Common parameters of Algorithm 2 in Sec. 5.2 are: cost-to-go c = −0.1 and positive
+reward Rpos = 1.0.
+The parameters for training without symmetries (nSym = 0) in Sec. 6.2 are:
+• 2x2x2 cube, HTM: α = 0.25, γ = 1.0, ϵ = 0.0, λ = 0.0, no output sigmoid. N-tuples:
+60 7-tuples created by random walk.
+TCL activated with transfer function TC-id,
+TC-Init= 10−4 and rec-weight-change accumulation. 3,000,000 training episodes.
+pmax = 13, Etrain = 16, Eeval = 50.
+Agent filename in GBG: 2x2x2_STICKER2_AT/TCL4-p13-ET16-3000k-60-7t-stub.agt.zip
+• 2x2x2 cube, QTM: same as 2x2x2 cube, HTM, but with pmax = 16, Etrain = 20.
+Agent filename in GBG: 2x2x2_STICKER2_QT/TCL4-p16-ET20-3000k-60-7t-stub.agt.zip
+• 3x3x3 cube, HTM: same as 2x2x2 cube, HTM, but with 120 7-tuples created by
+random walk, pmax = 9, Etrain = 13.
+Agent filename in GBG: 3x3x3_STICKER2_AT/TCL4-p9-ET13-3000k-120-7t-stub.agt.zip
+• 3x3x3 cube, QTM: same as 3x3x3 cube, HTM, but with pmax = 13, Etrain = 16.
+Agent filename in GBG: 3x3x3_STICKER2_QT/TCL4-p13-ET16-3000k-120-7t-stub.agt.zip
+The agent files given in the list above are just stubs, i.e. agents that are initialized with
+the correct parameters but not yet trained. This is because a trained agent can require up
+to 80 MB disk space, which is too much for GitHub. Instead, a user of GBG may load such
+a stub agent, train it (takes between 10-40 minutes) and save it to local disk.
+When evaluating in Sec. 6.2 the trained agents with different MCTS wrappers, we test
+in each case whether cPUCT = 1.0 or 10 is better. In most cases, cPUCT = 1.0 is better,
+but for (2x2x2, QTM, 800 iterations) and for (3x3x3, HTM, 100 iterations) cPUCT = 10.0 is
+the better choice.
+The parameters for training with symmetries (nSym = 8, 16, 24) in Sec. 6.4 are:
+• 3x3x3 cube, QTM: same as 3x3x3 cube, QTM in Sec. 6.2, but with nsym = 8, 16, 24.
+Agent filename in GBG: 3x3x3_STICKER2_QT/TCL4-p13-ET16-3000k-120-7t-nsym08-stub.agt.zip,
+3x3x3_STICKER2_QT/TCL4-p13-ET16-3000k-120-7t-nsym16-stub.agt.zip,
+3x3x3_STICKER2_QT/TCL4-p13-ET16-3000k-120-7t-nsym24-stub.agt.zip.
+42
+
+Again, the agent filenames are just stubs, i.e. agents that are initialized with the correct
+parameters but not yet trained. As above, a user of GBG may load such a stub agent, train
+it (which takes in the symmetry case between 5.4h and 13h, see Table 9) and save it to
+local disk.
+For further details and experiment shell scripts, see also the associated Papers-with-
+Code repository https://github.com/WolfgangKonen/PapersWithCodeRubiks.
+43
+
diff --git a/AtFLT4oBgHgl3EQfxTCi/content/tmp_files/load_file.txt b/AtFLT4oBgHgl3EQfxTCi/content/tmp_files/load_file.txt
new file mode 100644
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+page_content='d/TR-Rubiks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='pdf  Towards Learning Rubik’s Cube with  N-tuple-based Reinforcement Learning  Wolfgang Konen  Technical Report,  Computer Science Institute,  TH Köln,  University of Applied Sciences,  Germany  wolfgang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='konen@th-koeln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='de  Sep 2022,  last update Jan 2023  Abstract  This work describes in detail how to learn and solve the Rubik’s cube game (or  puzzle) in the General Board Game (GBG) learning and playing framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We cover  the cube sizes 2x2x2 and 3x3x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We describe in detail the cube’s state representation,  how to transform it with twists, whole-cube rotations and color transformations and  explain the use of symmetries in Rubik’s cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Next, we discuss different n-tuple  representations for the cube, how we train the agents by reinforcement learning and  how we improve the trained agents during evaluation by MCTS wrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  We present results for agents that learn Rubik’s cube from scratch, with and without  MCTS wrapping, with and without symmetries and show that both, MCTS wrapping  and symmetries, increase computational costs, but lead at the same time to much  better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We can solve the 2x2x2 cube completely, and the 3x3x3 cube in the  majority of the cases for scrambled cubes up to p = 15 (QTM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We cannot yet reliably  solve 3x3x3 cubes with more than 15 scrambling twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Although our computational costs are higher with MCTS wrapping and with sym-  metries than without, they are still considerably lower than in the approaches of McAleer  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2018, 2019) and Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) who provide the best Rubik’s cube  learning agents so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
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+page_content='  39  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3  STICKER2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
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+page_content='  39  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='4  Adjacency Sets  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
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+page_content='  40  C Hyperparameters  40  3  1  Introduction  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1  Motivation  Game learning and game playing is an interesting test bed for strategic decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Games usually have large state spaces, and they often require complex pattern recognition  and strategic planning capabilities to decide which move is the best in a certain situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  If algorithms learn a game (or, even better, a variety of different games) just by self-play,  given no other knowledge than the game rules, it is likely that they perform also well on  other problems of strategic decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  In recent years, reinforcement learning (RL) and deep neural networks (DNN) achieved  superhuman capabilities in a number of competitive games (Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=',  2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This success has been a product of the combination of reinforcement learning,  deep learning and Monte Carlo Tree Search (MCTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' However, current deep reinforcement  learning (DRL) methods struggle in environments with a high number of states and a small  number of reward states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  (a)  (b)  Figure 1: (a) Scrambled 3x3x3 Rubik’s Cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (b) 2x2x2 cube in the middle of a twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  The Rubik’s cube puzzle is an example of such an environment since the classical  3x3x3 cube has 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3 · 1019 states and only one state (the solved cube) has a reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' A  somewhat simpler puzzle is the 2x2x2 cube with 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='6 · 106 state and again only one reward  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Both cubes are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  The difficult task to learn from scratch how to solve arbitrary scrambled cubes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  without being taught by expert knowledge, whether from humans or from computerized  solvers) was not achievable with DRL methods for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Recently, the works of  McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2018, 2019) and Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) provided a breakthrough in that  direction (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1 and 7 for details): Their approach DAVI (Deep Approximate Value  Iteration) learned from scratch to solve arbitrary scrambled 3x3x3 cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  This work investigates whether TD-n-tuple learning with much lower computational de-  mands can solve (or partially solve) Rubik’s cube as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2  Overview  The General Board Game (GBG) learning and playing framework (Konen, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Konen and  Bagheri, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Konen, 2022) was developed for education and research in AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' GBG allows  applying the new algorithm easily to a variety of games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' GBG is open source and available  on GitHub1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The main contribution of this paper is to take the TD-n-tuple approach from  GBG (Scheiermann and Konen, 2022) that was also successful on other games (Othello,  ConnectFour) and to investigate this algorithm on various cube puzzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We will show that  it can solve the 2x2x2 cube perfectly and the 3x3x3 cube partly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' At the same time it has  drastically reduced computational requirements compared to McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We  will show that wrapping the base agent with an MCTS wrapper, as it was done by McAleer  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) and Scheiermann and Konen (2022), is essential to reach this success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  This work is at the same time an in-depth tutorial how to represent a cube and its  transformations within a computer program such that all types of cube operations can be  computed efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' As another important contribution we will show how symmetries  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='5, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='4) applied to cube puzzles can greatly increase sample efficiency and  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  The rest of this paper is organized as follows: Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 2 lays the foundation for Rubik’s  cube, its state representation, its transformations and its symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 3 we in-  troduce n-tuple systems and how they can be used to derive policies for game-playing  agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 4 defines and discusses several n-tuple representations for the cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 5  presents algorithms for learning the cube: first the DAVI algorithm of McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) and then our n-tuple-based TD learning (with extensions TCL and  MCTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 6 we present the results when applying our n-tuple-based TD learning  method to the 2x2x2 and the 3x3x3 cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 7 discusses related work and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 8 con-  cludes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='com/WolfgangKonen/GBG  5  2  Foundations  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1  Conventions and Symbols  We consider in this paper two well-known cube types, namely the 2x2x2 cube (pocket  cube) and the 3x3x3 cube (Rubik’s cube).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1  Color arrangement  Each cube consists of smaller cubies: 8 corner cubies for the 2x2x2 cube and 8 corner,  12 edge and 6 center cubies for the 3x3x3 cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' A corner cubie has 3 stickers of different  color on its 3 faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' An edge cubie has two, a center cubie has one sticker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  We enumerate the 6 cube faces with  (ULF) = (Up, Left, Front) and  (DRB) = (Down, Right, Back).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  We number the 6 colors with 0,1,2,3,4,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' My cube has these six colors  012 = wbo = (white,blue,orange) in the (ULF)-cubie2 and  345 = ygr = (yellow,green,red) in the opposing (DRB)-cubie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  The solved cube in default position has colors (012345) for the faces (ULFDRB), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' the  white color is at the Up face, blue at Left, orange as Front and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We can cut the cube  such that up- and bottom-face can be folded away and have a flattened representation as  shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  w  b  o  g  r  y  Figure 2: The face colors of the default cube in flattened representation  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2  Twist and Rotation Symbols  Twists of cube faces are denoted by uppercase letters U, L, F, D, R, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Each of these twists  means a 90◦ counterclockwise rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3 If U = U1 is a 90◦ rotation, then U2 is a 180◦  rotation and U3=U−1 is a 270◦ rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Whole-cube rotations are denoted by lowercase letters u, ℓ, f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (We do not need d, r, b  here, because d = u−1, r = ℓ−1 and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=')  Further symbols like fc[i], sℓ[i] that characterize a cube state will be explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3  Twist Types  Cube puzzles can have different twist types or twist metrics:  2We run through the faces of a cubie in counter-clockwise orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  3The rotation is counterclockwise when looking at the respective face  6  QTM (quarter turn metric): only quarter twists are allowed: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' U1 and U−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  HTM (half turn metric): quarter and half turns (twists) are allowed: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' U1, U2, U3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  By allowed we mean what counts as one move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' In QTM we can realize U2 via U U as  well, but it costs us 2 moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' In HTM, U2 counts as one move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  The twist type influences God’s number and the branching factor of the game, see  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2  Facts about Cubes  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1  2x2x2 Cube  The number of distinct states for the 2x2x2 pocket cube is (Wikipedia, 2022a)  8!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' · 37  24  = 7!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' · 36 = 3, 674, 160 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='6 · 106  (1)  Why this formula?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' — We have 8 cubies which we can place in 8!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' ways on the 8 cube  positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Each but the last cubie has the freedom to appear in 3 orientations, which gives  the factor 37 (the last cubie is then in a fixed orientation, the other two orientations would  yield illegal cube states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' – Each of these raw states has the (ygr)-cubie in any of the  24 possible positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Or, otherwise speaking, each truly different state appears in 24  whole-cube rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' To factor out the whole-cube rotations, we count only the states with  (ygr)-cubie in its default position (DRB) and divide the number of raw states by 24, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  God’s number: What is the minimal number of moves needed to solve any cube posi-  tion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' – For the 2x2x2 pocket cube, it is 11 in HTM (half-turn metric) and 14 in QTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Branching factor: 3 · 3 = 9 in HTM and 3 · 2 = 6 in QTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2  3x3x3 Cube  The number of distinct states for the 3x3x3 Cube is (Wikipedia, 2022b)  8!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' · 37 · 12!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' · 211  2  = 43, 252, 003, 274, 489, 856, 000 ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3 · 1019  (2)  Why this formula?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' – We have 8 corner cubies which we can place in 8!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' ways on the 8  cube positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Each but the last cubie has the freedom to appear in 3 orientations, which  gives the factor 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We have 12 edge cubies which we can place in 12!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' ways on the edge  positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Each but the last cubie has the freedom to appear in 2 orientations, which gives  the factor 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The division by 2 stems from the fact, that neither alone two corner cubies  may be swapped nor alone two edge cubies may be swapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Instead, the number of such  swaps must be even (factor 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  God’s Number: What is the minimal number of moves needed to solve any cube  position?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' — For the 3x3x3 Rubik’s Cube, it is 20 in HTM (half-turn metric) and 26 in QTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  This is a result from Rokicki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2014), see also http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='cube20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='org/qtm/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Branching factor: 6 · 3 = 18 in HTM and 6 · 2 = 12 in QTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  7  3  2  0  1  5  4  8  11  18  17  23  22  6  7  9  10  19  16  20  21  14  13  15  12  Figure 3: Sticker numbering for the 2x2x2 cube  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3  The Cube State  A cube should be represented by objects in GBG in such a way that  (a) cube states that are equivalent are represented by identical objects  (b) if two cube states are equivalent, it should be easy to check this by comparing their  objects  (c) cube transformations are easy to carry out on these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Condition (a) means that if two twist sequences lead to the same cube state (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' U−1  and UUU), this should result also in identical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Condition (b) means, that the equality  should be easy to check, given the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' That is, a cube should not be represented by  its twist sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  A cube state is in GBG represented by abstract class CubeState and has two describ-  ing members  fc[i]  =  fcol[i]  (3)  sℓ[i]  =  sloc[i]  (4)  fc[i] = fcol[i] denotes the face color at sticker location i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The color is one out of  0,1,2,3,4,5 for the colors w,b,o,y,g,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  sℓ[i] = sloc[i] contains the sticker location of the sticker which is in position i for the  solved cube d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Members fc and sℓ are vectors with 24 (2x2x2 cube) or 48 (3x3x3 cube) elements  where i denotes the ith sticker location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  The stickers are numbered in a certain way which is detailed in Figures 3 and 4 for the  flattened representations of the 2x2x2 and 3x3x3 cube, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  In principle, one of the two members fc and sℓ would be sufficient to characterize a  state, since the fcol-sloc-relation  fc[sℓ[i]] = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='fc[i]  (5)  holds, where d denotes the default cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This is because sℓ[i] transports the sticker i  of the default cube d to location sℓ[i], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' it has the color d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='fc[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' That is, we can easily  8  6  5  4  7  3  0  1  2  10  9  8  16  23  22  36  35  34  46  45  44  11  15  17  21  37  33  47  43  12  13  14  18  19  20  38  39  32  40  41  42  28  27  26  29  25  30  31  24  Figure 4: Sticker numbering for the 3x3x3 cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We do not number the center cubies, they  stay invariant under twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
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+page_content='Table 1: The three relevant twists for the 2x2x2 cube  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
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+page_content='14 15 10 11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='12 13 21 22  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='16 17 18 19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='F−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='T −1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='18 19 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
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+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
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+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='16 17 13 14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='20 21 22 23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='calculate fc given sℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' With some more effort, it is also possible to calculate sℓ given fc (see  Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Although one of these members fc and sℓ would be sufficient, we keep both  because this allows to better perform assertions or cross checks during transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Sometime we need the inverse function s−1  ℓ [i]: Which sticker is at location i?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' It is easy  to calculate s−1  ℓ  given sℓ with the help of the relation:  s−1  ℓ [sℓ[i]] = i  (6)  (Note that it is not possible to invert fc, because the face coloring function is not bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=')  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='4  Transformations  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1  Twist Transformations  Each basic twist is a counterclockwise4 rotation of a face by 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Table 1 shows the 2x2x2  transformation functions for three basic twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Each twist transformation can be coded in  two forms:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' T[i] (forward transformation): Which is the new location for the sticker being at i  before the twist?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  4The rotation is counterclockwise when looking at this face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  9  2  1  3  0  23  22  5  4  8  11  18  17  6  7  9  10  19  16  20  21  14  13  15  12  Figure 5: The default 2x2x2 cube after twist U1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' T −1[i] (inverse transformation): Which is the (parent) location of the sticker that lands  in i after the twist?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Example (read off from column 0 of Table 1): The L-twist transports sticker at 0 to 22:  T[0] = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The (parent) sticker being at location 9 before the L-twist comes to location 0  after the twist: T −1[0] = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Likewise, for the U-twist we have T[0] = 1 and T −1[0] = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We  show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 5 the default cube after twist U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  How can we apply a twist transformation to a cube state programmatically?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' – We  denote with f′  c and s′  ℓ the new states for fc and sℓ after transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The following  relations allow to calculate the transformed cube state:  f′  c[i]  =  fc[T −1[i]]  (7)  s′  ℓ[s−1  ℓ [i]]  =  T[i]  (8)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (7) says: The new color for sticker 0 is the color of the sticker which moves into  location 0 (fc[9] in the case of an L-twist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' To explain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (8), we first note that s−1  ℓ [i] is the  sticker being at i before the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Then, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (8) says: „The new location for the  sticker being at i before the transformation is T[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='“ For example, the L-twist transports the  current sticker at location 0 to the new location T[0] = 22, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' s′  ℓ[0] = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  For the 2x2x2 cube, these 3 twists U, L, F are sufficient, because D=U−1, R=L−1,  B=F−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This is because the 2x2x2 cube has no center cubies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' For the 3x3x3 cube, we  need all 6 twists U, L, F, D, R, B because this cube has center cubies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  In any case, we will show in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2 that only one row in Table 1 or Table 2, say T  for the U-twist, has to be known or established ’by hand’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' All other twists and their inverses  can be calculated programmatically with the help of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (9)-(15) that will be derived in  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='Table 2: The U twist for the 3x3x3 cube  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
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+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
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+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='10 11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='12 13 14 15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='16 17 18 19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='20 21 22 23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='U twist T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
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+page_content='22 23 16 11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='12 13 14 15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='36 17 18 19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='20 21 34 35  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='24 25 26 27  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='28 29 30 31  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='32 33 34 35  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='36 37 38 39  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='40 41 42 43  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='44 45 46 47  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='U twist T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='24 25 26 27  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='28 29 30 31  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='32 33 44 45  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='46 37 38 39  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='40 41 42 43  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='10 47  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='Table 3: Two basic whole-cube rotations for the 2x2x2 cube  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
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+page_content='15 12 13 14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='21 22 23 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='Normalizing the 2x2x2 Cube  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='As stated above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' the 3 twists U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' F are sufficient  for the 2x2x2 cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Therefore, the (DRB)-cubie will never leave its place, whatever the  twist sequence formed by U, L, F is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The (DRB)-cubie has the stickers (12, 16, 20), and we  can check in Table 1 that columns (12, 16, 20) are always invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' If we have an arbitrary  initial 2x2x2 cube state, we can normalize it by applying a whole-cube rotation such that  the (ygr)-cubie moves to the (DRB)-location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Normalizing the 3x3x3 Cube  In the case of the 3x3x3 cube, all center cubies  will be not affected by any twist sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Therefore, we normalize a 3x3x3 cube state  by applying initially a whole-cube rotation such that the center cubies are in their normal  position (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' white up, blue left and so on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2  Whole-Cube Rotations (WCR)  Each basic whole-cube rotation (WCR) is a counterclockwise rotation of the whole cube  around the u, l, f-axis by 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Table 3 shows two of the 2x2x2 transformation functions for  basic whole-cube rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Each rotation can be coded in two forms:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' T[i] (forward transformation): Which is the new location for the sticker being at i  before the twist?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' T −1[i] (inverse transformation): Which is the (parent) location of the sticker that lands  in i after the twist?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Besides the basic rotation u there is also u2 (180◦) and u3 = u−1 (270◦ = −90◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  All whole-cube rotations can be generated from these two forward rotations u and f:  First, we calculate the inverse transformations via  T −1[T[i]] = i  (9)  where T is a placeholder for u or f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Next, we calculate the missing base rotation ℓ (counter-  clockwise around the left face) as  ℓ = fuf−1  (10)  We use here the programm-code-oriented notation „first trafo first“: Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (10) reads as  „first f, then u, then f−1“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='5  5In programm code the relation would read cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='fTr(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='uTr().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='fTr(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This is „first trafo first“, because  each transformation is applied to the cube state object to the left and returns the transformed cube state object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='Table 4: All 24 whole-cube rotations (in first-trafo-first notation)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='number  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='first rotation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='∗ u0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='∗ u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='∗ u2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='∗ u3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='00-03  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='id (white up)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='id  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='u  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='u2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='u3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='04-07  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f (green up)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='fu  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='fu2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='fu3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='08-11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f2 (yellow up)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f2u  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f2u2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f2u3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='12-15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f−1 (blue up)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f−1u  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f−1u2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f−1u3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='16-19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='ℓ (orange up)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='ℓ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='ℓu  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='ℓu2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='ℓu3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='20-23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='ℓ−1 (red up)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='ℓ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='ℓ−1u  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='ℓ−1u2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='ℓ−1u3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='Table 5: The 24 inverse whole-cube rotations (in first-trafo-first notation)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='number  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='first rotation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='∗ u0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='∗ u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='∗ u2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='∗ u3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='00-03  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='id (white up)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='id  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='u3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='u2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='04-07  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f (green up)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='ℓu3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='fu2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='ℓ−1u  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='08-11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f2 (yellow up)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f2u  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f2u2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f2u3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='12-15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f−1 (blue up)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='ℓ−1u3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f−1u2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='ℓu  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='16-19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='ℓ (orange up)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='ℓ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f−1u3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='ℓu2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='fu  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='20-23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='ℓ−1 (red up)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='ℓ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='fu−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='ℓ−1u2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='f−1u  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='The other basic whole-cube rotations d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' b are not needed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' because d = u−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' r = ℓ−1  and b = f−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  The basic whole-cube rotations are rotations of the whole cube around just one axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  But there are also composite whole-cube rotations which consists of a sequence of basic  rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  How many different (composite) rotations are there for the cube?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' – A little thought  reveals that there are 24 of them: To be specific, we consider the default cube where we  have 4 rotations with the white face up, 4 with the blue face up, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' In total we have  6 · 4 = 24 rotations since there are 6 faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Table 4 lists all of them, togehter with the WCR  numbering convention used in GBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Sometimes we need the inverse whole-cube rotations which are given in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' In  this table, we read for example from the element with number 5, that the WCR with key 5  (which is fu according to Table 4) has the inverse WCR ℓu3 such that  fu ℓu3 = id  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  For convenience, we list in Table 6 the <Key, InverseKey> relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' For example, the  trafo with Key=5 (fu) has the inverse trafo with InverseKey=19 (ℓu3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Note that there are  10 whole-cube rotations which are their own inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Generating all twists from U twist  With the help of WCRs we can generate the other  twists from the U twist only: We simply rotate the face that we want to twist to the up-face,  12  Table 6: Whole-cube rotations: <Key, InverseKey> relation  key  0 1 2 3  4  5  6  7  8 9 10 11  12 13 14 15  16 17 18 19  20 21 22 23  inv key  0 3 2 1  12 19 6 21  8 9 10 11  4  23 14 17  20 15 18 05  16  7  22 13  apply the U twist and rotate back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This reads in first-trafo-first notation:  L = f−1Uf  (11)  F = ℓUℓ−1  (12)  D = f2Uf2  (13)  R = fUf−1  (14)  B = ℓ−1Uℓ  (15)  Thus, given the U twist from Table 1 or Table 2 and the basic WCRs given in Table 3 and  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (10), we can calculate all other forward transformations with the help of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (11)–(15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Then, all inverse transformations are calculable with the help of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3  Color Transformations  Color transformations are special transformations that allow to discover non-trivial symmet-  ric (equivalent) states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  One way to describe a color transformation is to select a valid color permutation and to  paint each sticker with the new color according to this color permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This is of course  nothing one can do with a real cube without destroying or altering it, but it is a theoretical  concept leading to an equivalent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Another way of looking at it is to record the twist sequence that leads from the default  cube to a certain scrambled cube state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Then we go back to the default cube, make at  first a whole-cube rotation (leading to a color-transformed default cube) and then apply the  recorded twist sequence to the color-transformed default cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  In any case, the transformed cube will be usually not in its normal position, so we apply  finally a normalizing operation to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  What are valid color permutations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' – These are permutations of the cube colors reach-  able when applying one of the available 24 WCRs (Table 4) to the default cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' For exam-  ple, if we apply WCR f (number 04) to the default cube, we get  g  w  o  y  r  b  Figure 6: The color transformation according to WCR f (number 04)  that is, g (green) is the new color for each up-sticker that was w (white) before and so  on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The colors o and r remain untouched under this color permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' [However, other  transformations like fu, fu2 and fu3 will change every color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=']  13  2  1  3  0  23  22  5  4  8  11  18  17  6  7  9  10  19  16  20  21  14  13  15  12  Figure 7: The cube of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 5 before color transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  16  19  17  18  20  23  2  1  11  10  13  12  3  0  8  9  14  15  21  22  4  7  5  6  8  2  9  1  4  7  14  11  18  17  23  0  5  6  15  10  19  16  20  3  21  13  22  12  (a)  (b)  Figure 8: The cube of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 7 with color transformation from Fig 6: (a) before normalization,  (b) after normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  How can we apply a color transformation to a cube state programmatically?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' – We  denote with f′ and s′  ℓ the new states for f and sℓ after transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  The following  relations allow to calculate the transformed cube state:  f′  c[i]  =  c[fc[i]]  (16)  s′  ℓ[s−1  ℓ [i]]  =  T[i]  (17)  where c[] is the 6-element color trafo vector (holding the new colors for current colors 0:w,  1:b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=', 5:r) and T is the 24- or 48-element vector of the WCR that produces this color  transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (16) is simple: If a certain sticker has color 0 (w, white) before the color  transformation, then it will get the new color c[0], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 4 (g, green), after the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (17) looks complicated, but it has a similar meaning as in the twist trafo: Take i = 0 as  example: The new place for the sticker being at 0 before the trafo (and coming from s−1  ℓ [0])  is T[0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Therefore, we write the number T[0] into s′  ℓ[s−1  ℓ [0]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  A color transformation example is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 7 is just a replication  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 5 showing a default cube after U1 twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The color transformation number 04 applied  to the cube of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 7 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 8 (a)-(b) in two steps:  (a) The stickers are re-painted and re-numbered (white becomes green, blue becomes  white and so on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The structure of coloring is the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Now the (DRB)-  cubie is no longer the (ygr)-cubie, it does not carry the numbers (12,16,20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  14  (b) We apply the proper WCR that brings the (ygr)-cubie back to the (DRB)-location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Compared to (a), each 4-sticker cube face is just rotated to another face, but not  changed internally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We can check that the (DRB)-location now carries again the num-  bers (12,16,20), as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 7 and as it should for a normalized cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='5  Symmetries  Symmetries are transformations of the game state (and the attached action, if applicable)  that lead to equivalent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  That is, if s is a certain state with value V (s), then all  states ssym being symmetric to s have the same value V (ssym) = V (s) because they are  equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Equivalent means: If s can be solved by a twist sequence of length n, then  ssym can be solved by an equivalent twist sequence of same length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  In the case of Rubik’s cube, all whole-cube rotations (WCRs) are symmetries because  they do not change the value of a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' But whole-cube rotations are ’trivial’ symmetries  because they are usually factored out by the normalization of the cube: After 2x2x2 cube  normalization, which brings the (ygr)-cubie in a certain position, or after 3x3x3 cube nor-  malization, which brings the center cubies in certain faces, all WCR-symmetric states are  transformed to the same state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Non-trivial symmetries are all color transformations (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3): In general, color trans-  formations transform a state s to a truly different state ssym, even after cube normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='6  Since there are 24 color transformations in Rubik’s cube, there are also 24 non-trivial sym-  metries (including self).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Symmetries are useful to learn to solve Rubik’s cube for two reasons: (a) to accelerate  learning and (b) to smooth an otherwise noisy value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  (a) Accelerated learning: If a state s (or state-action pair) is observed, not only the  weights activated by that state are updated, but also the weights of all symmetric states  ssym, because they have the same V (ssym) = V (s) and thus the same reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' In this  way, a single observed sample is connected with more weight updates (better sample  efficiency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  (b) Smoothed value function: By this we mean that the value function V (s) is replaced  by  V (sym)(s) =  1  |Fs|  �  s′∈Fs  V (s′)  (18)  where Fs is the set of states being symmetric to s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' If V (s) were the ideal value function,  both terms V (s) and V (sym)(s) would be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='7 But in a real n-tuple network, V (s)  is non-ideal due to n-tuple-noise (cross-talk from other states that activate the same  n-tuple LUT entries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' If we average over the symmetric states s′ ∈ Fs, the noise will be  dampened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  6In rare cases – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' for the solved cube – the transformed state may be identical to s or to another  symmetry state, but this happens seldom for sufficiently scrambled cubes, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  7because all V (s′) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (18) are the same for an ideal V  15  The downside of symmetries is their computational cost: In the case of Rubik’s cube,  the calculation of color transformations is a costly operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' On the other hand, the number  of necessary training episodes to reach a certain performance may be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' In the  end, the use of symmetries may pay off, because the total training time may be reduced as  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' In any case, we will have a better sample efficiency, since we learn more from each  observed state or state-action pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Secondly, the smoothing effect introduced with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (18)  can lead to better overall performance, because the smoothed value function provides a  better guidance on the path towards the solved cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  In order to balance computation time, GBG offers the option to select with nSym the  number of symmetries actually used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' If we specify for example nSym = 8 in GBG’s Rubik’s  cube implementation, then the state itself and 8 – 1 = 7 random other (non-id) color trans-  formations will be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The resulting set Fs of 8 states is then used for weight update  and value function computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  3  N-Tuple Systems  N-tuple systems coupled with TD were first applied to game learning by Lucas (2008), al-  though n-tuples were already introduced by Bledsoe and Browning (1959) for character  recognition purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The remarkable success of n-tuples in learning to play Othello (Lu-  cas, 2008) motivated other authors to benefit from this approach for a number of other  games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  The main goal of n-tuple systems is to map a highly non-linear function in a low di-  mensional space to a high dimensional space where it is easier to separate ‘good’ and  ‘bad’ regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This can be compared to the kernel trick of support-vector machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' An  n-tuple is defined as a sequence of n cells of the board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Each cell can have m positional  values representing the possible states of that cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='8 Therefore, every n-tuple will have a  (possibly large) look-up table indexed in form of an n-digit number in base m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Each entry  corresponds to a feature and carries a trainable weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' An n-tuple system is a system  consisting of k n-tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' As an example we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 9 an n-tuple system consisting of  four 8-tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Let Θ be the vector of all weights θi of the n-tuple system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='9 The length of this vector  may be large number, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' mnk, if all k n-tuples have the same length n and each cell  has m positional values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Let Φ(s) be a binary vector of the same length representing the  feature occurences in state s (that is, Φi(s) = 1 if in state s the cell of a specific n-tuple as  indexed by i has the positional value as indexed by i, Φi(s) = 0 else).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The value function  of the n-tuple network given state s is  V (s) = σ (Φ(s) · Θ)  (19)  with transfer function σ which may be a sigmoidal function or simply the identity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  8A typical example is a 2-player board game, where we usually have 3 positional values {0: empty, 1:  player1, 2: player2 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' But other, user-defined values are possible as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  9The index i indexes three qualities: an n-tuple, a cell in this n-tuple and a positional value for this cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  16  Figure 9: Example n-tuples: We show 4 random-walk 8-tuples on a 6x7 board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The tuples are  selected manually to show that not only snake-like shapes are possible, but also bifurcations  or cross shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Tuples may or may not be symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  An agent using this n-tuple system derives a policy from the value function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (19)  as follows: Given state s and the set A(s) of available actions in state s, it applies with a  forward model f every action a ∈ A(s) to state s, yielding the next state s′ = f(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Then  it selects the action that maximizes V (s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Each time a new agent is constructed, all n-tuples are either created in fixed, user-  defined positions and shapes, or they are formed by random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' In a random walk, all  cells are placed randomly with the constraint that each cell must be adjacent10 to at least  one other cell in the n-tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Agent training proceeds in the TD-n-tuple algorithm as follows: Let s′ be the actual  state generated by the agent and let s be the previous state generated by this agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' TD(0)  learning adapts the value function with model parameters Θ through (Sutton and Barto,  1998)  Θ ← Θ + αδ∇ΘV (s)  (20)  Here, α is the learning rate and V is in our case the n-tuple value function of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' δ is  the usual TD error (Sutton and Barto, 1998) after the agent has acted and generated s′:  δ = r + γV (s′) − V (s)  (21)  where the sum of the first two terms, reward r plus the discounted value γV (s′), is the  desirable target for V (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  10The form of adjacency, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 4- or 8-point neighborhood or any other (might be cell-dependent) form of  adjacency, is user-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  17  0  3  5  6  6  5  6  3  4  4  5  6  54  N-Tuple Representions for the Cube  In order to apply n-tuples to cubes, we have to define a board in one way or the other on  which we can place the n-tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This is not as straightforward as in other board games, but  we are free to invent abstract boards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Once we have defined a board, we can number the  board cells k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' , K−1 and translate a cube state into a BoardVector: A BoardVector  b is a vector of K non-negative integer numbers bk ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' , Nk − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Each k represents  a board cell and every board cell k has a predefined number Nk of position values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='11  A BoardVector is useful to calculate the feature occurence vector Φ(s) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (19) for  a given n-tuple set: If an n-tuple contains board cell k, then look into bk to get the position  value for this cell k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Set Φi(s) = 1 for that index i that indexes this n-tuple cell and this  position value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  In the following we present different options for boards and BoardVectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We do this  mainly for the 2x2x2 cube, because it is somewhat simpler to explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' But the same ideas  apply to the 3x3x3 cube as well, they are just a little bit longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Therefore, we defer the  lengthy details of the 3x3x3 cube to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1  CUBESTATE  A natural way to translate the cube state into a board is to use the flattened representation  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 11 as the board and extract from it the 24-element vector b, according to the given  numbering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The kth element bk represents a certain cubie face location and gets a number  from {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' , 5} according to its current face color fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The solved cube is for example  represented by b = [0000 1111 2222 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 5555].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  This representation CUBESTATE is what the BoardVecType CUBESTATE in our GBG-  implementation means: Each board vector is a copy of fcol, the face colors of all cubie  faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' fcol is also the vector that uniquely defines each cube state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' An upper bound of  possible combinations for b is 624 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='7 · 1018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' If we factor out the (DRB)-cubie, which  always stays at its home position, we can reduce this to 21 board cells with 6 positional  values, leading to 621 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1 · 1016 weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Both numbers are of course way larger than  the true number of distinct states (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1) which is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='6 · 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This is because most of  the combinations are dead weights in the n-tuple LUTs, they will never be activated during  game play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  The dead weights occur because many combinations are not realizable, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' three  white faces in one cubie or any of the 63 − 8 · 3 = 192 cubie-face-color combinations that  are not present in the real cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The problem is that the dead weights are scattered in a  complicated way among the active weights and it is thus not easy to factor them out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2  STICKER  McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) had the interesting idea for the 3x3x3 cube that 20 stickers (cubie  faces) are enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' To characterize the full 3x3x3 cube, we need only one (not 2 or 3) sticker  11In GBG package ntuple2 (base for agent TDNTuple3Agt), all Nk have to be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  In package  ntuple4( base for agent TDNTuple4Agt), numbers Nk may be different for different k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  18  (a) Top view  (b) Bottom view  Figure 10: The sticker representation used to reduce dimensionality: Stickers that are used  are shown in white, whereas ignored stickers are dark blue (from McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  3  2  0  1  5  4  8  11  18  17  23  22  6  7  9  10  19  16  20  21  14  13  15  12  Figure 11: Tracked stickers for the 2x2x2 cube (white), while ignored stickers are blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  for every of the 20 cubies, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This is because the location of one sticker  uniquely defines the location and orientation of that cubie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We name this representation  STICKER in GBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Translated to the 2x2x2 cube, this means that 8 stickers are enough because we have  only 8 cubies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We may for example track the 4 top stickers 0,1,2,3 plus the 4 bottom  stickers 12,13,14,15 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 11 and ignore the 16 other stickers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Since we always  normalize the cube such that the (DRB)-cubie with sticker 12 stays in place, we can reduce  this even more to 7 stickers (all but sticker 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  How to lay out this representation as a board?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' – McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) create a rect-  angular one-hot-encoding board with 7 × 21 = 147 cells (7 rows for the stickers and 21  columns for the locations) carrying only 0’s and 1’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This is fine for the approach of McAleer  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019), where they use this board as input for a DNN, but not so nice for n-tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Without constraints, such a board amounts to 2147 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='7 · 1044 combinations, which is  unpleasantly large (much larger than in CUBESTATE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='12  STICKER has more dead weights than CUBESTATE, so it seems like a step back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' But  the point is, that the dead weights are better structured: If for example sticker 0 appears at  column 1 then this column and the two other columns for the same cubie are automatically  12A possible STICKER BoardVector for the default cube would read b = [1000000 0100000 0010000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' ],  meaning that location 0 has the first sticker, location 1 has the second sticker, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' In any STICKER  BoardVector there are only 7 columns carrying exactly one 1, the other carry only 0’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Every row carries exactly  one 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  19  Table 7: The correspondence corner location ↔ STICKER2 for the solved cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The yellow  colored cells show the location of the 7 (2x2x2) and 8 (3x3x3) corner stickers that we track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  2x2x2  location  0 1 2 3  4  5  6  7  8  9  10 11  12 13 14 15  16 17 18 19  20 21 22 23  3x3x3  location  0 2 4 6  8 10 12 14  16 18 20 22  24 26 28 30  32 34 36 38  40 42 44 46  STICKER2  corner  a b c d  a  d  h  g  a  g  f  b  e  f  g  h  e  c  b  f  e  h  d  c  face ID  1 1 1 1  2  3  2  3  3  2  3  2  1  1  1  1  2  2  3  2  3  3  2  3  forbidden for all other stickers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Likewise, if sticker 1 is placed in another column, another set  of 3 columns is forbidden, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We can use this fact to form a much more compact  representation STICKER2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3  STICKER2  As the analysis in the preceding section has shown, the 21 location columns of STICKER  cannot carry the tracked stickers in arbitrary combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Each cubie (represented by 3  columns in STICKER) carries only exactly one sticker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We can make this fact explicit by  choosing another representation for the 21 locations:  corner location = (corner cubie, face ID).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  That is, each location is represented by a pair: corner cubie a,b,c,d,f,g,h (we number the  top cubies with letters a,b,c,d and the bottom cubies with letters e,f,g,h and omit e because  it corresponds to the (DRB)-cubie) and a face ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' To number the faces with a face ID, we  follow the convention that we start at the top (bottom) face with face ID 1 and then move  counter-clockwise around the corner cubie to visit the other faces (2,3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Table 7 shows the  explicit numbering in this new representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  To represent a state as board vector we use now a much smaller board shown in  Table 8: Each cell in the first row has 7 position values (the letters) and each cell in the  second row has 3 position values (the face IDs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We show in Table 8 the board vector for  the default cube, b = [abcdfgh 1111111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Representation STICKER2 allows for 77 · 37 =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='8 · 109 combinations in total, which is much smaller than STICKER and CUBESTATE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Table 8: STICKER2 board representation for the default 2x2x2 cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' For the BoardVector,  cells are numbered row-by-row from 0 to 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  corner  a  b  c  d  f  g  h  7 positions  face ID  1  1  1  1  1  1  1  3 positions  STICKER2 has some dead weights remaining, because the combinations can carry  the same letter multiple times, which is not allowed for a real cube state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' But this rate of  dead weights is tolerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  It turns out that STICKER2 is in all aspects better than CUBESTATE or STICKER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Therefore, we will only report the results for STICKER2 in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  20  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='4  Adjacency Sets  To create n-tuples by random walk, we need adjacency sets (sets of neighbors) to be  defined for every board cell k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  For CUBESTATE, the board is the flattened representation of the 2x2x2 cube (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  The adjacency set is defined as the 4-point neighborhood, where two stickers are neigh-  bors if they share a common edge on the cube, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' are neighbors on the cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  For STICKER2, the board consists of 16 cells shown in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Here, the adjacency  set for cell k contains all other cells different from k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Again, the details of ideas similar to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='4, but now for the 3x3x3 cube, are  shown in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1–B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  5  Learning the Cube  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1  McAleer and Agostinelli  The works of McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2018, 2019) and Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) contain up to now  the most advanced methods for learning to solve the cube from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  (2019) introduces the cost-to-go function for a general Marko decision process  J(s) = min  a∈A(s)  �  s′  P a(s, s′)  �  ga(s, s′) + γJ(s′)  �  (22)  where P a(s, s′) is the probability of transitioning from state s to s′ by taking action a and  ga(s, s′) is the cost for this transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' In the Rubik’s cube case, we have deterministic tran-  sitions, that is s′ = f(s, a) is deterministically prescribed by a forward model f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Therefore,  the sum reduces to one term and we specialize to γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Furthermore, we set ga(s, s′) = 1,  because only the length of the solution path counts, so that we get the simpler equation  J(s) = min  a∈A(s)  �  1 + J(s′)  �  with  s′ = f(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  (23)  Here, A(s) is the set of available actions in state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We additionally set J(s∗) = 0 if s∗  is the solved cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' To better understand Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (23) we look at a few examples: If s1 is a state  one twist away from s∗, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (23) will find this twist and set J(s1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' If s2 is a state two  twists away from s∗ and all one-twist states have already their correct labels J(s1) = 1,  then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (23) will find the twist leading to a s1 state and set J(s2) = 1 + 1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' While  iterations proceed, more and more states (being further away from s∗) will be correctly  labeled, once their preceding states are correctly labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' In the end we should ideally  have  J(sn) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  However, the number of states for Rubik’s cube is too large to store them all in tabular  form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Therefore, McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) and Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) approximate J(s) with  a deep neural network (DNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' To train such a network in the Rubik’s cube case, they  21  Algorithm 1 DAVI algorithm (from Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Input: B: batch size, K:  maximum number of twists, M: training iterations, C: how often to check for convergence,  ϵ: error threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Output: Θ, the trained neural network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  1: function DAVI(B, K, M, C, ϵ)  2:  Θ ← INITIALIZENETWORKPARAMETERS  3:  ΘC ← Θ  4:  for m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' , M do  5:  X ←GENERATESCRAMBLEDSTATES(B, K)  ▷ B scrambled cubes  6:  for xi ∈ X do  7:  yi ← mina∈A(s) [1 + jΘC(f(xi, a))]  ▷ cost-to-go function, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (23)  8:  (Θ, loss) ← TRAIN(jΘ, X, y)  ▷ loss = MSE(jΘ(xi), yi)  9:  if (m mod C = 0 & loss < ϵ) then  10:  ΘC ← Θ  11:  return Θ  introduce Deep Approximate Value Iteration (DAVI)13 shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The network  output jΘ(s) is trained in line 8 to approximate the (unknown) cost-to-go J(s) for every  state s = xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The main trick of DAVI is, as Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) write: „For learning to  occur, we must train on a state distribution that allows information to propagate from the  goal state to all the other states seen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Our approach for achieving this is  simple: each training state xi is obtained by randomly scrambling the goal state ki times,  where ki is uniformly distributed between 1 and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' During training, the cost-to-go function  first improves for states that are only one move away from the goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The cost-to-go  function then improves for states further away as the reward signal is propagated from the  goal state to other states through the cost-to-go function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='“  Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) use in Algorithm 1 two sets of parameters to train the DNN: the  parameters Θ being trained and the parameters ΘC used to obtain improved estimates  of the cost-to-go function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' If they did not use this two separate sets, performance often  „saturated after a certain point and sometimes became unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Updating ΘC only after  the error falls below a threshold ϵ yields better, more stable, performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='“ (Agostinelli  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=', 2019) To train the DNN, they used M = 1 000 000 iterations, each with batch size  B = 10 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Thus, the trained DNN has seen ten billion cubes (1010) during training, which  is still only a small subset of the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3 · 1019 possible cube states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  The heuristic function of the trained DNN alone cannot solve 100% of the cube states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Especially for higher twist numbers ki, an additional solver or search algorithm is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  This is in the case of McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) a Monte Carlo Tree Search (MCTS), similar to  AlphaZero (Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=', 2017), which uses the DNN as the source for prior probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) use instead a variant of A∗-search, which is found to produce  solutions with a shorter path in a shorter runtime than MCTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  13More precisely, McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) use Autodidactic Iteration (ADI), a precursor to DAVI, very similar to  DAVI, just a bit more complicated to explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Therefore, we describe here only DAVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  22  Algorithm 2 TD-n-tuple algorithm for Rubik’s cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Input: pmax: maximum number of  twists, M: training iterations, Etrain: maximum episode length during training, c: nega-  tive cost-to-go, Rpos: positive reward for reaching the solved cube s∗, α: learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  jΘ(s): n-tuple network value prediction for state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Output: Θ, the trained n-tuple network  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  1: function TDNTUPLE(pmax, M, Etrain, c, Rpos)  2:  Θ ← INITIALIZENETWORKPARAMETERS  3:  for m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' , M do  4:  p ∼ U(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' , pmax)  ▷ Draw p uniformly random from {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' , pmax}  5:  s ← SCRAMBLESOLVEDCUBE(p)  ▷ start state  6:  for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' , Etrain do  7:  snew ← arg max  a∈A(s)  V (s′)  with  s′ = f(s, a)  and  8:  V (s′) = c +  � Rpos  if  s′ = s∗  jΘ(s′)  if  s′ ̸= s∗  9:  Train network jΘ with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (20) to bring V (s) closer to target T = V (snew):  V (s) ← V (s) + α(T − V (s))  10:  s ← snew  11:  if (s = s∗) then  12:  break  ▷ break out of k-loop  13:  return Θ  23  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2  N-Tuple-based TD Learning  To solve the Rubik’s cube in GBG we use an algorithm that is on the one hand inspired by  DAVI, but on the other hand more similar to traditional reinforcement learning schemes like  temporal difference (TD) learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' In fact, we want to use in the end the same TD-FARL  algorithm (Konen and Bagheri, 2021) that we use for all other GBG games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  We show in Algorithm 2 our method, that we will explain in the following, highlighting  also the similarities and dissimilarities to DAVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  First of all, instead of minimizing the positive cost-to-go as in DAVI, we maximize in  lines 7-8 a value function V (s′) with a negative cost-to-go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This maximization is functionally  equivalent, but more similar to the usual TD-learning scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The negative cost-to-go, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1, plays the role of the positive 1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Secondly, we replace the DNN of DAVI by the simpler-to-train n-tuple network jΘ with  STICKER2 representation as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  That is, each time jΘ(s′) is  requested, we first calculate for state s′ the BoardVector in STICKER2 representation,  then the occurence vector Φ(s′) and the value function V (s′) according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  The central equations for V (s′) in Algorithm 2, lines 7-8, work similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (23) in  DAVI: If s = s1 is a state one twist away from s∗, the local search in arg max V (s′) will find  this twist and the training step in line 9 moves V (s) closer to c+Rpos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='14 Likewise, neighbors  s2 of s1 will find s1 and thus move V (s2) closer to 2c + Rpos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Similar for s3, s4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' under  the assumption that a ’known’ state is in the neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We have a clear gradient on  the path towards the solved cube s∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' If there are no ’known’ states in the neighborhood  of sn, we get for V (sn) what the net maximally estimates for all those neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We pick  the neighbor with the highest estimate, wander around randomly until we hit a state with a  ’known’ neighbor or until we reach the limit Etrain of too many steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Note that Algorithm 2 is different from DAVI insofar that it follows the path s → s′ → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  as prescribed by the current V , which may lead to a state sequence ’wandering in the  unknown’ until Etrain is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' In contrast to that, DAVI generates many start states s0  drawn from the distribution of training set states and trains the network just on pairs (s0, T),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' they do just one step on the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We instead follow the full path, because we want  the training method for Rubik’s cube to be as similar as possible to the training method for  other GBG games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='15  Algorithm 2 is basically the same algorithm as GBG uses for other games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The only  differences are (i) the cube-specific start state selection borrowed from DAVI (a 1-twist start  state has the same probability as a 10-twist start state) and (ii) the cube-specific reward in  line 8 of Algorithm 2 with its negative cost-go-go c which is however a common element of  many RL rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Algorithm 2 currently learns with only one parameter vector Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' However, it could be  extended as in DAVI to two parameter vectors Θ and ΘC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The weight training step in line  14It is relevant, that Rpos is a positive number, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='0 (and not 0, as it was for DAVI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This is because we  start with an initial n-tuple network with all weights set to 0, so the initial response of the network to any state  is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Thus, if Rpos were 0, a one-twist state would see all its neighbors (including s∗) initially as responding  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='0 and would not learn the right transition to s∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' With Rpos = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='0 it will quickly find s∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  15We note in passing that we tested the DAVI variant with Etrain = 1 for our TD-n-tuple method as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  However, we found that this method gave much worse results, so we stick with our GBG method here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  24  9 is done with the help of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (20) for Θ using the error signal δ of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  There are two extra elements, TCL and MCTS, that complete our n-tuple-based TD  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' They are described in the next two subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1  Temporal Coherence Learning (TCL)  The TCL algorithm developed by Beal and Smith Beal and Smith (1999) is an extension  of TD learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' It replaces the global learning rate α with the weight-individual product  ααi for every weight θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Here, the adjustable learning rate αi is a free parameter set by a  pretty simple procedure: For each weight θi, two counters Ni and Ai accumulate the sum  of weight changes and the sum of absolute weight changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' If all weight changes have the  same sign, then αi = |Ni|/Ai = 1, and the learning rate stays at its upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' If weight  changes have alternating signs, then the global learning rate is probably too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' In this  case, αi = |Ni|/Ai → 0 for t → ∞, and the effective learning rate will be largely reduced  for this weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  In our previous work (Bagheri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=', 2015) we extended TCL to αi = g(|Ni|/Ai) where  g is a transfer function being either the identity function (standard TCL) or an exponential  function g(x) = eβ(x−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' It was shown in Bagheri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2015) that TCL with this exponential  transfer function leads to faster learning and higher win rates for the game ConnectFour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2  MCTS  We use Monte Carlo Tree Search (MCTS) (Browne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=', 2012) to augment our trained  network during testing and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This is the method also used by McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  (2019) and by AlphaGo Zero (Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=', 2017), but they use it also during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  MCTS builds iteratively a search tree starting with a tree containing only the start state  s0 as the root node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Until the iteration budget is exhausted, MCTS does the following: In  every iteration we start from the root node and select actions following the tree policy until  we reach a yet unexpanded leaf node sℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The tree policy is implemented in our MCTS  wrapper according to the UCB formula (Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=', 2017):  anew  =  arg max  a∈A(s)  �W(s, a)  N(s, a) + U(s, a)  �  (24)  U(s, a)  =  cpuctP(s, a)  �  ε + �  b∈A(s) N(s, b)  1 + N(s, a)  (25)  Here, W(s, a) is the accumulator for all backpropagated values that arrive along branch  a of the node that carries state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Likewise, N(s, a) is the visit counter and P(s, a) the prior  probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' A(s) is the set of actions available in state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' ε is a small positive constant for  the special case �  b N(s, b) = 0: It guarantees that in this special case the maximum of  U(s, a) is given by the maximum of P(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The prior probabilities P(s, a) are obtained  25  Algorithm 3 TD-n-tuple training algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Input: see Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Output: Θ: trained  n-tuple network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  1: function TDNTUPLETRAIN(pmax, M, Etrain, c, Rpos)  2:  Θ ← INITIALIZENETWORKPARAMETERS  3:  INITIALIZETCLPARAMETERS  ▷ Set TCL-accumulators Ni = Ai = 0, αi = 1 ∀i  4:  for m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' , M do  5:  Perform one m-iteration of Algorithm 2 with learning rates ααi instead of α  6:  Ni ← Ni + ∆θi and Ai ← Ai + |∆θi|  ▷ Update TCL-accumulators  7:  ▷ where ∆θi is the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (20)  8:  αi ← |Ni|/Ai  ∀i with Ai ̸= 0  9:  return Θ  Algorithm 4 Evaluation algorithm with MCTS solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Input: trained n-tuple network jΘ,  p: number of scrambling twists, B: batch size, Eeval: maximum episode length during  evaluation, I: number of MCTS-iterations, cPUCT : relative weight for U(s, a) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (24),  dmax: maximum MCTS tree depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Output: solved rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  1: function TDNTUPLEEVAL(jΘ, p, B, Eeval, I, cPUCT , dmax)  2:  X ←GENERATESCRAMBLEDCUBES(B, p)  ▷ B scrambled cubes  3:  Csolved ← 0  4:  for xi ∈ X do  5:  s ← xi  6:  for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' , Eeval do  7:  T ← PERFORMMCTSSEARCH(s, I, cPUCT , dmax, jΘ)  8:  a ← SELECTMOSTVISITEDACTION  9:  s ← f(s, a)  10:  if (s = s∗) then  11:  Csolved ← Csolved + 1  12:  break  ▷ break out of k-loop  13:  return Csolved/B  ▷ percentage solved  by sending the trained network’s values of all follow-up states s′ = f(s, a) with a ∈ A(s)  through a softmax function (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='16  Once an unexpanded leaf node sℓ is reached, the node is expanded by initializing  its accumulators: W(s, a) = N(s, a) = 0 and P(s, a) = ps′ where ps′ is the softmax-  squashed output jΘ(s′) of our n-tuple network for each state s′ = f(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The value of  the node is the network output of the best state jΘ(sbest) = maxs′ jΘ(s′) and this value is  backpropagated up the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  More details on our MCTS wrapper can be found in Scheiermann and Konen (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  16Note that the prior probabilities and the MCTS iteration are only needed at test time, so that we – different  to AlphaZero – do not need MCTS during self-play training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  26  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3  Method Summary  We summarize the different ingredients of our n-tuple-based TD learning method in Algo-  rithm 3 (training) and Algorithm 4 (evaluation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  In line 5 of Algorithm 3 we perform one m-iteration of Algorithm 2 which does an update  step for weight vector Θ, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' All weights of activated n-tuple entries get a weight  change ∆θi equal to the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (20) where the global α is replaced by ααi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Line 2 in Algorithm 4 generates a set X of B scrambled cube states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Line 7 builds for  each xi ∈ X an MCTS tree (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2) starting from root node xi and line 8 selects  the most visited action of the root node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' If the goal state s∗ is not found during Eeval k-loop  trials, this xi is considered as not being solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  6  Results  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1  Experimental setup  We use for all our GBG experiments the same RL method based on n-tuple systems and  TCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Only its hyperparameters are tuned to the specific game, as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We refer  to this method/agent as TCL-base whenever it alone is used for game playing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' If we wrap  such an agent by an MCTS wrapper with a given number of iterations, then we refer to this  as TCL-wrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  We investigate two variants of Rubik’s Cube: 2x2x2 and 3x3x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We trained all TCL  agents by presenting them M = 3 000 000 cubes scrambled with p random twists, where  p is chosen uniformly at random from {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' , pmax}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Here, pmax = 13 [16] for 2x2x2 and  pmax = 9 [13] for 3x3x3, where the first number is for HTM, while the second number  in square brackets is for QTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' With these pmax cube twists we cover the complete cube  space for 2x2x2, where God’s number (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2) is known to be 11 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' But we cover only  a small subset in the 3x3x3 case, where God’s number is known to be 20 [26] (Rokicki  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='17 We train 3 agents for each cube variant { 2x2x2, 3x3x3 } × { HTM, QTM }  to assess the variability of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  The hyperparameters of the agent for each cube variant were found by manual fine-  tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' For brevity, we defer the exact explanation and setting of all parameters to Ap-  pendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  We evaluate the trained agents for each p on 200 scrambled cubes that are created by  applying the given number p of random scrambling twists to a solved cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The agent now  tries to solve each scrambled cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' A cube is said to be unsolved during evaluation if the  agent cannot reach the solved cube in Eeval = 50 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='18  17We limit ourselves to pmax = 9 [13] in the 3x3x3 HTM [QTM ] case, because our network has not enough  capacity to learn all states of the 3x3x3 Rubik’s cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Experiments with higher twist numbers during training  did not improve the solved-rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  18During training, we use lower maximum episode lengths Etrain (see Appendix C) than Eeval = 50 in  order to reduce computation time (in the beginning, many episodes cannot be solved, and 50 would waste a  lot of computation time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' But Etrain is always at least pmax + 3 in order to ensure that the agent has a fair  chance to solve the cube and collect the reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  27  HTM  0%  25%  50%  75%  100%  1  3  5  7  9  11  13  scrambling twists  percentage solved  cubeWidth  2x2x2  3x3x3  iterMWrap  0  100  800  QTM  0%  25%  50%  75%  100%  1  3  5  7  9  11  13  15  scrambling twists  percentage solved  cubeWidth  2x2x2  3x3x3  iterMWrap  0  100  800  Figure 12: Percentage of solved cubes as a function of scrambling twists p for the trained  TD-N-tuple agent wrapped by MCTS wrapper with different numbers of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The red  curves are TCL-base without wrapper, the other colors show different forms of TCL-wrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Twist type is HTM (left) and QTM (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Each point is the average of 3 independently trained  agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2  Cube Solving with MCTS Wrapper, without Symmetries  The trained TD-N-tuple agents learn to solve the cubes to some extent, as the red curves  TCL-base in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 12 show, but they are in many cases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' p > pmax/2) far from being  perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' These are the results from training each agent for 3 million episodes, but the  results would not change considerably, if 10 million training episodes were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Scheiermann and Konen (2022) have shown, that the performance of agents, namely  TD-N-tuple agents, is largely improved, if the trained agents are wrapped during test, play  and evaluation by an MCTS wrapper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This holds for Rubik’s cube as well, as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 12 shows:  For the 2x2x2 cube, the non-wrapped agent TCL-base (red curve) is already quite good,  but with wrapping it becomes almost perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' For the 3x3x3 cube, the red curves are not  satisfactorily: the solved-rates are below 20% for p = 9 [13] in the HTM [QTM ] case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' But  at least MCTS wrapping boosts the solved-rates by a factor of 3 [QTM: from 16% to 48%]  or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='5 [HTM: from 10% to 45%].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  All these results are without incorporating symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  How symmetries affect the  solved-rates will be investigated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' But before this, we look in the next section at  the number of symmetries that effectively exist in a cube state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3  Number of Symmetric States  Not every cube state has 24 truly different symmetric states (24 = number of color sym-  metries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' For example in the solved cube, all color-symmetric states are the same (after  normalization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Thus, we have here only one truly different symmetric state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  However, we show in this section that for the majority of cube states the number of  truly different symmetric states is close to 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Two states are truly different if they are  not the same after the normalizing operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We generate a cube state by applying p  random scrambling twists to the default cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Now we apply all 24 color transformations  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3) to it and count the truly different states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 13 for  28  twistType: HTM  twistType: QTM  0  4  8  12  16  0  4  8  12  16  0  5  10  15  20  25  scrambling twists  NsymmetricStates  cubeWidth  2x2x2  3x3x3  Figure 13: Count of truly different symmetric states for cube states generated by p random  scrambling twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Each point is an average over 500 such states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  both cube sizes and both twist types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' For the 3x3x3 cube, the number of states quickly (for  p > 5) approaches the maximum N = 24, while for the 2x2x2 cube it is a bit slower: p > 4  or p > 8 is needed to surpass N = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  As a consequence, it makes sense to use 16 or even 24 symmetries when training  and evaluating cube agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Especially for scrambled states with higher p, the 24 color  transformations used to construct symmetric states will usually lead to 24 different states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='4  The Benefit of Symmetries  In order to investigate the benefits of symmetries, we first train a TCL agent with dif-  ferent numbers of symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' As described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='5, we select in each step nSym  = 0, 8, 16, 24 symmetric states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Which symmetric states are chosen is selected randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Symmetries are used (a) to update the weights for each symmetric state and (b) to build  with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (18) a smoothed value function which is used to decide about the next action dur-  ing training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' For 0, 8, 16, 24 symmetries, we train 3 agents each (3x3x3 cube, STICKER2,  QTM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The 3 agents differ due to their differently created random-walk n-tuple sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 14 shows the learning curves for different nSym = 0, 8, 16, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' It is found that agents  with nSym > 0 learn faster and achieve a higher asymptotic solved rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Next, we evaluate each of the trained agents by trying to solve for each p ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' , 15}  (scrambling twists) 200 different scrambled cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' During evaluation, we use again the  same nSym as in training to form a smoothed value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We compare in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 15 different  29  50%  60%  70%  80%  90%  0e+00  1e+06  2e+06  3e+06  episodes  percentage solved  nSym  24  16  8  0  Figure 14: Learning curves for different numbers nSym = 0, 8, 16, 24 of symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Shown is  the solved rate of (3x3x3, QTM) cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The solved rate is the average over all twist numbers  p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' , 13 with 200 testing cubes for each p and over 3 agents with different random-walk  n-tuple sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  symmetry results, both without wrapping (TCL-base, red curves) and with MCTS-wrapped  agents using 100 (green) or 800 (blue) iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' It is clearly visible that MCTS wrapping  has a large effect, as it was also the case in Fig 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' But in addition to that, the use of  symmetries leads for each agent, wrapped or not, to a substantial increase in solved-rates  (a surplus of 10-20%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' It is remarkable, that even for p=14 or 15 a solved rate above or  near 50% can be reached19 by the combination (nSym=16, 800 MCTS iterations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Surprisingly, it seems that with wrapping it is only important whether we use symme-  tries, not how many, since the difference between nSym = 8, 16, 24 is only marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' For  800 MCTS iterations, the solved rate for nSym = 24 is in most cases even smaller than that  for nSym = 8, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This is surprising because it would have been expected that also with  wrapping a larger nSym should lead to a smoother value function and thus should in theory  produce larger solved rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' – Note that this is not a contradiction to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 14, because the  learning curves were obtained without wrapping and the red TCL-base curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 15  (again without wrapping) show the same positive trend with increasing nSym20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The red  curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 15 show approximately the same average solved rates as the asymptotic  values in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='5  Computational Costs  Table 9 shows the computational costs when training and testing with symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' All  computations were done on a single CPU Intel i7-9850H @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='60GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' If we subtract the  19p is above pmax=13, the maximum twist number used during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  20i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' nSym= 24 is for every p clearly better than nSym= 16  30  QTM  3x3x3  0%  25%  50%  75%  100%  1  3  5  7  9  11  13  15  scrambling twists  percentage solved  nSym  0  8  16  24  iterMWrap  0  100  800  Figure 15: With symmetries: Percentage of solved cubes (3x3x3, QTM) as a function of  scrambling twists p for TD-N-tuple agents trained and evaluated with different numbers of  symmetries nSym and wrapped by MCTS wrappers with different iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The red curves  are TCL-base (without wrapper), the other colors show different forms of TCL-wrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The  solved rates are the average over 200 testing cubes for each p and over 3 agents with differ-  ent random-walk n-tuple sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  computational costs for nsym= 0, computation time increases more or less linearly with  iter and roughly linearly with nSym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Computation times for nSym= 24 are approximately  10x larger than those for nSym= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Computation times are dependent on the solved rate: If a cube with p = 13 is solved,  the episode takes normally 12-15 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' If the cube is not solved, the episode needs 50  steps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' a factor of 3-4 more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Thus, the numbers in Table 9 should be taken only as  rough indication of the trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Bottom line: Training time through symmetries increases by a factor of 13/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='5 = 26  (nSym= 24) and testing time increases through 800 MCTS iterations by a factor of about  3130/8 ≈ 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Training with symmetries takes between 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='4h and 13h on a normal CPU, depending  on the number of symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This is much less than the 44h on a 32-core server with 3  GPUs that were used by McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' But it also does not reach the same quality  as McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  31  Table 9: Computation times with symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' All numbers are for 3x3x3 cube, STICKER2  and QTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Training: 3 million self-play episodes, w/o MCTS in the training loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Testing: 200  scrambled cubes with p = 13, agents wrapped by MCTS wrapper with iter iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  nSym  training  testing  [hours]  [seconds]  iter  0  100  400  800  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='5  48  196  390  8  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='0  241  877  1400  16  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='5  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3  464  1380  2330  24  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='0  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='0  550  1760  3130  7  Related Work  Ernö Rubik invented Rubik’s cube in 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Rubik’s cube has gained worldwide popularity  with many human-oriented algorithms being developed to solve the cube from arbitrary  scrambled start states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' By ’human-oriented’ we mean algorithms that are simple to mem-  orize for humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' They usually will find long, suboptimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' For a long time it was  an unsolved question what is the minimal number of moves (God’s Number) needed to  solve any given cube state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The early work of Thistlethwaite (1981) put an upper bound on  this number with his 52-move algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This was one of the first works to systematically  use group theory as an aid to solve Rubik’s cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Later, several authors have gradually  reduced the upper bound 52 (Joyner, 2014), until Rokicki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2014) could prove in 2014  for the 3x3x3 cube that God’s Number is 20 in HTM and 26 in QTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Computer algorithms to solve Rubik’s cube rely often on hand-engineered features and  group theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' One popular solver for Rubik’s cube is the two-phase algorithm of Kociemba  (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' A variant of A∗ heuristic search was used by Korf (1991), along with a pattern  database heuristic, to find the shortest possible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  The problem of letting a computer learn to solve Rubik’s cube turned out to be much  harder: Irpan (2016) experimented with different neural net baseline architectures (LSTM  gave for him reportedly best results) and tried to boost them with AdaBoost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' However, he  had only for scrambling twist ≤ 7 solved rates of better than 50% and the baseline turned  out to be better than the boosted variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Brunetto and Trunda (2017) found somewhat  better results with a DNN, they could solve cube states with 18 twists with a rate above  50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' But they did not learn from scratch because they used an optimal solver based  on Kociemba (2015) to generate training examples for the DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2016) tried  to learn Rubik’s cube by genetic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' However, their learned solver could only  reliably solve cubes with up to 5 scrambling twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  A breakthrough in learning to solve Rubik’s cube are the works of McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2018,  2019) and Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019): With Autodidactic Iteration (ADI) and Deep Approxi-  mate Value Iteration (DAVI) they were able to learn from scratch to solve Rubik’s cube in  QTM for arbitrary scrambling twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Their method has been explained in detail already  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1, so we highlight here only their important findings: McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) only  needs to inspect less than 4000 cubes with its trained network DeepCube when solving  32  for a particular cube, while the optimal solver of Korf (1991) inspects 122 billion different  nodes, so Korf’s method is much slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) extended the work of McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) by replacing the  MCTS solver with a batch-weighted A∗ solver which is found to produce shorter solution  paths and have shorter run times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' At the same time, Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) applied their  agent DeepCubeA successfully to other puzzles like LightsOut, Sokoban, and the 15-, 24-,  35- and 48-puzzle21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' DeepCubeA could solve all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  The deep network used by McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) and Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) were  trained without human knowledge or supervised input from computerized solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The  network of McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) had over 12 million weights and was trained for 44 hours  on a 32-core server with 3 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The network of McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) has seen 8 billion  cubes during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' – Our approach started from scratch as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' It required much less  computational effort (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='4h training time on a single standard CPU for nSym=8, see  Table 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' It can solve the 2x2x2 cube completely, but the 3x3x3 cube only partly (up to 15  scrambling twists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Each trained agent for the 3x3x3 cube has seen 48 million scrambled  cubes22 during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  8  Summary and Outlook  We have presented new work on how to solve Rubik’s cube with n-tuple systems, reinforce-  ment learning and an MCTS solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The main ideas were already presented in Scheier-  mann and Konen (2022) but only for HTM and up to p = 9 twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Here we extended  this work to QTM as well and presented all the details of cube representation and n-tuple  learning algorithms necessary to reproduce our Rubik’s cube results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' As a new aspect,  we added cube symmetries and studied their effect on solution quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We found that the  use of symmetries boosts the solved rates by 10-20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Based on this, we could increase  for QTM the number of scrambling twists where at least 45% of the cubes are solved from  p = 13 without symmetries to p = 15 with symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  We cannot solve the 3x3x3 cube completely, as McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) and Agostinelli  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' But our solution is much less computational demanding than their ap-  proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
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+page_content=' Agostinelli, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
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+page_content=' Shmakov, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Baldi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
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+page_content=' Nature Machine Intelligence, 1(8):356–363, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
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+page_content=' Pocket Cube, 2022a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' URL https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='org/wiki/Pocket_Cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  retrieved Aug-17-2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 7  Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Rubik’s Cube, 2022b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  URL https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='org/wiki/Rubik’s_  Cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' retrieved Aug-17-2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 7  36  Appendix  A  Calculating sloc from fcol  Given the face colors fc (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (3)) of a transformed cube, how can we calculate the trans-  formed sticker locations sℓ (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (4))?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  This problem seems ill-posed at first sight, because a certain face color, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' white,  appears multiple times in fc and it is not possible to tell from the appearance of white alone  to which sticker location sℓ it corresponds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' But with a little more effort, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' by looking at the  neighbors of the white sticker, we can solve the problem, as we show in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1  2x2x2 cube  All cubies of the 2x2x2 cube are corner cubies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We track for each cubie exactly one sticker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  This can be for example the set  B = {0, 1, 2, 3, 12, 13, 14, 15}  of 8 stickers, which is the same as the set of tracked stickers shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  For each s ∈ B:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Build the cubie that contains s as the first sticker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='23  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Locate the cubie in fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' That is, find a location in fc with the same color as the 1st  cubie face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' If found, check if the neighbor to the right24 has the color of the 2nd cubie  face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' If yes, check if its neighbor to the right has the color of the 3rd cubie face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' If  yes, we have located the cubie in fc and we return it, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' its three sticker locations  C = [a, b, c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Having located the cubie, we can infer three elements of sℓ:  sℓ[s]  =  C[0]  (26)  sℓ[R[s]]  =  C[1]  (27)  sℓ[R[R[s]]]  =  C[2]  (28)  Here R[s] is the right neighbor of sticker s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' R[R[s]]] is the left neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  In total, we have located 8 × 3 = 24 stickers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' the whole transformation for sℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='25  23We know for example from looking at the default cube in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 11 that sticker s = 0 is part of the 0-8-4-cubie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  24By neighbor to the right we mean the next sticker when we march in clockwise orientation around the  actual cubie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  25The relevant GBG source code is in CubeState.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='locate and CubeState2x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='apply_sloc_slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  37  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2  3x3x3 cube  The 3x3x3 cube has 8 corner cubies and 12 edge cubies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We track for each cubie exactly  one sticker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This can be for the corners the set  B = {0, 2, 4, 6, 24, 26, 28, 30}  and for the edges the set  E = {1, 3, 5, 7, 25, 27, 29, 31, 11, 15, 21, 33}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  We do for the corner set B the same as we did for the 2x2x2 cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  For each element s ∈ E of the edge set:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Build the edge cubie cE that contains s as the first sticker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Locate the cubie in fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' That is, find an edge location in fc with the same color as the  1st cubie face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' If found, check if the other sticker of that cubie has the same color as  the other sticker of cE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' If yes, we have located the edge cubie in fc and we return it,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' its two stickers C = [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Having located the cubie, we can infer two elements of sℓ:  sℓ[s]  =  C[0]  (29)  sℓ[O[s]]  =  C[1]  (30)  Here O[s] is the other sticker of the edge cubie that has sticker s as first sticker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  In total, we have located  8 × 3 + 12 × 2 = 48  stickers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' the whole transformation for sℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='26  B  N-Tuple Representations for the 3x3x3 Cube  In this appendix we describe the n-tuple representations of the cube, analogously to the  2x2x2 cube Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 4, but now for the 3x3x3 cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1  CUBESTATE  A natural way to translate the cube state into a board is to use the flattened representation  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 4 as the board and extract from it the 48-element vector b, according to the given  numbering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The kth element bk represents a certain cubie face location and gets a number  from {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' , 5} according to its current face color fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The solved cube is for example  represented by b = [00000000 11111111 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 55555555].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  This representation CUBESTATE is what the BoardVecType CUBESTATE in our GBG-  implementation means: Each board vector is a copy of fcol, the face colors of all cubie  faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' An upper bound of possible combinations for b is 648 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2 · 1032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This is much  larger than the true number of distinct states (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2) which is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3 · 1019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  26The relevant GBG source code is in CubeState.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='locate, CubeState3x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='locate_edge and CubeState3x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='apply_sloc_slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  38  Table 10: The correspondence edge location ↔ STICKER2 for the solved cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The yellow  colored cells show the location of the 12 edge stickers that we track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  3x3x3  location  1 3 5  7  9 11 13 15  17 19 21 23  25 27 29 31  33 35 37 39  41 43 45 47  STICKER2  edge  A B C D  D  G  K  E  E  J  F  A  I  J  K  L  H  B  F  I  L  G  C  H  face ID  1 1 1  1  2  2  2  2  1  1  1  1  1  1  1  1  2  2  2  2  1  1  1  1  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2  STICKER  McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) had the interesting idea for the 3x3x3 cube that 20 stickers (cubie  faces) are enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' To characterize the 3x3x3 cube, we need according to McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  (2019) only one (not 2 or 3) sticker for every of the 20 cubies, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This  is because the location of one sticker uniquely defines the location and orientation of that  cubie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' We name this representation STICKER in GBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  We track the 4 top corner stickers 0,2,4,6 plus the 4 bottom corner stickers 24,26,28,30  plus one sticker for each of the 12 edge stickes as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 10, in total 20 stickers and  ignore the 28 other stickers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  How to lay out this representation as a board?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' – McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) create a rect-  angular one-hot-encoding board with 20 × 24 = 480 cells (20 rows for the stickers and  24 columns for the locations27) carrying only 0’s and 1’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This is fine for the approach of  McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019), where they use this board as input for a DNN, but not so nice for  n-tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Without constraints, such a board amounts to 2480 ≈ 10145 combinations, which  is unpleasantly large (much larger than in CUBESTATE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='28  Another possibility to lay out the board: Specify 20 board cells (the stickers) with 24  position values each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This amounts to 2420 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='0 · 1027 combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3  STICKER2  Analogously to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3, we represent the 24 corner locations and 24 edge locations as:  corner location = (corner cubie, face ID),  edge location = (edge cubie, face ID).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  That is, each corner location is represented by a corner cubie a,b,c,d,e,f,g,h and by a face  ID 1,2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Table 7 shows the explicit numbering in this new representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Additionally,  each edge location is represented by an edge cubie A,B,C,D,E,F,G,H,I,J,K,L29 and by a  face ID 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Convention for face ID numbering of edge cubies: For top- and bottom-layer  edge cubies, it is 1 for U and D stickers, 2 else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The face ID for middle-layer edge cubies is  1 for F and B stickers, 2 else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Table 10 shows the explicit numbering in this representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  The corresponding board consists of 8 + 8 + 12 +12 = 40 cells shown in Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  The 8 cell pairs in the first two rows code the locations of the tracked corner stickers  278 · 3 for the corner stickers and 12 · 2 for the edge stickers  28McAleer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (2019) do not need a weight for every of the 2480 possible states, as the n-tuple network  would need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Instead they need only 480 · 4096 = 2 · 106 weights to the first hidden layer having 4096 neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  294 U-stickers, 4 D-sticker, 4 middle-layer stickers (2F, 2B)  39  0,2,4,6,24,26,28,30, see Table 7 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The 12 cell pairs in the last two rows code the  location of the tracked edge stickers 1,3,5,7,17,21,43,47,25,27,29,31, see Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This  n-tuple coding requires tuple cells with varying number of position values and leads to  88 · 38 · 1212 · 212 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='0 · 1027  combinations in representation STICKER2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='30  Table 11: STICKER2 board representation for the default 3x3x3 cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' For the BoardVector,  cells are numbered row-by-row from 0 to 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  corner  a  b  c  d  e  f  g  h  8 positions  face ID  1  1  1  1  1  1  1  1  3 positions  edge  A  B  C  D  E  F  G  H  I  J  K  L  12 positions  face ID  1  1  1  1  1  1  1  1  1  1  1  1  2 positions  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='4  Adjacency Sets  To create n-tuples by random walk, we need to define adjacency sets (sets of neighbors)  for every board cell k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  For CUBESTATE, the board is the flattened representation of the 3x3x3 cube (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  The adjacency set is defined as the 4-point neighborhood, where two stickers are neigh-  bors if they are neighbors (share a common edge) on the cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  For STICKER2, the board consists of 40 cells shown in Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Since it matters for  the corner stickers mostly where the other corner stickers are and for the edge stickers  mostly where the other edge stickers are, it is reasonable to form two adjacency subsets  S1 = {00, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' , 15} and S2 = {16, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' , 39} and to define the adjacency set  Adj(k) = Si \\ {k}  for each k ∈ Si, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  C  Hyperparameters  In this appendix we list all parameter settings for the GBG agents used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Pa-  rameters were manually tuned with two goals in mind: (a) to reach high-quality results and  (b) to reach stable (robust) performance when conducting multiple training runs with differ-  ent random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The agents listed further down are the best-so-far agents found (best  among all agents that learn from scratch by self-play).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  The detailed meaning of RL parameters is explained in Konen and Bagheri (2021):  30This is, by the way, identical to (8·3)8 ·(12·2)12 = 24(8+12) = 2420 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='0·1027, the same number we had  above in the second mode of STICKER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' But STICKER2 has the advantage that the combinations are spread  over more board cells (40) than in STICKER (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' By having more board cells with fewer position values, the  n-tuples can better represent the relationships between cube states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  40  Algorithms 2, 5 and 7 in Konen and Bagheri (2021) explain parameters α (learning  rate), γ (discount factor), ϵ (exploration rate) and output sigmoid σ (either identity or  tanh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='3 explains our eligibility method, parameters are: eligibility trace factor λ,  horizon cut ch, eligibility trace type ET (normal) or RESET (reset on random move).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  If not otherwise stated, we use in this paper λ = 0 (no eligibility traces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' For λ = 0,  horizon cut ch and eligibity trace type are irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' If λ > 0, their defaults ch = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1  and trace type ET apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='5 explains our TCL method (also summarized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Parameters  of TCL are: TC-Init (initialization constant for counters), TC transfer function (TC-id  or TC-EXP), β (exponential factor in case of TC-EXP), TC accumulation type (delta  or recommended weight-change).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Another branch of our algorithm is the MCTS wrapper, which can be used to wrap  TD-N-tuple agents during evaluation and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' MCTS wrapping is briefly explained in  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The precise algorithm for MCTS wrapping is explained in detail in (Scheiermann  and Konen, 2022, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' II-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='31 Parameters of MCTS are:  cPUCT : relative weight for the prior probabilities of the wrapped agent in relation to  the value that the wrapper estimates  dmax: maximum depth of the MCTS tree, if -1: no maximum depth  UseSoftMax: boolean, whether to use SoftMax normalization for the priors or not  UseLastMCTS: boolean, whether to re-use the MCTS from the previous move within  an episode or not  Further parameter explanations:  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 4 in this document explains n-tuples, parameters are: number of n-tuples, length  of n-tuples, and n-tuple creation mode (fixed, random walk, random points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='5 in this document explains symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' If parameter nSym = 0, do not use  symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' If nSym > 0, use this number nSym of symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' In the Rubik’s cube  case, nSym is a number between 0 and 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  LearnFromRM: whether to learn from random moves or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (Does not apply here,  because we use in Rubiks’s cube always ϵ = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' we have no random moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=')  ChooseStart-01: whether to start episodes from different 1-ply start states or always  from the default start state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' (Does not apply here, because we start in Rubik’s cube  never from the default cube, but always from the p-twisted cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=')  31As (Scheiermann and Konen, 2022, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' IV-E) shows, the MCTS wrapper may be used as well during  training, but due to large computation times needed for this, we do not follow that route in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  41  Etrain: maximum episode length during training, if -1: no maximum length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Eeval: maximum episode length during evaluation and play, if -1: no maximum length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  All agents were trained with no MCTS wrapper inside the training loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' The hyper-  parameters of the agent for each cube variant were found by manual fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' See  also (Konen, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  In the following, we list the precise settings for all agents used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' If not stated  otherwise, these common settings apply to all agents: sigmoid σ = id, LearnFromRM =  false, ChooseStart-01 = false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Wrapper settings during test and evaluation: MCTS wrapper  with cPUCT = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='0, dmax = 50, UseSoftMax = true, UseLastMCTS = true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Common parameters of Algorithm 2 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2 are: cost-to-go c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='1 and positive  reward Rpos = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  The parameters for training without symmetries (nSym = 0) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2 are:  2x2x2 cube, HTM: α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='25, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='0, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='0, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='0, no output sigmoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' N-tuples:  60 7-tuples created by random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  TCL activated with transfer function TC-id,  TC-Init= 10−4 and rec-weight-change accumulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 3,000,000 training episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  pmax = 13, Etrain = 16, Eeval = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Agent filename in GBG: 2x2x2_STICKER2_AT/TCL4-p13-ET16-3000k-60-7t-stub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='agt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='zip  2x2x2 cube, QTM: same as 2x2x2 cube, HTM, but with pmax = 16, Etrain = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Agent filename in GBG: 2x2x2_STICKER2_QT/TCL4-p16-ET20-3000k-60-7t-stub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='agt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='zip  3x3x3 cube, HTM: same as 2x2x2 cube, HTM, but with 120 7-tuples created by  random walk, pmax = 9, Etrain = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Agent filename in GBG: 3x3x3_STICKER2_AT/TCL4-p9-ET13-3000k-120-7t-stub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='agt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='zip  3x3x3 cube, QTM: same as 3x3x3 cube, HTM, but with pmax = 13, Etrain = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Agent filename in GBG: 3x3x3_STICKER2_QT/TCL4-p13-ET16-3000k-120-7t-stub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='agt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='zip  The agent files given in the list above are just stubs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' agents that are initialized with  the correct parameters but not yet trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' This is because a trained agent can require up  to 80 MB disk space, which is too much for GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' Instead, a user of GBG may load such  a stub agent, train it (takes between 10-40 minutes) and save it to local disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  When evaluating in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2 the trained agents with different MCTS wrappers, we test  in each case whether cPUCT = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='0 or 10 is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' In most cases, cPUCT = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='0 is better,  but for (2x2x2, QTM, 800 iterations) and for (3x3x3, HTM, 100 iterations) cPUCT = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='0 is  the better choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  The parameters for training with symmetries (nSym = 8, 16, 24) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='4 are:  3x3x3 cube, QTM: same as 3x3x3 cube, QTM in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='2, but with nsym = 8, 16, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  Agent filename in GBG: 3x3x3_STICKER2_QT/TCL4-p13-ET16-3000k-120-7t-nsym08-stub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='agt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='zip,  3x3x3_STICKER2_QT/TCL4-p13-ET16-3000k-120-7t-nsym16-stub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='agt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='zip,  3x3x3_STICKER2_QT/TCL4-p13-ET16-3000k-120-7t-nsym24-stub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='agt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='zip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  42  Again, the agent filenames are just stubs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' agents that are initialized with the correct  parameters but not yet trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content=' As above, a user of GBG may load such a stub agent, train  it (which takes in the symmetry case between 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='4h and 13h, see Table 9) and save it to  local disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  For further details and experiment shell scripts, see also the associated Papers-with-  Code repository https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='com/WolfgangKonen/PapersWithCodeRubiks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
+page_content='  43' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFLT4oBgHgl3EQfxTCi/content/2301.12167v1.pdf'}
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+VANISHING OF QUARTIC AND SEXTIC TWISTS OF
+L-FUNCTIONS
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+Abstract. Let E be an elliptic curve over Q.
+We conjecture
+asymptotic estimates for the number of vanishings of L(E, 1, χ) as
+χ varies over all primitive Dirichlet characters of orders 4 and 6.
+Our conjectures about these families come from conjectures about
+random unitary matrices as predicted by the philosophy of Katz-
+Sarnak. We support our conjectures with numerical evidence.
+Earlier work by David, Fearnley and Kisilevsky formulates anal-
+ogous conjectures for characters of any odd prime order. In the
+composite order case, however, we need to justify our use of random
+matrix theory heuristics by analyzing the equidistribution of the
+squares of normalized Gauss sums. Along the way we introduce the
+notion of totally order ℓ characters to quantify how quickly quartic
+and sextic Gauss sums become equidistributed. Surprisingly, the
+rate of equidistribution in the full family of quartic (sextic, resp.)
+characters is much slower than in the sub-family of totally quar-
+tic (sextic, resp.) characters. A conceptual explanation for this
+phenomenon is that the full family of order ℓ twisted elliptic curve
+L-functions, with ℓ even and composite, is a mixed family with
+both unitary and orthogonal aspects.
+1. Introduction
+Vanishings of elliptic curve L-functions at the value s = 1 (normalized
+so that the functional equation relates s and 2−s) is central to a great
+deal of modern number theory. For instance, if an L-function associated
+to an elliptic curve vanishes at s = 1, then the BSD conjecture predicts
+that the curve will have infinitely many rational points.
+Additionally, statistical questions about how often L-functions within
+a family vanish at the central value have also been of broad interest.
+For example, it is expected (as first conjectured by Chowla [Cho87])
+that, for all primitive Dirichlet characters χ, we have L(χ, 1/2) ̸= 0.
+A fruitful way of studying such questions has been to model L-functions
+using random matrices.
+For example, in [CKRS00] Conrey, Keat-
+ing, Rubinstein and Snaith consider the family of twisted L-functions
+1
+arXiv:2301.05329v1  [math.NT]  12 Jan 2023
+
+2
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+L(f, s, χd) associated to a modular form f of weight k and quadratic
+characters χd. They show that the random matrix theory model pre-
+dicts that infinitely many values L(f, s, χd) are zero when the weight
+of f is 2 or 4, but that only finitely many of the values are zero when
+the weight is at least 6.
+Another example, due to David, Fearnley and Kisilevsky [DFK04,
+DFK07], instead uses the random matrix model to give conjectural
+asymptotics for the number of vanishings of elliptic curve L-functions
+twisted by families of Dirichlet characters of a fixed order. In particu-
+lar, they predict that for an elliptic curve E, the values L(E, 1, χ) are
+zero infinitely often if χ has order 3 or 5, but for characters χ with a
+fixed prime order ℓ ≥ 7, only finitely many values L(E, 1, χ) are zero.
+In recent work, inspired by the conjectures of [DFK04, DFK07], Mazur
+and Rubin [MR21] use statistical properties of modular symbols to
+heuristically estimate the probability that L(E, 1, χ) vanishes. Their
+Conjecture 11.1 implies that for an elliptic curve E over Q, there
+should be only finitely many characters χ of a fixed order ℓ such that
+L(E, 1, χ) = 0 and ϕ(ℓ) > 4. This further implies the following: Let E
+be an elliptic curve over Q and let be F/Q an infinite abelian exten-
+sion such that Gal(F/Q) has only finitely many characters of orders 2,
+3 and 5. Then E(F) is finitely generated. Finally, for an elliptic curve
+E defined over Q, their Proposition 3.2 relates the (order of) vanishing
+of L(E, 1, χ) to the growth in rank of E over a finite abelian extension
+F/Q. In particular, if BSD holds for E over both Q and F, then
+rank(E(F)) = rank(E(Q)) +
+�
+χ:Gal(F/Q)→C×
+ords=1L(E, s, χ).
+1.1. Notation and statement of the Main Conjecture. We fix
+the following notation. See Definition 3.1 for the definition of totally
+order ℓ characters but, roughly speaking, these are order ℓ characters
+that, when factored, have all their factors also of order ℓ. Set
+Ψℓ = {primitive Dirichlet characters χ of order ℓ}
+Ψtot
+ℓ
+= {χ ∈ Ψℓ that are totally order ℓ}
+Ψ′
+ℓ = {χ ∈ Ψℓ with cond(χ) prime}.
+Note that Ψ′
+ℓ ⊆ Ψtot
+ℓ
+⊆ Ψℓ.
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+3
+Along the way we will need to estimate the number of characters in
+each family and so we define:
+Ψℓ(X) = {χ ∈ Ψℓ : cond(χ) ≤ X}
+Ψtot
+ℓ (X) = {χ ∈ Ψtot
+ℓ
+: cond(χ) ≤ X}
+Ψ′
+ℓ(X) = {χ ∈ Ψ′
+ℓ : cond(χ) ≤ X}.
+For an elliptic curve E over Q we also define:
+FΨℓ,E = {L(E, s, χ) : χ ∈ Ψℓ}
+FΨℓ,E(X) = {L(E, s, χ) ∈ FΨℓ,E : χ ∈ Ψℓ(X)}.
+We also define FΨtot
+ℓ
+,E and FΨtot
+ℓ
+,E(X) analogously for Ψtot
+ℓ
+in place of
+Ψℓ; we do the same with Ψ′
+ℓ, as well. Finally, let
+VΨℓ,E(X) = {L(E, s, χ) ∈ FΨℓ,E(X) : L(E, 1, χ) = 0}
+VΨtot
+ℓ
+,E(X) = {L(E, s, χ) ∈ FΨtot
+ℓ
+,E(X) : L(E, 1, χ) = 0}
+VΨ′
+ℓ,E(X) = {L(E, s, χ) ∈ FΨ′
+ℓ,E(X) : L(E, 1, χ) = 0}.
+With this notation, we make the following conjecture.
+Conjecture 1.1. Let E be an elliptic curve. Then, there exist con-
+stants bE,4 and bE,6 so that
+|VΨ4,E(X)| ∼ bE,4X1/2 log5/4 X
+and
+|VΨ6,E(X)| ∼ bE,6X1/2 log9/4 X
+as X → ∞.
+Moreover, if we restrict only to those twists by totally quartic or totally
+sextic characters, then there exist constants btot
+E,4 and btot
+E,6 such that
+|VΨtot
+4 ,E(X)| ∼ btot
+E,4X1/2 log1/4 X
+and
+|VΨtot
+6 ,E(X)| ∼ btot
+E,6X1/2 log1/4 X
+as X → ∞.
+Finally, if we restrict only to those twists by characters of prime con-
+ductor, then there exist constants b′
+E,4 and b′
+E,6 such that
+|VΨ′
+4,E(X)| ∼ b′
+E,4X1/2 log−3/4 X
+and
+|VΨ′
+6,E(X)| ∼ b′
+E,6X1/2 log−3/4 X
+as X → ∞.
+
+4
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+In particular, we conjecture that families of elliptic curve L-functions
+twisted by quartic and sextic characters vanish infinitely often at the
+central value.
+To assist the reader in comparing the powers of log X in the above
+asymptotics, we point out here that for ℓ = 4, |Ψ4(X)| is roughly log X
+times as large as |Ψtot
+4 (X)|, which in turn is roughly log X times as
+large as |Ψ′
+4(X)|. For ℓ = 6, then |Ψ6(X)|/|Ψtot
+6 (X)| ≍ (log X)2, and
+|Ψtot
+6 (X)|/|Ψ′
+6(X)| ≍ log X. Hence, in each of the three families with
+a given value of ℓ, the proportion of vanishing twists has the same
+order of magnitude. See Proposition 3.6, Lemma 3.7, Proposition 3.8,
+and Lemma 3.9 below for asymptotics of the underlying families of
+characters.
+1.2. Outline of the paper. There are two main ingredients needed
+to be able to apply random matrix theory predictions to our families of
+twists. The first is a discretization for the central values. As described
+in Section 2.1 this can be done for curves E satisfying certain technical
+conditions as described in [WW20]. We need this discretization in order
+to approximate the probability that L(E, 1, χ) vanishes.
+The second ingredient is a proper identification of the symmetry type
+of the family, which is largely governed by the distribution of the sign of
+the functional equation within the family (see Section 4 of [CFK+05]).
+This directly leads to an investigation around the equidistribution of
+squares of Gauss sums of quartic and sextic characters, which has con-
+nections to the theory of metaplectic automorphic forms [Pat87].
+See Section 3.1 for a thorough discussion.
+It is a subtle feature that the families of twists of elliptic curve L-
+functions by the characters in Ψtot
+ℓ
+and Ψ′
+ℓ have unitary symmetry
+type, but for composite even values of ℓ, the twists by Ψℓ should be
+viewed as a mixed family. To elaborate on this point, consider the case
+that ℓ = 4, and first note that a character χ ∈ Ψ4 factors uniquely
+as a totally quartic character times a quadratic character of relatively
+prime conductors. The totally quartic family has a unitary symmetry,
+but the family of twists of an elliptic curve by quadratic characters has
+orthogonal symmetry. This tension between the totally quartic aspect
+and the quadratic aspect is what leads to the mixed symmetry type.
+The situation is analogous to the family L(E, 1 + it, χd); if t = 0 and
+d varies then one has an orthogonal family, while if d is fixed and t
+varies, then one has a unitary family. See [SY10] for more discussion
+on this family.
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+5
+Another interesting feature of these families is that Ψℓ(X) is larger
+than Ψtot
+ℓ (X) by a logarithmic factor. For instance, when ℓ = 4, then
+Ψtot
+4 (X) grows linearly in X (see Proposition 3.6 below), and of course
+Ψ2(X) also grows linearly in X. Similarly to how the average size of the
+divisor function is log X, this indicates that |Ψ4(X)| grows like X log X
+(see Lemma 3.7 below).
+The rest of the paper is organized as follows.
+In the next section
+we give the necessary background and notation for L-functions and
+their central values and discuss the discretization we use in the paper.
+In the subsequent section we estimate some sums involving quartic
+and sextic characters and discuss totally quartic and sextic characters
+in more detail. In the final section, we motivate the asymptotics in
+Conjecture 1.1 and provide numerical evidence that supports them.
+Acknowledgments. We thank David Farmer and Brian Conrey for
+helpful conversations. This research was done using services provided
+by the OSG Consortium [PPK+07, SBH+09], which is supported by the
+National Science Foundation awards #2030508 and #1836650. This
+material is based upon work supported by the National Science Foun-
+dation under agreement No. DMS-2001306 (M.Y.). Any opinions, find-
+ings and conclusions or recommendations expressed in this material are
+those of the authors and do not necessarily reflect the views of the Na-
+tional Science Foundation.
+2. L-functions and central values
+Let E be an elliptic curve defined over Q of conductor NE. The L-
+function of E is given by the Euler product
+L(E, s) =
+�
+p∤NE
+�
+1 − ap
+ps +
+1
+p2s−1
+�−1 �
+p|NE
+�
+1 − ap
+ps
+�−1
+=
+�
+n≥1
+an
+ns .
+The modularity theorem [BCDT01, TW95, Wil95] implies that L(E, s)
+has an analytic continuation to all of C and satisfies the functional
+equation
+Λ(E, s) =
+� √NE
+2π
+�s
+Γ(s)L(E, s) = wEΛ(E, 2 − s)
+where the sign of the functional equation is wE = ±1 and is the eigen-
+value of the Fricke involution. Let χ be a primitive character and let
+cond(χ) be its conductor and suppose that cond(χ) is coprime to the
+
+6
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+conductor NE of the curve. The twisted L-function has Dirichlet series
+L(E, s, χ) =
+�
+n≥1
+anχ(n)
+ns
+and the functional equation (cf. [IK04, Prop. 14.20])
+Λ(E, s, χ) =
+�
+cond(χ)√NE
+2π
+�s
+Γ(s)L(E, s, χ)
+= wEχ(NE)τ(χ)2
+cond(χ)
+Λ(E, 2 − s, χ),
+(2.1)
+where τ(χ) = �
+r∈Z/mZ χ(r)e2πir/m is the Gauss sum and m = cond(χ).
+2.1. Discretization. To justify our Conjecture 1.1, we need a condi-
+tion that allows us to deduce that L(E, 1, χ) = 0, for a given E and
+χ of order ℓ. In particular, we show that L(E, 1, χ) is discretized (see
+Lemma 4.2) and so there exists a constant cE,ℓ such that |L(E, 1, χ)| <
+cE,ℓ/
+�
+cond(χ) implies L(E, 1, χ) = 0. In this section we prove the
+results necessary for the discretization.
+Let E be an elliptic curve over Q with conductor NE.
+Let χ be a
+nontrivial primitive Dirichlet character of conductor m and order ℓ.
+Set ϵ = {±1} = χ(−1) depending on whether χ is an even or odd
+character. Let Ω+(E) and Ω−(E) denote the real and imaginary periods
+of E, respectively, with Ω+(E) > 0 and Ω−(E) ∈ iR>0.
+The algebraic L-value is defined by
+(2.2)
+Lalg(E, 1, χ) := L(E, 1, χ) · m
+τ(χ)Ωϵ(E)
+= ϵ · L(E, 1, χ)τ(χ)
+Ωϵ(E)
+While it has been known for some time that algebraic L-values are
+algebraic numbers, recent work of Weirsema and Wuthrich [WW20]
+characterizes conditions on E and χ which guarantee integrality. In
+particular, under the assumption that the Manin constant c0(E) = 1,
+if the conductor m is not divisible by any prime of additive reduction for
+E, then Lalg(E, 1, χ) ∈ Z[ζℓ] is an algebraic integer [WW20, Theorem
+2]. For a given curve E, we will avoid the finitely many characters χ
+for which Lalg(E, 1, χ) fails to be integral.
+Proposition 2.1. Let χ be a primitive Dirichlet character of odd order
+ℓ and conductor m. Then
+Lalg(E, 1, χ) =
+�
+χ(NE)(ℓ+1)/2 nE(χ),
+if wE = 1,
+(ζℓ − ζ−1
+ℓ )−1 χ(NE)(ℓ+1)/2 nE(χ)
+if wE = −1,
+for some algebraic integer nE(χ) ∈ Z[ζℓ + ζ−1
+ℓ ] = Z[ζℓ] ∩ R.
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+7
+Proposition 2.2. Let χ be a primitive Dirichlet character of even
+order ℓ and conductor m. Then Lalg(E, 1, χ) = kE nE(χ) where nE(χ)
+is some algebraic integer in Z[ζℓ +ζ−1
+ℓ ] = Z[ζℓ]∩R and kE is a constant
+depending only on the curve E. In particular, when wE = 1 we have
+kE =
+�
+�
+�
+�
+�
+(1 + χ(NE))
+if χ(NE) ̸= −1
+ζℓ/4
+ℓ
+,
+if 4 | ℓ and χ(NE) = −1
+(ζℓ − ζ−1
+ℓ )
+if 4 ∤ ℓ and χ(NE) = −1.
+Proof of Prop 2.1 and Prop 2.2. Since E is defined over Q, we have
+L(E, 1, χ) = L(E, 1, χ). Using the functional equation, we obtain
+Lalg(E, 1, χ) = ϵ · L(E, 1, χ)τ(χ)
+Ωϵ(E)
+= ϵ · wE χ(NE) τ(χ)τ(χ)2
+m · Ωϵ(E)
+L(E, 1, χ)
+= wE χ(NE) τ(χ)
+Ωϵ(E)
+L(E, 1, χ)
+= wEχ(NE) ϵ · τ(χ)L(E, 1, χ)
+Ωϵ(E)
+= wEχ(NE) Lalg(E, 1, χ)
+Thus Lalg(E, 1, χ) is a solution to the equation z = wEχ(NE)z. Note
+that if z1, z2 ∈ Z[ζℓ] are two distinct solutions to this equation, then
+z1/z1 = z2/z2 so that z1/z2 = z1/z2 = (z1/z2), hence z1/z2 ∈ R. Thus
+Lalg(E, 1, χ) = αz with α ∈ Z[ζℓ] ∩ R = Z[ζℓ + ζ−1
+ℓ ] and z ∈ Z[ζℓ].
+Suppose that wE = 1. When ℓ is odd, we can take z = χ(NE)
+ℓ+1
+2 . Now
+suppose that ℓ is even. If χ(NE) ̸= −1, since χ(NE) = ζr
+ℓ for some
+1 ≤ r ≤ ℓ, we may take z = (1+χ(NE)). Indeed, we have wEχ(NE)z =
+ζr
+ℓ (1 + ζℓ−r
+ℓ
+) = ζr
+ℓ + 1 = z. If 4 | ℓ and χ(NE) = −1 = ζℓ/2
+ℓ
+, we take
+z = ζℓ/4
+ℓ
+. Finally, if 4 ∤ ℓ and χ(NE) = −1 take z = ζℓ−ζ−1
+ℓ
+= 2i Im(ζℓ).
+When wE = −1 and ℓ is odd, we may take z = (ζℓ − ζ−1
+ℓ )−1χ(NE)
+ℓ+1
+2 .
+When ℓ is even, if χ(NE) = −1 then we may take z = ζℓ + ζ−1
+ℓ
+=
+2 Re(ζℓ), and if χ(NE) ̸= −1 then we make take z = 1 − χ(NE).
+□
+Remark 2.3. We note that for ℓ even, |kE| ≤ 2.
+It is clear that
+|ζℓ/4
+ℓ
+| = 1 and |2i Im(ζℓ)| ≤ 2.
+Observe |(1 + χ(NE)| ≤ 2, by the
+triangle inequality.
+
+8
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+Note that since L(E, 1, χ) vanishes if and only if nE(χ) does, we may
+interpret the integers nE(χ) as a discretization of the special values
+L(E, 1, χ). This is similar to the case of cubic characters considered in
+[DFK04] since Q(ζ3)+ = Q, as opposed to characters of prime order ℓ ≥
+5 where further steps were needed to find an appropriate discretization
+[DFK07].
+3. Estimates for Dirichlet characters
+In this section we discuss various aspects of Dirichlet characters of
+order 4 and 6. A necessary condition for a family of L-functions to
+be modeled by the family of unitary matrices is that the signs must
+be uniformly distributed on the unit circle. From (2.1), L(E, s, χ) has
+sign wEχ(NE) τ(χ)2
+cond(χ); we will largely focus on the distribution of the
+square of the Gauss sums, viewing the extra factor χ(NE) as a minor
+perturbation. To obtain our estimates for the number of vanishings
+|VΨℓ,E(X)| (respectively, |VΨ′
+ℓ,E(X)| and |VΨtot
+ℓ
+,E(X)|) we must estimate
+the size of Ψℓ(X) (respectively, Ψ′
+ℓ(X) and Ψtot
+ℓ (X)) as well as the size
+of an associated sum. We also discuss the family of totally quartic
+and sextic characters to explain some phenomena we observed in our
+computations.
+3.1. Distributions of Gauss sums. Patterson [Pat87], building on
+work of Heath-Brown and Patterson [HBP79] on the cubic case, showed
+that the normalized Gauss sum τ(χ)/
+�
+cond(χ) is uniformly distributed
+on the circle for χ varying in each of Ψtot
+ℓ
+and Ψ′
+ℓ. This result was
+first announced in [PHH81]; see [BE81] for an excellent summary of
+this and other work related to the distributions of Gauss sums. Patter-
+son’s method moreover shows that the argument of τ(χ)χ(k) is equidis-
+tributed for any fixed nonzero integer k, and hence so is the argument
+of τ(χ)2χ(k).
+For the case of quartic and sextic characters with arbitrary conductors,
+there do not appear to be any results in the literature that imply their
+Gauss sums are uniformly distributed. In Figure 1 we see the distri-
+butions of Gauss sums of characters of orders 3 through 9 of arbitrary
+conductor up to 200000. We included characters of order 4 and 6 since
+those examples are the focus of the paper; we included characters of or-
+ders 3, 5, and 7 as consistency checks (in [DFK04, DFK07] the authors
+rely on them being uniformly distributed); and we included composite
+orders 8 and 9 to see if something similar happens in those cases as
+happens in the quartic case. In all cases but the quartic case, we see
+that the distributions of the angles of the signs appear to be uniformly
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+9
+Figure 1. Each histogram represents the distribution
+of the argument of the τ(χ)2/cond(χ) for characters of
+order 3 through 9, from top left to bottom right. Each
+histogram is made by calculating the Gauss sums of char-
+acters in Ψℓ of each conductor up to 200000.
+distributed. The quartic distribution has two obvious peaks that we
+discuss below, in Remark 3.17.
+The images in Figure 1 suggest that the family of matrices that best
+models the vanishing of L(E, 1, χ) is unitary in every case except possi-
+bly the case of quartic characters. Nevertheless, in Section 3.4 we show
+that the squares of the quartic Gauss sums are indeed equidistributed,
+despite what the data suggest. Indeed, we prove that the squares of
+the sextic and quartic Gauss sums are equidistributed, allowing us to
+apply the heuristics from random matrix theory as in Section 4.
+3.2. Totally quartic and sextic characters. Much of the back-
+ground material in this section can be found with proofs in [IR90,
+Ch. 9].
+Definition 3.1. Let χ be a primitive Dirichlet character of conductor
+q and order ℓ. For prime p, let vp be the p-adic valuation, so that
+q = �
+p pvp(q). We correspondingly factor χ = �
+p χ(p), where χ(p) has
+conductor pvp(q). We say that χ is totally order ℓ if each χp is exact
+
+5+00
+5000
+4000
+DORE
+2400
+1400
+E-
+-1
+0ADODE
+24000
+E-
+-2
+-1
+0
+1DOS
+4400
+3000
+DOZ
+1400
+2
+-1
+0
+112000
+40000
+E-
+-2
+-1 
+0
+34400
+DOSE
+DODE
+2500
+DOZ
+1500
+1400
+500
+0
+E-
+-1
+i
+2
+350000
+40000
+ADODE
+DO
+4000
+E-
+-2
+-1
+0
+i217500
+15000
+12500
+14000
+DOSr
+5000
+2500
+E-
+i210
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+order ℓ.
+By convention we also consider the trivial character to be
+totally order ℓ for every ℓ.
+3.2.1. Quartic characters. The construction of quartic characters uses
+the arithmetic in Z[i]. The ring Z[i] has class number 1, unit group
+{±1, ±i}, and discriminant −4. We say α ∈ Z[i] with (α, 2) = 1 is
+primary if α ≡ 1 (mod (1+i)3). Any odd element in Z[i] has a unique
+primary associate, which comes from the fact that the unit group in
+the ring Z[i]/(1+i)3 may be identified with {±1, ±i}. An odd prime p
+splits as p = ππ if and only if p ≡ 1 (mod 4). Given π with N(π) = p,
+define the quartic residue symbol [ α
+π] for α ∈ Z[i] with (α, π) = 1,
+by [ α
+π] ∈ {±1, ±i} and [ α
+π] ≡ α
+p−1
+4
+(mod π). The map χπ(α) = [ α
+π]
+from (Z[i]/(π))× to {±1, ±i} is a character of order 4.
+If α ∈ Z,
+then [ α
+π]2 ≡ α
+p−1
+2
+≡ ( α
+p) (mod π). Therefore, χ2
+π(α) = (α
+p), showing
+in particular that the restriction of the quartic residue symbol to Z
+defines a primitive quartic Dirichlet character of conductor p.
+Lemma 3.2. Every primitive totally quartic character of odd conductor
+is of the form χβ, where β = π1 . . . πk is a product of distinct primary
+primes, (β, 2β) = 1, and where
+(3.1)
+χβ(α) =
+�α
+β
+�
+=
+k
+�
+i=1
+� α
+πi
+�
+.
+The totally quartic primitive characters of even conductor are of the
+form χ2χβ where χ2 is one of four quartic characters of conductor 24,
+and χβ is totally quartic of odd conductor.
+Proof. We begin by classifying the quartic characters of odd prime-
+power conductor. If p ≡ 3 (mod 4), there is no quartic character of
+conductor pa, since φ(pa) = pa−1(p − 1) ̸≡ 0 (mod 4). Since φ(p) =
+p − 1, if p ≡ 1 (mod 4), there are two distinct quartic characters of
+conductor p, namely, χπ and χπ, where p = ππ. There are no primitive
+quartic characters modulo pj for j ≥ 2.
+To see this, suppose χ is
+a character of conductor pj, and note that χ(1 + pj−1) ̸= 1, while
+χ(1 + pj−1)p = χ(1 + pj) = 1, so χ(1 + pj−1) is a nontrivial pth root of
+unity. Since p is odd, χ(1+pj−1) is not a 4th root of unity, so χ cannot
+be quartic and primitive.
+By the above classification, a primitive totally quartic character χ of
+odd conductor must factor over distinct primes pi ≡ 1 (mod 4), and
+the p-part of χ must be χπ or χπ, where ππ = p. We may assume that
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+11
+π and π are primary primes. Hence χ factors as �
+i χπi. The property
+that β := π1 . . . πk is squarefree is equivalent to the condition that the
+πi are distinct. Moreover, the property (β, β) = 1 is equivalent to that
+πiπi = pi ≡ 1 (mod 4), for all i. Hence, every quartic character of odd
+conductor arises uniquely in the form (3.1).
+Next we treat p = 2. There are four primitive quartic characters of
+conductor 24, since (Z/(24))× ≃ Z/(2) × Z/(4). We claim there are no
+primitive quartic characters of conductor 2j, with j ̸= 4. For j ≤ 3 or
+j = 5 this is a simple finite computation. For j ≥ 6, one can show this
+as follows. First, χ(1 + 2j−1) = −1, since χ2(1 + 2j−1) = χ(1 + 2j) = 1,
+and primitivity shows χ(1+2j−1) ̸= 1. By a similar idea, χ(1+2j−2)2 =
+χ(1 + 2j−1) = −1, so χ(1 + 2j−2) = ±i. We finish the claim by noting
+χ2(1 + 2j−3) = χ(1 + 2j−2) = ±i, so χ(1 + 2j−3) is a square-root of
+±i, and hence χ is not quartic. With the claim established, we easily
+obtain the final sentence of the lemma.
+□
+Example 3.3. The first totally quartic primitive character of compos-
+ite conductor has conductor 65. While there are 8 quartic primitive
+characters of conductor 65, the LMFDB labels of the totally quartic
+ones are 65.18, 65.47, 65.8, and 65.57.
+3.2.2. Sextic characters. The construction of sextic characters uses the
+arithmetic in the Eisenstein integers Z[ω], where ω = e2πi/3. The ring
+Z[ω] has class number 1, unit group {±1, ±ω, ±ω2}, and discriminant
+−3. We say α ∈ Z[ω] with (α, 3) = 1 is primary1 if α ≡ 1 (mod 3).
+Warning: our usage of primary is consistent with [HBP79], but conflicts
+with the definition of [IR90]. However, it is easy to translate since α is
+primary in our sense if and only if −α is primary in the sense of [IR90].
+Any element in Z[ω] coprime to 3 has a unique primary associate, which
+comes from the fact that the unit group in the ring Z[ω]/(3) may be
+identified with {±1, ±ω, ±ω2}. An unramified prime p ∈ Z splits as
+p = ππ if and only if p ≡ 1 (mod 3). Given π with N(π) = p, define
+the cubic residue symbol ( α
+π)3 for α ∈ Z[ω] by ( α
+π)3 ∈ {1, ω, ω2} and
+( α
+π)3 ≡ α
+p−1
+3
+(mod π). The map χπ(α) = ( α
+π)3 from (Z[ω]/(π))× to
+{1, ω, ω2} is a character of order 3. The restriction of χπ to Z induces a
+primitive cubic Dirichlet character of conductor p. Note that χπ = χ−π.
+Motivated by the fact that a sextic character factors as a cubic times
+a quadratic, we next discuss the classification of cubic characters.
+1We remark that the usage of primary is context-dependent, and that since we
+do not mix quartic and sextic characters, we hope there will not be any ambiguity
+
+12
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+Lemma 3.4. Every primitive cubic Dirichlet character of conductor
+coprime to 3 is of the form χβ, where β = π1 . . . πk is a product of
+distinct primary primes, (β, 3β) = 1, and where
+(3.2)
+χβ(α) =
+�α
+β
+�
+3 =
+k
+�
+i=1
+� α
+πi
+�
+3.
+The cubic primitive characters of conductor divisible by 3 are of the
+form χ3χβ where χ3 is one of two cubic characters of conductor 32,
+and χβ is cubic of conductor coprime to 3.
+Proof. The classification of such characters with conductor coprime to
+3 is given by [BY10, Lemma 2.1], so it only remains to treat cubic
+characters of conductor 3j. The primitive character of conductor 3 is
+not cubic. Next, the group (Z/(9))× is cyclic of order 6, generated by
+2. There are two cubic characters, determined by χ(2) = ω±1. Next
+we argue that there is no primitive cubic character of conductor 3j
+with j ≥ 3. For this, we first observe that χ(1 + 3j−1) = ω±1, since
+primitivity implies χ(1 + 3j−1) ̸= 1, and χ(1 + 3j−1)3 = χ(1 + 3j) = 1.
+Next we have χ(1 + 3j−2)3 = χ(1 + 3j−1) = ω±1, so χ(1 + 3j−2) is a
+cube-root of ω±1. Therefore, χ cannot be cubic.
+□
+3.3. Counting characters. To start, we count all the quartic and sex-
+tic characters of conductor up to some bound and in each family. Such
+counts were found for arbitrary order in [FMS10] by Finch, Martin and
+Sebah, but since we are interested in only quartic and sextic charac-
+ters, in which case the proofs simplify, we prove the results we need.
+Moreover, we need other variants for which we cannot simply quote
+[FMS10], so we will develop a bit of machinery that will be helpful for
+these other questions as well.
+We begin with a lemma based on the Perron formula.
+Lemma 3.5. Suppose that a(n) is a multiplicative function such that
+|a(n)| ≤ dk(n), the k-fold divisor function, for some k ≥ 0. Let Z(s) =
+�
+n≥1 a(n)n−s, for Re(s) > 1. Suppose that for some integer j ≥ 0,
+(s−1)jZ(s) has a analytic continuation to a region of the form {σ+it :
+σ > 1 −
+c
+log(2+|t|)}, for some c > 0. In addition, suppose that Z(s) is
+bounded polynomially in log (2 + |t|) in this region. Then
+(3.3)
+�
+n≤X
+a(n) = XPj−1(log X) + O(X(log X)−100),
+for Pj−1 some polynomial of degree ≤ j − 1 (interpreted as 0, if j = 0).
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+13
+The basic idea is standard, yet we were unable to find a suitable refer-
+ence.
+Proof sketch. One begins by a use of the quantitative Perron formula,
+for which a convenient reference is [MV07, Thm. 5.2]. This implies
+(3.4)
+�
+n≤X
+a(n) =
+1
+2πi
+� σ0+iT
+σ0−iT
+Z(s)Xsds
+s + R,
+where R is a remainder term, and we take σ0 = 1+
+c
+log X . Using [MV07,
+Cor. 5.3] and standard bounds on mean values of dk(n), one can show
+R ≪ X
+T Poly(log X). Next one shifts the contour of integration to the
+line 1 − c/2
+log T . The pole (if it exists) of Z(s) leads to a main term of the
+form XPj−1(log X), as desired. The new line of integration is bounded
+by
+(3.5)
+Poly(log T)X1− c/2
+log T .
+Choosing log T = (log X)1/2 gives an acceptable error term.
+□
+3.3.1. Quartic characters. Let Ψtot,odd
+4
+(X) ⊆ Ψtot
+4 (X) denote the sub-
+set of characters with odd conductor.
+Proposition 3.6. For some constants Ktot
+4 , Ktot,odd
+4
+> 0, we have
+(3.6)
+|Ψtot
+4 (X)| ∼ Ktot
+4 X,
+and
+|Ψtot,odd
+4
+(X)| ∼ Ktot,odd
+4
+X.
+Moreover,
+(3.7)
+|Ψ′
+4(X)| ∼
+X
+log X .
+Proof. By Lemma 3.2,
+(3.8)
+|Ψtot,odd
+4
+(X)| =
+�
+0̸=(β)⊆Z[i]
+(β,2β)=1
+β squarefree
+N(β)≤X
+1,
+and
+(3.9)
+|Ψtot
+4 (X)| = |Ψtot,odd
+4
+(X)| + 4|Ψtot,odd
+4
+(2−4X)|.
+
+14
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+To show (3.6), it suffices to prove the asymptotic formula for |Ψtot,odd
+4
+(X)|.
+In view of Lemma 3.5, it will suffice to understand the Dirichlet series
+(3.10)
+Z4(s) =
+�
+0̸=(β)⊆Z[i]
+(β,2β)=1
+β squarefree
+1
+N(β)s =
+�
+π̸=π
+(π,2)=1
+(1 + N(π)−s) =
+�
+p≡1 (mod 4)
+(1 + p−s)2.
+Let χ4 be the primitive character modulo 4, so that ζ(s)L(s, χ4) =
+ζQ[i](s). Then
+(3.11) Z4(s) = ζQ[i](s)
+�
+p
+(1 − p−s)(1 − χ4(p)p−s)
+�
+p≡1 (mod 4)
+(1 + p−s)2,
+which can be simplified as
+(3.12)
+Z4(s) = ζQ[i](s)ζ−1(2s)(1 + 2−s)−1
+�
+p≡1 (mod 4)
+(1 − p−2s).
+Therefore, Z4(s) has a simple pole at s = 1, and its residue is a positive
+constant. Moreover, the standard analytic properties of ζQ[i](s) let us
+apply Lemma 3.5, giving the result.
+The asymptotic on Ψ′
+4(X) follows from the prime number theorem in
+arithmetic progressions, since there are two quartic characters of prime
+conductor p ≡ 1 (mod 4), and none with p ≡ 3 (mod 4).
+□
+Lemma 3.7. We have
+(3.13)
+|Ψ4(X)| = K4X log X + O(X),
+for some K4 > 0
+Proof. Every primitive quartic character factors uniquely as χ4χ2 with
+χ4 totally quartic of conductor q4 > 1 and χ2 quadratic of conductor
+q2, with (q4, q2) = 1. It is convenient to drop the condition q4 > 1,
+thereby including the quadratic characters; this is allowable since the
+number of quadratic characters is O(X), which is acceptable for the
+claimed error term.
+The Dirichlet series for |Ψ4(X)|, modified to include the quadratic char-
+acters, is
+(3.14)
+Zall
+4 (s) =
+�
+0̸=(β)⊆Z[i]
+(β,2β)=1
+β squarefree
+1
+N(β)s
+�
+q2∈Z≥1
+(q2,2N(β))=1
+1
+qs
+2
+.
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+15
+A calculation with Euler products shows Zall
+4 (s) = ζQ[i](s)ζ(s)A(s),
+where A(s) is given by an absolutely convergent Euler product for
+Re(s) > 1/2. Since Zall
+4 (s) has a double pole at s = 1, this shows the
+claim, using Lemma 3.5.
+□
+3.3.2. Sextic characters. Next we turn to the sextic case. The proof of
+the following proposition is similar to the proof of Proposition 3.6 and
+so we omit it here.
+Proposition 3.8. For some Ktot
+6
+> 0, we have
+(3.15)
+|Ψtot
+6 (X)| ∼ Ktot
+6 X,
+and
+|Ψ′
+6(X)| ∼
+X
+log X .
+A primitive totally sextic character factors uniquely as a primitive cubic
+character (with odd conductor, since 2 ̸≡ 1 (mod 3)), times the Jacobi
+symbol of the same modulus as the cubic character.
+In general, a
+primitive sextic character factors uniquely as χ6χ3χ2 of modulus q6q3q2,
+pairwise coprime, with χ6 totally sextic of conductor q6, χ3 cubic of
+conductor q3, and χ2 quadratic of conductor q2.
+Lemma 3.9. We have |Ψ6(X)| = K6X(log X)2+O(X log X), for some
+K6 > 0.
+Proof. Write χ = χ6χ3χ2 as above. Note that membership in Ψ6(X)
+requires q6 > 1, which is an unpleasant condition when working with
+Euler products. However, the number of χ = χ3χ2, i.e., with χ6 = 1,
+is O(X log X), so we may drop the condition q6 > 1 when estimating
+|Ψ6(X)|.
+For simplicity, we count the characters with q2 odd and (q6q3, 3) =
+1; the general case follows similar lines. The Dirichlet series for this
+counting function is
+Zall
+6 (s) =
+�
+0̸=(β6)⊆Z[ω]
+(β6,3β6)=1
+β6 squarefree
+1
+N(β6)s
+�
+0̸=(β3)⊆Z[ω]
+(β3,3β3)=1
+β3 squarefree
+(N(β3),N(β6)=1
+1
+N(β3)s
+�
+q2∈Z≥1
+(q2,2N(β3β6))=1
+1
+qs
+2
+.
+A calculation with Euler products shows Zall
+6 (s) = ζQ[ω](s)2ζ(s)A(s),
+where A(s) is given by an absolutely convergent Euler product for
+Re(s) > 1/2. Since Zall
+6 (s) has a triple pole at s = 1, this shows the
+claim, using Lemma 3.5.
+□
+
+16
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+3.4. Equidistribution of Gauss sums. We first focus on the quartic
+case, and then turn to the sextic case.
+3.4.1. Quartic characters. The following standard formula can be found
+as [IK04, (3.16)].
+Lemma 3.10. Suppose that χ = χ1χ2 has conductor q = q1q2, with
+(q1, q2) = 1, and χi of conductor qi. Then
+(3.16)
+τ(χ1χ2) = χ2(q1)χ1(q2)τ(χ1)τ(χ2).
+Corollary 3.11. Let notation be as in Lemma 3.10. Suppose that χ is
+totally quartic and q is odd. Then
+(3.17)
+τ(χ1χ2)2 = τ(χ1)2τ(χ2)2.
+Proof. By Lemma 3.10, we will obtain the formula provided χ2
+2(q1)χ2
+1(q2) =
+1. Note that χ2
+i is the Jacobi symbol, so χ2
+2(q1)χ2
+1(q2) = ( q1
+q2)( q2
+q1) = 1,
+by quadratic reciprocity, using that q1 ≡ q2 ≡ 1 (mod 4).
+□
+Lemma 3.12. Suppose π ∈ Z[i] is a primary prime, with N(π) = p ≡
+1 (mod 4). Let χπ(x) = [ x
+π] be the quartic residue symbol. Then
+(3.18)
+τ(χπ)2 = −χπ(−1)√pπ.
+More generally, if β is primary, squarefree, with (β, 2β) = 1, then
+(3.19)
+τ(χβ)2 = µ(β)χβ(−1)
+�
+N(β)β.
+Proof. The formula for χπ follows from [IR90, Thm.1 (Chapter 8),
+Prop. 9.9.4]. The formula for general β follows from Corollary 3.11
+and Lemma 3.2.
+□
+Lemma 3.13. Suppose that χ = χ2χ4 is a primitive quartic character
+with odd conductor q, with χ2 quadratic of conductor q2, χ4 totally
+quartic of conductor q4, and with q2q4 = q.
+Then
+(3.20)
+τ(χ)2 =
+�−q4
+q2
+�
+q2τ(χβ)2.
+Proof. By Lemma 3.10, we have τ(χ)2 = χ2(q4)2χ4(q2)2τ(χ2)2τ(χ4)2.
+To simplify this, note χ2(q4)2 = 1, χ2
+4(q2) = (q2
+q4) = (q4
+q2), and τ(χ2)2 =
+ϵ2
+q2q2 = ( −1
+q2 )q2.
+□
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+17
+Our next goal is to express τ(χβ)2 in terms of a Hecke Grossencharacter.
+Define
+(3.21)
+λ∞(α) = α
+|α|,
+α ∈ Z[i], α ̸= 0.
+Next define a particular character λ1+i : R× → S1, where R = Z[i]/(1+
+i)3, by
+(3.22)
+λ1+i(ik) = i−k,
+k ∈ {0, 1, 2, 3}.
+This indeed defines a character since R× ≃ Z/4Z, generated by i. For
+α ∈ Z[i], (α, 1 + i) = 1, define
+(3.23)
+λ((α)) = λ1+i(α)λ∞(α).
+For this to be well-defined, we need that the right hand side of (3.23)
+is constant on units in Z[i]. This is easily seen, since λ∞(ik) = ik =
+λ1+i(ik)−1. Therefore, λ defines a Hecke Grossencharacter, as in [IK04,
+Section 3.8]. Moreover, we note that
+(3.24)
+τ(χβ)2
+N(β) = µ(β)
+�
+2
+N(β)
+�
+λ((β))
+since this agrees with (3.19) for β primary, and is constant on units.
+According to [IK04, Theorem 3.8], the Dirichlet series
+(3.25)
+L(s, λk) =
+�
+0̸=(β)⊆Z[i]
+λ((β))k
+N(β)s ,
+(k ∈ Z),
+defines an L-function having analytic continuation to s ∈ C with no
+poles except for k = 0. The same statement holds when twisting λk by
+a finite-order character.
+For k ∈ Z, define the Dirichlet series
+(3.26)
+Z(k, s) =
+�
+0̸=(β)⊆Z[i]
+(β,2β)=1
+β squarefree
+(τ(χβ)2/N(β))k
+N(β)s
+,
+Re(s) > 1.
+Proposition 3.14. Let δk = −1 for k odd, and δk = +1 for k even.
+We have
+(3.27) Z(k, s) = A(k, s)L(s, (λ · χ2)k)δk,
+where
+χ2(β) =
+�
+2
+N(β)
+�
+,
+and where A(k, s) is given by an Euler product absolutely convergent
+for Re(s) > 1/2.
+
+18
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+In particular, the zero free region (as in [IK04, Theorem 5.35]) implies
+that Z(k, s) is analytic in a region of the type postulated in Lemma
+3.5. Moreover, the proof of [MV07, Theorem 11.4] shows that Z(k, s)
+is bounded polynomially in log(2 + |t|) in this region.
+Proof. The formula (3.24) shows that Z(k, s) has an Euler product of
+the form
+(3.28)
+Z(k, s) =
+�
+(π)̸=(π)
+(1 + (−1)k χk
+2(π)λk((π))
+N(π)s
+).
+This is an Euler product over the split primes in Z[i]. We extend this
+to include the primes p ≡ 3 (mod 4) as well, with N(π) = p2. It is
+convenient to define χ2(1 + i) = 0, so we can freely extend the product
+to include the ramified prime 1 + i. In all, we get
+(3.29)
+Z(k, s) =
+� �
+p
+(1 − χk
+2(p)λk(p)
+N(p)s
+)
+�−δk �
+p
+(1 + O(p−2s)).
+Note the product over p is L(s, (λ · χ2)k)δk, as claimed.
+□
+According to Weyl’s equidistribution criterion [IK04, Ch. 21.1], a se-
+quence of real numbers θn, 1 ≤ n ≤ N is equidistributed modulo 1 if
+and only if �
+n≤N e(kθn) = o(N) for each integer k ̸= 0. We apply
+this to e(θn) = (τ(χ)2/q), whence e(kθn) = (τ(χ)2/q)k. Due to the
+twisted multiplicativity formula (3.16), the congruence class in which
+2k lies modulo ℓ may have a simplifying effect on τ(χ)2k. For instance,
+when ℓ = 4, then k even leads to a simpler formula than k odd. This
+motivates treating these cases separately. As a minor simplification,
+below we focus on the sub-family of characters of odd conductor. The
+even conductor case is only a bit different.
+Corollary 3.15. The Gauss sums τ(χ)2/q for χ totally quartic of odd
+conductor q, equidistribute on the unit circle.
+Proof. The complex numbers τ(χ)2/q lie on the unit circle.
+Weyl’s
+equidistribution criterion says that these normalized squared Gauss
+sums equidistribute on the unit circle provided
+(3.30)
+�
+0̸=(β)⊆Z[i]
+(β,2β)=1
+β squarefree
+N(β)≤X
+(τ(χβ)2/N(β))k = o(X),
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+19
+Figure 2. This histogram represents the distribution
+of the argument of the τ(χ)2/cond(χ) for totally quartic
+characters. Each histogram is made by calculating the
+Gauss sums of characters of each order up to prime and
+composite conductor 300000.
+for each nonzero integer k. In turn, this bound is implied by Propo-
+sition 3.14, using the zero-free region for the Hecke Grossencharacter
+L-functions in [IK04, Theorem 5.35].
+□
+To contrast this, we will show that the normalized Gauss sums τ(χ)2/q,
+with χ ranging over all quartic characters, equidistribute slowly. More
+precisely, we have the following result.
+Proposition 3.16. Let k ∈ 2Z, k ̸= 0. There exists ck ∈ C such that
+(3.31)
+�
+q≤X
+(q,2)=1
+�
+χ:χ4=1
+cond(χ)=q
+(τ(χ)2/q)k = ckX + o(X).
+Remark 3.17. Recall from Lemma 3.7 that the total number of such
+characters grows like X log X, so Proposition 3.16 shows that the rate
+of equidistribution is only O((log X)−1) here. In contrast, in the family
+of totally quartic characters, the GRH would imply a rate of equidis-
+tribution of the form O(X−1/2+ε). This difference in rates of equidis-
+tribution is supported by Figure 2 in which we see that the arguments
+
+5+00
+4000
+DODE
+DOZ
+1000
+E-
+-1
+020
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+of squares of the Gauss sums of totally quartic characters quickly con-
+verge to being uniformly distributed, as compared to the Gauss sums
+of all quartic characters.
+In addition, one can derive a similar result when restricting to χ ∈
+Ψ4(X), simply by subtracting off the contribution from the quadratic
+characters alone.
+Proof. As in Lemma 3.13, write χ = χ2χ4, with χ2 quadratic and χ4
+totally quartic. Then τ(χ)4/(q1q2)2 = τ(χ4)4/q2
+2. The analog of Z(k, s),
+using k even to simplify, is
+(3.32)
+Zall(k, s) =
+�
+0̸=(β)⊆Z[i]
+(β,2β)=1
+β squarefree
+τ(χβ)2k/N(β)k
+N(β)s
+�
+q2∈Z≥1
+(q2,2N(β))=1
+1
+qs
+2
+.
+Referring to the calculation in Proposition 3.14, we obtain
+(3.33)
+Zall(k, s) = ζ(s)L(s, λk)A(s),
+where A(s) is an Euler product absolutely convergent for Re(s) > 1/2.
+Since this generating function has a simple pole at s = 1, we deduce
+Proposition 3.16.
+□
+As mentioned above, in order to deduce equidistribution, by Weyl’s
+equidistribution criterion, we also need to consider odd values of k in
+(3.31). This is more technical than the case for even k, so we content
+ourselves with a conjecture.
+Conjecture 3.18. For each odd k, there exists δ > 0 such that
+(3.34)
+�
+q≤X
+(q,2)=1
+�
+χ:χ4=1
+cond(χ)=q
+(τ(χ)2/q)k ≪k,δ X1−δ.
+Remark 3.19. By Lemma 3.13 and (3.24), this problem reduces to
+understanding sums of the rough shape
+�
+β,q2
+q2N(β)≤X
+�−N(β)
+q2
+�
+µ(β)
+�
+2
+N(β)
+�
+λ((β))k,
+where we have omitted many of the conditions on β and q2. In the
+range where q2 is very small, the GRH gives cancellation in the sum
+over β. Conversely, in the range where N(β) is very small, the GRH
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+21
+gives cancellation in the sum over q2. This discussion indicates that
+Conjecture 3.18 follows from GRH, with any δ < 1/4.
+Unconditionally, one can deduce some cancellation using the zero-free
+region for the β-sum (with q2 very small), and a subconvexity bound
+for the q2-sum (with N(β) very small). In the range where both q2
+and N(β) have some size, then Heath-Brown’s quadratic large sieve
+[HB95] gives some cancellation.
+Since we logically do not need an
+unconditional proof of equidistribution, we omit the details for brevity.
+Remark 3.20. Conjecture 3.18 and Proposition 3.16 together imply
+that the squares of the quartic Gauss sums do equidistribute in the full
+family Ψ4(X).
+3.4.2. Sextic characters. Now we turn to the sextic Gauss sums.
+Lemma 3.21. Suppose that χ is totally sextic of conductor q, and say
+χ = χ2χ3 with χ2 quadratic and χ3 cubic, each of conductor q. Suppose
+χ3 = χβ, as in Lemma 3.4. Then
+(3.35)
+τ(χ) = µ(q)χ3(2)τ(χ2)τ(χ3)βq−1.
+Proof. By [IK04, (3.18)], τ(χ2)τ(χ3) = J(χ2, χ3)τ(χ), where J(χ2, χ3)
+is the Jacobi sum.
+It is easy to show using the Chinese remainder
+theorem that if χ2 = �
+p χ(p)
+2
+and χ3 = �
+p χ(p)
+3 , then
+(3.36)
+J(χ2, χ3) =
+�
+p
+J(χ(p)
+2 , χ(p)
+3 ).
+The Jacobi sum for characters of prime conductor can be evaluated
+explicitly using the following facts. By [Lem00, Prop. 4.30],
+(3.37)
+J(χ(p)
+2 , χ(p)
+3 ) = χ(p)
+3 (22)J(χ(p)
+3 , χ(p)
+3 ).
+Suppose that χ(p)
+3
+= χπ, where ππ = p, and π is primary. Then [IR90,
+Ch. 9, Lem. 1] implies J(χπ, χπ) = −π. (Warning: they state the
+value π instead of −π, but recall their definition of primary is opposite
+our convention. Also recall that χπ = χ−π.) Gathering the formulas,
+we obtain
+(3.38)
+τ(χ2)τ(χ3) = τ(χ)χ3(2)2 �
+πi|β
+(−πi) = τ(χ)χ3(2)2µ(q)β.
+Rearranging this and using ββ = q completes the proof.
+□
+
+22
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+Corollary 3.22. Let conditions be as in Lemma 3.21. Then
+(3.39)
+τ(χ)2/q = χ3(4)
+�−1
+q
+�
+τ(χβ)2β
+2/q2.
+Patterson [Pat78] showed that τ(χβ)/√q is uniformly distributed on
+the unit circle, as χβ ranges over primitive cubic characters. The same
+method gives equidistribution after multiplication by a Hecke Grossen-
+character, and so similarly to the quartic case above, we deduce:
+Corollary 3.23 (Patterson). The Gauss sums τ(χ)2/q, for χ totally
+sextic of conductor q, equidistribute on the unit circle.
+In light of Corollary 3.22, Proposition 3.16, and Conjecture 3.18, it
+seems reasonable to conjecture that the points τ(χ)2/q are equidis-
+tributed on the unit circle, as χ varies over all sextic characters. To
+see a limitation in the rate of equidistribution, it is convenient to con-
+sider τ(χ)6/q3, which is multiplicative for χ sextic. For q ≡ 1 (mod 4),
+and χ = χ2 quadratic, we have τ(χ2)2/q = 1, so the quadratic part is
+constant. For χ cubic and q ≡ 1 (mod 4),
+(3.40)
+τ(χβ)6/q3 = µ(β)τ(χβ)3β
+3 = q−1β
+2,
+which is nearly a Hecke Grossencharacter. A similar formula holds for
+χ totally sextic, namely
+(3.41)
+τ(χ)6/q3 = q−4β
+8.
+Therefore, carrying out the same steps as in Proposition 3.16 shows
+that
+(3.42)
+�
+q≤X
+q≡1 (mod 4)
+�
+χ∈Ψ6
+cond(χ)=q
+�
+τ(χ)6/q3�k
+= CkX + o(X).
+This is less of an obstruction than in the quartic case, since here the
+rate of equidistribution is O((log X)−2) instead of O((log X)−1), due to
+the fact that |Ψ6(X)| is approximately log X times as large as |Ψ4(X)|.
+Similarly to the discussion of the quartic case in Remarks 3.19 and
+3.20, we make the following conjecture without further explanation.
+Conjecture 3.24. The Gauss sums τ(χ)2/q, for χ ranging in Ψ6(X),
+equidistribute on the unit circle.
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+23
+3.5. Estimates for quartic and sextic characters. In order to ap-
+ply the random matrix theory conjectures, we need variants on Propo-
+sition 3.6, Lemma 3.7, Proposition 3.8, and Lemma 3.9, as follows.
+Lemma 3.25. For primitive Dirichlet characters χ of order ℓ we have
+for ℓ = 4 and ℓ = 6 that
+(3.43)
+�
+χ∈Ψℓ(X)
+1
+�
+cond(χ)
+∼ 2Kℓ
+√
+X(log X)d(ℓ)−2,
+and
+(3.44)
+�
+χ∈Ψtot
+ℓ
+(X)
+1
+�
+cond(χ)
+∼ 2Ktot
+ℓ
+√
+X,
+�
+χ∈Ψ′
+ℓ(X)
+1
+�
+cond(χ)
+∼ 2
+√
+X
+log X .
+Proof. These estimates follow from a straightforward application of
+partial summation or from a minor modification of Lemma 3.5 since
+the generating Dirichlet series for one of these sums has its pole at
+s = 1/2 instead of at s = 1.
+□
+4. Random matrix theory: Conjectural asymptotic
+behavior
+This section closely follows the exposition of §3 of [DFK04] and §4 of
+[DFK07].
+Let U(N) be the set of unitary N×N matrices with complex coefficients
+which forms a probability space with respect to the Haar measure.
+For a family of L-functions with symmetry type U(N), Katz and Sar-
+nak conjectured that the statistics of the low-lying zeros should agree
+with those of the eigenangles of random matrices in U(N) [KS99]. Let
+PA(λ) = det(A − λI) be the characteristic polynomial of A. Keating
+and Snaith [KS00] suggest that the distribution of the values of the L-
+functions at the critical point is related to the value distribution of the
+characteristic polynomials |PA(1)| with respect to the Haar measure on
+U(N).
+For any s ∈ C we consider the moments
+MU(s, N) :=
+�
+U(N)
+|PA(1)|s dHaar
+
+24
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+for the distribution of |PA(1)| in U(N) with respect to the Haar mea-
+sure. In [KS00], Keating and Snaith proved that
+(4.1)
+MU(s, N) =
+N
+�
+j=1
+Γ(j)Γ(j + s)
+Γ2(j + s/2) ,
+so that MU(s, N) is analytic for Re(s) > −1 and has meromorphic
+continuation to the whole complex plane. The probability density of
+|PA(1)| is given by the Mellin transform
+pU(x, N) =
+1
+2πi
+�
+Re(s)=c
+MU(s, N)x−s−1 ds,
+for some c > −1.
+In the applications to the vanishing of twisted L-functions we consider
+in this paper, we are only interested in small values of x where the value
+of pU(x, N) is determined by the first pole of MU(s, N) at s = −1. More
+precisely, for x ≤ N −1/2, one can show that
+pU(x, N) ∼ G2(1/2)N 1/4
+as N → ∞,
+where G(z) is the Barnes G-function with special value [Bar99]
+G(1/2) = exp
+�3
+2ζ′(−1) − 1
+4 log π + 1
+24 log 2
+�
+.
+We will now consider the moments for the special values of twists of
+L-functions.
+We then define, for any s ∈ C, the following sum of
+evaluations at s = 1 of L-functions primitive order ℓ characters of
+conductor less than X:
+(4.2)
+ME(s, X) =
+1
+#FΨℓ,E(X)
+�
+L(E,s,χ)∈FΨℓ,E(X)
+|L(E, 1, χ)|s.
+Then, since the families of twists of order ℓ are expected to have unitary
+symmetry, we have
+Conjecture 4.1 (Keating and Snaith Conjecture for twists of order
+ℓ). With the notation as above,
+ME(s, X) ∼ aE(s/2)MU(s, N)
+as N = 2 log X → ∞,
+where aE(s/2) is an arithmetic factor depending only on the curve E.
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+25
+From Conjecture 4.1, the probability density for the distribution of the
+special values |L(E, 1, χ)| for characters of order ℓ is
+pE(x, X)
+=
+1
+2πi
+�
+Re(s)=c
+ME(s, X)x−s−1 ds
+(4.3)
+∼
+1
+2πi
+�
+Re(s)=c
+aE(s/2)MU(s, N)x−s−1 ds
+(4.4)
+as N = 2 log X → ∞. As above, when x ≤ N −1/2, the value of pE(x, X)
+is determined by the residue of MU(s, N) at s = −1, thus it follows
+from (4.4) that for x ≤ (2 log X)−1/2,
+(4.5)
+pE(x, X) ∼ 21/4aE(−1/2)G2(1/2) log1/4(X)
+as X → ∞.
+We now use the probability density of the random matrix model with
+the properties of the integers nE(χ) to obtain conjectures for the van-
+ishing of the L-values |L(E, 1, χ)|. When χ is either quartic or sextic,
+the discretization nE(χ) is a rational integer since Z[ζℓ] ∩ R = Z when
+ℓ = 4 or 6.
+Lemma 4.2. Let χ be a primitive Dirichlet character of order ℓ = 4
+or 6. Then
+|L(E, 1, χ)| =
+cE,ℓ
+�
+cond(χ)
+|nE(χ)|,
+where cE,ℓ is a nonzero constant which depends only on the curve E
+and ℓ.
+Proof. By rearranging equation (2.2) we obtain
+|L(E, 1, χ)| =
+����
+Ωϵ(E) τ(χ) kE nE(χ)
+cond(χ)
+���� = |Ωϵ(E) kE nE(χ)|
+�
+cond(χ)
+= cE,ℓ|nE(χ)|
+�
+cond(χ)
+,
+where the nonzero constant kE is that of Proposition 2.2.
+□
+We write
+(4.6)
+Prob{|L(E, 1, χ)| = 0} = Prob{|L(E, 1, χ)| < B(cond(χ))},
+for some function B(cond(χ)) of the character. By Lemma 4.2 we may
+take B(cond(χ)) =
+cE,ℓ
+�
+cond(χ)
+. Note that since cE,ℓ ̸= 0, if
+|nE(χ)|cE,ℓ
+�
+cond(χ)
+<
+cE,ℓ
+�
+cond(χ)
+,
+then |nE(χ)| < 1 and hence must vanish since |nE(χ)| ∈ Z≥0.
+
+26
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+Using (4.5), we have
+Prob{|L(E, 1, χ)| = 0}
+=
+� B(cond(χ))
+0
+21/4aE(−1/2)G2(1/2) log1/4(X) dx
+=
+21/4aE(−1/2)G2(1/2) log1/4(X)B(cond(χ))
+Summing the probabilities gives
+|VΨℓ,E(X)| = 21/4cE,kaE(−1/2)G2(1/2) log1/4(X)
+�
+cond(χ)≤X
+1
+�
+cond(χ)
+.
+Thus, by the analysis in §3.3, we have
+|VΨ4,E(X)| ∼ 25/4cE,4K4aE(−1/2)G2(1/2) log1/4(X)
+√
+X log X
+∼ bE,4X1/2 log5/4 X
+and
+|VΨ6,E(X)| ∼ 25/4cE,6K6aE(−1/2)G2(1/2) log1/4(X)
+√
+X(log X)2
+∼ bE,6X1/2 log9/4 X
+as X → ∞.
+Moreover, if we restrict to those characters that are totally quartic or
+sextic, we get the following estimates
+|VΨtot
+4 ,E(X)| ∼ 25/4cE,4Ktot
+4 aE(−1/2)G2(1/2) log1/4(X)
+√
+X
+∼ btot
+E,4X1/2 log1/4 X
+and
+|VΨtot
+6 ,E(X)| ∼ 25/4cE,6Ktot
+6 aE(−1/2)G2(1/2)
+∼ btot
+E,6X1/2 log1/4 X
+as X → ∞.
+Finally, if we restrict only to those twists by characters of prime con-
+ductor, we conclude
+|VΨ′
+4,E(X)| ∼ 25/4cE,4aE(−1/2)G2(1/2) log1/4(X)
+√
+X
+log X
+∼ b′
+E,4X1/2 log−3/4 X
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+27
+and
+|VΨ′
+6,E(X)| ∼ 25/4cE,6aE(−1/2)G2(1/2) log1/4(X)
+√
+X
+log X
+∼ b′
+E,6X1/2 log−3/4 X
+as X → ∞.
+4.1. Computations. Here we provide numerical evidence for Conjec-
+ture 1.1. The computations of the Conrey labels for the characters were
+done in SageMath [Sag21] and the computations of the L-functions were
+done in PARI/GP [PAR22]. The L-function computations were done
+in a distributed way on the Open Science Grid. For each curve, we
+generated a PARI/GP script to calculate a twisted L-function for each
+primitive character of order 4 and 6, and then combined the results into
+one file at the end. The combined wall time of all the computations
+was more than 50 years. The code and data are available at [BR23].
+In Figure 3 we plot the points
+(X, X1/2 log5/4 X
+|VΨ4,11.a.1(X)|), (X, X1/2 log−3/4 X
+|VΨ′
+4,11.a.1(X)| ), (X,
+X1/2 log1/4 X
+|VΨtot
+4
+,11.a.1(X)|)
+that provides a comparison between the predicted vanishings of L(E, 1, χ)
+for quartic characters and for the curve 11.a.1. In Figure 4 we plot the
+analogous points for the same curve but for sextic twists. In Figure 5
+we plot the points
+(X, X1/2 log−3/4 X
+|VΨ′
+4,37.a.1(X)| ), (X, X1/2 log−3/4 X
+|VΨ′
+6,37.a.1(X)| )
+Even though we are most interested in the families of all quartic and
+sextic twists, we include the families of twists of prime conductor be-
+cause there are far fewer such characters and so we can calculate the
+number of vanishings up to a much larger X. We include the fami-
+lies of twists by totally quartic and sextic characters to highlight the
+transition between the family of prime conductors and the family of all
+conductors.
+References
+[Bar99]
+E.W. Barnes. The theory of the G-function. Quart. J. Math., 31:264–314,
+1899.
+[BCDT01] Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor.
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+Bulletin of the American Mathematical Society, 5(2):107–129, 1981.
+
+28
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+(a)
+The
+ratio
+of
+predicted
+vanishings
+to
+empirical
+van-
+ishings
+of
+twists
+of
+the curve 11.a.1 by
+quartic characters of
+conductor ≤ 700000.
+(b) The ratio of pre-
+dicted
+vanishings
+to
+empirical
+vanishings
+of twists of the curve
+11.a.1
+by
+quartic
+characters
+of
+prime
+conductor ≤ 2000000.
+(c) The ratio of pre-
+dicted
+vanishings
+to
+empirical
+vanishings
+of twists of the curve
+11.a.1
+by
+totally
+quartic characters of
+conductor ≤ 700000.
+Figure 3. Verification of Conjecture 1.1 for quartic
+twists of 11.a.1.
+(a) The ratio of pre-
+dicted
+vanishings
+to
+empirical vanishings of
+twists
+of
+the
+curve
+11.a.1 by sextic char-
+acters of conductor ≤
+300000.
+(b) The ratio of pre-
+dicted
+vanishings
+to
+empirical vanishings of
+twists
+of
+the
+curve
+11.a.1 by sextic char-
+acters of prime con-
+ductor ≤ 2000000.
+(c) The ratio of pre-
+dicted
+vanishings
+to
+empirical vanishings of
+twists
+of
+the
+curve
+11.a.1 by totally sex-
+tic characters of con-
+ductor ≤ 300000.
+Figure 4. Verification of Conjecture 1.1 for sextic
+twists of 11.a.1.
+[BR23]
+Jen Berg and Nathan C. Ryan. Code and data for quartic and sextic
+twists of elliptic curve L-functions. http://eg.bucknell.edu/~ncr006/
+quartic-sextic-twists-website/, 2023.
+[BY10]
+Stephan Baier and Matthew P. Young. Mean values with cubic characters.
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+Sarvadaman Chowla. The Riemann hypothesis and Hilbert’s tenth prob-
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+0.38
+0.36 
+tE'O
+0.32
+0.30
+0.28 
+0.26
+0.D0
+0.25
+0.50
+0.75
+LDo
+125
+150
+175
+20o
+1e614
+12
+LD
+0.B 
+0.6 
+0
+ 11
+9 -
+8 -
+1 
+61
+50000
+150000
+DO
+250000
+3000000.45
+0.40 -
+0.35 
+0.30 
+0.25
+0.D0
+0.25
+0.50
+0.75
+Lio
+125
+150
+175
+200
+1e616
+15
+14
+13
+12
+11
+LD
+0
+DODS
+10dC0
+150000
+240000
+25000
+3+0dC04.5
+4.D
+3.5-
+3.0
+25 -
+20
+15
+LD
+   VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+29
+(a) The ratio of pre-
+dicted
+vanishings
+to
+empirical
+vanishings
+of twists of the curve
+37.a.1
+by
+quartic
+characters
+of
+prime
+conductor ≤ 2000000.
+(b) The ratio of pre-
+dicted
+vanishings
+to
+empirical vanishings of
+twists
+of
+the
+curve
+37.a.1 by sextic char-
+acters of prime con-
+ductor ≤ 2000000.
+Figure 5. Verification of parts of Conjecture 1.1 for
+twists of 37.a.1.
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+0.0O
+0.25
+0.50
+0.75
+125
+150
+175
+2b0
+1e6SLEO
+0.350
+0.325
+0.300 -
+0.275
+0.250 
+0.225
+0.200
+0.175
+0.DO
+0.25
+0.50
+0.75
+LDo
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+2o
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+sion 9.4), 2021. https://www.sagemath.org.
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+12(5):1097–1116, 2010.
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+of elliptic curves, 2020.
+Email address: jsb047@bucknell.edu
+Email address: nathan.ryan@bucknell.edu
+Department of Mathematics, Bucknell University, Lewisburg, PA 17837
+Email address: myoung@math.tamu.edu
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+31
+Department of Mathematics, Texas A&M University, College Station,
+TX 77843-3368
+
diff --git a/DdE4T4oBgHgl3EQf6A7h/content/tmp_files/load_file.txt b/DdE4T4oBgHgl3EQf6A7h/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..dddc8a6146475ca88730f1983aa2c93f4ca94a94
--- /dev/null
+++ b/DdE4T4oBgHgl3EQf6A7h/content/tmp_files/load_file.txt
@@ -0,0 +1,977 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf,len=976
+page_content='VANISHING OF QUARTIC AND SEXTIC TWISTS OF  L-FUNCTIONS  JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' YOUNG  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Let E be an elliptic curve over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  We conjecture  asymptotic estimates for the number of vanishings of L(E, 1, χ) as  χ varies over all primitive Dirichlet characters of orders 4 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Our conjectures about these families come from conjectures about  random unitary matrices as predicted by the philosophy of Katz-  Sarnak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' We support our conjectures with numerical evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Earlier work by David, Fearnley and Kisilevsky formulates anal-  ogous conjectures for characters of any odd prime order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' In the  composite order case, however, we need to justify our use of random  matrix theory heuristics by analyzing the equidistribution of the  squares of normalized Gauss sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Along the way we introduce the  notion of totally order ℓ characters to quantify how quickly quartic  and sextic Gauss sums become equidistributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Surprisingly, the  rate of equidistribution in the full family of quartic (sextic, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=')  characters is much slower than in the sub-family of totally quar-  tic (sextic, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=') characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' A conceptual explanation for this  phenomenon is that the full family of order ℓ twisted elliptic curve  L-functions, with ℓ even and composite, is a mixed family with  both unitary and orthogonal aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Introduction  Vanishings of elliptic curve L-functions at the value s = 1 (normalized  so that the functional equation relates s and 2−s) is central to a great  deal of modern number theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' For instance, if an L-function associated  to an elliptic curve vanishes at s = 1, then the BSD conjecture predicts  that the curve will have infinitely many rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Additionally, statistical questions about how often L-functions within  a family vanish at the central value have also been of broad interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  For example, it is expected (as first conjectured by Chowla [Cho87])  that, for all primitive Dirichlet characters χ, we have L(χ, 1/2) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  A fruitful way of studying such questions has been to model L-functions  using random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  For example, in [CKRS00] Conrey, Keat-  ing, Rubinstein and Snaith consider the family of twisted L-functions  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='05329v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='NT]  12 Jan 2023  2  JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' YOUNG  L(f, s, χd) associated to a modular form f of weight k and quadratic  characters χd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' They show that the random matrix theory model pre-  dicts that infinitely many values L(f, s, χd) are zero when the weight  of f is 2 or 4, but that only finitely many of the values are zero when  the weight is at least 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Another example, due to David, Fearnley and Kisilevsky [DFK04,  DFK07], instead uses the random matrix model to give conjectural  asymptotics for the number of vanishings of elliptic curve L-functions  twisted by families of Dirichlet characters of a fixed order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' In particu-  lar, they predict that for an elliptic curve E, the values L(E, 1, χ) are  zero infinitely often if χ has order 3 or 5, but for characters χ with a  fixed prime order ℓ ≥ 7, only finitely many values L(E, 1, χ) are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  In recent work, inspired by the conjectures of [DFK04, DFK07], Mazur  and Rubin [MR21] use statistical properties of modular symbols to  heuristically estimate the probability that L(E, 1, χ) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Their  Conjecture 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1 implies that for an elliptic curve E over Q, there  should be only finitely many characters χ of a fixed order ℓ such that  L(E, 1, χ) = 0 and ϕ(ℓ) > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' This further implies the following: Let E  be an elliptic curve over Q and let be F/Q an infinite abelian exten-  sion such that Gal(F/Q) has only finitely many characters of orders 2,  3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Then E(F) is finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Finally, for an elliptic curve  E defined over Q, their Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='2 relates the (order of) vanishing  of L(E, 1, χ) to the growth in rank of E over a finite abelian extension  F/Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' In particular, if BSD holds for E over both Q and F, then  rank(E(F)) = rank(E(Q)) +  �  χ:Gal(F/Q)→C×  ords=1L(E, s, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Notation and statement of the Main Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' We fix  the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' See Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1 for the definition of totally  order ℓ characters but, roughly speaking, these are order ℓ characters  that, when factored, have all their factors also of order ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Set  Ψℓ = {primitive Dirichlet characters χ of order ℓ}  Ψtot  ℓ  = {χ ∈ Ψℓ that are totally order ℓ}  Ψ′  ℓ = {χ ∈ Ψℓ with cond(χ) prime}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Note that Ψ′  ℓ ⊆ Ψtot  ℓ  ⊆ Ψℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS  3  Along the way we will need to estimate the number of characters in  each family and so we define:  Ψℓ(X) = {χ ∈ Ψℓ : cond(χ) ≤ X}  Ψtot  ℓ (X) = {χ ∈ Ψtot  ℓ  : cond(χ) ≤ X}  Ψ′  ℓ(X) = {χ ∈ Ψ′  ℓ : cond(χ) ≤ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  For an elliptic curve E over Q we also define:  FΨℓ,E = {L(E, s, χ) : χ ∈ Ψℓ}  FΨℓ,E(X) = {L(E, s, χ) ∈ FΨℓ,E : χ ∈ Ψℓ(X)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  We also define FΨtot  ℓ  ,E and FΨtot  ℓ  ,E(X) analogously for Ψtot  ℓ  in place of  Ψℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' we do the same with Ψ′  ℓ, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Finally, let  VΨℓ,E(X) = {L(E, s, χ) ∈ FΨℓ,E(X) : L(E, 1, χ) = 0}  VΨtot  ℓ  ,E(X) = {L(E, s, χ) ∈ FΨtot  ℓ  ,E(X) : L(E, 1, χ) = 0}  VΨ′  ℓ,E(X) = {L(E, s, χ) ∈ FΨ′  ℓ,E(X) : L(E, 1, χ) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  With this notation, we make the following conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Let E be an elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Then, there exist con-  stants bE,4 and bE,6 so that  |VΨ4,E(X)| ∼ bE,4X1/2 log5/4 X  and  |VΨ6,E(X)| ∼ bE,6X1/2 log9/4 X  as X → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Moreover, if we restrict only to those twists by totally quartic or totally  sextic characters, then there exist constants btot  E,4 and btot  E,6 such that  |VΨtot  4 ,E(X)| ∼ btot  E,4X1/2 log1/4 X  and  |VΨtot  6 ,E(X)| ∼ btot  E,6X1/2 log1/4 X  as X → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Finally, if we restrict only to those twists by characters of prime con-  ductor, then there exist constants b′  E,4 and b′  E,6 such that  |VΨ′  4,E(X)| ∼ b′  E,4X1/2 log−3/4 X  and  |VΨ′  6,E(X)| ∼ b′  E,6X1/2 log−3/4 X  as X → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  4  JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' YOUNG  In particular, we conjecture that families of elliptic curve L-functions  twisted by quartic and sextic characters vanish infinitely often at the  central value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  To assist the reader in comparing the powers of log X in the above  asymptotics, we point out here that for ℓ = 4, |Ψ4(X)| is roughly log X  times as large as |Ψtot  4 (X)|, which in turn is roughly log X times as  large as |Ψ′  4(X)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' For ℓ = 6, then |Ψ6(X)|/|Ψtot  6 (X)| ≍ (log X)2, and  |Ψtot  6 (X)|/|Ψ′  6(X)| ≍ log X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Hence, in each of the three families with  a given value of ℓ, the proportion of vanishing twists has the same  order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' See Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='6, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='7, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='8,  and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='9 below for asymptotics of the underlying families of  characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Outline of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' There are two main ingredients needed  to be able to apply random matrix theory predictions to our families of  twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The first is a discretization for the central values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' As described  in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1 this can be done for curves E satisfying certain technical  conditions as described in [WW20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' We need this discretization in order  to approximate the probability that L(E, 1, χ) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  The second ingredient is a proper identification of the symmetry type  of the family, which is largely governed by the distribution of the sign of  the functional equation within the family (see Section 4 of [CFK+05]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  This directly leads to an investigation around the equidistribution of  squares of Gauss sums of quartic and sextic characters, which has con-  nections to the theory of metaplectic automorphic forms [Pat87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  See Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1 for a thorough discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  It is a subtle feature that the families of twists of elliptic curve L-  functions by the characters in Ψtot  ℓ  and Ψ′  ℓ have unitary symmetry  type, but for composite even values of ℓ, the twists by Ψℓ should be  viewed as a mixed family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' To elaborate on this point, consider the case  that ℓ = 4, and first note that a character χ ∈ Ψ4 factors uniquely  as a totally quartic character times a quadratic character of relatively  prime conductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The totally quartic family has a unitary symmetry,  but the family of twists of an elliptic curve by quadratic characters has  orthogonal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' This tension between the totally quartic aspect  and the quadratic aspect is what leads to the mixed symmetry type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  The situation is analogous to the family L(E, 1 + it, χd);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' if t = 0 and  d varies then one has an orthogonal family, while if d is fixed and t  varies, then one has a unitary family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' See [SY10] for more discussion  on this family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS  5  Another interesting feature of these families is that Ψℓ(X) is larger  than Ψtot  ℓ (X) by a logarithmic factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' For instance, when ℓ = 4, then  Ψtot  4 (X) grows linearly in X (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='6 below), and of course  Ψ2(X) also grows linearly in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Similarly to how the average size of the  divisor function is log X, this indicates that |Ψ4(X)| grows like X log X  (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='7 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  In the next section  we give the necessary background and notation for L-functions and  their central values and discuss the discretization we use in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  In the subsequent section we estimate some sums involving quartic  and sextic characters and discuss totally quartic and sextic characters  in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' In the final section, we motivate the asymptotics in  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1 and provide numerical evidence that supports them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' We thank David Farmer and Brian Conrey for  helpful conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' This research was done using services provided  by the OSG Consortium [PPK+07, SBH+09], which is supported by the  National Science Foundation awards #2030508 and #1836650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' This  material is based upon work supported by the National Science Foun-  dation under agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' DMS-2001306 (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Any opinions, find-  ings and conclusions or recommendations expressed in this material are  those of the authors and do not necessarily reflect the views of the Na-  tional Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' L-functions and central values  Let E be an elliptic curve defined over Q of conductor NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The L-  function of E is given by the Euler product  L(E, s) =  �  p∤NE  �  1 − ap  ps +  1  p2s−1  �−1 �  p|NE  �  1 − ap  ps  �−1  =  �  n≥1  an  ns .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  The modularity theorem [BCDT01, TW95, Wil95] implies that L(E, s)  has an analytic continuation to all of C and satisfies the functional  equation  Λ(E, s) =  � √NE  2π  �s  Γ(s)L(E, s) = wEΛ(E, 2 − s)  where the sign of the functional equation is wE = ±1 and is the eigen-  value of the Fricke involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Let χ be a primitive character and let  cond(χ) be its conductor and suppose that cond(χ) is coprime to the  6  JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' YOUNG  conductor NE of the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The twisted L-function has Dirichlet series  L(E, s, χ) =  �  n≥1  anχ(n)  ns  and the functional equation (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' [IK04, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='20])  Λ(E, s, χ) =  �  cond(χ)√NE  2π  �s  Γ(s)L(E, s, χ)  = wEχ(NE)τ(χ)2  cond(χ)  Λ(E, 2 − s, χ),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1)  where τ(χ) = �  r∈Z/mZ χ(r)e2πir/m is the Gauss sum and m = cond(χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' To justify our Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1, we need a condi-  tion that allows us to deduce that L(E, 1, χ) = 0, for a given E and  χ of order ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' In particular, we show that L(E, 1, χ) is discretized (see  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='2) and so there exists a constant cE,ℓ such that |L(E, 1, χ)| <  cE,ℓ/  �  cond(χ) implies L(E, 1, χ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' In this section we prove the  results necessary for the discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Let E be an elliptic curve over Q with conductor NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Let χ be a  nontrivial primitive Dirichlet character of conductor m and order ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Set ϵ = {±1} = χ(−1) depending on whether χ is an even or odd  character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Let Ω+(E) and Ω−(E) denote the real and imaginary periods  of E, respectively, with Ω+(E) > 0 and Ω−(E) ∈ iR>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  The algebraic L-value is defined by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='2)  Lalg(E, 1, χ) := L(E, 1, χ) · m  τ(χ)Ωϵ(E)  = ϵ · L(E, 1, χ)τ(χ)  Ωϵ(E)  While it has been known for some time that algebraic L-values are  algebraic numbers, recent work of Weirsema and Wuthrich [WW20]  characterizes conditions on E and χ which guarantee integrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' In  particular, under the assumption that the Manin constant c0(E) = 1,  if the conductor m is not divisible by any prime of additive reduction for  E, then Lalg(E, 1, χ) ∈ Z[ζℓ] is an algebraic integer [WW20, Theorem  2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' For a given curve E, we will avoid the finitely many characters χ  for which Lalg(E, 1, χ) fails to be integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Let χ be a primitive Dirichlet character of odd order  ℓ and conductor m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Then  Lalg(E, 1, χ) =  �  χ(NE)(ℓ+1)/2 nE(χ),  if wE = 1,  (ζℓ − ζ−1  ℓ )−1 χ(NE)(ℓ+1)/2 nE(χ)  if wE = −1,  for some algebraic integer nE(χ) ∈ Z[ζℓ + ζ−1  ℓ ] = Z[ζℓ] ∩ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS  7  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Let χ be a primitive Dirichlet character of even  order ℓ and conductor m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Then Lalg(E, 1, χ) = kE nE(χ) where nE(χ)  is some algebraic integer in Z[ζℓ +ζ−1  ℓ ] = Z[ζℓ]∩R and kE is a constant  depending only on the curve E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' In particular, when wE = 1 we have  kE =  �  �  �  �  �  (1 + χ(NE))  if χ(NE) ̸= −1  ζℓ/4  ℓ  ,  if 4 | ℓ and χ(NE) = −1  (ζℓ − ζ−1  ℓ )  if 4 ∤ ℓ and χ(NE) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Proof of Prop 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1 and Prop 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Since E is defined over Q, we have  L(E, 1, χ) = L(E, 1, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Using the functional equation, we obtain  Lalg(E, 1, χ) = ϵ · L(E, 1, χ)τ(χ)  Ωϵ(E)  = ϵ · wE χ(NE) τ(χ)τ(χ)2  m · Ωϵ(E)  L(E, 1, χ)  = wE χ(NE) τ(χ)  Ωϵ(E)  L(E, 1, χ)  = wEχ(NE) ϵ · τ(χ)L(E, 1, χ)  Ωϵ(E)  = wEχ(NE) Lalg(E, 1, χ)  Thus Lalg(E, 1, χ) is a solution to the equation z = wEχ(NE)z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Note  that if z1, z2 ∈ Z[ζℓ] are two distinct solutions to this equation, then  z1/z1 = z2/z2 so that z1/z2 = z1/z2 = (z1/z2), hence z1/z2 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Thus  Lalg(E, 1, χ) = αz with α ∈ Z[ζℓ] ∩ R = Z[ζℓ + ζ−1  ℓ ] and z ∈ Z[ζℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Suppose that wE = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' When ℓ is odd, we can take z = χ(NE)  ℓ+1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Now  suppose that ℓ is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' If χ(NE) ̸= −1, since χ(NE) = ζr  ℓ for some  1 ≤ r ≤ ℓ, we may take z = (1+χ(NE)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Indeed, we have wEχ(NE)z =  ζr  ℓ (1 + ζℓ−r  ℓ  ) = ζr  ℓ + 1 = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' If 4 | ℓ and χ(NE) = −1 = ζℓ/2  ℓ  , we take  z = ζℓ/4  ℓ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Finally, if 4 ∤ ℓ and χ(NE) = −1 take z = ζℓ−ζ−1  ℓ  = 2i Im(ζℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  When wE = −1 and ℓ is odd, we may take z = (ζℓ − ζ−1  ℓ )−1χ(NE)  ℓ+1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  When ℓ is even, if χ(NE) = −1 then we may take z = ζℓ + ζ−1  ℓ  =  2 Re(ζℓ), and if χ(NE) ̸= −1 then we make take z = 1 − χ(NE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' We note that for ℓ even, |kE| ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  It is clear that  |ζℓ/4  ℓ  | = 1 and |2i Im(ζℓ)| ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Observe |(1 + χ(NE)| ≤ 2, by the  triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  8  JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' YOUNG  Note that since L(E, 1, χ) vanishes if and only if nE(χ) does, we may  interpret the integers nE(χ) as a discretization of the special values  L(E, 1, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' This is similar to the case of cubic characters considered in  [DFK04] since Q(ζ3)+ = Q, as opposed to characters of prime order ℓ ≥  5 where further steps were needed to find an appropriate discretization  [DFK07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Estimates for Dirichlet characters  In this section we discuss various aspects of Dirichlet characters of  order 4 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' A necessary condition for a family of L-functions to  be modeled by the family of unitary matrices is that the signs must  be uniformly distributed on the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1), L(E, s, χ) has  sign wEχ(NE) τ(χ)2  cond(χ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' we will largely focus on the distribution of the  square of the Gauss sums, viewing the extra factor χ(NE) as a minor  perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' To obtain our estimates for the number of vanishings  |VΨℓ,E(X)| (respectively, |VΨ′  ℓ,E(X)| and |VΨtot  ℓ  ,E(X)|) we must estimate  the size of Ψℓ(X) (respectively, Ψ′  ℓ(X) and Ψtot  ℓ (X)) as well as the size  of an associated sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' We also discuss the family of totally quartic  and sextic characters to explain some phenomena we observed in our  computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Distributions of Gauss sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Patterson [Pat87], building on  work of Heath-Brown and Patterson [HBP79] on the cubic case, showed  that the normalized Gauss sum τ(χ)/  �  cond(χ) is uniformly distributed  on the circle for χ varying in each of Ψtot  ℓ  and Ψ′  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' This result was  first announced in [PHH81];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' see [BE81] for an excellent summary of  this and other work related to the distributions of Gauss sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Patter-  son’s method moreover shows that the argument of τ(χ)χ(k) is equidis-  tributed for any fixed nonzero integer k, and hence so is the argument  of τ(χ)2χ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  For the case of quartic and sextic characters with arbitrary conductors,  there do not appear to be any results in the literature that imply their  Gauss sums are uniformly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' In Figure 1 we see the distri-  butions of Gauss sums of characters of orders 3 through 9 of arbitrary  conductor up to 200000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' We included characters of order 4 and 6 since  those examples are the focus of the paper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' we included characters of or-  ders 3, 5, and 7 as consistency checks (in [DFK04, DFK07] the authors  rely on them being uniformly distributed);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' and we included composite  orders 8 and 9 to see if something similar happens in those cases as  happens in the quartic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' In all cases but the quartic case, we see  that the distributions of the angles of the signs appear to be uniformly  VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS  9  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Each histogram represents the distribution  of the argument of the τ(χ)2/cond(χ) for characters of  order 3 through 9, from top left to bottom right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Each  histogram is made by calculating the Gauss sums of char-  acters in Ψℓ of each conductor up to 200000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The quartic distribution has two obvious peaks that we  discuss below, in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  The images in Figure 1 suggest that the family of matrices that best  models the vanishing of L(E, 1, χ) is unitary in every case except possi-  bly the case of quartic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Nevertheless, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='4 we show  that the squares of the quartic Gauss sums are indeed equidistributed,  despite what the data suggest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Indeed, we prove that the squares of  the sextic and quartic Gauss sums are equidistributed, allowing us to  apply the heuristics from random matrix theory as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Totally quartic and sextic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Much of the back-  ground material in this section can be found with proofs in [IR90,  Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Let χ be a primitive Dirichlet character of conductor  q and order ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' For prime p, let vp be the p-adic valuation, so that  q = �  p pvp(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' We correspondingly factor χ = �  p χ(p), where χ(p) has  conductor pvp(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' We say that χ is totally order ℓ if each χp is exact  5+00  5000  4000  DORE  2400  1400  E-  1  0ADODE  24000  E-  2  1  0  1DOS  4400  3000  DOZ  1400  2  1  0  112000  40000  E-  2  1  0  34400  DOSE  DODE  2500  DOZ  1500  1400  500  0  E-  1  i  2  350000  40000  ADODE  DO  4000  E-  2  1  0  i217500  15000  12500  14000  DOSr  5000  2500  E-  i210  JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' YOUNG  order ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  By convention we also consider the trivial character to be  totally order ℓ for every ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Quartic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The construction of quartic characters uses  the arithmetic in Z[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The ring Z[i] has class number 1, unit group  {±1, ±i}, and discriminant −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' We say α ∈ Z[i] with (α, 2) = 1 is  primary if α ≡ 1 (mod (1+i)3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Any odd element in Z[i] has a unique  primary associate, which comes from the fact that the unit group in  the ring Z[i]/(1+i)3 may be identified with {±1, ±i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' An odd prime p  splits as p = ππ if and only if p ≡ 1 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Given π with N(π) = p,  define the quartic residue symbol [ α  π] for α ∈ Z[i] with (α, π) = 1,  by [ α  π] ∈ {±1, ±i} and [ α  π] ≡ α  p−1  4  (mod π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The map χπ(α) = [ α  π]  from (Z[i]/(π))× to {±1, ±i} is a character of order 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  If α ∈ Z,  then [ α  π]2 ≡ α  p−1  2  ≡ ( α  p) (mod π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Therefore, χ2  π(α) = (α  p), showing  in particular that the restriction of the quartic residue symbol to Z  defines a primitive quartic Dirichlet character of conductor p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Every primitive totally quartic character of odd conductor  is of the form χβ, where β = π1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' πk is a product of distinct primary  primes, (β, 2β) = 1, and where  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1)  χβ(α) =  �α  β  �  =  k  �  i=1  � α  πi  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  The totally quartic primitive characters of even conductor are of the  form χ2χβ where χ2 is one of four quartic characters of conductor 24,  and χβ is totally quartic of odd conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' We begin by classifying the quartic characters of odd prime-  power conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' If p ≡ 3 (mod 4), there is no quartic character of  conductor pa, since φ(pa) = pa−1(p − 1) ̸≡ 0 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Since φ(p) =  p − 1, if p ≡ 1 (mod 4), there are two distinct quartic characters of  conductor p, namely, χπ and χπ, where p = ππ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' There are no primitive  quartic characters modulo pj for j ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  To see this, suppose χ is  a character of conductor pj, and note that χ(1 + pj−1) ̸= 1, while  χ(1 + pj−1)p = χ(1 + pj) = 1, so χ(1 + pj−1) is a nontrivial pth root of  unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Since p is odd, χ(1+pj−1) is not a 4th root of unity, so χ cannot  be quartic and primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  By the above classification, a primitive totally quartic character χ of  odd conductor must factor over distinct primes pi ≡ 1 (mod 4), and  the p-part of χ must be χπ or χπ, where ππ = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' We may assume that  VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS  11  π and π are primary primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Hence χ factors as �  i χπi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The property  that β := π1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' πk is squarefree is equivalent to the condition that the  πi are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Moreover, the property (β, β) = 1 is equivalent to that  πiπi = pi ≡ 1 (mod 4), for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Hence, every quartic character of odd  conductor arises uniquely in the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Next we treat p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' There are four primitive quartic characters of  conductor 24, since (Z/(24))× ≃ Z/(2) × Z/(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' We claim there are no  primitive quartic characters of conductor 2j, with j ̸= 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' For j ≤ 3 or  j = 5 this is a simple finite computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' For j ≥ 6, one can show this  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' First, χ(1 + 2j−1) = −1, since χ2(1 + 2j−1) = χ(1 + 2j) = 1,  and primitivity shows χ(1+2j−1) ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' By a similar idea, χ(1+2j−2)2 =  χ(1 + 2j−1) = −1, so χ(1 + 2j−2) = ±i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' We finish the claim by noting  χ2(1 + 2j−3) = χ(1 + 2j−2) = ±i, so χ(1 + 2j−3) is a square-root of  ±i, and hence χ is not quartic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' With the claim established, we easily  obtain the final sentence of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  □  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The first totally quartic primitive character of compos-  ite conductor has conductor 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' While there are 8 quartic primitive  characters of conductor 65, the LMFDB labels of the totally quartic  ones are 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='18, 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='47, 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='8, and 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Sextic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The construction of sextic characters uses the  arithmetic in the Eisenstein integers Z[ω], where ω = e2πi/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The ring  Z[ω] has class number 1, unit group {±1, ±ω, ±ω2}, and discriminant  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' We say α ∈ Z[ω] with (α, 3) = 1 is primary1 if α ≡ 1 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Warning: our usage of primary is consistent with [HBP79], but conflicts  with the definition of [IR90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' However, it is easy to translate since α is  primary in our sense if and only if −α is primary in the sense of [IR90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Any element in Z[ω] coprime to 3 has a unique primary associate, which  comes from the fact that the unit group in the ring Z[ω]/(3) may be  identified with {±1, ±ω, ±ω2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' An unramified prime p ∈ Z splits as  p = ππ if and only if p ≡ 1 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Given π with N(π) = p, define  the cubic residue symbol ( α  π)3 for α ∈ Z[ω] by ( α  π)3 ∈ {1, ω, ω2} and  ( α  π)3 ≡ α  p−1  3  (mod π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The map χπ(α) = ( α  π)3 from (Z[ω]/(π))× to  {1, ω, ω2} is a character of order 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The restriction of χπ to Z induces a  primitive cubic Dirichlet character of conductor p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Note that χπ = χ−π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Motivated by the fact that a sextic character factors as a cubic times  a quadratic, we next discuss the classification of cubic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  1We remark that the usage of primary is context-dependent, and that since we  do not mix quartic and sextic characters, we hope there will not be any ambiguity  12  JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' YOUNG  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Every primitive cubic Dirichlet character of conductor  coprime to 3 is of the form χβ, where β = π1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' πk is a product of  distinct primary primes, (β, 3β) = 1, and where  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='2)  χβ(α) =  �α  β  �  3 =  k  �  i=1  � α  πi  �  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  The cubic primitive characters of conductor divisible by 3 are of the  form χ3χβ where χ3 is one of two cubic characters of conductor 32,  and χβ is cubic of conductor coprime to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The classification of such characters with conductor coprime to  3 is given by [BY10, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1], so it only remains to treat cubic  characters of conductor 3j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The primitive character of conductor 3 is  not cubic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Next, the group (Z/(9))× is cyclic of order 6, generated by  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' There are two cubic characters, determined by χ(2) = ω±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Next  we argue that there is no primitive cubic character of conductor 3j  with j ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' For this, we first observe that χ(1 + 3j−1) = ω±1, since  primitivity implies χ(1 + 3j−1) ̸= 1, and χ(1 + 3j−1)3 = χ(1 + 3j) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Next we have χ(1 + 3j−2)3 = χ(1 + 3j−1) = ω±1, so χ(1 + 3j−2) is a  cube-root of ω±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Therefore, χ cannot be cubic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Counting characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' To start, we count all the quartic and sex-  tic characters of conductor up to some bound and in each family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Such  counts were found for arbitrary order in [FMS10] by Finch, Martin and  Sebah, but since we are interested in only quartic and sextic charac-  ters, in which case the proofs simplify, we prove the results we need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Moreover, we need other variants for which we cannot simply quote  [FMS10], so we will develop a bit of machinery that will be helpful for  these other questions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  We begin with a lemma based on the Perron formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Suppose that a(n) is a multiplicative function such that  |a(n)| ≤ dk(n), the k-fold divisor function, for some k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Let Z(s) =  �  n≥1 a(n)n−s, for Re(s) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Suppose that for some integer j ≥ 0,  (s−1)jZ(s) has a analytic continuation to a region of the form {σ+it :  σ > 1 −  c  log(2+|t|)}, for some c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' In addition, suppose that Z(s) is  bounded polynomially in log (2 + |t|) in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Then  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='3)  �  n≤X  a(n) = XPj−1(log X) + O(X(log X)−100),  for Pj−1 some polynomial of degree ≤ j − 1 (interpreted as 0, if j = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS  13  The basic idea is standard, yet we were unable to find a suitable refer-  ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' One begins by a use of the quantitative Perron formula,  for which a convenient reference is [MV07, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' This implies  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='4)  �  n≤X  a(n) =  1  2πi  � σ0+iT  σ0−iT  Z(s)Xsds  s + R,  where R is a remainder term, and we take σ0 = 1+  c  log X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Using [MV07,  Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='3] and standard bounds on mean values of dk(n), one can show  R ≪ X  T Poly(log X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Next one shifts the contour of integration to the  line 1 − c/2  log T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The pole (if it exists) of Z(s) leads to a main term of the  form XPj−1(log X), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The new line of integration is bounded  by  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='5)  Poly(log T)X1− c/2  log T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Choosing log T = (log X)1/2 gives an acceptable error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Quartic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Let Ψtot,odd  4  (X) ⊆ Ψtot  4 (X) denote the sub-  set of characters with odd conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' For some constants Ktot  4 , Ktot,odd  4  > 0, we have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='6)  |Ψtot  4 (X)| ∼ Ktot  4 X,  and  |Ψtot,odd  4  (X)| ∼ Ktot,odd  4  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Moreover,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='7)  |Ψ′  4(X)| ∼  X  log X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='2,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='8)  |Ψtot,odd  4  (X)| =  �  0̸=(β)⊆Z[i]  (β,2β)=1  β squarefree  N(β)≤X  1,  and  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='9)  |Ψtot  4 (X)| = |Ψtot,odd  4  (X)| + 4|Ψtot,odd  4  (2−4X)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  14  JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' YOUNG  To show (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='6), it suffices to prove the asymptotic formula for |Ψtot,odd  4  (X)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  In view of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='5, it will suffice to understand the Dirichlet series  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='10)  Z4(s) =  �  0̸=(β)⊆Z[i]  (β,2β)=1  β squarefree  1  N(β)s =  �  π̸=π  (π,2)=1  (1 + N(π)−s) =  �  p≡1 (mod 4)  (1 + p−s)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Let χ4 be the primitive character modulo 4, so that ζ(s)L(s, χ4) =  ζQ[i](s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Then  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='11) Z4(s) = ζQ[i](s)  �  p  (1 − p−s)(1 − χ4(p)p−s)  �  p≡1 (mod 4)  (1 + p−s)2,  which can be simplified as  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='12)  Z4(s) = ζQ[i](s)ζ−1(2s)(1 + 2−s)−1  �  p≡1 (mod 4)  (1 − p−2s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Therefore, Z4(s) has a simple pole at s = 1, and its residue is a positive  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Moreover, the standard analytic properties of ζQ[i](s) let us  apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='5, giving the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  The asymptotic on Ψ′  4(X) follows from the prime number theorem in  arithmetic progressions, since there are two quartic characters of prime  conductor p ≡ 1 (mod 4), and none with p ≡ 3 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' We have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='13)  |Ψ4(X)| = K4X log X + O(X),  for some K4 > 0  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Every primitive quartic character factors uniquely as χ4χ2 with  χ4 totally quartic of conductor q4 > 1 and χ2 quadratic of conductor  q2, with (q4, q2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' It is convenient to drop the condition q4 > 1,  thereby including the quadratic characters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' this is allowable since the  number of quadratic characters is O(X), which is acceptable for the  claimed error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  The Dirichlet series for |Ψ4(X)|, modified to include the quadratic char-  acters, is  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='14)  Zall  4 (s) =  �  0̸=(β)⊆Z[i]  (β,2β)=1  β squarefree  1  N(β)s  �  q2∈Z≥1  (q2,2N(β))=1  1  qs  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS  15  A calculation with Euler products shows Zall  4 (s) = ζQ[i](s)ζ(s)A(s),  where A(s) is given by an absolutely convergent Euler product for  Re(s) > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Since Zall  4 (s) has a double pole at s = 1, this shows the  claim, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Sextic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Next we turn to the sextic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The proof of  the following proposition is similar to the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='6 and  so we omit it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' For some Ktot  6  > 0, we have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='15)  |Ψtot  6 (X)| ∼ Ktot  6 X,  and  |Ψ′  6(X)| ∼  X  log X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  A primitive totally sextic character factors uniquely as a primitive cubic  character (with odd conductor, since 2 ̸≡ 1 (mod 3)), times the Jacobi  symbol of the same modulus as the cubic character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  In general, a  primitive sextic character factors uniquely as χ6χ3χ2 of modulus q6q3q2,  pairwise coprime, with χ6 totally sextic of conductor q6, χ3 cubic of  conductor q3, and χ2 quadratic of conductor q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' We have |Ψ6(X)| = K6X(log X)2+O(X log X), for some  K6 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Write χ = χ6χ3χ2 as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Note that membership in Ψ6(X)  requires q6 > 1, which is an unpleasant condition when working with  Euler products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' However, the number of χ = χ3χ2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=', with χ6 = 1,  is O(X log X), so we may drop the condition q6 > 1 when estimating  |Ψ6(X)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  For simplicity, we count the characters with q2 odd and (q6q3, 3) =  1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' the general case follows similar lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The Dirichlet series for this  counting function is  Zall  6 (s) =  �  0̸=(β6)⊆Z[ω]  (β6,3β6)=1  β6 squarefree  1  N(β6)s  �  0̸=(β3)⊆Z[ω]  (β3,3β3)=1  β3 squarefree  (N(β3),N(β6)=1  1  N(β3)s  �  q2∈Z≥1  (q2,2N(β3β6))=1  1  qs  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  A calculation with Euler products shows Zall  6 (s) = ζQ[ω](s)2ζ(s)A(s),  where A(s) is given by an absolutely convergent Euler product for  Re(s) > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Since Zall  6 (s) has a triple pole at s = 1, this shows the  claim, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  □  16  JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' YOUNG  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Equidistribution of Gauss sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' We first focus on the quartic  case, and then turn to the sextic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Quartic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The following standard formula can be found  as [IK04, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='16)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Suppose that χ = χ1χ2 has conductor q = q1q2, with  (q1, q2) = 1, and χi of conductor qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Then  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='16)  τ(χ1χ2) = χ2(q1)χ1(q2)τ(χ1)τ(χ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Let notation be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Suppose that χ is  totally quartic and q is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Then  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='17)  τ(χ1χ2)2 = τ(χ1)2τ(χ2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='10, we will obtain the formula provided χ2  2(q1)χ2  1(q2) =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Note that χ2  i is the Jacobi symbol, so χ2  2(q1)χ2  1(q2) = ( q1  q2)( q2  q1) = 1,  by quadratic reciprocity, using that q1 ≡ q2 ≡ 1 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Suppose π ∈ Z[i] is a primary prime, with N(π) = p ≡  1 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Let χπ(x) = [ x  π] be the quartic residue symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Then  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='18)  τ(χπ)2 = −χπ(−1)√pπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  More generally, if β is primary, squarefree, with (β, 2β) = 1, then  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='19)  τ(χβ)2 = µ(β)χβ(−1)  �  N(β)β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The formula for χπ follows from [IR90, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1 (Chapter 8),  Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The formula for general β follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='11  and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Suppose that χ = χ2χ4 is a primitive quartic character  with odd conductor q, with χ2 quadratic of conductor q2, χ4 totally  quartic of conductor q4, and with q2q4 = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Then  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='20)  τ(χ)2 =  �−q4  q2  �  q2τ(χβ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='10, we have τ(χ)2 = χ2(q4)2χ4(q2)2τ(χ2)2τ(χ4)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  To simplify this, note χ2(q4)2 = 1, χ2  4(q2) = (q2  q4) = (q4  q2), and τ(χ2)2 =  ϵ2  q2q2 = ( −1  q2 )q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  □  VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS  17  Our next goal is to express τ(χβ)2 in terms of a Hecke Grossencharacter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Define  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='21)  λ∞(α) = α  |α|,  α ∈ Z[i], α ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Next define a particular character λ1+i : R× → S1, where R = Z[i]/(1+  i)3, by  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='22)  λ1+i(ik) = i−k,  k ∈ {0, 1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  This indeed defines a character since R× ≃ Z/4Z, generated by i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' For  α ∈ Z[i], (α, 1 + i) = 1, define  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='23)  λ((α)) = λ1+i(α)λ∞(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  For this to be well-defined, we need that the right hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='23)  is constant on units in Z[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' This is easily seen, since λ∞(ik) = ik =  λ1+i(ik)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Therefore, λ defines a Hecke Grossencharacter, as in [IK04,  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Moreover, we note that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='24)  τ(χβ)2  N(β) = µ(β)  �  2  N(β)  �  λ((β))  since this agrees with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='19) for β primary, and is constant on units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  According to [IK04, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='8], the Dirichlet series  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='25)  L(s, λk) =  �  0̸=(β)⊆Z[i]  λ((β))k  N(β)s ,  (k ∈ Z),  defines an L-function having analytic continuation to s ∈ C with no  poles except for k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The same statement holds when twisting λk by  a finite-order character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  For k ∈ Z, define the Dirichlet series  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='26)  Z(k, s) =  �  0̸=(β)⊆Z[i]  (β,2β)=1  β squarefree  (τ(χβ)2/N(β))k  N(β)s  ,  Re(s) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Let δk = −1 for k odd, and δk = +1 for k even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  We have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='27) Z(k, s) = A(k, s)L(s, (λ · χ2)k)δk,  where  χ2(β) =  �  2  N(β)  �  ,  and where A(k, s) is given by an Euler product absolutely convergent  for Re(s) > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  18  JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' YOUNG  In particular, the zero free region (as in [IK04, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='35]) implies  that Z(k, s) is analytic in a region of the type postulated in Lemma  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Moreover, the proof of [MV07, Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='4] shows that Z(k, s)  is bounded polynomially in log(2 + |t|) in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='24) shows that Z(k, s) has an Euler product of  the form  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='28)  Z(k, s) =  �  (π)̸=(π)  (1 + (−1)k χk  2(π)λk((π))  N(π)s  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  This is an Euler product over the split primes in Z[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' We extend this  to include the primes p ≡ 3 (mod 4) as well, with N(π) = p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' It is  convenient to define χ2(1 + i) = 0, so we can freely extend the product  to include the ramified prime 1 + i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' In all, we get  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='29)  Z(k, s) =  � �  p  (1 − χk  2(p)λk(p)  N(p)s  )  �−δk �  p  (1 + O(p−2s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Note the product over p is L(s, (λ · χ2)k)δk, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  □  According to Weyl’s equidistribution criterion [IK04, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1], a se-  quence of real numbers θn, 1 ≤ n ≤ N is equidistributed modulo 1 if  and only if �  n≤N e(kθn) = o(N) for each integer k ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' We apply  this to e(θn) = (τ(χ)2/q), whence e(kθn) = (τ(χ)2/q)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Due to the  twisted multiplicativity formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='16), the congruence class in which  2k lies modulo ℓ may have a simplifying effect on τ(χ)2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' For instance,  when ℓ = 4, then k even leads to a simpler formula than k odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' This  motivates treating these cases separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' As a minor simplification,  below we focus on the sub-family of characters of odd conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The  even conductor case is only a bit different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The Gauss sums τ(χ)2/q for χ totally quartic of odd  conductor q, equidistribute on the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The complex numbers τ(χ)2/q lie on the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Weyl’s  equidistribution criterion says that these normalized squared Gauss  sums equidistribute on the unit circle provided  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='30)  �  0̸=(β)⊆Z[i]  (β,2β)=1  β squarefree  N(β)≤X  (τ(χβ)2/N(β))k = o(X),  VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS  19  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' This histogram represents the distribution  of the argument of the τ(χ)2/cond(χ) for totally quartic  characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Each histogram is made by calculating the  Gauss sums of characters of each order up to prime and  composite conductor 300000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  for each nonzero integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' In turn, this bound is implied by Propo-  sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='14, using the zero-free region for the Hecke Grossencharacter  L-functions in [IK04, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  □  To contrast this, we will show that the normalized Gauss sums τ(χ)2/q,  with χ ranging over all quartic characters, equidistribute slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' More  precisely, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Let k ∈ 2Z, k ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' There exists ck ∈ C such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='31)  �  q≤X  (q,2)=1  �  χ:χ4=1  cond(χ)=q  (τ(χ)2/q)k = ckX + o(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Recall from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='7 that the total number of such  characters grows like X log X, so Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='16 shows that the rate  of equidistribution is only O((log X)−1) here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' In contrast, in the family  of totally quartic characters, the GRH would imply a rate of equidis-  tribution of the form O(X−1/2+ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' This difference in rates of equidis-  tribution is supported by Figure 2 in which we see that the arguments  5+00  4000  DODE  DOZ  1000  E-  1  020  JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' YOUNG  of squares of the Gauss sums of totally quartic characters quickly con-  verge to being uniformly distributed, as compared to the Gauss sums  of all quartic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  In addition, one can derive a similar result when restricting to χ ∈  Ψ4(X), simply by subtracting off the contribution from the quadratic  characters alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' As in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='13, write χ = χ2χ4, with χ2 quadratic and χ4  totally quartic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Then τ(χ)4/(q1q2)2 = τ(χ4)4/q2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The analog of Z(k, s),  using k even to simplify, is  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='32)  Zall(k, s) =  �  0̸=(β)⊆Z[i]  (β,2β)=1  β squarefree  τ(χβ)2k/N(β)k  N(β)s  �  q2∈Z≥1  (q2,2N(β))=1  1  qs  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Referring to the calculation in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='14, we obtain  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='33)  Zall(k, s) = ζ(s)L(s, λk)A(s),  where A(s) is an Euler product absolutely convergent for Re(s) > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Since this generating function has a simple pole at s = 1, we deduce  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  □  As mentioned above, in order to deduce equidistribution, by Weyl’s  equidistribution criterion, we also need to consider odd values of k in  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' This is more technical than the case for even k, so we content  ourselves with a conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' For each odd k, there exists δ > 0 such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='34)  �  q≤X  (q,2)=1  �  χ:χ4=1  cond(χ)=q  (τ(χ)2/q)k ≪k,δ X1−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='13 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='24), this problem reduces to  understanding sums of the rough shape  �  β,q2  q2N(β)≤X  �−N(β)  q2  �  µ(β)  �  2  N(β)  �  λ((β))k,  where we have omitted many of the conditions on β and q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' In the  range where q2 is very small, the GRH gives cancellation in the sum  over β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Conversely, in the range where N(β) is very small, the GRH  VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS  21  gives cancellation in the sum over q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' This discussion indicates that  Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='18 follows from GRH, with any δ < 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Unconditionally, one can deduce some cancellation using the zero-free  region for the β-sum (with q2 very small), and a subconvexity bound  for the q2-sum (with N(β) very small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' In the range where both q2  and N(β) have some size, then Heath-Brown’s quadratic large sieve  [HB95] gives some cancellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Since we logically do not need an  unconditional proof of equidistribution, we omit the details for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='18 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='16 together imply  that the squares of the quartic Gauss sums do equidistribute in the full  family Ψ4(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Sextic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Now we turn to the sextic Gauss sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Suppose that χ is totally sextic of conductor q, and say  χ = χ2χ3 with χ2 quadratic and χ3 cubic, each of conductor q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Suppose  χ3 = χβ, as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Then  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='35)  τ(χ) = µ(q)χ3(2)τ(χ2)τ(χ3)βq−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' By [IK04, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='18)], τ(χ2)τ(χ3) = J(χ2, χ3)τ(χ), where J(χ2, χ3)  is the Jacobi sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  It is easy to show using the Chinese remainder  theorem that if χ2 = �  p χ(p)  2  and χ3 = �  p χ(p)  3 , then  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='36)  J(χ2, χ3) =  �  p  J(χ(p)  2 , χ(p)  3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  The Jacobi sum for characters of prime conductor can be evaluated  explicitly using the following facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' By [Lem00, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='30],  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='37)  J(χ(p)  2 , χ(p)  3 ) = χ(p)  3 (22)J(χ(p)  3 , χ(p)  3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Suppose that χ(p)  3  = χπ, where ππ = p, and π is primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Then [IR90,  Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' 9, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' 1] implies J(χπ, χπ) = −π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' (Warning: they state the  value π instead of −π, but recall their definition of primary is opposite  our convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Also recall that χπ = χ−π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=') Gathering the formulas,  we obtain  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='38)  τ(χ2)τ(χ3) = τ(χ)χ3(2)2 �  πi|β  (−πi) = τ(χ)χ3(2)2µ(q)β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Rearranging this and using ββ = q completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  □  22  JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' YOUNG  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Let conditions be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Then  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='39)  τ(χ)2/q = χ3(4)  �−1  q  �  τ(χβ)2β  2/q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Patterson [Pat78] showed that τ(χβ)/√q is uniformly distributed on  the unit circle, as χβ ranges over primitive cubic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The same  method gives equidistribution after multiplication by a Hecke Grossen-  character, and so similarly to the quartic case above, we deduce:  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='23 (Patterson).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The Gauss sums τ(χ)2/q, for χ totally  sextic of conductor q, equidistribute on the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  In light of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='22, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='16, and Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='18, it  seems reasonable to conjecture that the points τ(χ)2/q are equidis-  tributed on the unit circle, as χ varies over all sextic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' To  see a limitation in the rate of equidistribution, it is convenient to con-  sider τ(χ)6/q3, which is multiplicative for χ sextic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' For q ≡ 1 (mod 4),  and χ = χ2 quadratic, we have τ(χ2)2/q = 1, so the quadratic part is  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' For χ cubic and q ≡ 1 (mod 4),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='40)  τ(χβ)6/q3 = µ(β)τ(χβ)3β  3 = q−1β  2,  which is nearly a Hecke Grossencharacter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' A similar formula holds for  χ totally sextic, namely  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='41)  τ(χ)6/q3 = q−4β  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Therefore, carrying out the same steps as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='16 shows  that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='42)  �  q≤X  q≡1 (mod 4)  �  χ∈Ψ6  cond(χ)=q  �  τ(χ)6/q3�k  = CkX + o(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  This is less of an obstruction than in the quartic case, since here the  rate of equidistribution is O((log X)−2) instead of O((log X)−1), due to  the fact that |Ψ6(X)| is approximately log X times as large as |Ψ4(X)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Similarly to the discussion of the quartic case in Remarks 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='19 and  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='20, we make the following conjecture without further explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The Gauss sums τ(χ)2/q, for χ ranging in Ψ6(X),  equidistribute on the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS  23  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Estimates for quartic and sextic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' In order to ap-  ply the random matrix theory conjectures, we need variants on Propo-  sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='6, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='7, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='8, and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='9, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' For primitive Dirichlet characters χ of order ℓ we have  for ℓ = 4 and ℓ = 6 that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='43)  �  χ∈Ψℓ(X)  1  �  cond(χ)  ∼ 2Kℓ  √  X(log X)d(ℓ)−2,  and  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='44)  �  χ∈Ψtot  ℓ  (X)  1  �  cond(χ)  ∼ 2Ktot  ℓ  √  X,  �  χ∈Ψ′  ℓ(X)  1  �  cond(χ)  ∼ 2  √  X  log X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' These estimates follow from a straightforward application of  partial summation or from a minor modification of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='5 since  the generating Dirichlet series for one of these sums has its pole at  s = 1/2 instead of at s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Random matrix theory: Conjectural asymptotic  behavior  This section closely follows the exposition of §3 of [DFK04] and §4 of  [DFK07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Let U(N) be the set of unitary N×N matrices with complex coefficients  which forms a probability space with respect to the Haar measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  For a family of L-functions with symmetry type U(N), Katz and Sar-  nak conjectured that the statistics of the low-lying zeros should agree  with those of the eigenangles of random matrices in U(N) [KS99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Let  PA(λ) = det(A − λI) be the characteristic polynomial of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Keating  and Snaith [KS00] suggest that the distribution of the values of the L-  functions at the critical point is related to the value distribution of the  characteristic polynomials |PA(1)| with respect to the Haar measure on  U(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  For any s ∈ C we consider the moments  MU(s, N) :=  �  U(N)  |PA(1)|s dHaar  24  JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' YOUNG  for the distribution of |PA(1)| in U(N) with respect to the Haar mea-  sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' In [KS00], Keating and Snaith proved that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1)  MU(s, N) =  N  �  j=1  Γ(j)Γ(j + s)  Γ2(j + s/2) ,  so that MU(s, N) is analytic for Re(s) > −1 and has meromorphic  continuation to the whole complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The probability density of  |PA(1)| is given by the Mellin transform  pU(x, N) =  1  2πi  �  Re(s)=c  MU(s, N)x−s−1 ds,  for some c > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  In the applications to the vanishing of twisted L-functions we consider  in this paper, we are only interested in small values of x where the value  of pU(x, N) is determined by the first pole of MU(s, N) at s = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' More  precisely, for x ≤ N −1/2, one can show that  pU(x, N) ∼ G2(1/2)N 1/4  as N → ∞,  where G(z) is the Barnes G-function with special value [Bar99]  G(1/2) = exp  �3  2ζ′(−1) − 1  4 log π + 1  24 log 2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  We will now consider the moments for the special values of twists of  L-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  We then define, for any s ∈ C, the following sum of  evaluations at s = 1 of L-functions primitive order ℓ characters of  conductor less than X:  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='2)  ME(s, X) =  1  #FΨℓ,E(X)  �  L(E,s,χ)∈FΨℓ,E(X)  |L(E, 1, χ)|s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Then, since the families of twists of order ℓ are expected to have unitary  symmetry, we have  Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1 (Keating and Snaith Conjecture for twists of order  ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' With the notation as above,  ME(s, X) ∼ aE(s/2)MU(s, N)  as N = 2 log X → ∞,  where aE(s/2) is an arithmetic factor depending only on the curve E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS  25  From Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1, the probability density for the distribution of the  special values |L(E, 1, χ)| for characters of order ℓ is  pE(x, X)  =  1  2πi  �  Re(s)=c  ME(s, X)x−s−1 ds  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='3)  ∼  1  2πi  �  Re(s)=c  aE(s/2)MU(s, N)x−s−1 ds  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='4)  as N = 2 log X → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' As above, when x ≤ N −1/2, the value of pE(x, X)  is determined by the residue of MU(s, N) at s = −1, thus it follows  from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='4) that for x ≤ (2 log X)−1/2,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='5)  pE(x, X) ∼ 21/4aE(−1/2)G2(1/2) log1/4(X)  as X → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  We now use the probability density of the random matrix model with  the properties of the integers nE(χ) to obtain conjectures for the van-  ishing of the L-values |L(E, 1, χ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' When χ is either quartic or sextic,  the discretization nE(χ) is a rational integer since Z[ζℓ] ∩ R = Z when  ℓ = 4 or 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Let χ be a primitive Dirichlet character of order ℓ = 4  or 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Then  |L(E, 1, χ)| =  cE,ℓ  �  cond(χ)  |nE(χ)|,  where cE,ℓ is a nonzero constant which depends only on the curve E  and ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' By rearranging equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='2) we obtain  |L(E, 1, χ)| =  ����  Ωϵ(E) τ(χ) kE nE(χ)  cond(χ)  ���� = |Ωϵ(E) kE nE(χ)|  �  cond(χ)  = cE,ℓ|nE(χ)|  �  cond(χ)  ,  where the nonzero constant kE is that of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  □  We write  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='6)  Prob{|L(E, 1, χ)| = 0} = Prob{|L(E, 1, χ)| < B(cond(χ))},  for some function B(cond(χ)) of the character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='2 we may  take B(cond(χ)) =  cE,ℓ  �  cond(χ)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Note that since cE,ℓ ̸= 0, if  |nE(χ)|cE,ℓ  �  cond(χ)  <  cE,ℓ  �  cond(χ)  ,  then |nE(χ)| < 1 and hence must vanish since |nE(χ)| ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  26  JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' YOUNG  Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='5), we have  Prob{|L(E, 1, χ)| = 0}  =  � B(cond(χ))  0  21/4aE(−1/2)G2(1/2) log1/4(X) dx  =  21/4aE(−1/2)G2(1/2) log1/4(X)B(cond(χ))  Summing the probabilities gives  |VΨℓ,E(X)| = 21/4cE,kaE(−1/2)G2(1/2) log1/4(X)  �  cond(χ)≤X  1  �  cond(χ)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Thus, by the analysis in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='3, we have  |VΨ4,E(X)| ∼ 25/4cE,4K4aE(−1/2)G2(1/2) log1/4(X)  √  X log X  ∼ bE,4X1/2 log5/4 X  and  |VΨ6,E(X)| ∼ 25/4cE,6K6aE(−1/2)G2(1/2) log1/4(X)  √  X(log X)2  ∼ bE,6X1/2 log9/4 X  as X → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Moreover, if we restrict to those characters that are totally quartic or  sextic, we get the following estimates  |VΨtot  4 ,E(X)| ∼ 25/4cE,4Ktot  4 aE(−1/2)G2(1/2) log1/4(X)  √  X  ∼ btot  E,4X1/2 log1/4 X  and  |VΨtot  6 ,E(X)| ∼ 25/4cE,6Ktot  6 aE(−1/2)G2(1/2)  ∼ btot  E,6X1/2 log1/4 X  as X → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Finally, if we restrict only to those twists by characters of prime con-  ductor, we conclude  |VΨ′  4,E(X)| ∼ 25/4cE,4aE(−1/2)G2(1/2) log1/4(X)  √  X  log X  ∼ b′  E,4X1/2 log−3/4 X  VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS  27  and  |VΨ′  6,E(X)| ∼ 25/4cE,6aE(−1/2)G2(1/2) log1/4(X)  √  X  log X  ∼ b′  E,6X1/2 log−3/4 X  as X → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Here we provide numerical evidence for Conjec-  ture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The computations of the Conrey labels for the characters were  done in SageMath [Sag21] and the computations of the L-functions were  done in PARI/GP [PAR22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The L-function computations were done  in a distributed way on the Open Science Grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' For each curve, we  generated a PARI/GP script to calculate a twisted L-function for each  primitive character of order 4 and 6, and then combined the results into  one file at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The combined wall time of all the computations  was more than 50 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The code and data are available at [BR23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  In Figure 3 we plot the points  (X, X1/2 log5/4 X  |VΨ4,11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1(X)|), (X, X1/2 log−3/4 X  |VΨ′  4,11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1(X)| ), (X,  X1/2 log1/4 X  |VΨtot  4  ,11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1(X)|)  that provides a comparison between the predicted vanishings of L(E, 1, χ)  for quartic characters and for the curve 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' In Figure 4 we plot the  analogous points for the same curve but for sextic twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' In Figure 5  we plot the points  (X, X1/2 log−3/4 X  |VΨ′  4,37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1(X)| ), (X, X1/2 log−3/4 X  |VΨ′  6,37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1(X)| )  Even though we are most interested in the families of all quartic and  sextic twists, we include the families of twists of prime conductor be-  cause there are far fewer such characters and so we can calculate the  number of vanishings up to a much larger X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' We include the fami-  lies of twists by totally quartic and sextic characters to highlight the  transition between the family of prime conductors and the family of all  conductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  References  [Bar99]  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Barnes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The theory of the G-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Quart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=', 31:264–314,  1899.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  [BCDT01] Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  On the modularity of elliptic curves over Q: wild 3-adic exercises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Journal  of the American Mathematical Society, pages 843–939, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  [BE81]  Bruce C Berndt and Ronald J Evans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The determination of Gauss sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Bulletin of the American Mathematical Society, 5(2):107–129, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  28  JENNIFER BERG, NATHAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' RYAN, AND MATTHEW P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' YOUNG  (a)  The  ratio  of  predicted  vanishings  to  empirical  van-  ishings  of  twists  of  the curve 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1 by  quartic characters of  conductor ≤ 700000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  (b) The ratio of pre-  dicted  vanishings  to  empirical  vanishings  of twists of the curve  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1  by  quartic  characters  of  prime  conductor ≤ 2000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  (c) The ratio of pre-  dicted  vanishings  to  empirical  vanishings  of twists of the curve  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1  by  totally  quartic characters of  conductor ≤ 700000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Verification of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1 for quartic  twists of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  (a) The ratio of pre-  dicted  vanishings  to  empirical vanishings of  twists  of  the  curve  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1 by sextic char-  acters of conductor ≤  300000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  (b) The ratio of pre-  dicted  vanishings  to  empirical vanishings of  twists  of  the  curve  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1 by sextic char-  acters of prime con-  ductor ≤ 2000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  (c) The ratio of pre-  dicted  vanishings  to  empirical vanishings of  twists  of  the  curve  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1 by totally sex-  tic characters of con-  ductor ≤ 300000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Verification of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1 for sextic  twists of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  [BR23]  Jen Berg and Nathan C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Ryan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Code and data for quartic and sextic  twists of elliptic curve L-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' http://eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='bucknell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='edu/~ncr006/  quartic-sextic-twists-website/, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  [BY10]  Stephan Baier and Matthew P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Young.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Mean values with cubic characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Number Theory, 130(4):879–903, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  [CFK+05] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Brian Conrey, David W Farmer, Jon P Keating, Michael O Rubin-  stein, and Nina C Snaith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Integral moments of L-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Proceedings  of the London Mathematical Society, 91(1):33–104, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  [Cho87]  Sarvadaman Chowla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' The Riemann hypothesis and Hilbert’s tenth prob-  lem, volume 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' CRC Press, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
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+page_content='75  Lio  125  150  175  200  1e616  15  14  13  12  11  LD  0  DODS  10dC0  150000  240000  25000  3+0dC04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='D  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='5-  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='0  25 -  20  15  LD  VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS  29  (a) The ratio of pre-  dicted  vanishings  to  empirical  vanishings  of twists of the curve  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1  by  quartic  characters  of  prime  conductor ≤ 2000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  (b) The ratio of pre-  dicted  vanishings  to  empirical vanishings of  twists  of  the  curve  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1 by sextic char-  acters of prime con-  ductor ≤ 2000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Verification of parts of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1 for  twists of 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  [CKRS00] JB Conrey, JP Keating, MO Rubinstein, and NC Snaith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' On the fre-  quency of vanishing of quadratic twists of modular L-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' In Pro-  ceedings of the Millennial Conference on Number Theory, Urbana, Illinois,  21-26 May, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' AK Peters, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  [DFK04] Chantal David, Jack Fearnley, and Hershy Kisilevsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' On the vanishing of  twisted L-functions of elliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
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+page_content='  [DFK07] Chantal David, Jack Fearnley, and Hershy Kisilevsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Vanishing of L-  functions of elliptic curves over number fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' In Ranks of elliptic curves  and random matrix theory, volume 341 of London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Lecture Note  Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=', pages 247–259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Press, Cambridge, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  [FMS10] Steven Finch, Greg Martin, and Pascal Sebah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Roots of unity and  nullity modulo n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Proceedings of the American Mathematical Society,  138(8):2729–2743, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
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+page_content=' Acta  Arith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
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+page_content=' Analytic number theory, vol-  ume 53 of American Mathematical Society Colloquium Publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  American Mathematical Society, Providence, RI, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
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+page_content=' In J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
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+page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
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+page_content=' (JEMS),  12(5):1097–1116, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  [TW95]  Richard Taylor and Andrew Wiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Ring-theoretic properties of certain  Hecke algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content=' Annals of Mathematics, 141(3):553–572, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  [Wil95]  Andrew Wiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
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+page_content=' Annals  of mathematics, 141(3):443–551, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
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+page_content=' Integrality of twisted l-values  of elliptic curves, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='  Email address: jsb047@bucknell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
+page_content='edu  Email address: nathan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
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+page_content='edu  Department of Mathematics, Bucknell University, Lewisburg, PA 17837  Email address: myoung@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
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+page_content='edu  VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS  31  Department of Mathematics, Texas A&M University, College Station,  TX 77843-3368' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQf6A7h/content/2301.05329v1.pdf'}
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+Practical challenges in data-driven
+interpolation: dealing with noise, enforcing
+stability, and computing realizations
+Quirin Aumann∗
+Ion Victor Gosea†
+∗Max Planck Instiute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg,
+Germany.
+Email: aumann@mpi-magdeburg.mpg.de, ORCID: 0000-0001-7942-5703
+†Max Planck Instiute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg,
+Germany.
+Email: gosea@mpi-magdeburg.mpg.de, ORCID: 0000-0003-3580-4116
+Abstract: In this contribution, we propose a detailed study of interpolation-based data-
+driven methods that are of relevance in the model reduction and also in the systems and
+control communities. The data are given by samples of the transfer function of the under-
+lying (unknown) model, i.e., we analyze frequency-response data. We also propose novel
+approaches that combine some of the main attributes of the established methods, for ad-
+dressing particular issues.
+This includes placing poles and hence, enforcing stability of
+reduced-order models, robustness to noisy or perturbed data, and switching from different
+rational function representations. We mention here the classical state-space format and
+also various barycentric representations of the fitted rational interpolants. We show that
+the newly-developed approaches yield, in some cases, superior numerical results, when com-
+paring to the established methods. The numerical results include a thorough analysis of
+various aspects related to approximation errors, choice of interpolation points, or placing
+dominant poles, which are tested on some benchmark models and data-sets.
+Keywords: Data-driven methods, rational approximation, interpolatory methods, least
+squares fit, Loewner framework, frequency response data, pole placement, noisy measure-
+ments, Loewner and Cauchy matrices.
+Novelty statement: This note shows that by combining the features of established data-
+driven rational approximation methods based on interpolation (and/or least squares fit),
+one can devise methods that offer additional important advantages. These include stabil-
+ity enforcement by placing poles in an elegant and numerically stable manner, together
+with robustness to noisy data.
+1. Introduction
+Approximation of large-scale dynamical systems is pivotal for serving the scopes of efficient simulation
+and designing control laws in real-time. The technique for reducing the complexity of a system is known
+as model order reduction (MOR) [1, 5, 13, 14]. There exist a number of methodologies for reducing
+large-scale models, and each method is tailored to some specific applications (mostly, but not restricted
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+arXiv:2301.04906v1  [math.NA]  12 Jan 2023
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+2
+to mechanical and electrical engineering) and to achieving certain goals (stability, passivity or structure
+preservation), on top of the complexity reduction part. Data-driven MOR approaches are of particular
+importance when access to high-fidelity models is not explicitly granted. This means that a state-
+space formulation with access to internal variables is not available, yet input/output data are. Such
+methods circumvent the need to access an exact description of the original model and are applicable
+whenever classical projection-based MOR is not. Here, we mention system and control methodologies
+that are based on interpolation or least-square fit of data (i.e., frequency response measurements),
+such as vector fitting [31], the Loewner framework [41], or the AAA algorithm [42]. Methods that use
+time-domain data are also of interest, including the ones that require input-output data together with
+those which use snapshot data (access to the state evolution), such as the classical ones in [36,58,59],
+followed by [8], [47] or [54].
+We focus on interpolation-based or so-called moment matching (MM) methods which have emerged,
+were developed, and improved continuously in the last decades. The backbone of such methods rep-
+resent rational Krylov-type approaches together with the Sylvester matrix equation interpretation
+[1, 11, 20]. Apart from being computationally efficient and easy to implement, MM approaches have
+another advantage: they do not (necessarily) require access to a full-state realization of the original
+dynamical system. Hence, they can be viewed as data-driven methods. Here data are given by the
+moments of the system, i.e., samples of the underlying transfer function of the system (and of its
+derivatives) evaluated in a particular frequency range; for more details, we refer the readers to [5,8,41]
+and to Chapter 3 in [13]. The notion of a moment with respect to systems and control theory is related
+to the unique solution of a Sylvester matrix equation [20].
+The purpose of this note is twofold; first, we intend to review and to connect three important system
+theoretical model reduction methods based on interpolation that were introduced in the last 15 years:
+• The Loewner framework (LF) by Mayo and Antoulas from 2007 in [41];
+• The Astolfi framework (AF) from 2010 in [8];
+• The Adaptive Antoulas Anderson (AAA) algorithm by Nakatsukasa, Set´e and Trefethen from
+2018 in [42].
+Together, these three approaches were cited multiple times in various research publications, being
+arguably quite popular methods. However, until now, not too many connections between them were
+provided, neither in the automatic control, nor in the model reduction, or numerical analysis commu-
+nities. Together with the vector fitting algorithm (VF) in [31] (which is not based on interpolation,
+and is hence a purely optimization approach based on least-squares fitting), these methods repre-
+sent arguably the most prolific rational approximation schemes developed in the system and control
+community. However, VF is not the object of this study since it is not based on interpolation.
+The other scope of this note is to propose a new method that is based on the three methods
+enumerated above, and that addresses some of the shortcomings and challenges associated with them.
+Basically, the idea is to combine the attributes of each method, by following the steps below.
+• We make use of the order-revealing property of the LF (encoded by the rank of augmented
+Loewner matrices); additionally, the selection of interpolation points is done via a Loewner-CUR
+technique proposed in [38].
+• We utilize the elegant state-space parameterization of the LTI system proposed by the AF (after
+imposing k interpolation conditions); this is the backbone of the methods (we also show the
+connection between state-space forms and barycentric forms).
+• We use either the fitting step from AAA (that chooses free parameters to fit the un-interpolated
+data in a least square sense) or we impose pole placing (dominant poles are selected from those
+of the Loewner model); in both cases, a linear system of equations needs to be solved.
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+3
+In what follows, we consider a multiple-input multiple-output (MIMO) linear time-invariant (LTI)
+system ΣL of dimension n described by the following system of differential equations:
+ΣL :
+�
+˙x(t) = Ax(t) + Bu(t),
+y(t) = Cx(t),
+(1)
+with x(t) ∈ Rn as the state variable, u(t) ∈ Rm as the control inputs, and y(t) ∈ Rp as the observed
+outputs. Here, we have that A ∈ Cn×n, B ∈ Cn×m and C ∈ Cp×n. The transfer (matrix) function of
+the LTI system is given by H(s) ∈ Cp×n, with s ∈ C, as
+H(s) = C(sIn − A)−1B.
+(2)
+It is to be noted that, for m = p = 1, the system becomes single-input single-output (SISO). We will
+sometime switch between MIMO and SISO formats while presenting the methods covered in this note,
+since the latter allows a more easy exposition for some of the results shown here.
+Let si ∈ C \ σ(A), where σ(A) denotes the spectrum of matrix A ∈ Cn×n, i.e., the set of its
+eigenvalues.
+The j-moment of system ΣL at si is given by ηj(si) =
+(−1)j
+j!
+�
+dj
+dsj H(s)
+�
+s=si, for any
+integer j ⩾ 1. The 0-moment is obtained by sampling the transfer function H(s) in (2) at si, i.e.,
+η0 = H(si). In this contribution, we restrict the analysis to matching 0-moments, i.e., samples of
+the transfer function H(s), and not of its derivatives. However, all methodologies shown here can be
+expected to cope with this as well. Moreover, in practice, inferring 0-moments from time-domain data
+is usually a more straight-forward task; this is performed by exciting the system with harmonic inputs,
+and by applying spectral transformations to the outputs. Additionally, the inference of derivative
+values (of the transfer function) is typically susceptible to perturbations and it is more challenging to
+attain, from a numerical point of view.
+The paper is structured in the following way; after the introduction session sets up the stage, we
+propose a survey of three established interpolation-based methods in Section 2. Then, the proposed
+methodologies are developed in Section 3, with emphasis on the one-step approach that combines
+optimal selection of interpolation points (chosen using CUR-DEIM) with LS fit on the rest of the
+data, and also the pole placement method in barycentric form that enforces dominant poles from the
+Loewner data-driven model. Then, Section 4 illustrates the numerical aspects of applying the methods
+discussed/proposed in the previous two sections to a variety of test cases (various models and data
+sets). Finally, Section 5 presents the conclusions and the outlook into future research.
+2. A survey of established methods
+In this section we discuss three established data-driven methods for rational approximation (AF, LF,
+and AAA, as mentioned in the previous section).
+The data are samples of the transfer function
+corresponding to the underlying dynamical system, measured on a particular frequency grid. In what
+follows, we mention some state-of-the-art methodologies used to measure such data, i.e., frequency
+response data. Typically, such measurements can be produced in practice from experiments conducted
+in scientific laboratories using carefully calibrated machines, called spectrum analyzers (SAs). In this
+category we mention swept-tuned spectrum analyzers, scalar network analyzers (SNAs), and vector
+network analyzers (VNAs).
+The SNA is an instrument that measures microwave signals by converting them to a DC voltage
+using a diode detector. In a VNA, information regarding both the magnitude and the phase of a
+microwave signal is extracted.
+While there are different ways to perform such measurements, the
+method employed by commercial products (such as the Anritsu series described in [18]) of VNAs is
+to down-convert the signal to a lower intermediate frequency in a process called harmonic sampling.
+This signal can then be measured directly by a tuned receiver. Compared to the SNA, the VNA is
+a more powerful analyzer tool. The major difference is that the VNA can also measure the phase,
+and not only the amplitude. With this property enforced, then so-called scattering parameters (or
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+4
+S-parameters) can be processed. These can be used for identifying forward and reverse transmission
+and reflection characteristics. More details can be found in [18].
+The harmonic balance method (or HBM) [43], is an established methodology in the field of elec-
+tromagnetics. The HBM is used in many (if not most) commercial radio-frequency (RF) simulation
+tools. This is due to the fact that it has certain advantages over other common methods, namely
+modified nodal analysis (MNA), which makes it more appropriate to use for stiff problems and circuits
+containing transmission lines, nonlinearities and dispersive effects. More details can be found in the
+survey paper [49].
+2.1. The one-sided moment-matching approach in [8]
+The framework introduced by Astolfi in [8] (referred to as AF throughout the paper) deals with the
+problem of model reduction by moment matching. Although classically interpreted as a problem of
+interpolation of points in the complex plane, it has instead been recast as a problem of interpolation
+of steady-state responses. In the following we briefly review its application to linear systems. It is to
+be noted that the AF was steadily extended and applied to different scenarios (including nonlinear
+dynamical systems, pole-zero placement, and least-squares fit) [33–35,45,53,55].
+The moments of a linear system can be characterized in terms of the solution of Sylvester equations.
+By using this observation, it has been shown that the moments are in one-to-one relation with the
+steady-state output response of the interconnection between a signal generator and the original linear
+system.
+In what follows, for simplicity of exposition, it is assumed that ΣL is a minimal system (both fully
+controllable and fully observable). For exact definitions on minimality, controllability, or observability
+of LTI systems, we refer the reader to [1].
+Let k ⩽ n and S ∈ Ck×k a non-derogatory matrix (for which the characteristic and minimal
+polynomials coincide) with σ(S) ∩ σ(A) = ∅ and R ∈ C1×k so that (S, R) is observable. Consider the
+signal generator system Σsg described by the equations
+Σsg :
+�
+˙ω(t) = Sω(t),
+u(t) = Rω(t).
+(3)
+Then, the explicit solution of (3) can be written as ω(t) = eStω(0). Hence, the control input is written
+as u(t) = ReStω(0). In addition, the eigenvalues of S are called interpolation points.
+For a linear system ΣL, and interpolation points si ∈ C \ σ(A), for i = 1, . . . , k, consider a non-
+derogatory matrix S ∈ Rk×k. It follows that there exists a one-to-one relation between the moments
+of the system ΣL and
+1. the matrix CΠ, where Π is the (unique) solution of the Sylvester equation AΠ + BR = ΠS,
+for any row vector R ∈ R1×k so that (R, S) is observable;
+2. the steady-state response of the output y of the interconnection of system ΣL and the system
+Σsg, for any R and ω(0) such that the triplet (R, S, ω(0)) is minimal.
+More precisely, let ∆ ∈ Rk be a column vector containing k free parameters (denoted here by
+δ1, δ2, . . . , δk, with δi ̸= 0, 1 ≤ i ≤ k). Then, as stated in [8], the family of linear time-invariant
+systems that interpolates the moments of system ΣL at the eigenvalues of matrix S, is given by
+ˆΣ∆ :
+� ˙ˆx(t) = (S − ∆R)
+�
+��
+�
+= ˆA
+ˆx(t) + ∆
+����
+= ˆB
+u(t),
+ˆy(t) = CΠ
+����
+= ˆC
+ˆx(t),
+(4)
+where the matrices S and R are as before and Π is the unique solution of the Sylvester equation
+AΠ + BR = ΠS. Additionally, the condition σ(S) ∩ σ(S − ∆R) = ∅ needs to be enforced. It is
+to be noted that the free parameters explicitly enter the vector ˆB = ∆, but also the matrix ˆA, as
+ˆA = S − ∆R. Finally, ˆC = CΠ has fixed entries.
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+5
+The Sylvester matrix equation for the reduced-order system is written as ˆA ˆΠ+ ˆBR = ˆΠS. This can
+be explained by the fact that the reduced-order system matches the prescribed moments of the original
+system, hence the same format of the two equations. Without loss of generality, one can consider
+that the matrix ˆΠ is the identity matrix, i.e.
+ˆΠ = Ik (this can be achieved by applying similarity
+transformations). By replacing this value into the reduced-dimension Sylvester matrix equation above,
+the formula ˆA = S − ∆R directly follows.
+Afterwards, the free parameters collected in the vector ∆ can be chosen in order to enforce or impose
+additional conditions as mentioned in [8], such as: matching with imposing additional k interpolation
+conditions, matching with prescribed eigenvalues, matching with prescribed relative degree, matching
+with prescribed zeros, matching with a passivity constraint, matching with L2-gain, or matching with
+a compartmental constraint.
+An important aspect of the AF is the characterization of all, i.e., infinitely many families of reduced-
+order models that satisfy k prescribed interpolation conditions. This is done by explicitly computing
+such parameterized models, for which the free parameters are the variables entering in the vector ∆.
+The main parameterization developed here will be used as a “backbone” of the methods developed in
+Section 3.
+As stated in the original paper, the main advantage of the AF (characterization of moments in
+terms of steady-state responses) is that it allows the definition of moments for systems which do not
+admit a clear/immediate representation in terms of transfer function(s). Hence, the author provides
+as examples the case of linear time-varying systems, and the case of nonlinear systems. Moreover,
+it is stated in [8] that one disadvantage of the framework is that it requires the existence of steady-
+state responses. Consequently, the original system has to be (exponentially) stable. However, in most
+practical applications, this is a realistic requirement.
+2.2. The Loewner framework in [41]
+In this section we present a short summary of the Loewner framework (LF), as introduced in [41]. It
+is to mentioned that LF has its roots in the earlier work of [4], and that LF can be considered to be a
+double-sided moment-matching approach (as opposed to AF, which is one-sided).
+For a tutorial paper on LF for LTI systems, we refer the reader to [7], and for a recent extension
+that uses time-domain data, we refer the reader to [48]. The Loewner framework has been recently
+extended to certain classes of nonlinear systems, such as bilinear systems in [6], and quadratic-bilinear
+(QB) systems in [24], but also to linear parameter-varying systems in [28]. Additionally, issues such
+as stability preservation or enforcement, or passivity preservation, were tackled in the LF in [23, 29],
+for the former, and in [2,12], for the latter.
+The LF is based on processing frequency-domain measurements D = {(ωℓ, H(ωℓ)) , ℓ = 1, . . . , N}
+(with ωℓ ∈ R for 1 ≤ ℓ ≤ N) corresponding to evaluations of the transfer function of the underlying
+(unknown/hidden) dynamical system.
+The interpolation problem is formulated as shown below (for convenience of exposition, we show
+here only the SISO formulation). We are given data nodes and data points in the set D, partitioned
+into two disjoint subsets DL and DR, with DL ∪ DR = D and k + q = N, as
+right data : DL = {(λj, H(λj)) , j = 1, . . . , k}, and,
+left data : DR = {(µi, H(µi)) , i = 1, . . . , q},
+(5)
+and we seek to find a rational function ˆH(s), such that the following interpolation conditions hold:
+ˆH(µi) = H(µi) := vi,
+ˆH(λj) = H(λj) := wj.
+(6)
+The Loewner matrix L ∈ Cq×k and the shifted Loewner matrix Ls ∈ Cq×k play an important role in
+the LF, and are given by
+L(i,j) = vi − wj
+µi − λj
+,
+Ls(i,j) = µivi − λjwj
+µi − λj
+,
+(7)
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+6
+while the data vectors V ∈ Cq, WT ∈ Ck are given by
+V(i) = vi,
+W(j) = wj, for i = 1, . . . , q, j = 1, . . . , k.
+(8)
+Moreover, the following Sylvester matrix equations ([1, Ch. 6]) are satisfied by the Loewner and shifted
+Loewner matrices (here, 1q =
+�
+1
+· · ·
+1
+�T ∈ Cq)
+�
+ML − LΛ = V1T
+k − 1qW,
+MLs − LsΛ = MV1T
+k − 1qWΛ,
+(9)
+where M = diag(µ1, . . . , µq) and Λ = diag(λ1, . . . , λk) are diagonal matrices. The following relation
+holds true
+Ls = LΛ + V1T
+k = ML + 1qW.
+(10)
+The unprocessed Loewner surrogate model, provided that k = q, is composed of the matrices
+ˆE = −L,
+ˆA = −Ls,
+ˆB = V,
+ˆC = W,
+(11)
+and if the pencil (L, Ls) is regular, then the function ˆH(s) satisfying the interpolation conditions in
+(6) can be explicitly computed in terms of the matrices in (11), as ˆH(s) = ˆC(sˆE − ˆA)−1 ˆB.
+In practical applications (when processing a fairly large number of measurements), the pencil (Ls, L)
+is often singular. Hence, a post-processing step is required for the Loewner model in (11). In such
+cases, one needs to perform a singular value decomposition (SVD) of augmented Loewner matrices, to
+extract the dominant features and remove inherent redundancies in the data. By doing so, projection
+matrices X, Y ∈ Ck×r are obtained, as left, and respectively, right truncated singular vector matrices:
+[L Ls] = YS(1)
+r
+˜X
+H � L
+Ls
+�
+= ˜YS(2)
+r XH,
+(12)
+where S(1)
+r , S(2)
+r
+∈ Rr×r,
+Y ∈ Ck×r, X ∈ Cq×r, ˜Y ∈ C2q×r, ˜X ∈ Cr×2k. The truncation index r
+can be chosen as the numerical rank (based on a tolerance value τ > 0) or as the exact rank of the
+Loewner pencil (in exact arithmetic), depending on the application and data size. More details can be
+found in [7].
+The system matrices corresponding to a projected Loewner model of dimension r can be computed
+as follows:
+˜E = −XHLY,
+˜A = −XHLsY,
+˜B = XHV,
+˜C = WY.
+We note that MIMO extensions of the LF were already proposed in the original contribution [41].
+There, a tangential interpolation framework is considered. Instead of imposing interpolation of full
+p × m blocks, the authors prefer to interpolate the original transfer matrix function samples along
+certain vectors (or tangential directions). We also note that a first attempt of re-interpreting the LF
+in [41] as a one-sided method was made in [25]. In the latter, the main difference to the classical work
+in [4] was that a compression of the left (un-interpolated) data set was enforced. However, in [25], it
+was still unclear how to split the data, i.e., what the right data set should be (where interpolation is
+enforced). Finally, it is to be noted that the choice of interpolation points is crucial in the LF. An
+exhaustive study of different choices was proposed in [37], while a greedy strategy was proposed in
+[17], for scenarios in which limited experimental data are available.
+2.3. The AAA algorithm in [42]
+The AAA algorithm introduced in [42] is an adaptive and iterative extension of the interpolation
+method based on Loewner matrices, originally proposed in [4]. The main steps are as follows
+1. Express the fitted rational approximants in a barycentric representation, which represents a
+numerically stable way of expressing rational functions [15].
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+7
+Algorithm 1 The AAA algorithm.
+Require: A (discrete) set of sample points Γ ⊂ C with N points, function f (or the evaluations of f
+on the set Γ, i.e., the sample values), and an error tolerance ϵ > 0.
+Ensure: A rational approximant rn(s) of order (n, n) displayed in a barycentric form.
+1: Initialize j = 0, Γ(0) ← Γ, and r−1 ← N −1 �N
+i=1 f(γi).
+2: while |f(s) − rj−1(s)| > ϵ do
+3:
+Select a point zj ∈ Γ(j) for which |f(s) − rj−1(s)| attains a maximal value, where for j ≥ 1, it
+follows:
+rj−1(s) :=
+�j−1
+�
+k=0
+ω(j−1)
+k
+s − zk
+�−1 �j−1
+�
+k=0
+ω(j−1)
+k
+fk
+s − zk
+�
+.
+(13)
+4:
+if |f(zj) − rj−1(zj)| ≤ ε then
+5:
+Return rj−1.
+6:
+else
+7:
+fj ← f(zj) and Γ(j+1) ← Γ(j) \ {zj}.
+8:
+end if
+9:
+Find the weights ω(j) = [ω(j)
+0 , . . . , ω(j)
+j ] by solving a least squares problem over z ∈ Γ(j+1)
+j
+�
+k=0
+ω(j)
+k
+s − zk
+f(s) ≈
+j
+�
+k=0
+ω(j)
+k fk
+s − zk
+⇔
+�
+j
+�
+k=0
+f(s) − fk
+s − zk
+�
+ω(j)
+k
+≈ 0 ⇔ L(j)ω(j) = 0.
+(14)
+The solution of (14) is given by the (j + 1)th right singular vector of the Loewner matrix
+L(j) ∈ C(N−j−1)×(j+1).
+10:
+j ← j + 1.
+11: end while
+2. Select the next interpolation (support) points via a greedy scheme; basically, interpolation is
+enforced at the point where the (absolute or relative) error at the previous step is maximal.
+3. Compute the other variables (the so-called barycentric weights) in order to enforce least squares
+approximation on the non-interpolated data.
+In recent years, the AAA algorithm has proven to be an accurate, fast, and reliable rational ap-
+proximation tool with a fairly large range of applications. Here, we will mention only a few: nonlinear
+eigenvalue problems [39], MOR of parameterized linear dynamical systems [16], MOR of linear sys-
+tems with quadratic outputs [26], rational approximation of periodic functions [10], representation of
+conformal maps [22], rational approximation of matrix-valued functions [27], or signal processing with
+trigonometric rational functions [60]. The procedure is sketched in Algorithm 1.
+It is to be mentioned that a modified version of AAA that enforces real-valued and strictly-
+proper rational appoximants was recently proposed in [30]. There, the format of the function in (13)
+was modified by inserting a 1 into the denominator, as follows
+˜rj(s) :=
+�
+1 +
+j−1
+�
+k=0
+ω(j−1)
+k
+s − zk
+�−1 �j−1
+�
+k=0
+ω(j−1)
+k
+fk
+s − zk
+�
+.
+(15)
+Consequently, the equation in (14) becomes L(j)ω(j−1) = −f(j−1), where the vector f(j−1) ∈ Cj is
+given by f(j−1) =
+�f0
+f2
+· · ·
+fj−1
+�T. It is to be noted that ˜rj(s) in (15) is theoretically a rational
+approximant of order (j − 1, j), if we do not take into account pole/zero cancellations or any other
+zero cancellations of coefficients in the numerator or denominator.
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+8
+3. The proposed methodologies
+3.1. Skeleton of the main methods
+Similar to the methods reviewed in Section 2 we want to find an LTI system with a transfer function of
+the structure (1) that interpolates data provided as measurements H (si) , i = 1, . . . , k of the transfer
+function of the original system. We can directly put together an LTI parametrized model of dimension
+r = km, having km2 degrees of freedom with transfer function
+ˆH(s) = ˆC(sIr − ˆA)−1 ˆB,
+(16)
+with the underlying data concatenated to
+ˆC =
+�H(λ1)
+· · ·
+H(λk)�
+∈ Cp×r,
+(17)
+a matrix of weights ˆ
+Wi
+ˆB =
+�
+ˆ
+W
+H
+1
+· · ·
+ˆ
+W
+H
+k
+�H
+∈ Cr×m,
+(18)
+and ˆA ∈ Cr×r formed from a diagonal matrix populated with the interpolation points λi disturbed by
+ˆB, such that
+ˆA = Λ − ˆBR = diag (λ1, . . . , λk) ⊗ Im − ˆB
+�
+1T
+r ⊗ Im
+�
+.
+(19)
+Making use of the Woodbury matrix identity and denoting Λs = sIkm − Λ, the transfer function (16)
+can be rewritten as
+ˆH(s) = ˆCΛ−1
+s
+ˆB
+�
+Im + RΛ−1
+s
+ˆB
+�−1
+.
+(20)
+A complete derivation of (20) is given in Appendix A.1.
+In the single-input single-output case (m = p = 1, hence r = k), the barycentric weights reduce to
+scalars and the matrices for a ROM of structure (16) are given by
+ˆA = Λ − ˆBR ∈ Ck×k,
+ˆB =
+� ˆw1
+· · ·
+ˆwk
+�T ∈ Ck×1,
+ˆC =
+�H (λ1)
+· · ·
+H (λk)�
+∈ C1×k.
+(21)
+By inserting the formulae in (21) into (20), and using the notation hi := H (λi), leads to
+ˆCΛ−1
+s
+ˆB =
+k
+�
+i=1
+ˆwihi
+s − λi
+,
+�
+Im + RΛ−1
+s
+ˆB
+�−1
+=
+1
+1 + �k
+i=1
+ˆwi
+s − λi
+.
+(22)
+Hence, the transfer function of the model in (21) is given in barycentric representation by
+ˆH(s) =
+�k
+i=1
+ˆwihi
+s − λi
+1 + �k
+i=1
+ˆwi
+s − λi
+.
+(23)
+This can be performed analogously for a multi-input multi-output case (m = p, r = km). The first
+part of (20) becomes
+ˆCΛ−1
+s
+ˆB =
+�H(λ1)Im(s − λ1)−1
+· · ·
+H(λk)Im(s − λk)−1�
+�
+��
+ˆ
+W1
+...
+ˆ
+Wk
+�
+�� =
+k
+�
+i=1
+H(λi) ˆ
+Wi
+s − λi
+,
+(24)
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+9
+the second part
+�
+Im + RΛ−1
+s
+ˆB
+�−1
+=
+�
+�
+�
+�Im +
+�Im
+· · ·
+Im
+�
+�
+��
+Im(s − λ1)−1
+...
+Im(s − λk)−1
+�
+��
+−1 �
+��
+ˆ
+W1
+...
+ˆ
+Wk
+�
+��
+�
+�
+�
+�
+−1
+=
+�
+Im +
+k
+�
+i=1
+ˆ
+Wi
+s − λi
+�−1
+.
+(25)
+Consequently, the transfer function in (20) has also a barycentric form given by
+ˆH(s) =
+� k
+�
+i=1
+H(λi) ˆ
+Wi
+s − λi
+� �
+Im +
+k
+�
+i=1
+ˆ
+Wi
+s − λi
+�−1
+.
+(26)
+The transfer function is defined by the choice of the interpolation points and of the weights. The
+interpolation points can be chosen as dominant parts of the available data or based on their location in
+the frequency spectrum. The weights can be computed such that the data which are not interpolated,
+are approximated in an optimal way. Alternatively, the weights can be chosen to enforce poles at
+specific locations. In the following, we show different strategies for both choices.
+3.2. Automatic choice of interpolation points
+The approximation quality of a surrogate model of the form (16) is greatly influenced by the choice of
+the interpolation points λ. This choice is not always obvious, so automatic strategies are frequently
+employed. The Loewner framework uses the SVD to identify dominant subsets of the available data
+to enforce interpolation on. Alternatively, the AAA algorithm uses a greedy scheme to minimize the
+error between surrogate and original data. Another approach, originally introduced by [37], makes use
+of the CUR decomposition to extract interpolation points from a relevant subset of the available data.
+The CUR decomposition approximates a matrix A by a product of three low-rank matrices ˇA =
+ˇC ˇU ˇR, where ˇC and ˇR represent subsets of the columns respectively rows of A [40,56]. In our case the
+three matrices are only a byproduct, we are more interested in the interpolation points λ and µ that
+are associated to the columns and rows extracted as ˇC and ˇR. In combination with the skeleton for a
+realization described in Section 3.1, Algorithm 2 computes a surrogate model approximating a set of
+given transfer function data. We use the algorithm from [56] to compute the CUR decomposition and
+thus identify dominant parts of the original data set and their corresponding left and right interpolation
+points. Contrary to [37], we decompose the original Loewner matrix L rather than the augmented
+Loewner matrices
+�L
+Ls
+�
+and
+�
+LH
+LH
+s
+�H. Using all interpolation points obtained from the CUR
+decomposition would introduce redundant data into the surrogate. Therefore we choose only a subset
+of the interpolation points: either only the left points, only the right points, or every other entry from
+a concatenated and sorted vector of left and right points. Together with the data associated to the
+chosen interpolation points they are used to populate a rectangular Loewner matrix. We now need
+to compute weights for barycentric interpolation as described in the following section. After having
+obtained the weights, a surrogate model (16) can be computed from (17)–(19).
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+10
+Algorithm 2 LS-Loewner with CUR.
+Require: Transfer function samples {H (si)}N
+i=1, corresponding sampling points Ξ = {si}N
+i=1.
+Ensure: Surrogate model ˆH(s) = ˆC(sIr − ˆA)−1 ˆB.
+1: Partition data and compute Loewner matrix L as in (7).
+2: Compute CUR decomposition, such that L = ˇC ˇU ˇR with ˇC ∈ CN×k, ˇR ∈ Ck×N.
+3: Obtain interpolation points {λi}k
+i=1 , {µi}k
+i=1 corresponding to the columns and rows in ˇC, ˇR.
+4: Postprocess interpolation points to obtain ν = {ν}k
+i=1 and χ = Ξ \ ν.
+5: Populate a rectangular Loewner matrix L(i,j) = H(χi)−H(νj)
+χi−νj
+.
+6: Compute the weights Ω = −L† �
+H (ν1)H
+· · ·
+H (νk)H�H
+, where L† is the pseudo-inverse of L and
+Ω =
+�
+ˆ
+W
+H
+1
+· · ·
+ˆ
+W
+H
+k
+�H
+.
+7: Compute ˆA, ˆB, ˆC with (17)–(19).
+3.3. Computing the barycentric weights
+3.3.1. Least-squares approach
+The matrix-valued weights ˆ
+Wi can be computed similarly to AAA [27] by solving the minimization
+problem
+min
+ˆ
+Wi
+h
+�
+j=1
+�
+�
+� k
+�
+i=1
+H(λi) ˆ
+Wi
+sj − λi
+� �
+Im +
+k
+�
+i=1
+ˆ
+Wi
+sj − λi
+�−1
+− H(sj)
+�
+�
+2
+.
+(27)
+This solution can, for example, be obtained from an optimization in least-squares sense. The weights
+for the SISO case are computed analogously. Here, the matrix-values weights and transfer function
+values reduce to scalars.
+3.3.2. Pole placement
+The next step would be to take advantage of the degrees of freedom in the vector ˆB from (21), so
+that the ROM thus constructed has particular (stable) poles [21, 35, 46]. These will be denoted with
+ζ1, ζ2, . . . , ζk. The following derivations assume a SISO model. To enforce that this happens, we need
+to make sure that the matrix ζjIk − ˆA loses rank for all 1 ≤ j ≤ k. In what follows, we show how to
+enforce this property in an elegant, straightforward way. Remember that the transfer function of the
+parameterized AF model is given by:
+ˆH(s) =
+�k
+i=1
+ˆwihi
+s − λi
+1 + �k
+i=1
+ˆwi
+s − λi
+= N(s)
+D(s).
+(28)
+Now, let’s say we would like this transfer function to have k poles at the selected values ζj’s. Clearly,
+the condition is D(ζj) = 0 and hence we need to enforce:
+1 +
+k
+�
+i=1
+ˆwi
+ζj − λi
+= 0, ∀1 ≤ j ≤ k ⇔ Cζ,λ ˆB = −1k ⇔ ˆB = −C−1
+ζ,λ1k,
+(29)
+where Cζ,λ is a Cauchy matrix defined by: (Cζ,λ)i,j =
+1
+ζi−λj . Details on how to obtain the above
+expression by following the procedure in [3] are given in Appendix A.2. We note that placing poles
+is a difficult numerical problem which requires the inversion of a Cauchy matrix, which is highly
+ill-conditioned, by nature.
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+11
+Algorithm 3 Loewner framework with pole placement (LFPP).
+Require: Transfer function samples {H (si)}N
+i=1, corresponding sampling points Ξ = {si}N
+i=1, loca-
+tions for poles ζ = {ζi}k
+i=1, interpolation points λ = {λi}k
+i=1.
+Ensure: Surrogate model ˆH(s) = ˆC(sIr − ˆA)−1 ˆB.
+1: Compute ΣD from {H (si)}N
+i=1 and {si}N
+i=1 using the Loewner framework (cf. Section 2.2).
+2: ˆC ←
+�HD (λ1)
+· · ·
+HD (λk)�
+.
+3: ˆB ← −C−1
+ζ,λ1r.
+4: ˆA ← diag (λ1, . . . , λk) − ˆB1T
+k.
+Instead of doing this, we could solve Cζ,λ ˆB = −1r, without inverting the Cauchy matrix explicitly,
+i.e., by solving a linear systems of equations. Algorithm 3 summarizes this procedure in a data-driven
+context. The required underlying model is obtained from a set of transfer function evaluations by
+applying the Loewner framework. The method is illustrated for SISO systems, but can readily be
+extended to the MIMO case.
+3.4. Automatic choice of poles and interpolation points
+A reasonable choice of poles and interpolation points for Algorithm 3 is not always readily available,
+but the approximation of the surrogate is heavily influenced by this choice. In the following, we show
+an extension to Algorithm 3 which computes a surrogate model (16) without requiring sets of poles
+and interpolation points as input parameters. Algorithm 4 sketches the skeleton of such automatic
+algorithm. Similar to Algorithm 3 it employs the Loewner framework to obtain a realization of a
+surrogate interpolating the provided data.
+Subsequently, a generalized eigendecomposition of the
+Loewner realization of the original data is computed to find suitable locations for poles. From this
+is is possible to compute the dominance of all eigenvalues; for details, see, e.g. [51]. The algorithm
+now chooses the k most dominant eigenvalues as poles to enforce in the surrogate. It should be noted
+that only eigenvalues with negative real parts should be considered, if the stability of the surrogate is
+important. The required interpolation points can now be chosen similar to Algorithm 2 by computing
+a CUR decomposition and using the interpolation points associated to the rows or columns of the
+decomposition as interpolation points for the new surrogate.
+The approximation of dominant poles of the underlying model from data is less robust if the transfer
+function samples are disturbed by noise. This leads to a reduced approximation quality. For a better
+performance if applied to noisy data, Algorithm 4 can be modified as follows: To obtain poles which
+should be enforced, first choose manually the most prominent features in the transfer function, e.g.
+peaks, which should be approximated by the surrogate model.
+Now choose the eigenvalues which
+imaginary parts are closest to the frequencies, where the chosen features of the transfer function are
+located. The CUR decomposition also fails at extracting the most dominant rows and columns of
+the Loewner matrix if noisy data is assessed. Therefore another heuristic is employed to choose the
+interpolation points: Use the value si which corresponds to the lowest amplitude of the transfer function
+between the locations of two enforced poles. This leads to reasonable approximations, especially for
+lightly damped systems. Other approaches include choosing simply the middle between the location
+of two poles or specifying an offset between pole and interpolation point location.
+4. Numerical results
+In the following, we demonstrate the methods discussed in Section 3 by applying them on three
+benchmark examples available from the MOR-Wiki1:
+1http://modelreduction.org
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+12
+Algorithm 4 Loewner framework with automatic pole placement (LFaPP).
+Require: Transfer function samples {H (si)}N
+i=1, corresponding sampling points Ξ = {si}N
+i=1.
+Ensure: Surrogate model ˆH(s) = ˆC(sIr − ˆA)−1 ˆB.
+1: Compute ΣD from {H (si)}N
+i=1 and {si}N
+i=1 using the Loewner framework (cf. Section 2.2).
+2: Compute the generalized eigenvalue decompositions AX = EXα and YHA = αYHE for the
+matrices of right and left eigenvectors X, Y and the matrix of eigenvalues α = diag (α1, . . . , αn).
+3: Compute eigenvalue dominance di = |CY(:,i)αiX(:,i)HB|
+|ℜ(αi)|
+, i = 1, . . . , n and sort α accordingly
+4: Set ζ to the k most dominant eigenvalues.
+5: Compute CUR decomposition of L.
+6: Set λ to the k right or left interpolation points corresponding to the CUR decomposition.
+7: Compute surrogate as in Algorithm 3.
+ISS This system models the structural response of the Russian Service Module of the International
+Space Station (ISS) [52]. The model has n = 270 states, m = 3 inputs, and p = 3 outputs. The
+dataset used for the computations contains transfer function measurements at 400 logarithmically
+distributed points in the range
+�
+10−1, 102�
+· ı. The model is also part of the SLICOT benchmark
+collection [44].
+Flexible aircraft This system models lift and drag along the flexible wing of an aircraft. The system
+matrices are not available, we only have access to a dataset of 420 transfer functions samples
+at linearly distributed frequencies between 0.1 and 42.0 Hz. The original dataset has one input
+(the gust disturbance) and 92 outputs. For the following experiments, we choose the 91st output
+which corresponds to the first flexible mode [50]. The dataset is available from [57].
+Sound transmission This system models the sound transmission through a system of two brass plates
+with an air enclosure between them. The transfer function measures the sound pressure in an
+adjacent acoustic cavity. The geometry is based on [32]; the data—transfer function evaluations
+at 1000 linearly-distributed frequency values between 1 and 1000 Hz—is available from [9].
+We note that no tangential interpolation (as described in [41]) is applied for the MIMO model.
+Instead, the Loewner matrices are constructed in a block-wise manner. The case of tangential inter-
+polation, within the proposed approaches in this note, will be investigated in future works.
+We enforce realness of all surrogate models (all matrices contain only real entries) by applying the
+transformation described in [7]. For this, all data must be available in complex conjugate pairs. The
+required transformation matrix is given by
+J = Iℓ ⊗
+� 1
+√
+2
+� Im
+Im
+−ıIm
+ıIm
+��
+,
+(30)
+with ℓ = k
+2 and the real-valued quantities are obtained from ˆA
+(ℜ) = J ˆAJH, ˆB
+(ℜ) = J ˆB, and ˆC
+(ℜ) =
+ˆCJH.
+For some of the experiments we add artificial noise to the measurements, in order to obtain perturbed
+data. The modified measurements are given by
+ˇH (si) = H (si) (1 + Zi) , i = 1, . . . , n,
+(31)
+where Zi ∈ C is the ith sample drawn from a set of random numbers Z ∼ CN
+�
+µ, σ2�
+following a
+complex normal distribution with mean µ and standard deviation σ2. Here, the real and imaginary
+parts of Z are independent normally distributed variables [19].
+We assess the approximation error of the surrogate models with an approximated L∞ norm, because
+many surrogates have unstable poles and hence, the H∞ can not be computed. For a given reduced
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+13
+order r, the L∞ error in the considered frequency range ω ∈ [ωmin, ωmax] is approximated by
+ε(r) =
+max
+ω∈[ωmin,ωmax]
+���H(ωı) − ˆHr(ωı)
+���
+2
+max
+ω∈[ωmin,ωmax] ∥H(ωı)∥2
+≈
+���H − ˆHr
+���
+L∞
+∥H∥L∞
+.
+(32)
+Note that strategies to post-process surrogates to obtain stable models have been studied in [29].
+The numerical experiments have been conducted on a laptop equipped with an AMD Ryzen™
+7 PRO 5850U and 12 GB RAM running Linux Mint 21 as operating system. All algorithms have been
+implemented and run with MATLAB R2021b Update 2 (9.11.0.1837725).
+Code and data availability
+The data that support the findings of this study are openly available in Zenodo at
+doi:10.5281/zenodo.7490158
+under the BSD-2-Clause license, authored by Quirin Aumann and Ion Victor Gosea.
+4.1. Case of exact measurement data
+In the following, we compare the performance of the new approach LS-Loewner to the following estab-
+lished strategies:
+• Loewner-SVD: Truncate Loewner matrices populated with the complete dataset to order r using
+an SVD [7].
+• Loewner-CUR: Construct a purely interpolatory model of order r using all data points chosen by
+the CUR decomposition, similar to [37].
+• Modified AAA: Apply the strictly-proper variant of AAA [29] to the complete dataset to compute
+a reduced-order model of size r.
+We first consider the original MIMO ISS example and a SISO variant where we select the first input
+and output, respectively, from the MIMO system. To evaluate the overall performance of the different
+methods related to the size of a surrogate model, we compute the approximated L∞ errors for models
+with orders 6 ≤ r ≤ 60. The approximation error versus the dimension of the respective surrogate
+model is depicted in Figure 1 for all four methods.
+Since tangential interpolation was not employed here, the order of the MIMO surrogates rises by m
+for each additional interpolation point, i.e., r = km. This explains the lower accuracy of the MIMO
+surrogate. For the maximum reduced order r = 60, k = 20 interpolation points are considered. The
+errors of the SISO surrogates for r = 20, i.e., k = 20, is in a similar range as in the MIMO case. The
+SISO surrogates reach similar levels of approximation for all employed methods. In the MIMO case,
+Loewner-SVD performs best. This can be explained by the following observation: the other methods
+always consider the complete transfer function measurement H(λi) ∈ Cp×m per interpolation point,
+while Loewner-SVD extracts only the r most dominant singular vectors for projection, regardless of
+to which interpolation point they belong to. In turn, the other methods also consider probably less
+important parts of the data as long as one input/output combination of the respective sample is
+relevant for approximation. It can also be noted that LS-Loewner and Loewner-CUR perform very
+similar. This was expected, as both methods rely on the same interpolation points.
+All four methods are now employed to compute a surrogate model of size r = 108 to approximate
+the transfer function of the flexible aircraft model. The size of the surrogate model is determined by
+truncating all singular values τ < 1·10−6 of an underlying Loewner matrix.
+The transfer functions of all resulting models and their respective relative errors are given in Figure 2.
+Again, all methods succeed in computing a sufficiently accurate surrogate. However, the approximation
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+14
+10
+20
+30
+40
+50
+60
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+Reduced order r
+L∞ error
+SISO
+10
+20
+30
+40
+50
+60
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+Reduced order r
+L∞ error
+MIMO
+LS-Loewner
+Loewner-SVD
+Loewner-CUR
+Modified AAA
+Figure 1: The approximated L∞ errors of reduced-order models of order r computed from the ISS
+data. Left: SISO with first input and output, respectively. Right: MIMO with three inputs
+and three outputs m = p = 3 (the number of interpolation points is k = r
+m).
+quality of Loewner-CUR is noticeably worse than that of the other three methods. Given that both
+Loewner-CUR and LS-Loewner use the same interpolation points, the weights computed from the least
+squares problem show a better performance compared to the partitioning approach used in Loewner-
+CUR.
+4.2. Perturbed measurement data
+Analyzing measurement data perturbed by noise is a challenging task for interpolatory methods, such
+as the Loewner framework and the AAA algorithm (as pointed out in, e.g., [27]). In this experiment
+we investigate the effect of noise to the performance of the four methods described above and show,
+how enforcing poles and/or interpolation points can increase the approximation quality. In the first
+experiment we consider transfer function data from the ISS model perturbed by noise with mean µ = 0
+and standard deviation σ2 = 0.15. We employ LFaPP and enforce poles at ı[.77, 2, 4, 5.6, 9.33, 37.9]
+near peaks of the transfer function. The resulting real-valued surrogate model has order r = 12. The
+transfer functions of the surrogate model with enforced poles and reduced models computed from the
+same noisy data with LS-Loewner, Loewner-SVD, Loewner-CUR, and Modified AAA are given in Figure 3.
+Enforcing the poles near peaks in the transfer function of the underlying data allows the surrogate
+to capture the behavior of the original data in a wider frequency range than applying LS-Loewner,
+Loewner-SVD, and Loewner-CUR. The choice of the locations, in which vicinity the poles should be
+chosen is, however, not automatized. Figure 3 also shows the relative errors of all surrogate models
+referenced to the original data without noise. While the enforced poles all have a negative real part,
+the models computed from the variants of the LF and AAA exhibit unstable eigenvalues. Thus, pole
+placement can be seen also as a means to enforce stability of the surrogate models. Alternatively,
+a post-processing step can be added to enforce stable models (for both LF and AAA methods), as
+performed in [29].
+We now evaluate the performance of the algorithms by applying them to heavily distorted trans-
+fer function measurements of the sound transmission problem. Noise with a standard deviation of
+σ2 = 0.25 is considered and three algorithms are employed to compute surrogates: Loewner-SVD,
+LFPP (Algorithm 3), and LFaPP (Algorithm 4). We also test the modifications to LFaPP described in
+Section 3.4. These results are denoted by “LFaPP mod.”. For LFPP we enforce poles at the eigenval-
+ues of the underlying Loewner model which imaginary parts are near 2πı [72, 189, 392, 401, 706, 856].
+These locations correspond to characteristic peaks in the transfer function. Further, we choose the in-
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+15
+5
+10
+15
+20
+25
+30
+35
+40
+10−4
+10−2
+100
+Magnitude
+Original data
+LS-Loewner
+Loewner-SVD
+Loewner-CUR
+Modified AAA
+5
+10
+15
+20
+25
+30
+35
+40
+10−12
+10−9
+10−6
+10−3
+100
+Frequency [Hz]
+Relative error
+LS-Loewner
+Loewner-SVD
+Loewner-CUR
+Modified AAA
+Figure 2: Transfer function (top) and relative pointwise errors (bottom) for reduced-order models of
+size r = 108 for the aircraft model. The error is plotted only at frequencies which do not
+coincide to interpolation points of the respective method.
+terpolation points at 2πı [138, 339, 369, 569, 712, 954], which lie at the dips between the enforced poles.
+Loewner-SVD and LFaPP do not require input parameters in addition to the measured data. Figure 4
+shows the transfer function of the resulting surrogate models in comparison to the original and noisy
+underlying data. It can be observed, that the automatic approaches Loewner-SVD and LFaPP (mod.)
+cannot approximate the transfer function well after the first two peaks, i.e., for frequencies higher than
+200 Hz, while LFPP approximates the original data over the complete frequency range with decent
+accuracy. The importance of reasonable interpolation points can be seen in the difference of LFPP and
+LFaPP mod., which have the same poles. It should be noted that the surrogate model computed by
+Loewner-SVD has two unstable poles while the other three surrogate models are stable. It is, however,
+not always clear a priori how to choose the poles and interpolation points for LFPP in order to achieve
+the best approximation quality possible. In this example, the noise level is too high for one of the
+automatic approaches to yield reasonable dominant interpolation points or poles.
+5. Conclusion and outlook
+In this contribution, we have proposed an extensive study of interpolation-based data-driven ap-
+proaches for approximating the response of linear dynamical systems.
+All methods require input
+and output data, i.e., transfer function measurements, while direct access to the system operators or
+the states is not required. We showed different approaches how to achieve compact surrogate models
+approximating the input/output behavior of the original system and how to ensure various properties
+of the surrogate models, such as stability. Strategies how to work with noisy measurement data have
+also been addressed.
+A natural extension of the framework described here is to apply the ideas of tangential interpolation
+as a means of modeling a MIMO system from data. Here, the tangential directions need to be incorpo-
+rated in the parameterized one-sided realization. Further topics include enforcing different structures
+of the original model in the surrogate model, e.g., second-order or delay structures. It would also be
+interesting to study the possibility of placing certain stable poles while achieving interpolation in a
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+16
+10−1
+100
+101
+102
+10−5
+10−3
+10−1
+Magnitude
+Noisy data
+Original data
+LFaPP
+LS-Loewner
+Loewner-SVD
+Loewner-CUR
+Modified AAA
+10−1
+100
+101
+102
+10−4
+10−3
+10−2
+10−1
+100
+101
+102
+Frequency
+Relative error
+Noise
+LFaPP
+LS-Loewner
+Loewner-SVD
+Loewner-CUR
+Modified AAA
+Figure 3: Transfer function of a surrogate with enforced poles compared to the noisy and original
+transfer function values. The transfer function of a model r = 12 computed from Loewner-
+SVD is given for reference.
+least-squares sense. Application cases for the proposed methodology could include damping optimiza-
+tion. Here, a family of parameterized interpolants could be used to find optimal positions for viscous
+dampers in a structural system.
+A. Appendix
+A.1. The Woodbury matrix identity
+We can expand the right part of (19), such that:
+ˆA = Λ − ˆBR ⇒ sIkm − ˆA = sIkm − Λ
+�
+��
+�
+ˆ
+M
++
+�
+��
+ˆ
+W1
+...
+ˆ
+Wk
+�
+��
+� �� �
+ˆU
+�Im
+�
+����
+ˆT
+�Im
+· · ·
+Im
+�
+�
+��
+�
+ˆV
+.
+(33)
+The Woodbury matrix identity is as follows:
+�
+ˆM + ˆU ˆT ˆV
+�−1
+= ˆM
+−1 − ˆM
+−1 ˆU
+�
+ˆT
+−1 + ˆV ˆM
+−1 ˆU
+�−1 ˆV ˆM
+−1,
+where ˆM, ˆU, ˆT and ˆV are conformable matrices: ˆM is n × n, ˆT is k × k, ˆU is n × k, and ˆV is k × n.
+This can be derived using blockwise matrix inversion. By denoting with Λs = sIkm − Λ, then the first
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+17
+100
+200
+300
+400
+500
+600
+700
+800
+900
+1,000
+10−6
+10−2
+102
+Magnitude
+Noisy data
+Original data
+Loewner-SVD
+LFPP
+LFaPP
+LFaPP mod.
+100
+200
+300
+400
+500
+600
+700
+800
+900
+1,000
+10−3
+10−2
+10−1
+100
+101
+102
+103
+Frequency [Hz]
+Relative error
+Noise
+Loewner-SVD
+LFPP
+LFaPP
+LFaPP mod.
+Figure 4: Transfer function (top) and relative pointwise errors (bottom) as well as the added noise for
+reduced-order models of size r = 12 for the aircraft model.
+transfer function of the fitted model is written:
+ˆH(s) = ˆC
+�
+sI3m − ˆA
+�−1 ˆB = ˆC
+�
+Λs + ˆU ˆV
+�−1 ˆB
+= ˆCΛ−1
+s
+ˆB − ˆCΛ−1
+s
+ˆU
+�
+Im + ˆVΛ−1
+s
+ˆU
+�−1 ˆVΛ−1
+s
+ˆB
+= ˆCΛ−1
+s
+ˆB − ˆCΛ−1
+s
+ˆB
+�
+Im + RΛ−1
+s
+ˆB
+�−1
+RΛ−1
+s
+ˆB
+= ˆCΛ−1
+s
+ˆB
+�
+Im −
+�
+Im + ˆX
+�−1 ˆX
+�
+= ˆCΛ−1
+s
+ˆB
+�
+Im + ˆX
+�−1
+,
+(34)
+where ˆX = RΛ−1
+s
+ˆB. Hence, we arrive at (20) and the transfer function ˆH(s) can be written as follows:
+ˆH(s) = ˆCΛ−1
+s
+ˆB
+�
+Im + RΛ−1
+s
+ˆB
+�−1
+.
+(20)
+A.2. Pole placement as in [3]
+In order to enforce both prescribed poles and certain interpolation conditions in the ROM, we follow
+the derivations from [3].
+It is to be noted that this approach is intrusive, i.e., requires access to
+the system’s matrices. Hence, a descriptor model characterized in (generalized) state-space by the
+following equations
+ΣDes :
+�
+E ˙x(t) = Ax(t) + Bu(t),
+y(t) = Cx(t),
+(35)
+with corresponding transfer function HDes(s) = C(sE − A)−1B is considered to be given. For the
+(right) interpolation points λi, i = 1, . . . , k (where interpolation is imposed), and the desired poles to
+be placed, denoted with ζj’s, the author in [3] starts by finding a row vector Cζ ∈ C1×n so that:
+Cζ
+�
+(λ1E − A)−1B · · · (λkE − A)−1B
+�
+= 01×k.
+(36)
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+18
+Then, the next step is to choose projection matrices W, V ∈ Cn×k as
+WH =
+�
+��
+Cζ(ζ1E − A)−1
+...
+˜C(ζkE − A)−1
+�
+�� ,
+V =
+�(λ1E − A)−1B · · · (λkE − A)−1B�
+.
+(37)
+As explained in [3], the choice of WH above is explained by imposing the required poles for the
+reduced model, while V is chosen to match the interpolation conditions at the λi’s. Moreover, using
+these notations, it follows that ˜CV = 0. Next, put together the following matrices ˜E = WHEV,
+˜A =
+WHAV. Then, it follows that (s˜E − ˜A) loses rank when s ∈ {ζ1, . . . , ζr}. To show this, we simply
+write
+eT
+j (ζj ˜E − ˜A) = eT
+j WH(ζjE − A)V = Cζ(ζjE − A)−1(ζjE − A)V = CζV = 0.
+(38)
+Let Hζ(s) = Cζ(sE − A)−1B be a rational function in s and we note that ˆE and ˆA are a special type
+of diagonally scaled Cauchy matrices, with the following exact definition:
+˜Ei,j = −Cζ(ζiE − A)−1B − Cζ(λjE − A)−1B
+ζi − λj
+= − Hζ(ζi)
+ζi − λj
+˜Ai,j = −ζiCζ(ζiE − A)−1B − λjCζ(λjE − A)−1B
+ζi − λj
+= −ζiHζ(ζi)
+ζi − λj
+(39)
+From the definition in (39), it follows that ˜E = −D ˜BCζ,λ, where D ˜B = diag( ˜B) is a diagonal matrix.
+Similarly, it follows that ˜A = −ZD ˜BCζ,λ, where Z = diag(ζ1, . . . , ζk).
+Next, we write the other projected quantities as
+˜B = WHB =
+�Hζ(ζ1)
+· · ·
+Hζ(ζk)�T ,
+˜C = CV =
+�H(λ1)
+· · ·
+H(λk)�
+(40)
+Hence, the reduced-order linear descriptor system Σpp : (˜E, ˜A, ˜B, ˜C) matches k interpolation conditions
+and has the required poles.
+Next, we show that this model can be written equivalently in the AF format. We first note that
+ˆC = ˜C. For next step, provided that the matrix ˜E is non-singular, we remove it by incorporating
+it into the other matrices, as: ˘A = ˜E
+−1 ˜A,
+˘B = ˜E
+−1 ˜B,
+˘E = Ik,
+˘C = ˜C. We note that the two
+realizations of the interpolatory ROM, i.e., ( ˆA, ˆB, ˆC) in (21) and ( ˘A, ˘B, ˘C) introduced above, are
+actually identical. The reason for this is that ˘C = ˆC and the two ROMs match the same k moments.
+Hence, it also follows that ˘B = ˆB. Now, since ˘B = ˜E
+−1 ˜B and ˜E = −D ˜BCζ,λ, we can write that
+ˆB = −(D ˜BCζ,λ)−1 ˜B = −C−1
+ζ,λD−1
+˜B ˜B = −C−1
+ζ,λ1k.
+(41)
+Hence, the above choice of vector ˆB in (21) imposes the required poles.
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf,len=1324
+page_content='Practical challenges in data-driven  interpolation: dealing with noise, enforcing  stability, and computing realizations  Quirin Aumann∗  Ion Victor Gosea†  ∗Max Planck Instiute for Dynamics of Complex Technical Systems, Sandtorstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' 1, 39106 Magdeburg,  Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Email: aumann@mpi-magdeburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='de, ORCID: 0000-0001-7942-5703  †Max Planck Instiute for Dynamics of Complex Technical Systems, Sandtorstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' 1, 39106 Magdeburg,  Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Email: gosea@mpi-magdeburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='de, ORCID: 0000-0003-3580-4116  Abstract: In this contribution, we propose a detailed study of interpolation-based data-  driven methods that are of relevance in the model reduction and also in the systems and  control communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The data are given by samples of the transfer function of the under-  lying (unknown) model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=', we analyze frequency-response data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' We also propose novel  approaches that combine some of the main attributes of the established methods, for ad-  dressing particular issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  This includes placing poles and hence, enforcing stability of  reduced-order models, robustness to noisy or perturbed data, and switching from different  rational function representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' We mention here the classical state-space format and  also various barycentric representations of the fitted rational interpolants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' We show that  the newly-developed approaches yield, in some cases, superior numerical results, when com-  paring to the established methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The numerical results include a thorough analysis of  various aspects related to approximation errors, choice of interpolation points, or placing  dominant poles, which are tested on some benchmark models and data-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Keywords: Data-driven methods, rational approximation, interpolatory methods, least  squares fit, Loewner framework, frequency response data, pole placement, noisy measure-  ments, Loewner and Cauchy matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Novelty statement: This note shows that by combining the features of established data-  driven rational approximation methods based on interpolation (and/or least squares fit),  one can devise methods that offer additional important advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' These include stabil-  ity enforcement by placing poles in an elegant and numerically stable manner, together  with robustness to noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Introduction  Approximation of large-scale dynamical systems is pivotal for serving the scopes of efficient simulation  and designing control laws in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The technique for reducing the complexity of a system is known  as model order reduction (MOR) [1, 5, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' There exist a number of methodologies for reducing  large-scale models, and each method is tailored to some specific applications (mostly, but not restricted  Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2023-01-13  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='04906v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='NA]  12 Jan 2023  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Gosea: Data-driven interpolation: challenges and solutions  2  to mechanical and electrical engineering) and to achieving certain goals (stability, passivity or structure  preservation), on top of the complexity reduction part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Data-driven MOR approaches are of particular  importance when access to high-fidelity models is not explicitly granted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' This means that a state-  space formulation with access to internal variables is not available, yet input/output data are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Such  methods circumvent the need to access an exact description of the original model and are applicable  whenever classical projection-based MOR is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Here, we mention system and control methodologies  that are based on interpolation or least-square fit of data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=', frequency response measurements),  such as vector fitting [31], the Loewner framework [41], or the AAA algorithm [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Methods that use  time-domain data are also of interest, including the ones that require input-output data together with  those which use snapshot data (access to the state evolution), such as the classical ones in [36,58,59],  followed by [8], [47] or [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  We focus on interpolation-based or so-called moment matching (MM) methods which have emerged,  were developed, and improved continuously in the last decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The backbone of such methods rep-  resent rational Krylov-type approaches together with the Sylvester matrix equation interpretation  [1, 11, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Apart from being computationally efficient and easy to implement, MM approaches have  another advantage: they do not (necessarily) require access to a full-state realization of the original  dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Hence, they can be viewed as data-driven methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Here data are given by the  moments of the system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=', samples of the underlying transfer function of the system (and of its  derivatives) evaluated in a particular frequency range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' for more details, we refer the readers to [5,8,41]  and to Chapter 3 in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The notion of a moment with respect to systems and control theory is related  to the unique solution of a Sylvester matrix equation [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  The purpose of this note is twofold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' first, we intend to review and to connect three important system  theoretical model reduction methods based on interpolation that were introduced in the last 15 years:  The Loewner framework (LF) by Mayo and Antoulas from 2007 in [41];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  The Astolfi framework (AF) from 2010 in [8];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  The Adaptive Antoulas Anderson (AAA) algorithm by Nakatsukasa, Set´e and Trefethen from  2018 in [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Together, these three approaches were cited multiple times in various research publications, being  arguably quite popular methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' However, until now, not too many connections between them were  provided, neither in the automatic control, nor in the model reduction, or numerical analysis commu-  nities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Together with the vector fitting algorithm (VF) in [31] (which is not based on interpolation,  and is hence a purely optimization approach based on least-squares fitting), these methods repre-  sent arguably the most prolific rational approximation schemes developed in the system and control  community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' However, VF is not the object of this study since it is not based on interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  The other scope of this note is to propose a new method that is based on the three methods  enumerated above, and that addresses some of the shortcomings and challenges associated with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Basically, the idea is to combine the attributes of each method, by following the steps below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  We make use of the order-revealing property of the LF (encoded by the rank of augmented  Loewner matrices);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' additionally, the selection of interpolation points is done via a Loewner-CUR  technique proposed in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  We utilize the elegant state-space parameterization of the LTI system proposed by the AF (after  imposing k interpolation conditions);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' this is the backbone of the methods (we also show the  connection between state-space forms and barycentric forms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  We use either the fitting step from AAA (that chooses free parameters to fit the un-interpolated  data in a least square sense) or we impose pole placing (dominant poles are selected from those  of the Loewner model);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' in both cases, a linear system of equations needs to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2023-01-13  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Gosea: Data-driven interpolation: challenges and solutions  3  In what follows, we consider a multiple-input multiple-output (MIMO) linear time-invariant (LTI)  system ΣL of dimension n described by the following system of differential equations:  ΣL :  �  ˙x(t) = Ax(t) + Bu(t),  y(t) = Cx(t),  (1)  with x(t) ∈ Rn as the state variable, u(t) ∈ Rm as the control inputs, and y(t) ∈ Rp as the observed  outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Here, we have that A ∈ Cn×n, B ∈ Cn×m and C ∈ Cp×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The transfer (matrix) function of  the LTI system is given by H(s) ∈ Cp×n, with s ∈ C, as  H(s) = C(sIn − A)−1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (2)  It is to be noted that, for m = p = 1, the system becomes single-input single-output (SISO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' We will  sometime switch between MIMO and SISO formats while presenting the methods covered in this note,  since the latter allows a more easy exposition for some of the results shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Let si ∈ C \\ σ(A), where σ(A) denotes the spectrum of matrix A ∈ Cn×n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=', the set of its  eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  The j-moment of system ΣL at si is given by ηj(si) =  (−1)j  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  �  dj  dsj H(s)  �  s=si, for any  integer j ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The 0-moment is obtained by sampling the transfer function H(s) in (2) at si, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=',  η0 = H(si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' In this contribution, we restrict the analysis to matching 0-moments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=', samples of  the transfer function H(s), and not of its derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' However, all methodologies shown here can be  expected to cope with this as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Moreover, in practice, inferring 0-moments from time-domain data  is usually a more straight-forward task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' this is performed by exciting the system with harmonic inputs,  and by applying spectral transformations to the outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Additionally, the inference of derivative  values (of the transfer function) is typically susceptible to perturbations and it is more challenging to  attain, from a numerical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  The paper is structured in the following way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' after the introduction session sets up the stage, we  propose a survey of three established interpolation-based methods in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Then, the proposed  methodologies are developed in Section 3, with emphasis on the one-step approach that combines  optimal selection of interpolation points (chosen using CUR-DEIM) with LS fit on the rest of the  data, and also the pole placement method in barycentric form that enforces dominant poles from the  Loewner data-driven model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Then, Section 4 illustrates the numerical aspects of applying the methods  discussed/proposed in the previous two sections to a variety of test cases (various models and data  sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Finally, Section 5 presents the conclusions and the outlook into future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' A survey of established methods  In this section we discuss three established data-driven methods for rational approximation (AF, LF,  and AAA, as mentioned in the previous section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  The data are samples of the transfer function  corresponding to the underlying dynamical system, measured on a particular frequency grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' In what  follows, we mention some state-of-the-art methodologies used to measure such data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=', frequency  response data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Typically, such measurements can be produced in practice from experiments conducted  in scientific laboratories using carefully calibrated machines, called spectrum analyzers (SAs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' In this  category we mention swept-tuned spectrum analyzers, scalar network analyzers (SNAs), and vector  network analyzers (VNAs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  The SNA is an instrument that measures microwave signals by converting them to a DC voltage  using a diode detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' In a VNA, information regarding both the magnitude and the phase of a  microwave signal is extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  While there are different ways to perform such measurements, the  method employed by commercial products (such as the Anritsu series described in [18]) of VNAs is  to down-convert the signal to a lower intermediate frequency in a process called harmonic sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  This signal can then be measured directly by a tuned receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Compared to the SNA, the VNA is  a more powerful analyzer tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The major difference is that the VNA can also measure the phase,  and not only the amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' With this property enforced, then so-called scattering parameters (or  Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2023-01-13  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Gosea: Data-driven interpolation: challenges and solutions  4  S-parameters) can be processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' These can be used for identifying forward and reverse transmission  and reflection characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' More details can be found in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  The harmonic balance method (or HBM) [43], is an established methodology in the field of elec-  tromagnetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The HBM is used in many (if not most) commercial radio-frequency (RF) simulation  tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' This is due to the fact that it has certain advantages over other common methods, namely  modified nodal analysis (MNA), which makes it more appropriate to use for stiff problems and circuits  containing transmission lines, nonlinearities and dispersive effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' More details can be found in the  survey paper [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The one-sided moment-matching approach in [8]  The framework introduced by Astolfi in [8] (referred to as AF throughout the paper) deals with the  problem of model reduction by moment matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Although classically interpreted as a problem of  interpolation of points in the complex plane, it has instead been recast as a problem of interpolation  of steady-state responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' In the following we briefly review its application to linear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' It is to  be noted that the AF was steadily extended and applied to different scenarios (including nonlinear  dynamical systems, pole-zero placement, and least-squares fit) [33–35,45,53,55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  The moments of a linear system can be characterized in terms of the solution of Sylvester equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  By using this observation, it has been shown that the moments are in one-to-one relation with the  steady-state output response of the interconnection between a signal generator and the original linear  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  In what follows, for simplicity of exposition, it is assumed that ΣL is a minimal system (both fully  controllable and fully observable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' For exact definitions on minimality, controllability, or observability  of LTI systems, we refer the reader to [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Let k ⩽ n and S ∈ Ck×k a non-derogatory matrix (for which the characteristic and minimal  polynomials coincide) with σ(S) ∩ σ(A) = ∅ and R ∈ C1×k so that (S, R) is observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Consider the  signal generator system Σsg described by the equations  Σsg :  �  ˙ω(t) = Sω(t),  u(t) = Rω(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (3)  Then, the explicit solution of (3) can be written as ω(t) = eStω(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Hence, the control input is written  as u(t) = ReStω(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' In addition, the eigenvalues of S are called interpolation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  For a linear system ΣL, and interpolation points si ∈ C \\ σ(A), for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' , k, consider a non-  derogatory matrix S ∈ Rk×k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' It follows that there exists a one-to-one relation between the moments  of the system ΣL and  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' the matrix CΠ, where Π is the (unique) solution of the Sylvester equation AΠ + BR = ΠS,  for any row vector R ∈ R1×k so that (R, S) is observable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' the steady-state response of the output y of the interconnection of system ΣL and the system  Σsg, for any R and ω(0) such that the triplet (R, S, ω(0)) is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  More precisely, let ∆ ∈ Rk be a column vector containing k free parameters (denoted here by  δ1, δ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' , δk, with δi ̸= 0, 1 ≤ i ≤ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Then, as stated in [8], the family of linear time-invariant  systems that interpolates the moments of system ΣL at the eigenvalues of matrix S, is given by  ˆΣ∆ :  � ˙ˆx(t) = (S − ∆R)  �  ��  �  = ˆA  ˆx(t) + ∆  ����  = ˆB  u(t),  ˆy(t) = CΠ  ����  = ˆC  ˆx(t),  (4)  where the matrices S and R are as before and Π is the unique solution of the Sylvester equation  AΠ + BR = ΠS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Additionally, the condition σ(S) ∩ σ(S − ∆R) = ∅ needs to be enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' It is  to be noted that the free parameters explicitly enter the vector ˆB = ∆, but also the matrix ˆA, as  ˆA = S − ∆R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Finally, ˆC = CΠ has fixed entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2023-01-13  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Gosea: Data-driven interpolation: challenges and solutions  5  The Sylvester matrix equation for the reduced-order system is written as ˆA ˆΠ+ ˆBR = ˆΠS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' This can  be explained by the fact that the reduced-order system matches the prescribed moments of the original  system, hence the same format of the two equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Without loss of generality, one can consider  that the matrix ˆΠ is the identity matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  ˆΠ = Ik (this can be achieved by applying similarity  transformations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' By replacing this value into the reduced-dimension Sylvester matrix equation above,  the formula ˆA = S − ∆R directly follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Afterwards, the free parameters collected in the vector ∆ can be chosen in order to enforce or impose  additional conditions as mentioned in [8], such as: matching with imposing additional k interpolation  conditions, matching with prescribed eigenvalues, matching with prescribed relative degree, matching  with prescribed zeros, matching with a passivity constraint, matching with L2-gain, or matching with  a compartmental constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  An important aspect of the AF is the characterization of all, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=', infinitely many families of reduced-  order models that satisfy k prescribed interpolation conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' This is done by explicitly computing  such parameterized models, for which the free parameters are the variables entering in the vector ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  The main parameterization developed here will be used as a “backbone” of the methods developed in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  As stated in the original paper, the main advantage of the AF (characterization of moments in  terms of steady-state responses) is that it allows the definition of moments for systems which do not  admit a clear/immediate representation in terms of transfer function(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Hence, the author provides  as examples the case of linear time-varying systems, and the case of nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Moreover,  it is stated in [8] that one disadvantage of the framework is that it requires the existence of steady-  state responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Consequently, the original system has to be (exponentially) stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' However, in most  practical applications, this is a realistic requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The Loewner framework in [41]  In this section we present a short summary of the Loewner framework (LF), as introduced in [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' It  is to mentioned that LF has its roots in the earlier work of [4], and that LF can be considered to be a  double-sided moment-matching approach (as opposed to AF, which is one-sided).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  For a tutorial paper on LF for LTI systems, we refer the reader to [7], and for a recent extension  that uses time-domain data, we refer the reader to [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The Loewner framework has been recently  extended to certain classes of nonlinear systems, such as bilinear systems in [6], and quadratic-bilinear  (QB) systems in [24], but also to linear parameter-varying systems in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Additionally, issues such  as stability preservation or enforcement, or passivity preservation, were tackled in the LF in [23, 29],  for the former, and in [2,12], for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  The LF is based on processing frequency-domain measurements D = {(ωℓ, H(ωℓ)) , ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' , N}  (with ωℓ ∈ R for 1 ≤ ℓ ≤ N) corresponding to evaluations of the transfer function of the underlying  (unknown/hidden) dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  The interpolation problem is formulated as shown below (for convenience of exposition, we show  here only the SISO formulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' We are given data nodes and data points in the set D, partitioned  into two disjoint subsets DL and DR, with DL ∪ DR = D and k + q = N, as  right data : DL = {(λj, H(λj)) , j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' , k}, and,  left data : DR = {(µi, H(µi)) , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' , q},  (5)  and we seek to find a rational function ˆH(s), such that the following interpolation conditions hold:  ˆH(µi) = H(µi) := vi,  ˆH(λj) = H(λj) := wj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (6)  The Loewner matrix L ∈ Cq×k and the shifted Loewner matrix Ls ∈ Cq×k play an important role in  the LF, and are given by  L(i,j) = vi − wj  µi − λj  ,  Ls(i,j) = µivi − λjwj  µi − λj  ,  (7)  Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2023-01-13  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Gosea: Data-driven interpolation: challenges and solutions  6  while the data vectors V ∈ Cq, WT ∈ Ck are given by  V(i) = vi,  W(j) = wj, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' , q, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (8)  Moreover, the following Sylvester matrix equations ([1, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' 6]) are satisfied by the Loewner and shifted  Loewner matrices (here, 1q =  �  1  · ·  1  �T ∈ Cq)  �  ML − LΛ = V1T  k − 1qW,  MLs − LsΛ = MV1T  k − 1qWΛ,  (9)  where M = diag(µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' , µq) and Λ = diag(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' , λk) are diagonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The following relation  holds true  Ls = LΛ + V1T  k = ML + 1qW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (10)  The unprocessed Loewner surrogate model, provided that k = q, is composed of the matrices  ˆE = −L,  ˆA = −Ls,  ˆB = V,  ˆC = W,  (11)  and if the pencil (L, Ls) is regular, then the function ˆH(s) satisfying the interpolation conditions in  (6) can be explicitly computed in terms of the matrices in (11), as ˆH(s) = ˆC(sˆE − ˆA)−1 ˆB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  In practical applications (when processing a fairly large number of measurements), the pencil (Ls, L)  is often singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Hence, a post-processing step is required for the Loewner model in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' In such  cases, one needs to perform a singular value decomposition (SVD) of augmented Loewner matrices, to  extract the dominant features and remove inherent redundancies in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' By doing so, projection  matrices X, Y ∈ Ck×r are obtained, as left, and respectively, right truncated singular vector matrices:  [L Ls] = YS(1)  r  ˜X  H � L  Ls  �  = ˜YS(2)  r XH,  (12)  where S(1)  r , S(2)  r  ∈ Rr×r,  Y ∈ Ck×r, X ∈ Cq×r, ˜Y ∈ C2q×r, ˜X ∈ Cr×2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The truncation index r  can be chosen as the numerical rank (based on a tolerance value τ > 0) or as the exact rank of the  Loewner pencil (in exact arithmetic), depending on the application and data size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' More details can be  found in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  The system matrices corresponding to a projected Loewner model of dimension r can be computed  as follows:  ˜E = −XHLY,  ˜A = −XHLsY,  ˜B = XHV,  ˜C = WY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  We note that MIMO extensions of the LF were already proposed in the original contribution [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  There, a tangential interpolation framework is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Instead of imposing interpolation of full  p × m blocks, the authors prefer to interpolate the original transfer matrix function samples along  certain vectors (or tangential directions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' We also note that a first attempt of re-interpreting the LF  in [41] as a one-sided method was made in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' In the latter, the main difference to the classical work  in [4] was that a compression of the left (un-interpolated) data set was enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' However, in [25], it  was still unclear how to split the data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=', what the right data set should be (where interpolation is  enforced).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Finally, it is to be noted that the choice of interpolation points is crucial in the LF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' An  exhaustive study of different choices was proposed in [37], while a greedy strategy was proposed in  [17], for scenarios in which limited experimental data are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The AAA algorithm in [42]  The AAA algorithm introduced in [42] is an adaptive and iterative extension of the interpolation  method based on Loewner matrices, originally proposed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The main steps are as follows  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Express the fitted rational approximants in a barycentric representation, which represents a  numerically stable way of expressing rational functions [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2023-01-13  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Gosea: Data-driven interpolation: challenges and solutions  7  Algorithm 1 The AAA algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Require: A (discrete) set of sample points Γ ⊂ C with N points, function f (or the evaluations of f  on the set Γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=', the sample values), and an error tolerance ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Ensure: A rational approximant rn(s) of order (n, n) displayed in a barycentric form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  1: Initialize j = 0, Γ(0) ← Γ, and r−1 ← N −1 �N  i=1 f(γi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2: while |f(s) − rj−1(s)| > ϵ do  3:  Select a point zj ∈ Γ(j) for which |f(s) − rj−1(s)| attains a maximal value, where for j ≥ 1, it  follows:  rj−1(s) :=  �j−1  �  k=0  ω(j−1)  k  s − zk  �−1 �j−1  �  k=0  ω(j−1)  k  fk  s − zk  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (13)  4:  if |f(zj) − rj−1(zj)| ≤ ε then  5:  Return rj−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  6:  else  7:  fj ← f(zj) and Γ(j+1) ← Γ(j) \\ {zj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  8:  end if  9:  Find the weights ω(j) = [ω(j)  0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' , ω(j)  j ] by solving a least squares problem over z ∈ Γ(j+1)  j  �  k=0  ω(j)  k  s − zk  f(s) ≈  j  �  k=0  ω(j)  k fk  s − zk  ⇔  �  j  �  k=0  f(s) − fk  s − zk  �  ω(j)  k  ≈ 0 ⇔ L(j)ω(j) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (14)  The solution of (14) is given by the (j + 1)th right singular vector of the Loewner matrix  L(j) ∈ C(N−j−1)×(j+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  10:  j ← j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  11: end while  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Select the next interpolation (support) points via a greedy scheme;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' basically, interpolation is  enforced at the point where the (absolute or relative) error at the previous step is maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Compute the other variables (the so-called barycentric weights) in order to enforce least squares  approximation on the non-interpolated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  In recent years, the AAA algorithm has proven to be an accurate, fast, and reliable rational ap-  proximation tool with a fairly large range of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Here, we will mention only a few: nonlinear  eigenvalue problems [39], MOR of parameterized linear dynamical systems [16], MOR of linear sys-  tems with quadratic outputs [26], rational approximation of periodic functions [10], representation of  conformal maps [22], rational approximation of matrix-valued functions [27], or signal processing with  trigonometric rational functions [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The procedure is sketched in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  It is to be mentioned that a modified version of AAA that enforces real-valued and strictly-  proper rational appoximants was recently proposed in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' There, the format of the function in (13)  was modified by inserting a 1 into the denominator, as follows  ˜rj(s) :=  �  1 +  j−1  �  k=0  ω(j−1)  k  s − zk  �−1 �j−1  �  k=0  ω(j−1)  k  fk  s − zk  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (15)  Consequently, the equation in (14) becomes L(j)ω(j−1) = −f(j−1), where the vector f(j−1) ∈ Cj is  given by f(j−1) =  �f0  f2  · ·  fj−1  �T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' It is to be noted that ˜rj(s) in (15) is theoretically a rational  approximant of order (j − 1, j), if we do not take into account pole/zero cancellations or any other  zero cancellations of coefficients in the numerator or denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2023-01-13  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Gosea: Data-driven interpolation: challenges and solutions  8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The proposed methodologies  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Skeleton of the main methods  Similar to the methods reviewed in Section 2 we want to find an LTI system with a transfer function of  the structure (1) that interpolates data provided as measurements H (si) , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' , k of the transfer  function of the original system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' We can directly put together an LTI parametrized model of dimension  r = km, having km2 degrees of freedom with transfer function  ˆH(s) = ˆC(sIr − ˆA)−1 ˆB,  (16)  with the underlying data concatenated to  ˆC =  �H(λ1)  · ·  H(λk)�  ∈ Cp×r,  (17)  a matrix of weights ˆ  Wi  ˆB =  �  ˆ  W  H  1  · ·  ˆ  W  H  k  �H  ∈ Cr×m,  (18)  and ˆA ∈ Cr×r formed from a diagonal matrix populated with the interpolation points λi disturbed by  ˆB, such that  ˆA = Λ − ˆBR = diag (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' , λk) ⊗ Im − ˆB  �  1T  r ⊗ Im  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (19)  Making use of the Woodbury matrix identity and denoting Λs = sIkm − Λ, the transfer function (16)  can be rewritten as  ˆH(s) = ˆCΛ−1  s  ˆB  �  Im + RΛ−1  s  ˆB  �−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (20)  A complete derivation of (20) is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  In the single-input single-output case (m = p = 1, hence r = k), the barycentric weights reduce to  scalars and the matrices for a ROM of structure (16) are given by  ˆA = Λ − ˆBR ∈ Ck×k,  ˆB =  � ˆw1  · ·  ˆwk  �T ∈ Ck×1,  ˆC =  �H (λ1)  · ·  H (λk)�  ∈ C1×k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (21)  By inserting the formulae in (21) into (20), and using the notation hi := H (λi), leads to  ˆCΛ−1  s  ˆB =  k  �  i=1  ˆwihi  s − λi  ,  �  Im + RΛ−1  s  ˆB  �−1  =  1  1 + �k  i=1  ˆwi  s − λi  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (22)  Hence, the transfer function of the model in (21) is given in barycentric representation by  ˆH(s) =  �k  i=1  ˆwihi  s − λi  1 + �k  i=1  ˆwi  s − λi  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (23)  This can be performed analogously for a multi-input multi-output case (m = p, r = km).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The first  part of (20) becomes  ˆCΛ−1  s  ˆB =  �H(λ1)Im(s − λ1)−1  · ·  H(λk)Im(s − λk)−1�  �  ��  ˆ  W1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  ˆ  Wk  �  �� =  k  �  i=1  H(λi) ˆ  Wi  s − λi  ,  (24)  Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2023-01-13  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Gosea: Data-driven interpolation: challenges and solutions  9  the second part  �  Im + RΛ−1  s  ˆB  �−1  =  �  �  �  �Im +  �Im  · ·  Im  �  �  ��  Im(s − λ1)−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Im(s − λk)−1  �  ��  −1 �  ��  ˆ  W1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  ˆ  Wk  �  ��  �  �  �  �  −1  =  �  Im +  k  �  i=1  ˆ  Wi  s − λi  �−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (25)  Consequently, the transfer function in (20) has also a barycentric form given by  ˆH(s) =  � k  �  i=1  H(λi) ˆ  Wi  s − λi  � �  Im +  k  �  i=1  ˆ  Wi  s − λi  �−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (26)  The transfer function is defined by the choice of the interpolation points and of the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The  interpolation points can be chosen as dominant parts of the available data or based on their location in  the frequency spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The weights can be computed such that the data which are not interpolated,  are approximated in an optimal way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Alternatively, the weights can be chosen to enforce poles at  specific locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' In the following, we show different strategies for both choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Automatic choice of interpolation points  The approximation quality of a surrogate model of the form (16) is greatly influenced by the choice of  the interpolation points λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' This choice is not always obvious, so automatic strategies are frequently  employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The Loewner framework uses the SVD to identify dominant subsets of the available data  to enforce interpolation on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Alternatively, the AAA algorithm uses a greedy scheme to minimize the  error between surrogate and original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Another approach, originally introduced by [37], makes use  of the CUR decomposition to extract interpolation points from a relevant subset of the available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  The CUR decomposition approximates a matrix A by a product of three low-rank matrices ˇA =  ˇC ˇU ˇR, where ˇC and ˇR represent subsets of the columns respectively rows of A [40,56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' In our case the  three matrices are only a byproduct, we are more interested in the interpolation points λ and µ that  are associated to the columns and rows extracted as ˇC and ˇR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' In combination with the skeleton for a  realization described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='1, Algorithm 2 computes a surrogate model approximating a set of  given transfer function data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' We use the algorithm from [56] to compute the CUR decomposition and  thus identify dominant parts of the original data set and their corresponding left and right interpolation  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Contrary to [37], we decompose the original Loewner matrix L rather than the augmented  Loewner matrices  �L  Ls  �  and  �  LH  LH  s  �H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Using all interpolation points obtained from the CUR  decomposition would introduce redundant data into the surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Therefore we choose only a subset  of the interpolation points: either only the left points, only the right points, or every other entry from  a concatenated and sorted vector of left and right points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Together with the data associated to the  chosen interpolation points they are used to populate a rectangular Loewner matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' We now need  to compute weights for barycentric interpolation as described in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' After having  obtained the weights, a surrogate model (16) can be computed from (17)–(19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2023-01-13  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Gosea: Data-driven interpolation: challenges and solutions  10  Algorithm 2 LS-Loewner with CUR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Require: Transfer function samples {H (si)}N  i=1, corresponding sampling points Ξ = {si}N  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Ensure: Surrogate model ˆH(s) = ˆC(sIr − ˆA)−1 ˆB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  1: Partition data and compute Loewner matrix L as in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2: Compute CUR decomposition, such that L = ˇC ˇU ˇR with ˇC ∈ CN×k, ˇR ∈ Ck×N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  3: Obtain interpolation points {λi}k  i=1 , {µi}k  i=1 corresponding to the columns and rows in ˇC, ˇR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  4: Postprocess interpolation points to obtain ν = {ν}k  i=1 and χ = Ξ \\ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  5: Populate a rectangular Loewner matrix L(i,j) = H(χi)−H(νj)  χi−νj  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  6: Compute the weights Ω = −L† �  H (ν1)H  · ·  H (νk)H�H  , where L† is the pseudo-inverse of L and  Ω =  �  ˆ  W  H  1  · ·  ˆ  W  H  k  �H  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  7: Compute ˆA, ˆB, ˆC with (17)–(19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Computing the barycentric weights  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Least-squares approach  The matrix-valued weights ˆ  Wi can be computed similarly to AAA [27] by solving the minimization  problem  min  ˆ  Wi  h  �  j=1  �  �  � k  �  i=1  H(λi) ˆ  Wi  sj − λi  � �  Im +  k  �  i=1  ˆ  Wi  sj − λi  �−1  − H(sj)  �  �  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (27)  This solution can, for example, be obtained from an optimization in least-squares sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The weights  for the SISO case are computed analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Here, the matrix-values weights and transfer function  values reduce to scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Pole placement  The next step would be to take advantage of the degrees of freedom in the vector ˆB from (21), so  that the ROM thus constructed has particular (stable) poles [21, 35, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' These will be denoted with  ζ1, ζ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' , ζk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The following derivations assume a SISO model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' To enforce that this happens, we need  to make sure that the matrix ζjIk − ˆA loses rank for all 1 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' In what follows, we show how to  enforce this property in an elegant, straightforward way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Remember that the transfer function of the  parameterized AF model is given by:  ˆH(s) =  �k  i=1  ˆwihi  s − λi  1 + �k  i=1  ˆwi  s − λi  = N(s)  D(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (28)  Now, let’s say we would like this transfer function to have k poles at the selected values ζj’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Clearly,  the condition is D(ζj) = 0 and hence we need to enforce:  1 +  k  �  i=1  ˆwi  ζj − λi  = 0, ∀1 ≤ j ≤ k ⇔ Cζ,λ ˆB = −1k ⇔ ˆB = −C−1  ζ,λ1k,  (29)  where Cζ,λ is a Cauchy matrix defined by: (Cζ,λ)i,j =  1  ζi−λj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Details on how to obtain the above  expression by following the procedure in [3] are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' We note that placing poles  is a difficult numerical problem which requires the inversion of a Cauchy matrix, which is highly  ill-conditioned, by nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2023-01-13  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Gosea: Data-driven interpolation: challenges and solutions  11  Algorithm 3 Loewner framework with pole placement (LFPP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Require: Transfer function samples {H (si)}N  i=1, corresponding sampling points Ξ = {si}N  i=1, loca-  tions for poles ζ = {ζi}k  i=1, interpolation points λ = {λi}k  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Ensure: Surrogate model ˆH(s) = ˆC(sIr − ˆA)−1 ˆB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  1: Compute ΣD from {H (si)}N  i=1 and {si}N  i=1 using the Loewner framework (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2: ˆC ←  �HD (λ1)  · ·  HD (λk)�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  3: ˆB ← −C−1  ζ,λ1r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  4: ˆA ← diag (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' , λk) − ˆB1T  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Instead of doing this, we could solve Cζ,λ ˆB = −1r, without inverting the Cauchy matrix explicitly,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=', by solving a linear systems of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Algorithm 3 summarizes this procedure in a data-driven  context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The required underlying model is obtained from a set of transfer function evaluations by  applying the Loewner framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The method is illustrated for SISO systems, but can readily be  extended to the MIMO case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Automatic choice of poles and interpolation points  A reasonable choice of poles and interpolation points for Algorithm 3 is not always readily available,  but the approximation of the surrogate is heavily influenced by this choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' In the following, we show  an extension to Algorithm 3 which computes a surrogate model (16) without requiring sets of poles  and interpolation points as input parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Algorithm 4 sketches the skeleton of such automatic  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Similar to Algorithm 3 it employs the Loewner framework to obtain a realization of a  surrogate interpolating the provided data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Subsequently, a generalized eigendecomposition of the  Loewner realization of the original data is computed to find suitable locations for poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' From this  is is possible to compute the dominance of all eigenvalues;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' for details, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The algorithm  now chooses the k most dominant eigenvalues as poles to enforce in the surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' It should be noted  that only eigenvalues with negative real parts should be considered, if the stability of the surrogate is  important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The required interpolation points can now be chosen similar to Algorithm 2 by computing  a CUR decomposition and using the interpolation points associated to the rows or columns of the  decomposition as interpolation points for the new surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  The approximation of dominant poles of the underlying model from data is less robust if the transfer  function samples are disturbed by noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' This leads to a reduced approximation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' For a better  performance if applied to noisy data, Algorithm 4 can be modified as follows: To obtain poles which  should be enforced, first choose manually the most prominent features in the transfer function, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  peaks, which should be approximated by the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Now choose the eigenvalues which  imaginary parts are closest to the frequencies, where the chosen features of the transfer function are  located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The CUR decomposition also fails at extracting the most dominant rows and columns of  the Loewner matrix if noisy data is assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Therefore another heuristic is employed to choose the  interpolation points: Use the value si which corresponds to the lowest amplitude of the transfer function  between the locations of two enforced poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' This leads to reasonable approximations, especially for  lightly damped systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Other approaches include choosing simply the middle between the location  of two poles or specifying an offset between pole and interpolation point location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Numerical results  In the following, we demonstrate the methods discussed in Section 3 by applying them on three  benchmark examples available from the MOR-Wiki1:  1http://modelreduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='org  Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2023-01-13  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Gosea: Data-driven interpolation: challenges and solutions  12  Algorithm 4 Loewner framework with automatic pole placement (LFaPP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Require: Transfer function samples {H (si)}N  i=1, corresponding sampling points Ξ = {si}N  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Ensure: Surrogate model ˆH(s) = ˆC(sIr − ˆA)−1 ˆB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  1: Compute ΣD from {H (si)}N  i=1 and {si}N  i=1 using the Loewner framework (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2: Compute the generalized eigenvalue decompositions AX = EXα and YHA = αYHE for the  matrices of right and left eigenvectors X, Y and the matrix of eigenvalues α = diag (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' , αn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  3: Compute eigenvalue dominance di = |CY(:,i)αiX(:,i)HB|  |ℜ(αi)|  , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' , n and sort α accordingly  4: Set ζ to the k most dominant eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  5: Compute CUR decomposition of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  6: Set λ to the k right or left interpolation points corresponding to the CUR decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  7: Compute surrogate as in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  ISS This system models the structural response of the Russian Service Module of the International  Space Station (ISS) [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The model has n = 270 states, m = 3 inputs, and p = 3 outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The  dataset used for the computations contains transfer function measurements at 400 logarithmically  distributed points in the range  �  10−1, 102�  ı.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The model is also part of the SLICOT benchmark  collection [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Flexible aircraft This system models lift and drag along the flexible wing of an aircraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The system  matrices are not available, we only have access to a dataset of 420 transfer functions samples  at linearly distributed frequencies between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='1 and 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='0 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The original dataset has one input  (the gust disturbance) and 92 outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' For the following experiments, we choose the 91st output  which corresponds to the first flexible mode [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The dataset is available from [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Sound transmission This system models the sound transmission through a system of two brass plates  with an air enclosure between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The transfer function measures the sound pressure in an  adjacent acoustic cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The geometry is based on [32];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' the data—transfer function evaluations  at 1000 linearly-distributed frequency values between 1 and 1000 Hz—is available from [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  We note that no tangential interpolation (as described in [41]) is applied for the MIMO model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Instead, the Loewner matrices are constructed in a block-wise manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The case of tangential inter-  polation, within the proposed approaches in this note, will be investigated in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  We enforce realness of all surrogate models (all matrices contain only real entries) by applying the  transformation described in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' For this, all data must be available in complex conjugate pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The  required transformation matrix is given by  J = Iℓ ⊗  � 1  √  2  � Im  Im  −ıIm  ıIm  ��  ,  (30)  with ℓ = k  2 and the real-valued quantities are obtained from ˆA  (ℜ) = J ˆAJH, ˆB  (ℜ) = J ˆB, and ˆC  (ℜ) =  ˆCJH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  For some of the experiments we add artificial noise to the measurements, in order to obtain perturbed  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The modified measurements are given by  ˇH (si) = H (si) (1 + Zi) , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' , n,  (31)  where Zi ∈ C is the ith sample drawn from a set of random numbers Z ∼ CN  �  µ, σ2�  following a  complex normal distribution with mean µ and standard deviation σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Here, the real and imaginary  parts of Z are independent normally distributed variables [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  We assess the approximation error of the surrogate models with an approximated L∞ norm, because  many surrogates have unstable poles and hence, the H∞ can not be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' For a given reduced  Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2023-01-13  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Gosea: Data-driven interpolation: challenges and solutions  13  order r, the L∞ error in the considered frequency range ω ∈ [ωmin, ωmax] is approximated by  ε(r) =  max  ω∈[ωmin,ωmax]  ���H(ωı) − ˆHr(ωı)  ���  2  max  ω∈[ωmin,ωmax] ∥H(ωı)∥2  ≈  ���H − ˆHr  ���  L∞  ∥H∥L∞  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (32)  Note that strategies to post-process surrogates to obtain stable models have been studied in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  The numerical experiments have been conducted on a laptop equipped with an AMD Ryzen™  7 PRO 5850U and 12 GB RAM running Linux Mint 21 as operating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' All algorithms have been  implemented and run with MATLAB R2021b Update 2 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='1837725).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Code and data availability  The data that support the findings of this study are openly available in Zenodo at  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='7490158  under the BSD-2-Clause license, authored by Quirin Aumann and Ion Victor Gosea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Case of exact measurement data  In the following, we compare the performance of the new approach LS-Loewner to the following estab-  lished strategies:  Loewner-SVD: Truncate Loewner matrices populated with the complete dataset to order r using  an SVD [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Loewner-CUR: Construct a purely interpolatory model of order r using all data points chosen by  the CUR decomposition, similar to [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Modified AAA: Apply the strictly-proper variant of AAA [29] to the complete dataset to compute  a reduced-order model of size r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  We first consider the original MIMO ISS example and a SISO variant where we select the first input  and output, respectively, from the MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' To evaluate the overall performance of the different  methods related to the size of a surrogate model, we compute the approximated L∞ errors for models  with orders 6 ≤ r ≤ 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The approximation error versus the dimension of the respective surrogate  model is depicted in Figure 1 for all four methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Since tangential interpolation was not employed here, the order of the MIMO surrogates rises by m  for each additional interpolation point, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=', r = km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' This explains the lower accuracy of the MIMO  surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' For the maximum reduced order r = 60, k = 20 interpolation points are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The  errors of the SISO surrogates for r = 20, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=', k = 20, is in a similar range as in the MIMO case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The  SISO surrogates reach similar levels of approximation for all employed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' In the MIMO case,  Loewner-SVD performs best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' This can be explained by the following observation: the other methods  always consider the complete transfer function measurement H(λi) ∈ Cp×m per interpolation point,  while Loewner-SVD extracts only the r most dominant singular vectors for projection, regardless of  to which interpolation point they belong to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' In turn, the other methods also consider probably less  important parts of the data as long as one input/output combination of the respective sample is  relevant for approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' It can also be noted that LS-Loewner and Loewner-CUR perform very  similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' This was expected, as both methods rely on the same interpolation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  All four methods are now employed to compute a surrogate model of size r = 108 to approximate  the transfer function of the flexible aircraft model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The size of the surrogate model is determined by  truncating all singular values τ < 1·10−6 of an underlying Loewner matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  The transfer functions of all resulting models and their respective relative errors are given in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Again, all methods succeed in computing a sufficiently accurate surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' However, the approximation  Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2023-01-13  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Gosea: Data-driven interpolation: challenges and solutions  14  10  20  30  40  50  60  10−6  10−5  10−4  10−3  10−2  10−1  100  Reduced order r  L∞ error  SISO  10  20  30  40  50  60  10−6  10−5  10−4  10−3  10−2  10−1  100  Reduced order r  L∞ error  MIMO  LS-Loewner  Loewner-SVD  Loewner-CUR  Modified AAA  Figure 1: The approximated L∞ errors of reduced-order models of order r computed from the ISS  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Left: SISO with first input and output, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Right: MIMO with three inputs  and three outputs m = p = 3 (the number of interpolation points is k = r  m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  quality of Loewner-CUR is noticeably worse than that of the other three methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Given that both  Loewner-CUR and LS-Loewner use the same interpolation points, the weights computed from the least  squares problem show a better performance compared to the partitioning approach used in Loewner-  CUR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Perturbed measurement data  Analyzing measurement data perturbed by noise is a challenging task for interpolatory methods, such  as the Loewner framework and the AAA algorithm (as pointed out in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=', [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' In this experiment  we investigate the effect of noise to the performance of the four methods described above and show,  how enforcing poles and/or interpolation points can increase the approximation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' In the first  experiment we consider transfer function data from the ISS model perturbed by noise with mean µ = 0  and standard deviation σ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' We employ LFaPP and enforce poles at ı[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='77, 2, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='6, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='33, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='9]  near peaks of the transfer function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The resulting real-valued surrogate model has order r = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The  transfer functions of the surrogate model with enforced poles and reduced models computed from the  same noisy data with LS-Loewner, Loewner-SVD, Loewner-CUR, and Modified AAA are given in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Enforcing the poles near peaks in the transfer function of the underlying data allows the surrogate  to capture the behavior of the original data in a wider frequency range than applying LS-Loewner,  Loewner-SVD, and Loewner-CUR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The choice of the locations, in which vicinity the poles should be  chosen is, however, not automatized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Figure 3 also shows the relative errors of all surrogate models  referenced to the original data without noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' While the enforced poles all have a negative real part,  the models computed from the variants of the LF and AAA exhibit unstable eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Thus, pole  placement can be seen also as a means to enforce stability of the surrogate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Alternatively,  a post-processing step can be added to enforce stable models (for both LF and AAA methods), as  performed in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  We now evaluate the performance of the algorithms by applying them to heavily distorted trans-  fer function measurements of the sound transmission problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Noise with a standard deviation of  σ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='25 is considered and three algorithms are employed to compute surrogates: Loewner-SVD,  LFPP (Algorithm 3), and LFaPP (Algorithm 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' We also test the modifications to LFaPP described in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' These results are denoted by “LFaPP mod.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' For LFPP we enforce poles at the eigenval-  ues of the underlying Loewner model which imaginary parts are near 2πı [72, 189, 392, 401, 706, 856].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  These locations correspond to characteristic peaks in the transfer function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Further, we choose the in-  Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2023-01-13  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Gosea: Data-driven interpolation: challenges and solutions  15  5  10  15  20  25  30  35  40  10−4  10−2  100  Magnitude  Original data  LS-Loewner  Loewner-SVD  Loewner-CUR  Modified AAA  5  10  15  20  25  30  35  40  10−12  10−9  10−6  10−3  100  Frequency [Hz]  Relative error  LS-Loewner  Loewner-SVD  Loewner-CUR  Modified AAA  Figure 2: Transfer function (top) and relative pointwise errors (bottom) for reduced-order models of  size r = 108 for the aircraft model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The error is plotted only at frequencies which do not  coincide to interpolation points of the respective method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  terpolation points at 2πı [138, 339, 369, 569, 712, 954], which lie at the dips between the enforced poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Loewner-SVD and LFaPP do not require input parameters in addition to the measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Figure 4  shows the transfer function of the resulting surrogate models in comparison to the original and noisy  underlying data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' It can be observed, that the automatic approaches Loewner-SVD and LFaPP (mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=')  cannot approximate the transfer function well after the first two peaks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=', for frequencies higher than  200 Hz, while LFPP approximates the original data over the complete frequency range with decent  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The importance of reasonable interpolation points can be seen in the difference of LFPP and  LFaPP mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=', which have the same poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' It should be noted that the surrogate model computed by  Loewner-SVD has two unstable poles while the other three surrogate models are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' It is, however,  not always clear a priori how to choose the poles and interpolation points for LFPP in order to achieve  the best approximation quality possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' In this example, the noise level is too high for one of the  automatic approaches to yield reasonable dominant interpolation points or poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Conclusion and outlook  In this contribution, we have proposed an extensive study of interpolation-based data-driven ap-  proaches for approximating the response of linear dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  All methods require input  and output data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=', transfer function measurements, while direct access to the system operators or  the states is not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' We showed different approaches how to achieve compact surrogate models  approximating the input/output behavior of the original system and how to ensure various properties  of the surrogate models, such as stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Strategies how to work with noisy measurement data have  also been addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  A natural extension of the framework described here is to apply the ideas of tangential interpolation  as a means of modeling a MIMO system from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Here, the tangential directions need to be incorpo-  rated in the parameterized one-sided realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Further topics include enforcing different structures  of the original model in the surrogate model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=', second-order or delay structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' It would also be  interesting to study the possibility of placing certain stable poles while achieving interpolation in a  Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2023-01-13  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Gosea: Data-driven interpolation: challenges and solutions  16  10−1  100  101  102  10−5  10−3  10−1  Magnitude  Noisy data  Original data  LFaPP  LS-Loewner  Loewner-SVD  Loewner-CUR  Modified AAA  10−1  100  101  102  10−4  10−3  10−2  10−1  100  101  102  Frequency  Relative error  Noise  LFaPP  LS-Loewner  Loewner-SVD  Loewner-CUR  Modified AAA  Figure 3: Transfer function of a surrogate with enforced poles compared to the noisy and original  transfer function values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The transfer function of a model r = 12 computed from Loewner-  SVD is given for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  least-squares sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Application cases for the proposed methodology could include damping optimiza-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Here, a family of parameterized interpolants could be used to find optimal positions for viscous  dampers in a structural system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Appendix  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The Woodbury matrix identity  We can expand the right part of (19), such that:  ˆA = Λ − ˆBR ⇒ sIkm − ˆA = sIkm − Λ  �  ��  �  ˆ  M  +  �  ��  ˆ  W1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  ˆ  Wk  �  ��  � �� �  ˆU  �Im  �  ����  ˆT  �Im  · ·  Im  �  �  ��  �  ˆV  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (33)  The Woodbury matrix identity is as follows:  �  ˆM + ˆU ˆT ˆV  �−1  = ˆM  −1 − ˆM  −1 ˆU  �  ˆT  −1 + ˆV ˆM  −1 ˆU  �−1 ˆV ˆM  −1,  where ˆM, ˆU, ˆT and ˆV are conformable matrices: ˆM is n × n, ˆT is k × k, ˆU is n × k, and ˆV is k × n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  This can be derived using blockwise matrix inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' By denoting with Λs = sIkm − Λ, then the first  Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2023-01-13  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Gosea: Data-driven interpolation: challenges and solutions  17  100  200  300  400  500  600  700  800  900  1,000  10−6  10−2  102  Magnitude  Noisy data  Original data  Loewner-SVD  LFPP  LFaPP  LFaPP mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  100  200  300  400  500  600  700  800  900  1,000  10−3  10−2  10−1  100  101  102  103  Frequency [Hz]  Relative error  Noise  Loewner-SVD  LFPP  LFaPP  LFaPP mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Figure 4: Transfer function (top) and relative pointwise errors (bottom) as well as the added noise for  reduced-order models of size r = 12 for the aircraft model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  transfer function of the fitted model is written:  ˆH(s) = ˆC  �  sI3m − ˆA  �−1 ˆB = ˆC  �  Λs + ˆU ˆV  �−1 ˆB  = ˆCΛ−1  s  ˆB − ˆCΛ−1  s  ˆU  �  Im + ˆVΛ−1  s  ˆU  �−1 ˆVΛ−1  s  ˆB  = ˆCΛ−1  s  ˆB − ˆCΛ−1  s  ˆB  �  Im + RΛ−1  s  ˆB  �−1  RΛ−1  s  ˆB  = ˆCΛ−1  s  ˆB  �  Im −  �  Im + ˆX  �−1 ˆX  �  = ˆCΛ−1  s  ˆB  �  Im + ˆX  �−1  ,  (34)  where ˆX = RΛ−1  s  ˆB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Hence, we arrive at (20) and the transfer function ˆH(s) can be written as follows:  ˆH(s) = ˆCΛ−1  s  ˆB  �  Im + RΛ−1  s  ˆB  �−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (20)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Pole placement as in [3]  In order to enforce both prescribed poles and certain interpolation conditions in the ROM, we follow  the derivations from [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  It is to be noted that this approach is intrusive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=', requires access to  the system’s matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Hence, a descriptor model characterized in (generalized) state-space by the  following equations  ΣDes :  �  E ˙x(t) = Ax(t) + Bu(t),  y(t) = Cx(t),  (35)  with corresponding transfer function HDes(s) = C(sE − A)−1B is considered to be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' For the  (right) interpolation points λi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' , k (where interpolation is imposed), and the desired poles to  be placed, denoted with ζj’s, the author in [3] starts by finding a row vector Cζ ∈ C1×n so that:  Cζ  �  (λ1E − A)−1B · · · (λkE − A)−1B  �  = 01×k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (36)  Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  2023-01-13  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Aumann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Gosea: Data-driven interpolation: challenges and solutions  18  Then, the next step is to choose projection matrices W, V ∈ Cn×k as  WH =  �  ��  Cζ(ζ1E − A)−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  ˜C(ζkE − A)−1  �  �� ,  V =  �(λ1E − A)−1B · · · (λkE − A)−1B�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (37)  As explained in [3], the choice of WH above is explained by imposing the required poles for the  reduced model, while V is chosen to match the interpolation conditions at the λi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Moreover, using  these notations, it follows that ˜CV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Next, put together the following matrices ˜E = WHEV,  ˜A =  WHAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Then, it follows that (s˜E − ˜A) loses rank when s ∈ {ζ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' , ζr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' To show this, we simply  write  eT  j (ζj ˜E − ˜A) = eT  j WH(ζjE − A)V = Cζ(ζjE − A)−1(ζjE − A)V = CζV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (38)  Let Hζ(s) = Cζ(sE − A)−1B be a rational function in s and we note that ˆE and ˆA are a special type  of diagonally scaled Cauchy matrices, with the following exact definition:  ˜Ei,j = −Cζ(ζiE − A)−1B − Cζ(λjE − A)−1B  ζi − λj  = − Hζ(ζi)  ζi − λj  ˜Ai,j = −ζiCζ(ζiE − A)−1B − λjCζ(λjE − A)−1B  ζi − λj  = −ζiHζ(ζi)  ζi − λj  (39)  From the definition in (39), it follows that ˜E = −D ˜BCζ,λ, where D ˜B = diag( ˜B) is a diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Similarly, it follows that ˜A = −ZD ˜BCζ,λ, where Z = diag(ζ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' , ζk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Next, we write the other projected quantities as  ˜B = WHB =  �Hζ(ζ1)  · ·  Hζ(ζk)�T ,  ˜C = CV =  �H(λ1)  · ·  H(λk)�  (40)  Hence, the reduced-order linear descriptor system Σpp : (˜E, ˜A, ˜B, ˜C) matches k interpolation conditions  and has the required poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Next, we show that this model can be written equivalently in the AF format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' We first note that  ˆC = ˜C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' For next step, provided that the matrix ˜E is non-singular, we remove it by incorporating  it into the other matrices, as: ˘A = ˜E  −1 ˜A,  ˘B = ˜E  −1 ˜B,  ˘E = Ik,  ˘C = ˜C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' We note that the two  realizations of the interpolatory ROM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=', ( ˆA, ˆB, ˆC) in (21) and ( ˘A, ˘B, ˘C) introduced above, are  actually identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' The reason for this is that ˘C = ˆC and the two ROMs match the same k moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  Hence, it also follows that ˘B = ˆB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' Now, since ˘B = ˜E  −1 ˜B and ˜E = −D ˜BCζ,λ, we can write that  ˆB = −(D ˜BCζ,λ)−1 ˜B = −C−1  ζ,λD−1  ˜B ˜B = −C−1  ζ,λ1k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  (41)  Hence, the above choice of vector ˆB in (21) imposes the required poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content='  References  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfHwxv/content/2301.04906v1.pdf'}
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diff --git a/HdAzT4oBgHgl3EQfHvsI/content/tmp_files/2301.01048v1.pdf.txt b/HdAzT4oBgHgl3EQfHvsI/content/tmp_files/2301.01048v1.pdf.txt
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@@ -0,0 +1,1483 @@
+1
+A Theory of I/O-Efficient Sparse Neural Network
+Inference
+Niels Gleinig, Tal Ben-Nun, Torsten Hoefler
+{niels.gleinig, talbn, torsten.hoefler}@inf.ethz.ch
+ETH Zürich
+Abstract—As the accuracy of machine learning models in-
+creases at a fast rate, so does their demand for energy and
+compute resources. On a low level, the major part of these
+resources is consumed by data movement between different
+memory units. Modern hardware architectures contain a form
+of fast memory (e.g., cache, registers), which is small, and a slow
+memory (e.g., DRAM), which is larger but expensive to access. We
+can only process data that is stored in fast memory, which incurs
+data movement (input/output-operations, or I/Os) between the
+two units. In this paper, we provide a rigorous theoretical analysis
+of the I/Os needed in sparse feedforward neural network (FFNN)
+inference. We establish bounds that determine the optimal
+number of I/Os up to a factor of 2 and present a method that uses
+a number of I/Os within that range. Much of the I/O-complexity is
+determined by a few high-level properties of the FFNN (number
+of inputs, outputs, neurons, and connections), but if we want to
+get closer to the exact lower bound, the instance-specific sparsity
+patterns need to be considered. Departing from the 2-optimal
+computation strategy, we show how to reduce the number of I/Os
+further with simulated annealing. Complementing this result, we
+provide an algorithm that constructively generates networks with
+maximum I/O-efficiency for inference. We test the algorithms and
+empirically verify our theoretical and algorithmic contributions.
+In our experiments on real hardware we observe speedups of up
+to 45× relative to the standard way of performing inference.
+Index Terms—Neural Network Inference, Sparse Neural Net-
+works, I/O-Complexity, Simulated Annealing.
+I. INTRODUCTION
+Almost all modern computing systems deploy different
+memory units: Some are fast but small and some are large but
+slow. Data can only be processed in fast memory. As the fast
+memory is very limited in size, data needs to be moved between
+fast and slow memory over the course of a computation. These
+data movements are called I/Os (abbreviating input/output-
+operations). They are very expensive in time and energy. For
+example, on an NVIDIA V100 GPU, loading values from
+global memory costs up to 514.5× more clock cycles than
+one fused-multiply-add (FMA) operation between two 32-bit
+floating point values (or 14× if stored on an L1 cache) [1].
+In deep learning, and scientific computing in general, I/Os
+account for the major part of time and energy expenses [2].
+Hence, it is important to use I/Os efficiently by performing
+computations in a way that makes use of “data locality” and
+“data reuse”. We need to schedule the computational steps in
+such a way that accesses to the same data are close together,
+sparing eviction of values from fast memory (e.g., cache) and
+reading them again. Therefore, optimizing the implementation
+of DNN inference [3] and training [4] often requires tailor-
+made implementations for the available memory architectures
+in order to reach high utilization.
+In this paper, we analyze and optimize I/Os with a theoretical
+model that applies across the wide variety of hardware in which
+this phenomenon occurs. We consider exclusively sparse Feed-
+Forward Neural Networks (FFNNs) without shared connections.
+This model contains, but is not limited to, pruned Multi-Layer
+Perceptrons (MLP). We assume that the FFNN is given as a list
+of weighted edges in a Directed Acyclic Graph (DAG) together
+with one additional value for each vertex (being the input-value
+for input-neurons and the bias for non-input neurons). In this
+setting, the primary way to obtain I/O-efficiency is to “use”
+the connections in an efficient order. That is, when we use
+a connection, the value of the input neuron and the value of
+the partial sum of the output neuron should ideally already be
+in fast memory. Therefore, we need to find an order of the
+connections, in which the connections that have neurons in
+common, are close together.
+Finding an order which is optimal with respect to this
+property is a combinatorial problem. We will show that if
+we use a traditional computational order, which corresponds
+to a natural interpretation of FFNN inference (for example,
+seeing it as a sequence of matrix vector multiplications) we
+can be far from optimal: The number of read-I/Os could be
+up to twice the optimal number, and the number of write-I/Os
+could differ from the optimal number by an arbitrarily large
+factor.
+Computations in ML have some special features that we can
+use when aiming for I/O-efficiency. Unlike the research on
+I/O-efficient computing outside of ML, there are more degrees
+of freedom that we can exploit. For example, there are several
+acceptable solutions to a given problem: Two different FFNNs
+may produce different outputs, yet both can have a satisfying
+accuracy. It thus makes sense to adapt the neural architecture
+to the hardware. On the other hand, there is additional freedom
+on the side of the hardware, as it is not uncommon to search
+or even build hardware for specific ML applications [5], [6],
+[3].
+Hence, our investigation is directed towards answering the
+following questions: (1) If we are given an FFNN and a fast
+memory of a given size, what is the minimal number of I/Os
+needed to perform inference on this FFNN? (2) For a given
+FFNN, what is the smallest memory size with which we can
+perform inference with a minimal number of I/Os (i.e., we can
+avoid having to read and write intermediate results due to
+lack of memory)? and (3) For a fast memory of a given size,
+arXiv:2301.01048v1  [cs.DC]  3 Jan 2023
+
+2
+what are the FFNN architectures on which we can perform
+inference with a minimal number of I/Os?
+A. Background and related work
+a) I/O-complexity: There is an extensive line of theoret-
+ical research on I/O-efficient algorithms outside of machine
+learning. Hong & Kung’s seminal work [7] is among the
+first to investigate I/O-efficient algorithms formally. This work
+introduced a theoretical model called the red-blue pebble game
+and used this model to show how standard matrix multiplication
+or FFT should be scheduled to use an asymptotically minimal
+number of I/Os. This model has been extended and altered in
+various ways. These extensions have been used to establish
+lower bounds and matching upper bounds for problems such
+as sorting, permutation, and matrix transposition [8]; as well as
+sparse matrix dense vector multiplication [9], and various graph
+algorithms [10]. However, these solutions are typically only
+optimal up to constant multiplicative factors and often these
+factors are large or not exactly known. In fact, it has been shown
+that finding exactly optimal solutions to the red-blue pebble
+game (corresponding to computation schedules with an exact
+minimal number of data movements) is a PSPACE-complete
+problem [11]. Furthermore, this problem has been hypothesized
+to be even hard to approximate [11], since inapproximability
+is known for related pebble games [12], [13].
+There are also bounds on the I/O-complexity of general
+computations with large proportions of inputs [14]. These
+bounds apply to the problem considered in our paper. However,
+they are less fine-grained (they do not distinguish between
+read- and write-I/Os as we do) and generally less tight as the
+ones that we establish in this paper.
+In ML-specific algorithms, I/O-efficiency has been practically
+identified as a crucial performance factor. Performance model-
+ing of DNNs is usually derived from operation counts [15] and
+sums of fixed values obtained from benchmarking individual
+layers [16], [17]. As for I/O complexity of DNN evaluation,
+the only theoretical work apart from this one, to the best of
+our knowledge, is given by Demmel & Dinh [18], who provide
+communication bounds for convolutional and pooling layers.
+b) Network pruning and sparsity: DNN pruning (i.e.,
+removing certain edges or nodes from a network while
+preserving accuracy) has been extensively studied over the last
+decades [19], [20], [21]. Pruning is of much practical interest
+for edge computing devices, as it allows to drastically reduce
+the memory footprint and number of floating point operations.
+Dense (fully-connected) layers have been a prime target for
+pruning, as they tend to account for a major proportion of the
+memory size. For example, in AlexNet [22], the final dense
+layers contain 90% of the parameters. Also transformers [23],
+which are currently very popular and achieve state-of-the-art
+results in NLP tasks, include FFNNs that comprise a large
+proportion of the overall size. For example, BERT [23] includes
+several FFNNs of depth 2 and weight matrices of dimensions
+1024 × 4096 and 4096 × 1024 and the major part of BERTs
+parameters and compute time for inference come from these
+FFNNs [24].
+There are different ways to prune networks, e.g., by a given
+weight threshold [25], [26], the top-k values [27], [28], or
+by way of gradual weight elimination [19]. Regardless of the
+method, pruning dense layers leads to sparse and unstructured
+networks, for which data-locality is more difficult to obtain.
+c) Hardware Architectures: There has been work on
+hardware-based optimizations for inference, especially for
+the class of sparse FFNNs, where Sze et al. [3] provide a
+detailed survey. Specific approaches include EDEN [29], which
+uses approximate DRAM for improving energy-efficiency;
+Bhardwaj et al. [30], who consider communication in inference
+in a distributed setting; and EIE [5] and Eyeriss [31] runs
+inference on a compressed CNN directly, using accelerated
+sparse matrix-vector multiplications and run-length encoding,
+respectively. While these approaches improve certain aspects of
+DNN processing, as the network size grows, such architectures
+ultimately also resort to multi-level memory hierarchies, which
+our approach addresses.
+B. Contributions
+Our main theoretical contributions are a collection of
+theorems and propositions that analyze the I/Os in FFNN
+inference. Theorem 1 provides bounds on the number of
+I/Os that are needed to perform inference on a given FFNN-
+architecture. In Proposition 1, we show that these bounds are
+optimal in the sense that none of them could be tightened by
+multiplying with any factor other than 1. The lower and upper
+bounds on the total number of I/Os differ by a multiplicative
+factor of at most 2. The proof of Theorem 1 is constructive
+and shows how a computation can be done to use a number of
+I/Os within this 2-optimal range. Departing from 2-optimality,
+our goal is to get closer to the exact optimum by reordering
+the connections beneficially. We approach this problem with
+Simulated Annealing. In Section V we present a method for
+generating FFNNs, which according to Theorem 2 completely
+characterizes the architectures on which we can perform
+inference with a minimal number of I/Os for a given memory
+size (i.e., exactly matching the lower bound). Hence, this
+method can be used as a powerful tool to co-design neural
+networks and hardware. As a corollary of this theorem we
+obtain an upper bound on the smallest possible memory size
+that allows us to perform inference with maximal I/O-efficiency
+(that is, without having to write or read any temporary values).
+We test the algorithms on random FFNNs across a wide range
+of parameter settings (density, depth, width, memory sizes).
+II. MODEL AND NOTATIONS
+We assume that initially all FFNN parameters and all input
+values are laid out in slow memory. These data are exactly the
+union of the following three types of data:
+• The weighted connections.
+• One bias value for each non-input neuron.
+• One input value for each input neuron.
+Each connection is described by an independent parameter
+triple (i, j, wij), where i is the input neuron, j the output
+neuron, and wij the weight (there is no reuse or sharing of
+weights). We denote the number of connections as W, the
+number of neurons N, the number of input neurons I, and the
+number of output neurons S. In our theoretical analysis, we
+
+3
+assume the connections (the entire triples describing them), the
+numerical values at the neurons (biases, partial sums, outputs
+of the activation functions), and all other numerical values
+involved in the computation are data types of the same size.
+Hence, the entire problem size is given by the total number of
+weights, biases, and input neurons: W +(N −I)+I = W +N.
+We have a fast memory that can hold at any given time a number
+of values (of this data type) that is given by the parameter M,
+where we assume M ≥ 3. In this model the single parameter
+M defines the whole architecture: fast memory of size M
+and slow memory of unlimited size. We can perform arbitrary
+computations “for free” on data that is stored in fast memory,
+including application of activation functions to those values.
+In particular, if a connection, the value of its input neuron,
+and the partial sum of its output neuron (which we consider
+initially the bias) are stored in fast memory, then we can add
+the product of the input value and the connection weight to
+the partial sum of the output neuron (and we assume that no
+additional memory is required for this operation). If we want
+to do a computation that requires a value that is currently only
+stored in slow memory, we first need to move this value from
+slow to fast memory. This movement counts as 1 read-I/O.
+Furthermore, if our memory is full, we first need to free up
+space before we can read new values. To do this, we can
+delete values. However, if we need a value again in future
+computation steps or if it is the value of an output neuron,
+we have to write the value to slow memory before deleting
+it. Deletions are for free whereas each write operation counts
+as 1 write-I/O. Initially, all N + W parameters are stored in
+slow memory. The number of I/Os of a computation is the sum
+of read-I/Os and write-I/Os. The goal is to perform inference
+computation, namely compute and store (i.e., write back to
+slow memory) the value of all output-neurons, with as few
+I/Os as possible. We formally define the optimum that we aim
+for:
+Definition 1. For a given FFNN N and fast memory size
+M, we let I/Os(N, M) ∈ N denote the minimum number
+of overall I/Os that are needed to perform inference on
+N with a memory size M, where the minimum is taken
+over all possible computation strategies (i.e., sequences of
+computation-, read-, and write- steps, that solve this problem).
+Let rI/Os(N, M) ∈ N denote the minimum number of read-
+I/Os and wI/Os(N, M) ∈ N denote the minimum number of
+write-I/Os needed.
+A. How can we describe inference-computations in this model?
+We introduce the concept of eviction policy, and show that
+a computation corresponds to an eviction policy together with
+a topological order of the connections. An eviction policy is a
+set of rules or instructions that specify how we evict data from
+fast memory. That is, when the fast memory is full and we
+need to evict a value to create free space, the eviction policy
+determines which of the values to evict. LRU (least-recently-
+used) is the eviction policy defined by always evicting the
+value that has been used least recently among all values in
+fast memory. RR (round-robin) is the eviction policy defined
+by having a pointer specifying the value to be evicted next
+and moving this pointer one place to the right whenever we
+evict a value (and moving the pointer again to the first place
+of the memory when we reach the end). The eviction policy
+MIN (also known as Belady’s optimal replacement algorithm)
+is defined by always evicting the value that will be referenced
+farthest in the future (if there are values that will not be used
+again, any of those is evicted). It has been shown that MIN
+uses the minimal number of I/Os for a given sequence of
+computation steps [32]. Notice that while it is difficult to
+implement MIN eviction policies for general computations,
+in the case of FFNN inference it is trivial to implement it
+offline once we fixed a topological order in which we process
+the weights (always evict the activations adjacent to weights
+farthest away in the given topological order).
+We assume that an efficient eviction policy is provided. That
+is, when we evict a value that is either (1) already stored
+in slow memory (for example, when we read the value of a
+computed neuron to use it for its outgoing connections, but do
+not change it), or (2) a value that we will not need again in the
+future (a computed non-output-neuron has been used already
+for computing all neurons that depend on it), we simply delete
+it from fast memory, without spending a write-I/O.
+Now, let e1, e2, . . . , eW be a topological order of the
+connections of the neural network (that is, whenever ei and
+ej are connections for which the output-neuron of ei is the
+input-neuron of ej, we have i < j). Together with an eviction
+policy, a topological order of the connections gives rise to an
+inference-computation in a natural way, shown by Algorithm
+1.
+Algorithm 1 Inference Algorithm
+1: Input: A topological order of the connections e1, . . . , eW
+2: Output: Values of the output neurons
+3: for i = 1 to W do
+4:
+Read the connection ei = (a, b, w);
+5:
+if value of the input neuron, na, is not in fast memory
+then
+6:
+Read na (possibly, first evicting one value, if fast
+memory is full);
+7:
+end if
+8:
+if partial sum of the output neuron, nb, is not in fast
+memory then
+9:
+Read nb (possibly, first evicting one value, if fast
+memory is full);
+10:
+end if
+11:
+Update nb = nb + w · na
+12:
+if there is no connection after ei with output-neuron nb
+then
+13:
+Apply the activation function: nb = f(nb);
+14:
+end if
+15: end for
+Notice that this pseudocode generalizes all “standard
+ways” of performing inference. For example, matrix-vector-
+multiplication based inference would correspond to orders that
+start with all connections from the first layer, followed by
+the connections of the second layer and so on. Yet, it also
+
+4
+allows us to perform inference on FFNN-architectures given
+by any possible DAG (including those with very “chaotic” skip
+connections) and not just those that are layered.
+As for optimizing I/Os, it gives us more flexibility, as it
+allows us to employ any possible topological order of the
+connections, including those that do not correspond to layer-
+after-layer computations. For example, it allows us to start
+computing neurons of the layers i+1, i+2, . . . even when not
+all neurons of the i-th layer have been computed. This can save
+I/Os, because when we finish computing a neuron from the
+i-th layer, we can directly reuse it to start computing neurons
+of the (i + 1)-st layer, instead of storing it and reading it again
+later. Since for each connection we perform a computation for
+which we need the value of the input neuron and the partial
+sum at its output neuron, we would like to find an order in
+which connections that have neurons in common, are clustered
+together.
+III. BOUNDS FOR INFERENCE
+In this section we present generic bounds on the I/O-
+complexities of inference. They serve as guidelines for the
+more instance-specific optimizations that we consider later on.
+Theorem 1. Let N be a connected FFNN and assume M ≥ 3.
+Then the optimal number of I/Os for inference satisfies
+W + N + S ≤ I/Os(N, M) ≤ 2 · (W + N − I).
+(1)
+The optimal number of read-I/Os for this problem satisfies
+W + N ≤ rI/Os(N, M) ≤ 2 · W + N − I.
+(2)
+The optimal number of write-I/Os for this problem satisfies
+S ≤ wI/Os(N, M) ≤ N − I.
+(3)
+Proof. The weights, biases and input values have a total size
+of N + W. Since we cannot perform inference without having
+read all of these data at least once, we obtain the lower bound
+for the number of read-I/Os. Likewise, we cannot perform
+inference with less than S write-I/Os, because (by definition
+of the inference problem) we need to write the values of all S
+output-neurons. Hence, the lower bound for the total number
+of I/Os follows by adding the lower bounds for the read- and
+write-I/Os.
+Now it remains to show that we can do inference using
+no more than N + 2 · W − I read-I/Os, N − I write-I/Os
+and 2 · (N + W − I) overall I/Os. To achieve this, we fix
+a topological order of the non-input neurons: n1, . . . , nN−I.
+Then, we reorder the connections in such a way that their
+output neurons appear in the order of this topological order
+(notice that this is also a topological order of the connections;
+see Figure 1 for an illustration of this association of topological
+orderings). Notice that this order is naturally partitioned into
+intervals: It begins with an interval of connections ending in
+n1, followed by the connections ending in n2 and so on.
+We now show that performing inference in this order with
+our inference-algorithm (Algorithm 1 from the main paper) and
+a MIN eviction policy, costs at most 2W + N − I read-I/Os
+and N − I write-I/Os. As we start reading the connections, we
+Fig. 1: Associated topological orderings of neurons and
+connections.
+spend 1 read-I/O to read the bias of n1 and then at most 2 read-
+I/Os for each of the connections that end on n1 (1 for reading
+the input neuron and 1 for reading the connection itself). Once
+we passed through this interval of connections that end on n1,
+we apply the activation to finish the computation of neuron n1.
+Then we continue with the interval of connections that end
+on n2. Also in this interval of connections as well as in all
+following intervals of connections, we spend 1 read-I/O for
+the bias of the output-neuron and at most 2 on each of the
+connections. Since there is exactly one interval of connections
+for each of the N − I non-input neurons, we spend at most
+N − I read-I/Os to read the biases and at most additional
+2W for the connections and their input-neurons, adding up to
+2W + N − I read-I/Os.
+Now we count the write-I/Os deployed in this computation.
+Since all connections that end in the same neuron follow
+each other, we always compute neurons without having to
+write temporary values (once we start computing a neuron all
+of the following computation steps are also directed towards
+computing this neuron, and we only start computing another
+neuron once the previous is finished). From this it follows, that
+if we spend a write-I/O on some neuron, then we are writing a
+fully computed neuron value to slow memory. And hence, if we
+read and evict this value again, we do not spend another write-
+I/O on this value (since the fully computed value is already
+stored in slow memory, the efficient eviction policy will evict
+this value by deleting it). Therefore, we spend at most one write-
+I/O for each of the non-input neurons. Since there are N − I
+non-input neurons, we have overall at most N − I write-I/Os.
+Adding the read- and write-I/Os, we conclude that the total
+number of I/Os is at most N+2·W−I+N−I = 2·(N+W−I).
+Notice that in both Inequalities 1 and 2, the term on the
+right hand side is at most twice as large as the term on the
+left hand side. The next Proposition establishes that none of
+the generic bounds given by this theorem could be tightened
+
+4
+C
+6
+35
+any further by multiplying it with a constant other than 1.
+Proposition 1. As generic bounds that depend only on W, N, I,
+and S, the bounds in Theorem 1 are tight: For each one of
+them and for any ϵ > 0, there are instances for which the true
+value differs from the bound by a multiplicative factor that lies
+in [1 − ϵ, 1 + ϵ].
+To prove this Proposition, we need to show that for each of
+the bounds of Theorem 1, there are instances that are arbitrarily
+close to the bound (close in terms of multiplicative factors).
+The first lemma establishes that there are instances that
+exactly attain all lower bounds of Proposition 1 (and hence,
+are obviously arbitrarily close).
+Lemma 1. On any FFNN N = [L1, L2, . . . , Ld] in which any
+two consecutive layers Li, Li+1 have together at most M − 1
+neurons, we can do inference, with a number of read-, write-,
+and total I/Os that matches the lower bounds given by Theorem
+1.
+Proof. We prove this by describing a computation strategy that
+achieves this.
+Read the |L1| input values into fast memory. Also initialize
+in fast memory |L2| partial sums with the biases of the neurons
+in L2. Since |L1| + |L2| ≤ M − 1, we have in fast memory
+free space for at least one more value. We use this free space
+to iterate over the weights between L1 and L2 and perform
+the following steps for each one of them:
+1) Read the weight.
+2) Multiply it with the value of its input neuron (which is
+already in fast memory).
+3) Add this product to the partial sum of its output neuron.
+4) Delete the weight from fast memory.
+Once this iteration is finished, we apply the activation function
+to the partial sums of L2, which finishes the computation of the
+neurons in this layer. Now we do not need the values from L1
+anymore and can delete them from fast memory. This gives us
+enough free space to read the biases of L3, and compute this
+layer in the same way. We proceed like this, computing layer
+after layer. Once we finish the computation of the neuron-vales
+of the last layer, we write those S values back to slow memory.
+Notice that we read each of the N −I biases, I input values,
+and W weights exactly once. Hence we use N + W read-I/Os.
+We only use write-I/Os to write the S output values. Hence,
+altogether we use N + W + S I/Os.
+Alternatively, we could have proved the previous Lemma
+by verifying that an FFNN with the mentioned properties,
+can be constructed with the Compact Growth method that we
+introduce later. The next lemma shows that there are instances
+that use a number of read- and total I/Os that are arbitrarily
+close to their upper bound.
+Lemma 2. For every ϵ > 0, there exist FFNNs N (of arbitrary
+size) that have I/Os(N, M) > (1 − ϵ) · 2 · (W + N − I) and
+rI/Os(N, M) > (1 − ϵ) · (2 · W + N − I).
+Proof. This is the case for architectures, where a large
+proportion of the neurons are input neurons. For example,
+in a tree with I input neurons, all connected to a single
+output neuron, we have I/Os(N, M) = ·2 · (W + N − I)
+and rI/Os(N, M) = 2 · W + N − I.
+The next lemma shows that there are instances that use
+a number of write-I/Os that is arbitrarily close to the upper
+bound.
+Lemma 3. For every ϵ > 0, there exist FFNNs N (of arbitrary
+size) that have wI/Os(N, M) > (1 − ϵ) · (N − I).
+Proof. Notice that this inequality is satisfied by any FFNN
+for which all neurons are either input or output neurons. But
+to give less trivial examples, we will construct FFNNs with
+hidden neurons that satisfy this inequality.
+For M, h, w ∈ N+, consider the FFNN that has I input
+neurons, S output neurons, and one hidden layer with h neurons.
+Clearly, this FFNN requires at least S write-I/Os. So, for any
+parameter configuration I, h, S with S > h·(1−ϵ)/ϵ, we have
+wI/Os(N, M) > (1 − ϵ) · (N − I).
+The three previous lemmata provide the tight extremal cases
+that bring us into the position to prove Proposition 1. Although
+the constructed architectures in the proofs of these lemmata
+were “artificial”, they suffice to show that generic bounds that
+only depend on I, S, N, and W, cannot be tighter (tighter by a
+constant multiplicative factor other than 1) than our bounds. Yet,
+notice that, despite being “artificial”, these extremal cases are
+more than just “a few cornercases”. In fact, the constructions
+in each of these proofs could have been chosen to be arbitrarily
+large.
+Proof of Proposition 1. Lemma 1 shows that the lower bound
+on the number of read-, write-, and total I/Os is tight. Lemma
+2 shows that the upper bound on the number of read-I/Os and
+total I/Os is tight. Lemma 3 shows that the upper bound on
+the number of write-I/Os is tight.
+Notice that none of the bounds depends on M. This is
+surprising, because for most computational problems, I/O-
+complexities depend strongly on the memory size1. As we
+increase M, we gain more flexibility that we can use to avoid
+I/Os. Also surprising is the fact that these bounds do not depend
+on specific properties of the network, except for its sizes. This
+does not mean that the overall I/O-complexity does not depend
+on these quantities (memory size and FFNN architecture),
+but from the tightness of our bounds we conclude that the
+dependence is only moderate.
+For write-I/Os, this is very different. The bounds for write-
+I/Os given by Theorem 1 are less tight. In fact, they can
+be arbitrarily loose (despite being optimal in the sense of
+Proposition 1). The problem for write-I/Os is that, it is simply
+not possible to give interesting bounds that only depend on
+W, N, I, and S but no other factors such as the connection
+order, the memory size M, and the FFNN architecture (this
+impossibility is implied by Proposition 1). The following
+Proposition shows that inference in a layer-after-layer fashion
+can be arbitrarily more expensive than using an optimal order.
+1For example, in the case of matrix multiplication, the dependency on M
+is of the order O(1/
+√
+M) [7].
+
+6
+Proposition 2. For every c ∈ N and every memory size M ∈ N,
+there exists an FFNN NM,c, such that performing inference
+in a layer-after-layer fashion, requires at least c times more
+write-I/Os than optimal.
+Proof. Consider a sparse FFNN that has c+2 layers: 1 neuron
+in the input layer, 2M neurons in each of the c hidden layers,
+and one neuron in the output layer. Let each hidden neuron
+have exactly one incoming and one outgoing connection, let the
+input-neuron be connected to each neuron of the first hidden
+layer, and let the output-neuron have incoming connections
+from all neurons of the last hidden layer (in other words, this
+FFNN has 2M chains of neurons of length c + 2 that meet in
+the input and output neuron). If we perform inference on this
+FFNN in a layer-after-layer fashion, we need the values of all
+2M neurons of each hidden layer to compute the next layer.
+Since our fast memory has only capacity for M neurons, we
+will need to store the other M neurons, requiring at least M
+writes for each hidden layer. Hence, if we perform inference
+on this FFNN in a layer-after-layer fashion, we would need to
+use at least M × c write-I/Os. Yet, if we compute chain after
+chain, a single write-I/O would suffice.
+IV. CONNECTION REORDERING: ADAPTING THE ORDER TO
+THE FFNN AND HARDWARE
+In this section we introduce Connection Reordering, which is
+a method to optimize the topological order of the connections
+for a given FFNN architecture and memory size M. This
+method depends on the following hyperparameters: The win-
+dow size ws ∈ N, the cooling rate σ ∈ R, and the number
+of iterations T ∈ N. The high-level idea of this method is
+based on Simulated Annealing [33]: Over T iterations, we
+perform random changes to the topological order (we call this
+creating neighbors) and either retain or discard the changes
+with a probability that depends on the quality of the old and
+the new order (called updating). Now, we fully specify this
+method by describing the processes of creating neighbors and
+updating.
+A. Creating neighbors
+We start with a topological order of the connections
+e1, e2, . . . , eW and let OldI/Os ∈ N denote the number of I/Os
+used in this topological order. The number of I/Os obviously
+depends on the memory size and eviction policy, but those are
+fixed throughout the execution of this algorithm and hence the
+number of I/Os depends only on the topological order of the
+connections.
+First, we choose uniformly at random one connection ei. We
+let w be an integer that we choose uniformly at random from
+{0, 1, . . . , ws − 1}. We consider the window of connections
+ei, ei+1, . . . , emin(i+w,W ). We choose the direction in which
+we move the connections from this window: Either left or right
+with probability 0.5.
+a) Case 1: Moving to the left: If we move the connections
+to the left, we start moving the leftmost connection ei of the
+window. We move ei to the left until we encounter another
+connection es that has the same input neuron as ei, or whose
+output neuron is equal to the input neuron of ei (if we
+never encounter such a connection, we move ei to the very
+beginning of the order). We insert ei right next to es so that
+. . . , es, ei, es+1, . . . is the new order. Note that this is again
+a topological order. Then we continue moving the second
+leftmost connection in the window in the same fashion. We
+continue like this until we moved all connections from this
+window.
+b) Case 2: Moving to the right: If we move the connec-
+tions to the right, we start moving the rightmost connection of
+the window. We move it until we encounter another connection
+ez, which has the same output neuron as ei or whose input
+neuron is equal to the output neuron of ei. We insert ei right
+before ez so that . . . , ez−1, ei, ez, . . . is the new order. Then
+we continue with the second rightmost connection from the
+window. We continue like this until we moved all connections
+from this window.
+B. Updating
+After we moved all connections from the window, we have
+a new topological order. We denote it ˜e1, ˜e2, . . . , ˜
+eW . We
+measure how many I/Os are used to perform inference in this
+new order and call this number newI/Os. If newI/Os<oldI/Os,
+we certainly update e1 = ˜e1, e2 = ˜e2, . . . , eW =
+˜
+eW and
+oldI/Os=newI/Os.
+If newI/Os≥oldI/Os, we either update or go back to the old
+order. We decide this at random, choosing to update with a
+probability 2−(newI/Os−oldI/Os)·tσ, where t is the iteration number.
+V. COMPACT GROWTH: ADAPTING FFNNS TO HARDWARE
+In this section we present a construction scheme that
+completely characterizes the FFNN architectures that allow
+inference with a minimal number of I/Os for a given memory
+size M. This answers the following question: For a given
+memory size M, which are the FFNN architectures on which
+we can do inference with a number of I/Os that matches the
+lower bound given by Theorem 1? Or equivalently: Which are
+the FFNNs on which we can do inference without having to
+write or read intermediate values due to lack of memory? The
+idea is to couple the construction of the FFNN closely to the
+steps in the inference computation. More precisely, we will
+build the FFNN by a sequence of steps of four different types.
+We introduce the concepts of pebbles and bags to reason
+about I/Os in our computation. Each pebble corresponds to
+one neuron. A pebble can be either gray (if it is not yet fully
+computed) or black (if it is fully computed and can be used by
+its outgoing connections). The bag represents the fast memory.
+A. The construction rules
+We start our construction with an “empty FFNN” (there
+are no neurons and no connections) and an empty bag that
+represents the content of our fast memory. We build up the
+FFNN by a sequence of construction steps of the following four
+types (for each type we write in parentheses the corresponding
+computation step into which it can be translated).
+1) Whenever we have less or equal than M − 2 pebbles in
+our bag, we can either add a gray or a black pebble to our
+
+7
+bag and simultaneously add an isolated neuron to our FFNN
+(reading a neuron into fast memory).
+2) When we have a black and a gray pebble in our bag, we
+can draw a connection in our FFNN, that starts at the neuron
+that corresponds to the black pebble and ends at the neuron
+that corresponds to the gray pebble. (multiply weight with the
+value of the input neuron and add it to the partial sum of the
+output neuron).
+3) We can turn a gray pebble into a black pebble (finish
+computation of the neuron by applying activation function).
+4) We can remove a black pebble from the bag (delete from
+fast memory).
+The following theorem summarizes the properties of FFNNs
+built by a sequence of the above steps.
+Theorem 2 (Compact growth). Let N be a connected FFNN
+with W weights and N neurons. Given a memory of size
+M ≥ 3, we can do inference on N with N + W read-I/Os
+and S write-I/Os if and only if N can be constructed by the
+compact growth scheme that we just described.
+Proof of Theorem 2. If we can build N by a sequence of
+these construction steps, then the corresponding steps in the
+parentheses, describe a valid sequence of computation steps for
+inference. By only allowing to insert pebbles when we have
+(strictly) less than M − 1 pebbles in the bag, we ensure that
+there are never more than M −1 pebbles in memory and hence
+there is one free space for a connection. Hence, this sequence
+of computation steps can be executed with a fast memory of
+size M without requiring any I/Os on temporary values. This
+proves the “if” part of the statement.
+Conversely, assume for a given N and M ≥ 3, we can
+order the connections in a way that allows us to perform
+inference with a minimal number of I/Os. Denote this order
+e1, e2, . . . , eW . We will show how we can translate the
+computation steps into a sequence of pebble-moves that creates
+the FFNN. According to Theorem 1, a minimal number of I/Os
+means that we perform W + N read-I/Os and S write-I/Os.
+Since this is the number of read-I/Os required to read all
+necessary parameters once, and write out the values of the
+output-neurons once, these I/Os only suffice to perform these
+necessary reads and writes. In particular, we cannot spend any
+more I/Os to write or read temporary values. Hence, for each
+neuron we will read one value at some moment (the bias or the
+input value), then have this value in fast memory for a certain
+interval of time during which we update it according to the
+progress of the computation, and then evict the fully computed
+neuron value when it is not needed anymore. After that we will
+not be able to use this value again. We translate this into our
+pebble construction: Putting a gray pebble into our bag, having
+it there for a certain amount of time, transforming it into a
+black pebble (once the neuron is fully computed), and then
+removing it (when the neuron value is not needed anymore
+and is evicted).
+During this interval, this optimal computation strategy must
+use all incoming and outgoing connections of this neuron: The
+incoming connections during the first part of the interval (when
+the neuron is not yet fully computed and hence the pebble
+is gray) and the outgoing connections during the second part.
+Further, any time we use a connection, the “other end” of
+the connection must also be in fast memory and hence the
+corresponding pebble must be in the bag (remember that we
+insert a pebble whenever we read a bias/input and remove
+the pebble when we evict the value). Also, any time we use
+a connection, the color of the pebble of the input neuron is
+already black (because we turn a pebble black as soon as
+the neuron is fully computed) and the pebble of the output
+neuron is still gray. Hence, as we use the connections on the
+computation side, we can draw the connections on the side of
+the pebble model.
+Due to the restricted memory size, we can assume without
+loss of generality that we have never more than M − 1 neuron
+values in fast memory (if we read an M-th value, there is no
+more space for a connection, and the computation is stuck
+until we remove one neuron value; but then we could just
+as well first remove this neuron value before reading another
+one). And hence, we can assume without loss of generality
+that we only read values when there are at most M − 2 values
+in fast memory. On the side of the pebble construction, this
+ensures that we only add pebbles, when there are at most
+M − 2 pebbles in the bag.
+We can apply this theorem to the problem of optimal
+hardware for a given FFNN. To do this, we consider the
+bandwidth of an FFNN, defined as the smallest k ∈ N for
+which there exists a topological order of the neurons, such
+that any two connected neurons are at most k steps apart in
+the topological order. We prove now that when FFNNs have
+bandwidth k, we can build them with compact growth, using
+M = k + 2.
+Corollary 1. Let N have bandwidth k ∈ N. If we have a
+memory size M ≥ k + 2, then we can perform inference on
+N without reading or writing any temporary values.
+Proof. Let N have bandwidth k. This means that there
+exists a topological order of the neurons n1, n2, . . . , nN,
+such that for any i
+∈
+{1, . . . , N}, all incoming con-
+nections
+to
+ni
+are
+among
+the
+k
+previous
+neurons
+{nmax(1,i−k), nmax(1,i−k)+1, . . . , ni−1}. So, we can build N
+with compact growth, by adding the pebbles according to the
+topological order of the neurons, and when the bag is full,
+removing the pebble that was added k + 1 steps ago. This
+ensures that when we add a pebble of a neuron n to the bag,
+the pebbles of all other neurons on which n directly depends
+are also in the bag and we can add all connections that go to
+n. Hence, with a memory size of k + 2, we can build N.
+VI. EXPERIMENTAL EVALUATION
+To demonstrate both the theoretical accuracy as well as
+the practical utility of our results, we run both simulated
+experiments counting I/Os and measure CPU performance
+of the reordered sparse neural networks compared with high-
+performance libraries.
+As inputs, we use two types of networks: MLPs with varying
+depth, breadth, and density; and fully-connected layers from
+
+8
+16,907 / 15,296 / 13,496
+(a) Neurons per Layer
+1,002 / 1,002 / 1,002
+(b) Depth
+10,169 / 8,773 / 6,973
+(c) Connection Density
+5
+100
+200
+300
+400
+500
+600
+Fast Memory Size (M)
+0
+20,000
+40,000
+60,000
+80,000
+100,000
+Total I/Os
+Initial
+Reordered
+Lower Bound
+(d) Fast Memory Size
+Fig. 2: Connection Reordering for varying sparse neural
+network properties. Unless otherwise stated, the network is a
+500-neuron wide, 4-layer MLP, with an edge density of 10%
+and M = 100. The lower bound is obtained from Theorem 1.
+When too small, I/Os for MLPs are annotated.
+the popular BERT [34] Transformer [23] neural network. For
+the former, we generate five randomly-sparse FFNNs (obtained
+by random edge sampling or using Compact Growth). For the
+latter, we use a pre-trained BERTLARGE neural network and
+perform magnitude pruning [21].
+A. Simulated Experiments
+To test Connection Reordering (CR) and Compact Growth
+(CG), we implement Algorithm 1 and cache simulation, along
+with LRU, RR, and MIN eviction policies. We run each
+experiment configuration with five random MLPs and BERT,
+reporting the median values and 95% nonparametric confidence
+intervals as error bars.
+1) Connection Reordering: We evaluate CR by optimizing
+the I/Os of random sparse FFNNs with varying structural and
+sparsity properties. Figure 2 shows four dimensions in which
+we vary a baseline FFNN: a 10% dense, 4-layer MLP with
+500 neurons in each layer and one output neuron, running
+inference with a fast memory size of 100 elements. We either
+vary density, width, depth, or fast memory size, while keeping
+the other parameters constant at their baseline value. We use a
+MIN eviction policy. Initially, we order the connections as in
+the proof of Theorem 1, so that we are guaranteed to not use
+more than twice the optimal number of I/Os. Then, we apply
+CR with T =1,000,000, σ = 0.2, and set ws to four times
+the average in-degree of the network. As we can see in these
+examples, CR reduces the total number of I/Os further by up
+to 43.5% and brings us up to 97.4% closer to the theoretical
+lower bound. Further, we can see that CR gives consistently
+large improvements across all parameter configurations (except
+for the cases where the initial I/O-complexity is already close
+to the lower bound).
+2) Compact Growth: We generate three FFNNs with Com-
+pact Growth as described in Appendix B. These FFNNs
+correspond to the values Mg = 100, 300, and 500. Then we
+count the I/Os used for these FFNNs with varying sizes of
+fast memory M. These numbers are shown in Figure 3. As
+predicted by Theorem 2, we can see for all three FFNNs that
+when M ≥ Mg, we use a minimal number of I/Os. In these
+cases, we directly achieve this minimal number of I/Os using
+the connection order produced by CG, without applying CR.
+When we additionally apply CR, we are also able to use
+a minimal number of I/Os in some ranges of memory size
+M with M < Mg. Of course this is not a contradiction to
+Theorem 2: The theorem says that we can achieve a minimal
+number of I/Os using a memory size M if and only if the
+FFNN can be built using Mg = M. But this does not imply
+that the same FFNN cannot be built with Mg < M.
+3) Cache Eviction Policies:
+Figure 4 shows the I/O-
+complexity evolution (over Simulated Annealing iterations)
+of random FFNNs, for the RR, LRU, and MIN cache eviction
+policies. Owing to the update scheme, we observe a decaying
+convergence rate towards a minimum value, where the majority
+of I/O reduction happens in the first 10,000 iterations. It is
+interesting to observe that upon optimization, both RR and
+LRU converge to similar I/Os, which indicates that CR can
+tune FFNNs for hardware architectures with specific caching
+mechanisms.
+4) Size of Fast Memory: In Figure 5, we show how the
+number of I/Os changes as we increase the size of the fast
+
+9
+5
+100 105 205 300 305 405 500 505 605 705 805 9051005
+Fast Memory Size (M)
+0
+2,000
+4,000
+6,000
+8,000
+10,000
+Total I/Os
+Initial
+Reordered
+Lower Bound
+(a) Mg = 100
+5
+100 105 205 300 305 405 500 505 605 705 805 9051005
+Fast Memory Size (M)
+0
+2,000
+4,000
+6,000
+8,000
+10,000
+12,000
+Total I/Os
+Initial
+Reordered
+Lower Bound
+(b) Mg = 300
+5
+100 105 205 300 305 405 500 505 605 705 805 9051005
+Fast Memory Size (M)
+0
+2,000
+4,000
+6,000
+8,000
+10,000
+12,000
+Total I/Os
+Initial
+Reordered
+Lower Bound
+(c) Mg = 500
+Fig. 3: Study of Compact Growth-generated FFNNs designed
+for different fast memory sizes Mg (1, 000 neurons with in-
+degree of 4 and one output-neuron with in-degree Mg).
+Round-Robin
+LRU
+Optimal (MIN)
+Lower Bound
+Fig. 4: I/Os for different cache eviction policies.
+memory, both before and after Connection Reordering (CR).
+As we can see, once the fast memory is sufficiently large,
+we use a number of I/Os that matches the theoretical lower
+bound provided by Theorem 1, with and without CR. But for
+insufficient memory size, we see that with CR we reduce I/Os
+and converge faster to the lower bound.
+Reordered
+Initial connection order
+Lower Bound
+Fig. 5: Study of different fast memory sizes (M) on random
+sparse FFNNs (3 layers of 500 neurons each followed by one
+output neuron, 1% density).
+5) Application to Transformer-model BERT: Transform-
+ers [23] are a DNN architecture built from so-called encoders
+and decoders. A characteristic ingredient of transformers are
+attention heads, which make them suitable for natural language
+processing and related tasks. A prominent example of a
+Transformer is BERT [34], which was introduced in 2019.
+BERT’s groundbreaking results made Transformers arguably
+the most popular and extensively investigated DNN architecture
+of the past years. The main challenge encountered when
+deploying transformers is the large size of these models and the
+resulting demand for compute resources [24]. BERT includes
+several FFNNs of depth 2 and weight matrices of dimensions
+1024 × 4096 and 4096 × 1024. The major part of BERTs
+parameters and compute time for inference come from these
+FFNNs [24]. This makes them a prime target for pruning.
+We took one of these FFNNs and pruned it by removing
+the connections with the weights of smallest absolute value.
+Figure 6 shows the I/O-counts and the I/O lower bounds
+for inference on this FFNN. We considered different levels
+of sparsity and different cache eviction policies. Further, we
+assumed a fast memory of size 100.
+B. Performance Experiments
+To validate the performance benefits of I/O reduction, we
+perform batched inference on an Intel CPU and measure the
+performance. For all experiments, we reorder the FFNNs with
+the simulated optimal (MIN) eviction policy, and then run the
+FFNN before and after reordering, as well as in the traditional,
+layer-based approach using MKL for sparse-dense matrix
+matrix multiplication (CSRMM). Using batched inference (as
+is performed in production environments) enables the use of
+SIMD instructions and to better saturate the memory bandwidth.
+We run each experiment 10 times, with a batch size of 128,
+reporting median values and error bars the shortest and longest
+execution time. Annotations show the speed-ups that we obtain
+with our methods (without and with reordering) relative to the
+layer-based inference approach.
+The hardware used for the experiments is a 32-core Intel
+Xeon Gold 6130 CPU (running at 2.10 GHz) with 1.5 TB
+
+10
+1%
+10%
+25%
+50%
+75%
+100%
+Density
+0
+2,500,000
+5,000,000
+7,500,000
+10,000,000
+12,500,000
+15,000,000
+Total I/Os, Optimal (Min)
+153607,
+139881,
+91057
+ 
+ 
+ 
+Initial
+Reordered
+Lower Bound
+(a) Optimal (MIN)
+1%
+10%
+25%
+50%
+75%
+100%
+Density
+0
+5,000,000
+10,000,000
+15,000,000
+Total I/Os, LRU
+173780,
+171279,
+91057
+ 
+ 
+ 
+Initial
+Reordered
+Lower Bound
+(b) LRU
+1%
+10%
+25%
+50%
+75%
+100%
+Density
+0
+5,000,000
+10,000,000
+15,000,000
+Total I/Os, Round-Robin
+173733,
+173376,
+91057
+ 
+ 
+ 
+Initial
+Reordered
+Lower Bound
+(c) Round-Robin
+Fig. 6: Connection Reordering for an MLP from BERTLARGE
+using different eviction policies, different edge densities and
+M = 100. The lower bound is obtained from Theorem 1.
+RAM. For software, we use Intel oneAPI MKL 2021.3.0 for
+CSRMM, and GCC 10.2.0 for our reordered networks.
+1) Random sparse FFNNs: Starting with the same baseline
+FFNN from the simulated experiments (a 10% dense, 4-layer
+MLP with 500 neurons in each layer and one output neuron)
+we measure the execution time of our methods varying density,
+width, or depth while keeping the other parameters constant at
+their baseline value.
+In Figure 7a we show the execution time for varying degrees
+of density. As we can see, the sparser the FFNN, the larger
+the speedup that we obtain with our method relative to the
+standard, layer-wise way of performing inference. For example,
+at density 0.1% we get a speed-up of about 45x. On the other
+hand, the more sparse the FFNN, the less additional speedup we
+gain by reordering. In fact, somewhat surprisingly, for density
+values below 1% the execution time is slightly lower without
+than with reordering, which we attribute to two causes: (a) the
+100.0%50.0% 25.0% 10.0% 5.0%
+2.5%
+1.0%
+0.5%
+0.1%
+Density
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+Execution Time (ms)
+1.4x,
+1.5x
+1.4x,
+1.6x
+1.7x,
+1.9x
+2.6x,
+2.8x
+4.1x,
+4.2x
+6.8x,
+6.8x
+14.6x,
+14.2x
+24.7x,
+24.1x
+45.6x,
+44.3x
+Based on Sparse Matrix Multiplication (MKL)
+Our Method Before Reordering
+Our Method After Reordering
+(a) Connection Density
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+Depth
+0
+2
+4
+6
+8
+Execution Time (ms)
+5.2x,
+5.7x
+3.2x,
+3.5x
+2.6x,
+2.8x
+2.2x,
+2.4x
+2.0x,
+2.2x
+1.8x,
+2.0x
+1.8x,
+1.9x
+1.7x,
+1.8x
+1.6x,
+1.8x
+1.6x,
+1.7x
+1.5x,
+1.7x
+1.5x,
+1.6x
+Based on Sparse Matrix Multiplication (MKL)
+Our Method Before Reordering
+Our Method After Reordering
+(b) Depth
+250
+500
+750
+1000
+1250
+1500
+Breadth
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+Execution Time (ms)
+7.1x,
+7.5x
+2.6x,
+2.8x
+1.8x,
+1.9x
+1.5x,
+1.7x
+1.4x,
+1.5x
+1.3x,
+1.4x
+Based on Sparse Matrix Multiplication (MKL)
+Our Method Before Reordering
+Our Method After Reordering
+(c) Breadth
+Fig. 7: Execution time for randomly-sparse FFNNs with
+different methods.
+relatively low reduction in I/Os (see Section VI-A1); and (b)
+the small size of the networks, which makes the performance
+benefit negligible due to the FFNNs fitting in L2 caches.
+Another effect shown in Figure 7a is speedup compared with
+MKL on 100% dense FFNNs. The reason for this phenomenon
+is that we use CSRMM in that case for consistency in measure-
+ments, rather than a dense matrix- dense matrix multiplication
+(GEMM), which explains the observed slowdown.
+In Figures 7b–7c we show the execution time as a function
+of the depth (Figure 7b) and breadth (Figure 7c) of the FFNN
+(while keeping the other structural parameters constant at their
+baseline values). In both cases we observe speedups that grow
+linearly with depth and breadth (and hence, with the size of
+
+11
+10.0%
+1.0%
+0.1%
+Density
+0
+5
+10
+15
+20
+25
+30
+35
+Execution Time (ms)
+0.8x,
+1.0x
+2.1x,
+2.1x
+11.7x,
+11.9x
+Based on Sparse Matrix Multiplication (MKL)
+Our Method Before Reordering
+Our Method After Reordering
+Fig. 8: Execution time for BERTLARGE with Connection
+Reordering and varying densities.
+the instance), however, with diminishing returns as the network
+grows in one dimension and not in the others.
+In sum, our MLP performance experiments align with the
+simulation results, indicating that reordering is beneficial in
+practice for sparse neural networks, especially for cases where
+density is low.
+2) BERT: In Figure 8 we show the execution times for
+inference on the previously introduced encoder MLP from
+BERTLARGE with varying degrees of density. In the case of
+MKL inference at 10% density we had one clear outlier (as
+defined by Tukey’s method [35]; one run took 106 ms while in
+the other 9 runs it took about 17 ms) which we removed from
+the data shown in this figure. The figure shows that runtime
+after reordering is always lower than before reordering, and
+confirms that even for realistic network modules with only two
+layers, there is a benefit in using our method, especially for
+low densities.
+VII. DISCUSSION
+A. Prefetching
+Modern systems support prefetching to accelerate I/Os which
+is not considered in the presented model. Notice however that
+the effect of prefetching is “orthogonal” to the optimizations
+considered in this paper as prefetching does not reduce the
+overall amount of I/Os but it only affects the time the I/Os
+take.
+B. Cost of using a given topological order
+The computations of “good” (that is, I/O-efficient) topo-
+logical orders are performed “offline” and once the order is
+determined and we perform inference, we do not need to look
+at it explicitly again as it is encoded in the way the connections
+are laid out. Hence, during inference there is no additional cost
+associated with processing the connections according to any
+given topological order.
+C. The term “I/O”
+The theory in the paper applies not only to caches, but local
+memory of any kind (including, e.g., registers). I/O refers to a
+load/store from a "slow" memory, which could be represented
+by off-chip memory, communicating with another node, or even
+scratch-pad memory. The work does neither assume nor rely
+on caches, and covers aspects such as how to stream memory
+efficiently into a chip.
+VIII. CONCLUSIONS
+We established tight bounds on the I/O-complexity of FFNN
+inference. We present a 2-optimal computation strategy and
+show how combinatorial optimizations on the order of the
+connections can lead to further improvements in I/Os. Given
+a fixed memory size, Compact Growth allows us to create
+FFNNs on which inference can be performed I/O-optimally:
+we do not need to read nor write any temporary values.
+Our work provides theoretical insight into the problem of
+mapping existing neural networks to hardware, such that energy-
+consumption and latency during deployment are minimized.
+Most importantly, these insights indicate that the improvements
+are achieved only when dispensing with the straightforward,
+layer-by-layer manner of processing DNNs. In turn, our
+proposed Connection Reordering overcomes this misconception,
+reducing the I/O costs significantly. In our experiments on
+real hardware we observe speedups of up to 45× relative to
+the standard (CSRMM-based) way of performing inference.
+Further, between our 2-optimal computation strategy and the
+strategy that is further improved by connection reordering, we
+observe speedups of up to 1.17×.
+ACKNOWLEDGEMENTS
+This work was supported by the EU Horizon project
+GLACIATION under grant agreement No. 101070141. T.B.N.
+is supported by the Swiss National Science Foundation (Am-
+bizione Project #185778). We also acknowledge the Swiss
+National Supercomputing Centre (CSCS) for support and access
+to computational resources.
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+APPENDIX
+A. How we generate random FFNNs
+We describe how we generate random FFNNs of width
+w, depth d, and sparsity p: For each non-output neuron, we
+determine how many outgoing connections it has, by drawing
+uniformly at random an integer k between 1 and max(1, ⌈2 ·
+p · #of neurons in the next layer − 1⌉). Then, we connect this
+neuron to k randomly chosen neurons of the next layer (k ≥ 1
+ensures that the FFNN is connected and the output neuron is
+connected to all neurons of the last hidden layer). Once we
+determined all connections, we order the connections layer-
+by-layer with respect to their output neuron (according to the
+proof of Theorem 1, this ensures that we have a connection
+layout that uses not more than twice the optimal number of
+I/Os).
+B. How we generate FFNNs with Compact Growth
+We start with Mg − 2 readily computed input neurons in the
+bag (in our experiments we used Mg = 100, 300, and 500).
+
+13
+Then we iterate from 1 to 1000, and in each iteration: 1) we
+add a new neuron 2) we draw incoming connections to this
+neuron from 5 other randomly chosen neurons from the bag 3)
+we remove the last of these 5 randomly chosen neurons from
+the bag. After the 1000 iterations, we add one more neuron
+(the output neuron) and draw connections from all remaining
+neurons in the bag to this output neuron.
+
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@@ -0,0 +1,922 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf,len=921
+page_content='1  A Theory of I/O-Efficient Sparse Neural Network  Inference  Niels Gleinig, Tal Ben-Nun, Torsten Hoefler  {niels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='gleinig, talbn, torsten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='hoefler}@inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='ch  ETH Zürich  Abstract—As the accuracy of machine learning models in-  creases at a fast rate, so does their demand for energy and  compute resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' On a low level, the major part of these  resources is consumed by data movement between different  memory units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Modern hardware architectures contain a form  of fast memory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=', cache, registers), which is small, and a slow  memory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=', DRAM), which is larger but expensive to access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We  can only process data that is stored in fast memory, which incurs  data movement (input/output-operations, or I/Os) between the  two units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' In this paper, we provide a rigorous theoretical analysis  of the I/Os needed in sparse feedforward neural network (FFNN)  inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We establish bounds that determine the optimal  number of I/Os up to a factor of 2 and present a method that uses  a number of I/Os within that range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Much of the I/O-complexity is  determined by a few high-level properties of the FFNN (number  of inputs, outputs, neurons, and connections), but if we want to  get closer to the exact lower bound, the instance-specific sparsity  patterns need to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Departing from the 2-optimal  computation strategy, we show how to reduce the number of I/Os  further with simulated annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Complementing this result, we  provide an algorithm that constructively generates networks with  maximum I/O-efficiency for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We test the algorithms and  empirically verify our theoretical and algorithmic contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  In our experiments on real hardware we observe speedups of up  to 45× relative to the standard way of performing inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Index Terms—Neural Network Inference, Sparse Neural Net-  works, I/O-Complexity, Simulated Annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' INTRODUCTION  Almost all modern computing systems deploy different  memory units: Some are fast but small and some are large but  slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Data can only be processed in fast memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' As the fast  memory is very limited in size, data needs to be moved between  fast and slow memory over the course of a computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' These  data movements are called I/Os (abbreviating input/output-  operations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' They are very expensive in time and energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' For  example, on an NVIDIA V100 GPU, loading values from  global memory costs up to 514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='5× more clock cycles than  one fused-multiply-add (FMA) operation between two 32-bit  floating point values (or 14× if stored on an L1 cache) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  In deep learning, and scientific computing in general, I/Os  account for the major part of time and energy expenses [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Hence, it is important to use I/Os efficiently by performing  computations in a way that makes use of “data locality” and  “data reuse”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We need to schedule the computational steps in  such a way that accesses to the same data are close together,  sparing eviction of values from fast memory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=', cache) and  reading them again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Therefore, optimizing the implementation  of DNN inference [3] and training [4] often requires tailor-  made implementations for the available memory architectures  in order to reach high utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  In this paper, we analyze and optimize I/Os with a theoretical  model that applies across the wide variety of hardware in which  this phenomenon occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We consider exclusively sparse Feed-  Forward Neural Networks (FFNNs) without shared connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  This model contains, but is not limited to, pruned Multi-Layer  Perceptrons (MLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We assume that the FFNN is given as a list  of weighted edges in a Directed Acyclic Graph (DAG) together  with one additional value for each vertex (being the input-value  for input-neurons and the bias for non-input neurons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' In this  setting, the primary way to obtain I/O-efficiency is to “use”  the connections in an efficient order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' That is, when we use  a connection, the value of the input neuron and the value of  the partial sum of the output neuron should ideally already be  in fast memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Therefore, we need to find an order of the  connections, in which the connections that have neurons in  common, are close together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Finding an order which is optimal with respect to this  property is a combinatorial problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We will show that if  we use a traditional computational order, which corresponds  to a natural interpretation of FFNN inference (for example,  seeing it as a sequence of matrix vector multiplications) we  can be far from optimal: The number of read-I/Os could be  up to twice the optimal number, and the number of write-I/Os  could differ from the optimal number by an arbitrarily large  factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Computations in ML have some special features that we can  use when aiming for I/O-efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Unlike the research on  I/O-efficient computing outside of ML, there are more degrees  of freedom that we can exploit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' For example, there are several  acceptable solutions to a given problem: Two different FFNNs  may produce different outputs, yet both can have a satisfying  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' It thus makes sense to adapt the neural architecture  to the hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' On the other hand, there is additional freedom  on the side of the hardware, as it is not uncommon to search  or even build hardware for specific ML applications [5], [6],  [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Hence, our investigation is directed towards answering the  following questions: (1) If we are given an FFNN and a fast  memory of a given size, what is the minimal number of I/Os  needed to perform inference on this FFNN?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' (2) For a given  FFNN, what is the smallest memory size with which we can  perform inference with a minimal number of I/Os (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=', we can  avoid having to read and write intermediate results due to  lack of memory)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' and (3) For a fast memory of a given size,  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='01048v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='DC]  3 Jan 2023  2  what are the FFNN architectures on which we can perform  inference with a minimal number of I/Os?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Background and related work  a) I/O-complexity: There is an extensive line of theoret-  ical research on I/O-efficient algorithms outside of machine  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Hong & Kung’s seminal work [7] is among the  first to investigate I/O-efficient algorithms formally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' This work  introduced a theoretical model called the red-blue pebble game  and used this model to show how standard matrix multiplication  or FFT should be scheduled to use an asymptotically minimal  number of I/Os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' This model has been extended and altered in  various ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' These extensions have been used to establish  lower bounds and matching upper bounds for problems such  as sorting, permutation, and matrix transposition [8];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' as well as  sparse matrix dense vector multiplication [9], and various graph  algorithms [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' However, these solutions are typically only  optimal up to constant multiplicative factors and often these  factors are large or not exactly known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' In fact, it has been shown  that finding exactly optimal solutions to the red-blue pebble  game (corresponding to computation schedules with an exact  minimal number of data movements) is a PSPACE-complete  problem [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Furthermore, this problem has been hypothesized  to be even hard to approximate [11], since inapproximability  is known for related pebble games [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  There are also bounds on the I/O-complexity of general  computations with large proportions of inputs [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' These  bounds apply to the problem considered in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' However,  they are less fine-grained (they do not distinguish between  read- and write-I/Os as we do) and generally less tight as the  ones that we establish in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  In ML-specific algorithms, I/O-efficiency has been practically  identified as a crucial performance factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Performance model-  ing of DNNs is usually derived from operation counts [15] and  sums of fixed values obtained from benchmarking individual  layers [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' As for I/O complexity of DNN evaluation,  the only theoretical work apart from this one, to the best of  our knowledge, is given by Demmel & Dinh [18], who provide  communication bounds for convolutional and pooling layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  b) Network pruning and sparsity: DNN pruning (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=',  removing certain edges or nodes from a network while  preserving accuracy) has been extensively studied over the last  decades [19], [20], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Pruning is of much practical interest  for edge computing devices, as it allows to drastically reduce  the memory footprint and number of floating point operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Dense (fully-connected) layers have been a prime target for  pruning, as they tend to account for a major proportion of the  memory size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' For example, in AlexNet [22], the final dense  layers contain 90% of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Also transformers [23],  which are currently very popular and achieve state-of-the-art  results in NLP tasks, include FFNNs that comprise a large  proportion of the overall size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' For example, BERT [23] includes  several FFNNs of depth 2 and weight matrices of dimensions  1024 × 4096 and 4096 × 1024 and the major part of BERTs  parameters and compute time for inference come from these  FFNNs [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  There are different ways to prune networks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=', by a given  weight threshold [25], [26], the top-k values [27], [28], or  by way of gradual weight elimination [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Regardless of the  method, pruning dense layers leads to sparse and unstructured  networks, for which data-locality is more difficult to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  c) Hardware Architectures: There has been work on  hardware-based optimizations for inference, especially for  the class of sparse FFNNs, where Sze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' [3] provide a  detailed survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Specific approaches include EDEN [29], which  uses approximate DRAM for improving energy-efficiency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Bhardwaj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' [30], who consider communication in inference  in a distributed setting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' and EIE [5] and Eyeriss [31] runs  inference on a compressed CNN directly, using accelerated  sparse matrix-vector multiplications and run-length encoding,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' While these approaches improve certain aspects of  DNN processing, as the network size grows, such architectures  ultimately also resort to multi-level memory hierarchies, which  our approach addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Contributions  Our main theoretical contributions are a collection of  theorems and propositions that analyze the I/Os in FFNN  inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Theorem 1 provides bounds on the number of  I/Os that are needed to perform inference on a given FFNN-  architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' In Proposition 1, we show that these bounds are  optimal in the sense that none of them could be tightened by  multiplying with any factor other than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The lower and upper  bounds on the total number of I/Os differ by a multiplicative  factor of at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The proof of Theorem 1 is constructive  and shows how a computation can be done to use a number of  I/Os within this 2-optimal range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Departing from 2-optimality,  our goal is to get closer to the exact optimum by reordering  the connections beneficially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We approach this problem with  Simulated Annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' In Section V we present a method for  generating FFNNs, which according to Theorem 2 completely  characterizes the architectures on which we can perform  inference with a minimal number of I/Os for a given memory  size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=', exactly matching the lower bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Hence, this  method can be used as a powerful tool to co-design neural  networks and hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' As a corollary of this theorem we  obtain an upper bound on the smallest possible memory size  that allows us to perform inference with maximal I/O-efficiency  (that is, without having to write or read any temporary values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  We test the algorithms on random FFNNs across a wide range  of parameter settings (density, depth, width, memory sizes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' MODEL AND NOTATIONS  We assume that initially all FFNN parameters and all input  values are laid out in slow memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' These data are exactly the  union of the following three types of data:  The weighted connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  One bias value for each non-input neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  One input value for each input neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Each connection is described by an independent parameter  triple (i, j, wij), where i is the input neuron, j the output  neuron, and wij the weight (there is no reuse or sharing of  weights).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We denote the number of connections as W, the  number of neurons N, the number of input neurons I, and the  number of output neurons S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' In our theoretical analysis, we  3  assume the connections (the entire triples describing them), the  numerical values at the neurons (biases, partial sums, outputs  of the activation functions), and all other numerical values  involved in the computation are data types of the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Hence, the entire problem size is given by the total number of  weights, biases, and input neurons: W +(N −I)+I = W +N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  We have a fast memory that can hold at any given time a number  of values (of this data type) that is given by the parameter M,  where we assume M ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' In this model the single parameter  M defines the whole architecture: fast memory of size M  and slow memory of unlimited size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We can perform arbitrary  computations “for free” on data that is stored in fast memory,  including application of activation functions to those values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  In particular, if a connection, the value of its input neuron,  and the partial sum of its output neuron (which we consider  initially the bias) are stored in fast memory, then we can add  the product of the input value and the connection weight to  the partial sum of the output neuron (and we assume that no  additional memory is required for this operation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' If we want  to do a computation that requires a value that is currently only  stored in slow memory, we first need to move this value from  slow to fast memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' This movement counts as 1 read-I/O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Furthermore, if our memory is full, we first need to free up  space before we can read new values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' To do this, we can  delete values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' However, if we need a value again in future  computation steps or if it is the value of an output neuron,  we have to write the value to slow memory before deleting  it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Deletions are for free whereas each write operation counts  as 1 write-I/O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Initially, all N + W parameters are stored in  slow memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The number of I/Os of a computation is the sum  of read-I/Os and write-I/Os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The goal is to perform inference  computation, namely compute and store (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=', write back to  slow memory) the value of all output-neurons, with as few  I/Os as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We formally define the optimum that we aim  for:  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' For a given FFNN N and fast memory size  M, we let I/Os(N, M) ∈ N denote the minimum number  of overall I/Os that are needed to perform inference on  N with a memory size M, where the minimum is taken  over all possible computation strategies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=', sequences of  computation-, read-, and write- steps, that solve this problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Let rI/Os(N, M) ∈ N denote the minimum number of read-  I/Os and wI/Os(N, M) ∈ N denote the minimum number of  write-I/Os needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' How can we describe inference-computations in this model?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  We introduce the concept of eviction policy, and show that  a computation corresponds to an eviction policy together with  a topological order of the connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' An eviction policy is a  set of rules or instructions that specify how we evict data from  fast memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' That is, when the fast memory is full and we  need to evict a value to create free space, the eviction policy  determines which of the values to evict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' LRU (least-recently-  used) is the eviction policy defined by always evicting the  value that has been used least recently among all values in  fast memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' RR (round-robin) is the eviction policy defined  by having a pointer specifying the value to be evicted next  and moving this pointer one place to the right whenever we  evict a value (and moving the pointer again to the first place  of the memory when we reach the end).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The eviction policy  MIN (also known as Belady’s optimal replacement algorithm)  is defined by always evicting the value that will be referenced  farthest in the future (if there are values that will not be used  again, any of those is evicted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' It has been shown that MIN  uses the minimal number of I/Os for a given sequence of  computation steps [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Notice that while it is difficult to  implement MIN eviction policies for general computations,  in the case of FFNN inference it is trivial to implement it  offline once we fixed a topological order in which we process  the weights (always evict the activations adjacent to weights  farthest away in the given topological order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  We assume that an efficient eviction policy is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' That  is, when we evict a value that is either (1) already stored  in slow memory (for example, when we read the value of a  computed neuron to use it for its outgoing connections, but do  not change it), or (2) a value that we will not need again in the  future (a computed non-output-neuron has been used already  for computing all neurons that depend on it), we simply delete  it from fast memory, without spending a write-I/O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Now, let e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' , eW be a topological order of the  connections of the neural network (that is, whenever ei and  ej are connections for which the output-neuron of ei is the  input-neuron of ej, we have i < j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Together with an eviction  policy, a topological order of the connections gives rise to an  inference-computation in a natural way, shown by Algorithm  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Algorithm 1 Inference Algorithm  1: Input: A topological order of the connections e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' , eW  2: Output: Values of the output neurons  3: for i = 1 to W do  4:  Read the connection ei = (a, b, w);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  5:  if value of the input neuron, na, is not in fast memory  then  6:  Read na (possibly, first evicting one value, if fast  memory is full);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  7:  end if  8:  if partial sum of the output neuron, nb, is not in fast  memory then  9:  Read nb (possibly, first evicting one value, if fast  memory is full);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  10:  end if  11:  Update nb = nb + w · na  12:  if there is no connection after ei with output-neuron nb  then  13:  Apply the activation function: nb = f(nb);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  14:  end if  15: end for  Notice that this pseudocode generalizes all “standard  ways” of performing inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' For example, matrix-vector-  multiplication based inference would correspond to orders that  start with all connections from the first layer, followed by  the connections of the second layer and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Yet, it also  4  allows us to perform inference on FFNN-architectures given  by any possible DAG (including those with very “chaotic” skip  connections) and not just those that are layered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  As for optimizing I/Os, it gives us more flexibility, as it  allows us to employ any possible topological order of the  connections, including those that do not correspond to layer-  after-layer computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' For example, it allows us to start  computing neurons of the layers i+1, i+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' even when not  all neurons of the i-th layer have been computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' This can save  I/Os, because when we finish computing a neuron from the  i-th layer, we can directly reuse it to start computing neurons  of the (i + 1)-st layer, instead of storing it and reading it again  later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Since for each connection we perform a computation for  which we need the value of the input neuron and the partial  sum at its output neuron, we would like to find an order in  which connections that have neurons in common, are clustered  together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' BOUNDS FOR INFERENCE  In this section we present generic bounds on the I/O-  complexities of inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' They serve as guidelines for the  more instance-specific optimizations that we consider later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Let N be a connected FFNN and assume M ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Then the optimal number of I/Os for inference satisfies  W + N + S ≤ I/Os(N, M) ≤ 2 · (W + N − I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  (1)  The optimal number of read-I/Os for this problem satisfies  W + N ≤ rI/Os(N, M) ≤ 2 · W + N − I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  (2)  The optimal number of write-I/Os for this problem satisfies  S ≤ wI/Os(N, M) ≤ N − I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  (3)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The weights, biases and input values have a total size  of N + W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Since we cannot perform inference without having  read all of these data at least once, we obtain the lower bound  for the number of read-I/Os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Likewise, we cannot perform  inference with less than S write-I/Os, because (by definition  of the inference problem) we need to write the values of all S  output-neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Hence, the lower bound for the total number  of I/Os follows by adding the lower bounds for the read- and  write-I/Os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Now it remains to show that we can do inference using  no more than N + 2 · W − I read-I/Os, N − I write-I/Os  and 2 · (N + W − I) overall I/Os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' To achieve this, we fix  a topological order of the non-input neurons: n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' , nN−I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Then, we reorder the connections in such a way that their  output neurons appear in the order of this topological order  (notice that this is also a topological order of the connections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  see Figure 1 for an illustration of this association of topological  orderings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Notice that this order is naturally partitioned into  intervals: It begins with an interval of connections ending in  n1, followed by the connections ending in n2 and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  We now show that performing inference in this order with  our inference-algorithm (Algorithm 1 from the main paper) and  a MIN eviction policy, costs at most 2W + N − I read-I/Os  and N − I write-I/Os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' As we start reading the connections, we  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' 1: Associated topological orderings of neurons and  connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  spend 1 read-I/O to read the bias of n1 and then at most 2 read-  I/Os for each of the connections that end on n1 (1 for reading  the input neuron and 1 for reading the connection itself).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Once  we passed through this interval of connections that end on n1,  we apply the activation to finish the computation of neuron n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Then we continue with the interval of connections that end  on n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Also in this interval of connections as well as in all  following intervals of connections, we spend 1 read-I/O for  the bias of the output-neuron and at most 2 on each of the  connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Since there is exactly one interval of connections  for each of the N − I non-input neurons, we spend at most  N − I read-I/Os to read the biases and at most additional  2W for the connections and their input-neurons, adding up to  2W + N − I read-I/Os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Now we count the write-I/Os deployed in this computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Since all connections that end in the same neuron follow  each other, we always compute neurons without having to  write temporary values (once we start computing a neuron all  of the following computation steps are also directed towards  computing this neuron, and we only start computing another  neuron once the previous is finished).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' From this it follows, that  if we spend a write-I/O on some neuron, then we are writing a  fully computed neuron value to slow memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' And hence, if we  read and evict this value again, we do not spend another write-  I/O on this value (since the fully computed value is already  stored in slow memory, the efficient eviction policy will evict  this value by deleting it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Therefore, we spend at most one write-  I/O for each of the non-input neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Since there are N − I  non-input neurons, we have overall at most N − I write-I/Os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Adding the read- and write-I/Os, we conclude that the total  number of I/Os is at most N+2·W−I+N−I = 2·(N+W−I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Notice that in both Inequalities 1 and 2, the term on the  right hand side is at most twice as large as the term on the  left hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The next Proposition establishes that none of  the generic bounds given by this theorem could be tightened  4  C  6  35  any further by multiplying it with a constant other than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' As generic bounds that depend only on W, N, I,  and S, the bounds in Theorem 1 are tight: For each one of  them and for any ϵ > 0, there are instances for which the true  value differs from the bound by a multiplicative factor that lies  in [1 − ϵ, 1 + ϵ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  To prove this Proposition, we need to show that for each of  the bounds of Theorem 1, there are instances that are arbitrarily  close to the bound (close in terms of multiplicative factors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  The first lemma establishes that there are instances that  exactly attain all lower bounds of Proposition 1 (and hence,  are obviously arbitrarily close).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' On any FFNN N = [L1, L2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' , Ld] in which any  two consecutive layers Li, Li+1 have together at most M − 1  neurons, we can do inference, with a number of read-, write-,  and total I/Os that matches the lower bounds given by Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We prove this by describing a computation strategy that  achieves this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Read the |L1| input values into fast memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Also initialize  in fast memory |L2| partial sums with the biases of the neurons  in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Since |L1| + |L2| ≤ M − 1, we have in fast memory  free space for at least one more value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We use this free space  to iterate over the weights between L1 and L2 and perform  the following steps for each one of them:  1) Read the weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  2) Multiply it with the value of its input neuron (which is  already in fast memory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  3) Add this product to the partial sum of its output neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  4) Delete the weight from fast memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Once this iteration is finished, we apply the activation function  to the partial sums of L2, which finishes the computation of the  neurons in this layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Now we do not need the values from L1  anymore and can delete them from fast memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' This gives us  enough free space to read the biases of L3, and compute this  layer in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We proceed like this, computing layer  after layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Once we finish the computation of the neuron-vales  of the last layer, we write those S values back to slow memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Notice that we read each of the N −I biases, I input values,  and W weights exactly once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Hence we use N + W read-I/Os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  We only use write-I/Os to write the S output values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Hence,  altogether we use N + W + S I/Os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Alternatively, we could have proved the previous Lemma  by verifying that an FFNN with the mentioned properties,  can be constructed with the Compact Growth method that we  introduce later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The next lemma shows that there are instances  that use a number of read- and total I/Os that are arbitrarily  close to their upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' For every ϵ > 0, there exist FFNNs N (of arbitrary  size) that have I/Os(N, M) > (1 − ϵ) · 2 · (W + N − I) and  rI/Os(N, M) > (1 − ϵ) · (2 · W + N − I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' This is the case for architectures, where a large  proportion of the neurons are input neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' For example,  in a tree with I input neurons, all connected to a single  output neuron, we have I/Os(N, M) = ·2 · (W + N − I)  and rI/Os(N, M) = 2 · W + N − I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  The next lemma shows that there are instances that use  a number of write-I/Os that is arbitrarily close to the upper  bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' For every ϵ > 0, there exist FFNNs N (of arbitrary  size) that have wI/Os(N, M) > (1 − ϵ) · (N − I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Notice that this inequality is satisfied by any FFNN  for which all neurons are either input or output neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' But  to give less trivial examples, we will construct FFNNs with  hidden neurons that satisfy this inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  For M, h, w ∈ N+, consider the FFNN that has I input  neurons, S output neurons, and one hidden layer with h neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Clearly, this FFNN requires at least S write-I/Os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' So, for any  parameter configuration I, h, S with S > h·(1−ϵ)/ϵ, we have  wI/Os(N, M) > (1 − ϵ) · (N − I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  The three previous lemmata provide the tight extremal cases  that bring us into the position to prove Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Although  the constructed architectures in the proofs of these lemmata  were “artificial”, they suffice to show that generic bounds that  only depend on I, S, N, and W, cannot be tighter (tighter by a  constant multiplicative factor other than 1) than our bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Yet,  notice that, despite being “artificial”, these extremal cases are  more than just “a few cornercases”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' In fact, the constructions  in each of these proofs could have been chosen to be arbitrarily  large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Lemma 1 shows that the lower bound  on the number of read-, write-, and total I/Os is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Lemma  2 shows that the upper bound on the number of read-I/Os and  total I/Os is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Lemma 3 shows that the upper bound on  the number of write-I/Os is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Notice that none of the bounds depends on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' This is  surprising, because for most computational problems, I/O-  complexities depend strongly on the memory size1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' As we  increase M, we gain more flexibility that we can use to avoid  I/Os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Also surprising is the fact that these bounds do not depend  on specific properties of the network, except for its sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' This  does not mean that the overall I/O-complexity does not depend  on these quantities (memory size and FFNN architecture),  but from the tightness of our bounds we conclude that the  dependence is only moderate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  For write-I/Os, this is very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The bounds for write-  I/Os given by Theorem 1 are less tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' In fact, they can  be arbitrarily loose (despite being optimal in the sense of  Proposition 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The problem for write-I/Os is that, it is simply  not possible to give interesting bounds that only depend on  W, N, I, and S but no other factors such as the connection  order, the memory size M, and the FFNN architecture (this  impossibility is implied by Proposition 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The following  Proposition shows that inference in a layer-after-layer fashion  can be arbitrarily more expensive than using an optimal order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  1For example, in the case of matrix multiplication, the dependency on M  is of the order O(1/  √  M) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  6  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' For every c ∈ N and every memory size M ∈ N,  there exists an FFNN NM,c, such that performing inference  in a layer-after-layer fashion, requires at least c times more  write-I/Os than optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Consider a sparse FFNN that has c+2 layers: 1 neuron  in the input layer, 2M neurons in each of the c hidden layers,  and one neuron in the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Let each hidden neuron  have exactly one incoming and one outgoing connection, let the  input-neuron be connected to each neuron of the first hidden  layer, and let the output-neuron have incoming connections  from all neurons of the last hidden layer (in other words, this  FFNN has 2M chains of neurons of length c + 2 that meet in  the input and output neuron).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' If we perform inference on this  FFNN in a layer-after-layer fashion, we need the values of all  2M neurons of each hidden layer to compute the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Since our fast memory has only capacity for M neurons, we  will need to store the other M neurons, requiring at least M  writes for each hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Hence, if we perform inference  on this FFNN in a layer-after-layer fashion, we would need to  use at least M × c write-I/Os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Yet, if we compute chain after  chain, a single write-I/O would suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' CONNECTION REORDERING: ADAPTING THE ORDER TO  THE FFNN AND HARDWARE  In this section we introduce Connection Reordering, which is  a method to optimize the topological order of the connections  for a given FFNN architecture and memory size M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' This  method depends on the following hyperparameters: The win-  dow size ws ∈ N, the cooling rate σ ∈ R, and the number  of iterations T ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The high-level idea of this method is  based on Simulated Annealing [33]: Over T iterations, we  perform random changes to the topological order (we call this  creating neighbors) and either retain or discard the changes  with a probability that depends on the quality of the old and  the new order (called updating).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Now, we fully specify this  method by describing the processes of creating neighbors and  updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Creating neighbors  We start with a topological order of the connections  e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' , eW and let OldI/Os ∈ N denote the number of I/Os  used in this topological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The number of I/Os obviously  depends on the memory size and eviction policy, but those are  fixed throughout the execution of this algorithm and hence the  number of I/Os depends only on the topological order of the  connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  First, we choose uniformly at random one connection ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We  let w be an integer that we choose uniformly at random from  {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' , ws − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We consider the window of connections  ei, ei+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' , emin(i+w,W ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We choose the direction in which  we move the connections from this window: Either left or right  with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  a) Case 1: Moving to the left: If we move the connections  to the left, we start moving the leftmost connection ei of the  window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We move ei to the left until we encounter another  connection es that has the same input neuron as ei, or whose  output neuron is equal to the input neuron of ei (if we  never encounter such a connection, we move ei to the very  beginning of the order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We insert ei right next to es so that  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' , es, ei, es+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' is the new order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Note that this is again  a topological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Then we continue moving the second  leftmost connection in the window in the same fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We  continue like this until we moved all connections from this  window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  b) Case 2: Moving to the right: If we move the connec-  tions to the right, we start moving the rightmost connection of  the window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We move it until we encounter another connection  ez, which has the same output neuron as ei or whose input  neuron is equal to the output neuron of ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We insert ei right  before ez so that .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' , ez−1, ei, ez, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' is the new order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Then  we continue with the second rightmost connection from the  window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We continue like this until we moved all connections  from this window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Updating  After we moved all connections from the window, we have  a new topological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We denote it ˜e1, ˜e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' , ˜  eW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We  measure how many I/Os are used to perform inference in this  new order and call this number newI/Os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' If newI/Os<oldI/Os,  we certainly update e1 = ˜e1, e2 = ˜e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' , eW =  ˜  eW and  oldI/Os=newI/Os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  If newI/Os≥oldI/Os, we either update or go back to the old  order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We decide this at random, choosing to update with a  probability 2−(newI/Os−oldI/Os)·tσ, where t is the iteration number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' COMPACT GROWTH: ADAPTING FFNNS TO HARDWARE  In this section we present a construction scheme that  completely characterizes the FFNN architectures that allow  inference with a minimal number of I/Os for a given memory  size M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' This answers the following question: For a given  memory size M, which are the FFNN architectures on which  we can do inference with a number of I/Os that matches the  lower bound given by Theorem 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Or equivalently: Which are  the FFNNs on which we can do inference without having to  write or read intermediate values due to lack of memory?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The  idea is to couple the construction of the FFNN closely to the  steps in the inference computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' More precisely, we will  build the FFNN by a sequence of steps of four different types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  We introduce the concepts of pebbles and bags to reason  about I/Os in our computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Each pebble corresponds to  one neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' A pebble can be either gray (if it is not yet fully  computed) or black (if it is fully computed and can be used by  its outgoing connections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The bag represents the fast memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The construction rules  We start our construction with an “empty FFNN” (there  are no neurons and no connections) and an empty bag that  represents the content of our fast memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We build up the  FFNN by a sequence of construction steps of the following four  types (for each type we write in parentheses the corresponding  computation step into which it can be translated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  1) Whenever we have less or equal than M − 2 pebbles in  our bag, we can either add a gray or a black pebble to our  7  bag and simultaneously add an isolated neuron to our FFNN  (reading a neuron into fast memory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  2) When we have a black and a gray pebble in our bag, we  can draw a connection in our FFNN, that starts at the neuron  that corresponds to the black pebble and ends at the neuron  that corresponds to the gray pebble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' (multiply weight with the  value of the input neuron and add it to the partial sum of the  output neuron).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  3) We can turn a gray pebble into a black pebble (finish  computation of the neuron by applying activation function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  4) We can remove a black pebble from the bag (delete from  fast memory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  The following theorem summarizes the properties of FFNNs  built by a sequence of the above steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Theorem 2 (Compact growth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Let N be a connected FFNN  with W weights and N neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Given a memory of size  M ≥ 3, we can do inference on N with N + W read-I/Os  and S write-I/Os if and only if N can be constructed by the  compact growth scheme that we just described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' If we can build N by a sequence of  these construction steps, then the corresponding steps in the  parentheses, describe a valid sequence of computation steps for  inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' By only allowing to insert pebbles when we have  (strictly) less than M − 1 pebbles in the bag, we ensure that  there are never more than M −1 pebbles in memory and hence  there is one free space for a connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Hence, this sequence  of computation steps can be executed with a fast memory of  size M without requiring any I/Os on temporary values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' This  proves the “if” part of the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Conversely, assume for a given N and M ≥ 3, we can  order the connections in a way that allows us to perform  inference with a minimal number of I/Os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Denote this order  e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' , eW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We will show how we can translate the  computation steps into a sequence of pebble-moves that creates  the FFNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' According to Theorem 1, a minimal number of I/Os  means that we perform W + N read-I/Os and S write-I/Os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Since this is the number of read-I/Os required to read all  necessary parameters once, and write out the values of the  output-neurons once, these I/Os only suffice to perform these  necessary reads and writes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' In particular, we cannot spend any  more I/Os to write or read temporary values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Hence, for each  neuron we will read one value at some moment (the bias or the  input value), then have this value in fast memory for a certain  interval of time during which we update it according to the  progress of the computation, and then evict the fully computed  neuron value when it is not needed anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' After that we will  not be able to use this value again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We translate this into our  pebble construction: Putting a gray pebble into our bag, having  it there for a certain amount of time, transforming it into a  black pebble (once the neuron is fully computed), and then  removing it (when the neuron value is not needed anymore  and is evicted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  During this interval, this optimal computation strategy must  use all incoming and outgoing connections of this neuron: The  incoming connections during the first part of the interval (when  the neuron is not yet fully computed and hence the pebble  is gray) and the outgoing connections during the second part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Further, any time we use a connection, the “other end” of  the connection must also be in fast memory and hence the  corresponding pebble must be in the bag (remember that we  insert a pebble whenever we read a bias/input and remove  the pebble when we evict the value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Also, any time we use  a connection, the color of the pebble of the input neuron is  already black (because we turn a pebble black as soon as  the neuron is fully computed) and the pebble of the output  neuron is still gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Hence, as we use the connections on the  computation side, we can draw the connections on the side of  the pebble model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Due to the restricted memory size, we can assume without  loss of generality that we have never more than M − 1 neuron  values in fast memory (if we read an M-th value, there is no  more space for a connection, and the computation is stuck  until we remove one neuron value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' but then we could just  as well first remove this neuron value before reading another  one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' And hence, we can assume without loss of generality  that we only read values when there are at most M − 2 values  in fast memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' On the side of the pebble construction, this  ensures that we only add pebbles, when there are at most  M − 2 pebbles in the bag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  We can apply this theorem to the problem of optimal  hardware for a given FFNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' To do this, we consider the  bandwidth of an FFNN, defined as the smallest k ∈ N for  which there exists a topological order of the neurons, such  that any two connected neurons are at most k steps apart in  the topological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We prove now that when FFNNs have  bandwidth k, we can build them with compact growth, using  M = k + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Let N have bandwidth k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' If we have a  memory size M ≥ k + 2, then we can perform inference on  N without reading or writing any temporary values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Let N have bandwidth k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' This means that there  exists a topological order of the neurons n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' , nN,  such that for any i  ∈  {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' , N}, all incoming con-  nections  to  ni  are  among  the  k  previous  neurons  {nmax(1,i−k), nmax(1,i−k)+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' , ni−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' So, we can build N  with compact growth, by adding the pebbles according to the  topological order of the neurons, and when the bag is full,  removing the pebble that was added k + 1 steps ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' This  ensures that when we add a pebble of a neuron n to the bag,  the pebbles of all other neurons on which n directly depends  are also in the bag and we can add all connections that go to  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Hence, with a memory size of k + 2, we can build N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' EXPERIMENTAL EVALUATION  To demonstrate both the theoretical accuracy as well as  the practical utility of our results, we run both simulated  experiments counting I/Os and measure CPU performance  of the reordered sparse neural networks compared with high-  performance libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  As inputs, we use two types of networks: MLPs with varying  depth, breadth, and density;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' and fully-connected layers from  8  16,907 / 15,296 / 13,496  (a) Neurons per Layer  1,002 / 1,002 / 1,002  (b) Depth  10,169 / 8,773 / 6,973  (c) Connection Density  5  100  200  300  400  500  600  Fast Memory Size (M)  0  20,000  40,000  60,000  80,000  100,000  Total I/Os  Initial  Reordered  Lower Bound  (d) Fast Memory Size  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' 2: Connection Reordering for varying sparse neural  network properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Unless otherwise stated, the network is a  500-neuron wide, 4-layer MLP, with an edge density of 10%  and M = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The lower bound is obtained from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  When too small, I/Os for MLPs are annotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  the popular BERT [34] Transformer [23] neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' For  the former, we generate five randomly-sparse FFNNs (obtained  by random edge sampling or using Compact Growth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' For the  latter, we use a pre-trained BERTLARGE neural network and  perform magnitude pruning [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Simulated Experiments  To test Connection Reordering (CR) and Compact Growth  (CG), we implement Algorithm 1 and cache simulation, along  with LRU, RR, and MIN eviction policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We run each  experiment configuration with five random MLPs and BERT,  reporting the median values and 95% nonparametric confidence  intervals as error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  1) Connection Reordering: We evaluate CR by optimizing  the I/Os of random sparse FFNNs with varying structural and  sparsity properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Figure 2 shows four dimensions in which  we vary a baseline FFNN: a 10% dense, 4-layer MLP with  500 neurons in each layer and one output neuron, running  inference with a fast memory size of 100 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We either  vary density, width, depth, or fast memory size, while keeping  the other parameters constant at their baseline value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We use a  MIN eviction policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Initially, we order the connections as in  the proof of Theorem 1, so that we are guaranteed to not use  more than twice the optimal number of I/Os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Then, we apply  CR with T =1,000,000, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='2, and set ws to four times  the average in-degree of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' As we can see in these  examples, CR reduces the total number of I/Os further by up  to 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='5% and brings us up to 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='4% closer to the theoretical  lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Further, we can see that CR gives consistently  large improvements across all parameter configurations (except  for the cases where the initial I/O-complexity is already close  to the lower bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  2) Compact Growth: We generate three FFNNs with Com-  pact Growth as described in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' These FFNNs  correspond to the values Mg = 100, 300, and 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Then we  count the I/Os used for these FFNNs with varying sizes of  fast memory M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' These numbers are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' As  predicted by Theorem 2, we can see for all three FFNNs that  when M ≥ Mg, we use a minimal number of I/Os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' In these  cases, we directly achieve this minimal number of I/Os using  the connection order produced by CG, without applying CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  When we additionally apply CR, we are also able to use  a minimal number of I/Os in some ranges of memory size  M with M < Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Of course this is not a contradiction to  Theorem 2: The theorem says that we can achieve a minimal  number of I/Os using a memory size M if and only if the  FFNN can be built using Mg = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' But this does not imply  that the same FFNN cannot be built with Mg < M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  3) Cache Eviction Policies:  Figure 4 shows the I/O-  complexity evolution (over Simulated Annealing iterations)  of random FFNNs, for the RR, LRU, and MIN cache eviction  policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Owing to the update scheme, we observe a decaying  convergence rate towards a minimum value, where the majority  of I/O reduction happens in the first 10,000 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' It is  interesting to observe that upon optimization, both RR and  LRU converge to similar I/Os, which indicates that CR can  tune FFNNs for hardware architectures with specific caching  mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  4) Size of Fast Memory: In Figure 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' we show how the  number of I/Os changes as we increase the size of the fast  9  5  100 105 205 300 305 405 500 505 605 705 805 9051005  Fast Memory Size (M)  0  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  Total I/Os  Initial  Reordered  Lower Bound  (a) Mg = 100  5  100 105 205 300 305 405 500 505 605 705 805 9051005  Fast Memory Size (M)  0  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  Total I/Os  Initial  Reordered  Lower Bound  (b) Mg = 300  5  100 105 205 300 305 405 500 505 605 705 805 9051005  Fast Memory Size (M)  0  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  Total I/Os  Initial  Reordered  Lower Bound  (c) Mg = 500  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' 3: Study of Compact Growth-generated FFNNs designed  for different fast memory sizes Mg (1, 000 neurons with in-  degree of 4 and one output-neuron with in-degree Mg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Round-Robin  LRU  Optimal (MIN)  Lower Bound  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' 4: I/Os for different cache eviction policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  memory, both before and after Connection Reordering (CR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  As we can see, once the fast memory is sufficiently large,  we use a number of I/Os that matches the theoretical lower  bound provided by Theorem 1, with and without CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' But for  insufficient memory size, we see that with CR we reduce I/Os  and converge faster to the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Reordered  Initial connection order  Lower Bound  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' 5: Study of different fast memory sizes (M) on random  sparse FFNNs (3 layers of 500 neurons each followed by one  output neuron, 1% density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  5) Application to Transformer-model BERT: Transform-  ers [23] are a DNN architecture built from so-called encoders  and decoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' A characteristic ingredient of transformers are  attention heads, which make them suitable for natural language  processing and related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' A prominent example of a  Transformer is BERT [34], which was introduced in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  BERT’s groundbreaking results made Transformers arguably  the most popular and extensively investigated DNN architecture  of the past years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The main challenge encountered when  deploying transformers is the large size of these models and the  resulting demand for compute resources [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' BERT includes  several FFNNs of depth 2 and weight matrices of dimensions  1024 × 4096 and 4096 × 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The major part of BERTs  parameters and compute time for inference come from these  FFNNs [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' This makes them a prime target for pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  We took one of these FFNNs and pruned it by removing  the connections with the weights of smallest absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Figure 6 shows the I/O-counts and the I/O lower bounds  for inference on this FFNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We considered different levels  of sparsity and different cache eviction policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Further, we  assumed a fast memory of size 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Performance Experiments  To validate the performance benefits of I/O reduction, we  perform batched inference on an Intel CPU and measure the  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' For all experiments, we reorder the FFNNs with  the simulated optimal (MIN) eviction policy, and then run the  FFNN before and after reordering, as well as in the traditional,  layer-based approach using MKL for sparse-dense matrix  matrix multiplication (CSRMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Using batched inference (as  is performed in production environments) enables the use of  SIMD instructions and to better saturate the memory bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  We run each experiment 10 times, with a batch size of 128,  reporting median values and error bars the shortest and longest  execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Annotations show the speed-ups that we obtain  with our methods (without and with reordering) relative to the  layer-based inference approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  The hardware used for the experiments is a 32-core Intel  Xeon Gold 6130 CPU (running at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='10 GHz) with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='5 TB  10  1%  10%  25%  50%  75%  100%  Density  0  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content='000  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='500,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  Total I/Os,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Optimal (Min)  153607,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  139881,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  91057  Initial  Reordered  Lower Bound  (a) Optimal (MIN)  1%  10%  25%  50%  75%  100%  Density  0  5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  Total I/Os,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' LRU  173780,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  171279,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  91057  Initial  Reordered  Lower Bound  (b) LRU  1%  10%  25%  50%  75%  100%  Density  0  5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='000  Total I/Os,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Round-Robin  173733,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  173376,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  91057  Initial  Reordered  Lower Bound  (c) Round-Robin  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' 6: Connection Reordering for an MLP from BERTLARGE  using different eviction policies, different edge densities and  M = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The lower bound is obtained from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' For software, we use Intel oneAPI MKL 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='0 for  CSRMM, and GCC 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='0 for our reordered networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  1) Random sparse FFNNs: Starting with the same baseline  FFNN from the simulated experiments (a 10% dense, 4-layer  MLP with 500 neurons in each layer and one output neuron)  we measure the execution time of our methods varying density,  width, or depth while keeping the other parameters constant at  their baseline value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  In Figure 7a we show the execution time for varying degrees  of density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' As we can see, the sparser the FFNN, the larger  the speedup that we obtain with our method relative to the  standard, layer-wise way of performing inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' For example,  at density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='1% we get a speed-up of about 45x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' On the other  hand, the more sparse the FFNN, the less additional speedup we  gain by reordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' In fact, somewhat surprisingly, for density  values below 1% the execution time is slightly lower without  than with reordering, which we attribute to two causes: (a) the  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content='3x,  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='4x  Based on Sparse Matrix Multiplication (MKL)  Our Method Before Reordering  Our Method After Reordering  (c) Breadth  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' 7: Execution time for randomly-sparse FFNNs with  different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  relatively low reduction in I/Os (see Section VI-A1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' and (b)  the small size of the networks, which makes the performance  benefit negligible due to the FFNNs fitting in L2 caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Another effect shown in Figure 7a is speedup compared with  MKL on 100% dense FFNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The reason for this phenomenon  is that we use CSRMM in that case for consistency in measure-  ments, rather than a dense matrix- dense matrix multiplication  (GEMM), which explains the observed slowdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  In Figures 7b–7c we show the execution time as a function  of the depth (Figure 7b) and breadth (Figure 7c) of the FFNN  (while keeping the other structural parameters constant at their  baseline values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' In both cases we observe speedups that grow  linearly with depth and breadth (and hence, with the size of  11  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content='1%  Density  0  5  10  15  20  25  30  35  Execution Time (ms)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='8x,  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content='1x,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='1x  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='7x,  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='9x  Based on Sparse Matrix Multiplication (MKL)  Our Method Before Reordering  Our Method After Reordering  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' 8: Execution time for BERTLARGE with Connection  Reordering and varying densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  the instance), however, with diminishing returns as the network  grows in one dimension and not in the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  In sum, our MLP performance experiments align with the  simulation results, indicating that reordering is beneficial in  practice for sparse neural networks, especially for cases where  density is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  2) BERT: In Figure 8 we show the execution times for  inference on the previously introduced encoder MLP from  BERTLARGE with varying degrees of density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' In the case of  MKL inference at 10% density we had one clear outlier (as  defined by Tukey’s method [35];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' one run took 106 ms while in  the other 9 runs it took about 17 ms) which we removed from  the data shown in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The figure shows that runtime  after reordering is always lower than before reordering, and  confirms that even for realistic network modules with only two  layers, there is a benefit in using our method, especially for  low densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' DISCUSSION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Prefetching  Modern systems support prefetching to accelerate I/Os which  is not considered in the presented model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Notice however that  the effect of prefetching is “orthogonal” to the optimizations  considered in this paper as prefetching does not reduce the  overall amount of I/Os but it only affects the time the I/Os  take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Cost of using a given topological order  The computations of “good” (that is, I/O-efficient) topo-  logical orders are performed “offline” and once the order is  determined and we perform inference, we do not need to look  at it explicitly again as it is encoded in the way the connections  are laid out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Hence, during inference there is no additional cost  associated with processing the connections according to any  given topological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The term “I/O”  The theory in the paper applies not only to caches, but local  memory of any kind (including, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=', registers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' I/O refers to a  load/store from a "slow" memory, which could be represented  by off-chip memory, communicating with another node, or even  scratch-pad memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' The work does neither assume nor rely  on caches, and covers aspects such as how to stream memory  efficiently into a chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' CONCLUSIONS  We established tight bounds on the I/O-complexity of FFNN  inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We present a 2-optimal computation strategy and  show how combinatorial optimizations on the order of the  connections can lead to further improvements in I/Os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Given  a fixed memory size, Compact Growth allows us to create  FFNNs on which inference can be performed I/O-optimally:  we do not need to read nor write any temporary values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Our work provides theoretical insight into the problem of  mapping existing neural networks to hardware, such that energy-  consumption and latency during deployment are minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Most importantly, these insights indicate that the improvements  are achieved only when dispensing with the straightforward,  layer-by-layer manner of processing DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' In turn, our  proposed Connection Reordering overcomes this misconception,  reducing the I/O costs significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' In our experiments on  real hardware we observe speedups of up to 45× relative to  the standard (CSRMM-based) way of performing inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Further, between our 2-optimal computation strategy and the  strategy that is further improved by connection reordering, we  observe speedups of up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='17×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  This work was supported by the EU Horizon project  GLACIATION under grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' 101070141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  is supported by the Swiss National Science Foundation (Am-  bizione Project #185778).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' We also acknowledge the Swiss  National Supercomputing Centre (CSCS) for support and access  to computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  REFERENCES  [1] Zhe Jia, Marco Maggioni, Benjamin Staiger, and Daniele P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Scarpazza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Dissecting the nvidia volta gpu architecture via microbenchmarking,  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Unat, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Dubey, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Hoefler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Shalf, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content=' Bianco,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content=' Chamberlain, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Cledat, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content=' Finkel, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Fuerlinger,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Hannig, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Jeannot, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Kamil, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Keasler, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content=' Leung,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Ltaief, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Maruyama, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content=' Pericás.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Trends in data  locality abstractions for hpc systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' IEEE Transactions on Parallel and  Distributed Systems, 28(10):3007–3020, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  [3] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Sze, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Chen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Yang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content='  Efficient processing of  deep neural networks: A tutorial and survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Proceedings of the IEEE,  105(12):2295–2329, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  [4] Tal Ben-Nun and Torsten Hoefler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Demystifying parallel and distributed  deep learning: An in-depth concurrency analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=',  52(4), August 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  [5] Song Han, Xingyu Liu, Huizi Mao, Jing Pu, Ardavan Pedram, Mark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Horowitz, and William J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Dally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Eie: Efficient inference engine on  compressed deep neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' In Proceedings of the 43rd International  Symposium on Computer Architecture, ISCA ’16, page 243–254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' IEEE  Press, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  12  [6] Norman P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Jouppi, Cliff Young, Nishant Patil, David Patterson, Gaurav  Agrawal, Raminder Bajwa, Sarah Bates, Suresh Bhatia, Nan Boden,  Al Borchers, Rick Boyle, Pierre-luc Cantin, Clifford Chao, Chris Clark,  Jeremy Coriell, Mike Daley, Matt Dau, Jeffrey Dean, Ben Gelb, Tara Vazir  Ghaemmaghami, Rajendra Gottipati, William Gulland, Robert Hagmann,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Richard Ho,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Doug Hogberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' John Hu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Robert Hundt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Dan Hurt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  Julian Ibarz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Aaron Jaffey,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Alek Jaworski,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Alexander Kaplan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Harshit  Khaitan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content=' Andy Koch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content=' Steve Lacy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  James Laudon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' James Law,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Diemthu Le,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Chris Leary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content=' Adriana Maggiore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Maire  Mahony,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content=' Jonathan Ross,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content=' Matthew Snelham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Jed Souter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Dan  Steinberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content=' Bo Tian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Horia  Toma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Erick Tuttle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Vijay Vasudevan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Richard Walter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Walter Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Eric  Wilcox,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' and Doe Hyun Yoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' In-datacenter performance analysis of a  tensor processing unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content=' Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content=' Society  for Industrial and Applied Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content='  Red-blue pebble game:  Complexity of computing the trade-off between cache size and memory  transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content=' Springer International  Publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content=' The red-blue pebble game on trees  and dags with large input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content=' Springer-Verlag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content=' Proceedings of the 2017 Conference on  Empirical Methods in Natural Language Processing, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+page_content='  [35] Torsten Hoefler and Roberto Belli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Scientific benchmarking of parallel  computing systems: Twelve ways to tell the masses when reporting  performance results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' In Proceedings of the International Conference for  High Performance Computing, Networking, Storage and Analysis, SC  ’15, New York, NY, USA, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  APPENDIX  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' How we generate random FFNNs  We describe how we generate random FFNNs of width  w, depth d, and sparsity p: For each non-output neuron, we  determine how many outgoing connections it has, by drawing  uniformly at random an integer k between 1 and max(1, ⌈2 ·  p · #of neurons in the next layer − 1⌉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Then, we connect this  neuron to k randomly chosen neurons of the next layer (k ≥ 1  ensures that the FFNN is connected and the output neuron is  connected to all neurons of the last hidden layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' Once we  determined all connections, we order the connections layer-  by-layer with respect to their output neuron (according to the  proof of Theorem 1, this ensures that we have a connection  layout that uses not more than twice the optimal number of  I/Os).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' How we generate FFNNs with Compact Growth  We start with Mg − 2 readily computed input neurons in the  bag (in our experiments we used Mg = 100, 300, and 500).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content='  13  Then we iterate from 1 to 1000, and in each iteration: 1) we  add a new neuron 2) we draw incoming connections to this  neuron from 5 other randomly chosen neurons from the bag 3)  we remove the last of these 5 randomly chosen neurons from  the bag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
+page_content=' After the 1000 iterations, we add one more neuron  (the output neuron) and draw connections from all remaining  neurons in the bag to this output neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfHvsI/content/2301.01048v1.pdf'}
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+High-Performance Deterministic Concurrency using
+Lingua Franca
+Christian Menard
+christian.menard@tu-dresden.de
+TU Dresden, Germany
+Marten Lohstroh
+marten@berkeley.edu
+UC Berkeley, USA
+Soroush Bateni
+soroush@berkely.edu
+UC Berkely, USA
+Matthew Chorlian
+mattchorlian@berkeley.edu
+UC Berkeley, USA
+Arthur Deng
+langxing.deng@berkeley.edu
+UC Berkeley, USA
+Peter Donovan
+peterdonovan@berkeley.edu
+UC Berkeley, USA
+Clément Fournier
+clement.fournier@tu-dresden.de
+TU Dresden, Germany
+Shaokai Lin
+shaokai@berkeley.edu
+UC Berkeley, USA
+Felix Suchert
+felix.suchert@tu-dresden.de
+TU Dresden, Germany
+Tassilo Tanneberger
+tassilo.tanneberger@tu-dresden.de
+TU Dresden, Germany
+Hokeun Kim
+hokeun@hanyang.ac.kr
+Hanyang University, South Korea
+Jeronimo Castrillon
+jeronimo.castrillon@tu-dresden.de
+TU Dresden, Germany
+Edward A. Lee
+eal@berkeley.edu
+UC Berkeley, USA
+Abstract
+Actor frameworks and similar reactive programming tech-
+niques are widely used for building concurrent systems. They
+promise to be efficient and scale well to a large number of
+cores or nodes in a distributed system. However, they also
+expose programmers to nondeterminism, which often makes
+implementations hard to understand, debug, and test. The
+recently proposed reactor model is a promising alternative
+that enables efficient deterministic concurrency. In this pa-
+per, we show that determinacy does neither imply a loss in
+expressivity nor in performance. To show this, we evaluate
+Lingua Franca (LF), a reactor-oriented coordination lan-
+guage that equips mainstream programming languages with
+a concurrency model that automatically takes advantage of
+opportunities to exploit parallelism that do not introduce
+nondeterminism. Our implementation of the Savina bench-
+mark suite demonstrates that, in terms of execution time, the
+runtime performance of LF programs even exceeds popular
+and highly optimized actor frameworks. We compare against
+Akka and CAF, which LF outperforms by 1.86𝑥 and 1.42𝑥,
+respectively.
+CCS Concepts: • Computing methodologies → Concur-
+rent programming languages; • Software and its engi-
+neering → Runtime environments; Source code gener-
+ation.
+Keywords: coordination, concurrency, determinism, perfor-
+mance
+1
+Introduction
+Theoreticians working on programming language semantics
+have long understood the value of determinism as well as the
+expressive power of nondeterminism in programming lan-
+guages. In practice, however, today, nondeterminism creeps
+into programming languages and frameworks not to benefit
+from its expressiveness, but rather because of a widespread
+perception that it is needed to get good performance on par-
+allel hardware. We show in this paper that it is not necessary
+to sacrifice determinism to achieve performance. We do this
+by focusing on actor frameworks, which have proved pop-
+ular and successful in many very demanding applications,
+but admit nondeterminism that is often not actually needed
+by their applications.
+Exploiting parallel hardware such as multicore machines
+to improve performance is only possible when programs
+expose concurrency. Common abstractions for concurrency
+include threads [18], remote procedure calls [63], publish-
+subscribe [27], service-oriented architectures [64], and ac-
+tors [1, 33]. Each of these models has its own merits, but
+they all introduce nondeterminism: situations where, for
+a given state and input, the behavior of a program is not
+uniquely defined. While nondeterminism can be useful in
+some applications, most programming tasks benefit from
+more repeatable behavior. Deterministic programs are easier
+to understand, debug, and test (for each test vector, there is
+one known-good response). For nondeterministic programs,
+problematic behaviors might be harder to discover because
+they may only occupy a small fraction of the state space [37].
+And reproducing failures can be extremely hard [40, 48, 61]
+arXiv:2301.02444v1  [cs.PL]  6 Jan 2023
+
+Menard & Lohstroh et al.
+because they might occur only when the system is under a
+specific amount of load [67].
+Determinism is a subtle concept [41]. Here, we focus on a
+particular form of determinism for programs, where a pro-
+gram is deterministic if, given the same inputs, it always
+produces the same outputs. This definition does not require
+that operations be performed in a particular order, and there-
+fore is not at odds with concurrency and parallel execution.
+It is possible, but often not easy, to achieve this form of deter-
+minism even when using nondeterministic abstractions such
+as threads, actors, and asynchronous remote procedure calls.
+For simple enough programs, such as a chain of actors, if
+communication is reliable, then execution will be determin-
+istic. Some of the benchmarks we compare against in this
+paper are deterministic in this way. As we will show, how-
+ever, even slightly more complex communication structures
+result in nondeterminism that can be difficult to correct.
+In this paper, we evaluate a language-based coordination
+approach to specifying concurrent software that preserves
+determinism by default and only admits nondeterminism
+when explicitly introduced by the programmer. The coordi-
+nation language Lingua Franca (LF) [53], which is based on
+a concurrent model of computation called reactors [50, 51],
+achieves this by analyzing program structure and ensuring
+that data dependencies are observed correctly at runtime.
+An LF program defines reactive software components called
+“reactors” and provides operators to compose them hierarchi-
+cally (through containment) and bilaterally (via connections).
+Because the language supports both deterministic and non-
+deterministic concurrency, it provides a fertile ground for
+exploring the impact of determinism on performance.
+The semantics of the deterministic subset of LF can be
+thought of as a deterministic variant of actors [1, 33, 52]. We
+show in this paper that it delivers performance comparable
+to popular nondeterministic realizations of actors on parallel
+hardware. Like Akka [66] and CAF [16]—the frameworks we
+compare against—LF orchestrates the execution of chunks
+of code written in conventional programming languages,
+allowing programmers to rely on the languages, libraries, and
+tools that they are comfortable with. Unlike the frameworks
+we compare against, LF is polyglot. It currently supports C,
+C++, Python, TypeScript, and Rust. This paper focuses on
+the runtime performance of the C++ target, which, as a core
+contribution of this paper, has been optimized to efficiently
+exploit concurrency on parallel hardware. Earlier work [53]
+has only reported preliminary performance indications of
+LF based on its C target, which is predominantly aimed at
+running on embedded systems.
+At the core of LF’s concurrency model is a logical model of
+time that gives a clear notion of simultaneity and avoids dead-
+locks using dependency analysis based on causality inter-
+faces [47]. It is this timed semantics that enables efficient de-
+terministic concurrency in LF. However, the benchmarks we
+compare against were created to evaluate actor frameworks,
+User A
+User B
+Account
+Deposit
+Withdrawal
+(a) Deposit and With-
+drawal sent by different
+users.
+User
+Account
+Deposit
+Withdrawal
+(b)
+Deposit
+and
+With-
+drawal
+sent
+by
+same
+user.
+User
+Proxy
+Account
+Deposit
+Withdrawal
+Withdrawal
+(c) Withdrawal sent via a proxy.
+Figure 1. Example actor programs that may expose nonde-
+terministic behavior.
+which have no temporal semantics. None of the benchmarks
+take advantage of the time-related features of LF; the tempo-
+ral semantics is only used to deliver determinism.
+Contributions. We show that the reactor-oriented par-
+adigm as implemented in Lingua Franca enables efficient
+exploitation of parallel hardware without relinquishing deter-
+minism. For this, we explain the mechanisms through which
+LF programs expose concurrency; we present a language
+extension that allows for the definition of scalable programs;
+and we introduce an optimized C++ runtime for LF that
+enables efficient parallel execution. We further present an ex-
+tensive evaluation based on the Savina benchmark suite [34],
+showing that our LF runtime outperforms Akka and CAF by
+1.86𝑥 and 1.42𝑥, respectively.
+Outline. We first motivate our work (Section 2) and then
+introduce LF (Section 3). We then go into detail about the
+concurrency in LF and introduce our optimized C++ runtime
+(Section 4). Next, we report benchmark results (Section 5),
+discuss related work (Section 6), and conclude (Section 7).
+2
+Motivation
+The actor model is widely accepted and deployed in produc-
+tion for its promise to allow programmers to easily express
+concurrency, provide high execution performance, and scale
+well to large datasets and complex applications. Moreover, in
+contrast to thread-based programs, actor semantics prevents
+low-level data races. However, like most message passing
+paradigms, actors expose the programmer to nondetermin-
+ism in the form of high-level data races [76], a problem that
+becomes considerably challenging to manage as the com-
+plexity of a program grows.
+Consider the simple example in Figure 1a. The Account
+actor manages the balance of a bank account that two users
+interact with. User A sends a deposit message increasing
+the account’s balance and User B sends a withdrawal mes-
+sage decreasing the account’s balance. If we assume that the
+
+High-Performance Deterministic Concurrency using Lingua Franca
+account : Account
+balance: float(0.0)
+1
+2
+userA : User
+offset: time(1 sec)
+value: float(20.0)
+(1 sec)
+userB : User
+offset: time(2 sec)
+value: float(-10.0)
+(2 sec)
+(a) Simple LF implementation.
+Proxy
+Account
+2
+1
+userB : User
+(2 sec)
+userA : User
+(1 sec)
+(b) Adding a proxy reactor.
+ProxyDelay
+2
+1
+L
+Account
+2
+1
+userB : User
+(2 sec)
+userA : User
+(1 sec)
+(c) Adding a proxy introducing a logical delay.
+Figure 2. Example LF implementation of the actor program shown in Figure 1a and variations thereof.
+balance is initialized to 0 and the account only grants a with-
+drawal if the resulting balance is not negative, then there are
+two possible behaviors. If A’s message is processed first, then
+the withdrawal is granted to B. If B’s message is processed
+first, then the withdrawal is denied. The actor model assigns
+no meaning to the ordering of messages. Therefore, there is
+no well-defined correct behavior for this example.
+The reader may object that for an application like that of
+Figure 1a, the order of transactions is intrinsically nonde-
+terministic, and any additional nondeterminism introduced
+by the software framework is inconsequential. However, if
+we focus on testability, we see that even identical inputs can
+yield different results, making testing more difficult. If we
+focus on consistency, the problem that different observers of
+the same events may see different behaviors becomes prob-
+lematic. In databases, it is common to assign time stamps
+to external inputs and to then treat those timestamps as a
+semantic property of the inputs and define the behavior of
+the database relative to those time stamps. We adopt this
+perspective in this paper, and rely on the definition of de-
+terminism given by Lee [41]: “determinism is a property of
+models, not of physical realizations,” and “A model is deter-
+ministic if given all the inputs that are provided to the model,
+the model defines exactly one possible behavior.” If we de-
+fine “inputs” in Figure 1a to be time-stamped user queries
+and “behavior” to be the sequence of actions taken by the
+Account, then it is reasonable to demand determinism.
+Consider Figure 1b, which has only one user. Even if this
+one user first sends a deposit and then a withdrawal mes-
+sage, the actor model does not guarantee that the receiving
+actor sees and processes the incoming messages in this order.
+While some actor frameworks, e.g., Akka and Erlang, guaran-
+tee in-order message delivery, others, e.g., AmbientTalk [77],
+expressly do not. Yet, even if the framework guarantees point-
+to-point in-order message delivery, this property is not tran-
+sitive. If we add a Proxy, as shown in Figure 1c, then we
+cannot make any assumptions about the order in which
+Account receives messages. This example further illustrates
+that composing actors can have unexpected side effects.
+Consequently, implementing solutions to practical concur-
+rency problems with actors can be challenging. Even seem-
+ingly simple concurrency problems like the one discussed
+above require high programming discipline, and solutions are
+typically difficult to maintain and tend to lack modularity. In
+addition, the inherent nondeterminism of actor frameworks
+makes it hard to verify such solutions. Erroneous behavior
+might only occur in a fraction of executions, and thus inte-
+gration tests cannot reliably detect such “Heisenbugs” [61].
+In a recent study, Bagherzadeh et al. [4] analyzed bugs
+in Akka programs that were discussed on StackOverflow or
+GitHub and determined that 14.6% of the bugs are caused by
+races. This makes high-level races the second most common
+cause of bugs in Akka programs after errors in the program
+logic. In a similar study of 12 actor-based production systems,
+Hedden and Zhao [32] determined that 3.2% of the reported
+bugs were caused by bad message ordering, 4.8% of bugs were
+caused by incorrect coordination mechanisms, 4.8% were
+caused by erroneous coordination at shutdown, and 2.4%
+of bugs were caused by erroneous coordination at startup.
+Note that these numbers only cover known bugs in their
+studied projects and, as noted by the authors, the majority of
+the reported message ordering bugs belonged to the Gatling
+project because it already incorporated a debugging tool
+called Bita [73] that is designed to detect such bugs. We
+suspect that there are more undetected bugs in projects that
+do not use specialized debugging tools.
+The actor community has addressed the inherent nondeter-
+minism of actors and the resulting bugs by introducing better
+tools for analyzing and debugging actor programs. This in-
+cludes TransPDOR [72], Bita [73], Actoverse [69], iDeA [55],
+CauDEr [39], and Multiverse debugging [75]. While these
+are valuable solutions, we argue that a programming model
+for expressing concurrent programs should provide deter-
+ministic semantics by default and allow the programmer to
+introduce nondeterminism only where it is desired and un-
+derstood to do no harm. In such cases, the aforementioned
+tools for nondeterministic behavior can still be utilized to
+debug the implementation.
+There are a number of ways to achieve deterministic con-
+currency, including Kahn process networks [35, 36], many fla-
+vors of dataflow models [21, 43, 62], physically asynchronous,
+logically synchronous models [68], synchronous-reactive
+languages [8, 24], and discrete-event systems [15, 23, 46, 79].
+Lohstroh et al. [53] compare the reactor model to each of
+these, showing that it has many of their best features and
+
+Menard & Lohstroh et al.
+fewer of their pitfalls. Lingua Franca builds on this reac-
+tor model because it is more expressive than some of the
+alternatives (e.g., Kahn networks) and is stylistically close to
+actors, which have proven effective in practice. In this paper,
+we show that the resulting determinism does not incur a
+performance penalty, but on the contrary, helps to achieve
+improved performance in most cases.
+3
+Introduction to Lingua Franca
+Lingua Franca (LF) builds on the relatively new reactor-
+oriented programming paradigm. Intuitively, we can describe
+reactors as deterministic actors with a discrete event execu-
+tion semantics and explicitly declared ports and connections.
+A logical timeline is used to order events and ensure a deter-
+ministic execution. As a polyglot language, LF incorporates
+code in a target programming language to implement the
+logic of each component. LF itself is only concerned with
+the coordination aspect of a program.
+In this section, we introduce the core concepts of reactors
+and LF. Note however, that a full discussion of LF includ-
+ing its syntax and tooling is beyond the scope of this paper.
+Instead, we base our discussion on LF’s diagrammatic repre-
+sentation of programswhich gets synthesized from LF source
+code automatically [78]. A complete introduction to LF’s tex-
+tual syntax is given by Lohstroh et al. [53].
+3.1
+LF by Example
+Figure 2a shows an LF implementation of the deposit/with-
+drawal example in Figure 1a. The program is assembled
+from three reactor instances userA, userB and account,
+shown as light gray boxes in the diagram. Note that userA
+and userB are instances of the same reactor class User and
+hence share the same structure and functionality. In the di-
+agram, black triangles denote ports. In this example, both
+users have an output port which is connected to a respective
+input port at the account. These ports and connections allow
+the users to send requests to the account.
+In LF, all computation is performed in reactive code seg-
+ments called reactions that are implemented in an arbitrary
+target language. In the diagram, reactions are represented
+by dark gray chevrons. All reactions must explicitly declare
+their triggers, other dependencies and potential effects.In
+the example in Figure 2a, both users define a reaction that
+is triggered by a timer. Timers are an LF construct used to
+produce events in regular intervals or once at a specific time.
+The timer of userA is configured to trigger an event one sec-
+ond after program startup; the timer of userB is configured
+to trigger an event two seconds after program startup. The
+corresponding reactions simply send a deposit or withdrawal
+request by setting the user’s output port.
+The Account reactor defines two reactions, one for each
+of its inputs. Both reactions will simply try to apply the
+requested change to the balance, which is stored in a state
+variable local to the reactor instance. Note that reactors may
+define arbitrary state variables which are accessible by all
+their own reactions (which does not include reactions of
+contained reaction). In addition to state variables, reactors
+may also define parameters which can be set at instantiation.
+This mechanism allows the User reactor to be reusable, as
+the precise time at which the timer triggers (offset) and
+the amount to withdraw or deposit (value) are configurable
+at instantiation time.
+The reader might notice that the separated reactions in
+account duplicate logic and are not a practical solution, in
+particular if there are many users. We choose this represen-
+tation to keep our exposition simple and avoid a detailed
+discussion of LF syntax. Indeed, in LF a single reaction can
+bind to an arbitrary number of upstream ports.
+When executed, the program will wait for 1 second before
+triggering the timer of the userA reactor and invoking its
+reaction. The event produced by this reaction will trigger
+reaction 1 in account, which is invoked immediately af-
+ter the first reaction completes. Two seconds after program
+startup, userB will react and subsequently trigger reaction
+2 in account. In this example, the deposit event (+20.0) oc-
+curs earlier than the withdrawal event (-10.0), and hence
+our execution semantics ensures that the account processes
+the deposit event before the withdrawal event, meaning the
+balance will not become negative. In a more realistic imple-
+mentation, the two users would generate events sporadically
+and have their reactions triggered not by a timer but a phys-
+ical action (see Section 3.3). However, using a timer greatly
+simplifies our exposition as we only have to consider a single
+logical timeline along which events are ordered. Moreover,
+such timers can be used to create regression tests that vali-
+date program execution with specific input timings.
+Note that even when the two events occur logically si-
+multaneously, meaning that both reactions in the account
+reactor are triggered at the same logical time, the resulting
+program will be deterministic. All reactions within the same
+tag are executed according to a well-defined precedence re-
+lation. In particular, any reactions within the same reactor
+are mutually exclusive and executed following the lexical
+declaration order of the reactions in LF code. This order is
+also reflected by the numbers displayed on the reactions in
+the diagrams in Figure 2. More details on the precedence
+relation of reactions are given in Section 4.1.
+To deliberately change the order in which events occur, a
+logical delay can be introduced in the program using a log-
+ical action, as shown in Figure 2c. In the diagram, actions
+are denoted by small white triangles. In contrast to ports,
+which allow relaying events logically instantaneously from
+one reactor to another, logical actions provide a mechanism
+for scheduling new events at a later (logical) time. Upon re-
+ceiving an input, reaction 2 of the ProxyDelay reactor is trig-
+gered, which schedules its logical action with a configurable
+
+High-Performance Deterministic Concurrency using Lingua Franca
+delay. This creates a new event which, when processed, trig-
+gers reaction 1 of the ProxyDelay reactor, which retrieves
+the original value and forwards it to its output port.
+If we assume that a delay of 2 seconds is used for schedul-
+ing the logical action, then the deposit message from userA
+will only arrive at the account 3 seconds after startup. Hence
+the deposit message will be processed after the withdrawal
+message from userB, causing B’s request to be denied.
+It is important to note that all of the discussed examples
+are deterministic, regardless of the physical execution times
+of reactions, as all events are unambiguously ordered along a
+single logical timeline. The physical timing of the events, on
+the other hand, will be approximate. The contribution of this
+paper is to show that such determinism does not necessarily
+reduce performance and is also useful for applications that
+have no need for explicit timing.
+3.2
+Logical and physical time
+All events have an associated tag, which is used to order
+events on a logical timeline at runtime. In time-sensitive
+applications, tags are not purely used for logical ordering
+but also relate to physical time. By default, the runtime only
+processes the events associated with a certain tag once the
+current physical time 𝑇 is greater than the time value of
+the tag 𝑡 (𝑇 > 𝑡). We say that logical time “chases” physical
+time. The relationship between physical and logical time in
+the reactor model gives logical delays a useful semantics
+and also permits the formulation of deadlines. This timed
+semantics is particularly useful for software that operates in
+cyber-physical systems. For a more in depth discussion of
+LF’s timed-semantics, the interested reader may refer to [54].
+If an application has no need for any physical time prop-
+erties, the concurrence of physical and logical time can be
+turned off; in this case, the tags are used only to preserve
+determinism, not to control timing. Moreover, LF program-
+mers are not required to explicitly control timing aspects of
+their programs. Delays can simply be omitted, for instance
+when scheduling an action, in which case the runtime will
+use the next available tag. In consequence, also untimed gen-
+eral purpose programs can benefit from the deterministic
+concurrency enabled by LF’s timed-semantics.
+3.3
+Asynchrony and deliberate nondeterminism
+The reactor model distinguishes logical actions and physical
+ones. A logical action is always scheduled synchronously
+with a delay relative to the current tag. A physical action
+may be scheduled from asynchronous contexts; its event
+is assigned a logical time based on the current reading of
+physical time. Physical actions are the key mechanism for
+handling sporadic inputs originating from physical processes
+(such as users initiating withdrawal or deposit requests).
+The assignment of tags to physical actions is nondeter-
+ministic in the sense that it is not defined by the program.
+However, once those tags are assigned, for example, to de-
+posit or withdrawal requests by a user, the processing of the
+events is deterministic and occurs in tag order. Hence, the
+tags assigned to externally initiated events are considered as
+part of the input, and given this input, the program remains
+deterministic. This approach draws a clear perimeter around
+the deterministic and therefore testable program logic while
+allowing it to interact with sporadic external inputs.
+Physical actions can also be used within the program it-
+self, for example, to nondeterministically assign a new tag
+to a message received from another reactor. In this usage,
+physical actions provide a means for deliberately introducing
+actor-like nondeterminism into a program.
+4
+Efficient Deterministic Concurrency
+LF programs are deterministic by default. This property is in-
+herited from the reactor model that LF implements. Lohstroh
+et al. [53] explain why reactors behave deterministically.
+Their argument can be adapted to the concrete context of
+the Lingua Franca language, but this is beyond the scope
+of this paper. Reactors are also concurrent, and, as we show
+in this paper, the exposed concurrency is sufficient for the
+runtime system to effectively exploit multi-core hardware to
+where it matches or exceeds the performance of fundamen-
+tally asynchronous and nondeterministic actor frameworks.
+In this section, we first show exactly how concurrency is ex-
+posed and then describe in more depth how our C++ runtime
+is implemented and how it utilizes parallel hardware.
+4.1
+Parallelism
+The use of statically declared ports and connections as the
+interfaces between reactors, as well as the declarations of
+reaction dependencies, distinguish reactors from more dy-
+namic models like actors or other asynchronous message
+passing frameworks where communication is purely based
+on addresses. While the fixed topology of reactor programs
+is less flexible and limits runtime adaptation, it also pro-
+vides two key advantages. First, it achieves a separation
+of concerns between the functionality of components and
+their composition. Second, it makes explicit at the interface
+level which dependencies exist between components. As a
+consequence, a dependency graph can be derived for any
+composition of reactors. The dependency graph is an acyclic
+precedence graph (APG) that organizes all reactions into
+a partial order that captures all scheduling constraints that
+must be observed to ensure that the execution of a reactor
+program yields deterministic results. Because this graph is
+valid irrespective of the contents of the code that executes
+when reactions are triggered, reactions can be treated as a
+black box. It is this property that enables the polyglot nature
+of LF and exposes the concurrency in the application.
+Figure 3 shows the dependency graph for the program
+given in Figure 2c. The solid arrows represent dependencies
+
+Menard & Lohstroh et al.
+that arise because one reaction (possibly) sends data to the
+other. The dashed arrows represent dependencies that arise
+because the two reactions belong to the same reactor. Analo-
+gous to the behaviors of actors, reactions of the same reactor
+are mutually exclusive. The execution order is well-defined
+and given by the lexical declaration order of the reactions in
+LF code. This order is also indicated by the numbers in the
+reaction labels in Figure 2.
+The dependency graph precisely defines in which order
+reactions need to be executed. Independent reactions may
+be executed in parallel without breaking determinism. For
+instance, the APG in Figure 3 tells us that reaction 1 of
+ProxyDelay and the reactions of userA and userB can all
+execute in parallel. Note that the dependency graph is re-
+quired to be acyclic as any cycle would violate causality. The
+LF compiler ensures that a valid program has an acyclic de-
+pendency graph. Any dependency cycles in LF programs can
+be resolved by introducing a logical action and using it to
+schedule a new event at a future tag.
+Since in LF all dependencies are statically declared, there is
+a lack of runtime agility compared to actors and similar mod-
+els. The reactor model compensates this with mutations
+that support runtime adaptations of the reactor topology and
+the implied dependency graph. However, LF does not fully
+implement mutations yet and a discussion of mutations is
+beyond the scope of this paper.
+4.2
+Scalable Connection Patterns
+Creating individual reactor instances, ports and connections
+becomes tedious for larger programs. To address this prob-
+lem, we introduce an extension to the LF syntax that allows
+to create multiple ports or reactor instances at once. Further,
+we introduce an overloading of LF’s connection operator
+to create multiple connections over multiports and banks
+at once. This mechanism allows realizing various complex
+connection patterns in a single line of code and, as it is fully
+parameterizable, allows LF programs to transparently scale
+to a given problem size without recompiling.
+Consider a simple fork-join program in LF:1
+1
+reactor Src(w: int (3)) {
+2
+output[w] out: int
+3 }
+4
+reactor Worker {
+5
+input in: int
+6
+output out: int
+7 }
+8
+reactor Sink(w: int (3)) {
+9
+input[w] in: int
+10 }
+11 main reactor(w: int (3)) {
+12
+src = new Src(w = w)
+13
+dst = new Sink(w = w)
+14
+wrk = new[w] Worker ()
+15
+src.out -> wrk.in
+16
+wrk.out -> dst.in
+17 }
+The program defines a Src, a Worker, and a Sink reactor and
+an unnamed main reactor that assembles the program. Worker
+defines two individual ports of type int called in and out. Src
+1Implementation details are omitted for brevity. Please refer to [53] for an
+introduction to LF syntax.
+userA
+ProxyDelay
+2
+ProxyDelay
+1
+Account
+1
+Account
+2
+userB
+Figure 3. Reaction graph for the program in Figure 2c.
+and Sink use our syntax extension to each define a multi-
+port of width w, where w is a parameter and defaults to 3. The
+main reactor creates all reactor instances and connections.
+Concretely, it creates two individual instances of Src and Sink
+and uses our syntax extension to create a bank of worker
+reactors of width w (line 14). The two connection statements
+(line 15, 16) establish w connections each, one for each pair of
+multiport and bank instance. The resulting connection pat-
+tern is illustrated in Figure 4a. Note that the precise number
+of workers is configurable via the w parameter of the main
+reactor, which can be specified when executing the program
+without recompiling. Hence the program can be configured
+to an arbitrary number of workers.
+In this example, the source reactor produces three sepa-
+rate values to be sent to the worker. Instead, if we want to
+broadcast a single value to all workers, then we can use the
+broadcast syntax (...)+. Configuring the source reactor to
+use a single output (by setting w=1 in line 12) and changing
+line 15 to (src.out)+ -> wrk.in creates the pattern in Figure 4b.
+Another common pattern that can be conveniently ex-
+pressed using LF syntax is a cascade composition, illustrated
+by the following program:
+1 main reactor(n: int (2)) {
+2
+src = new Src(w = 1)
+3
+dst = new Sink(w = 1)
+4
+wrk = new[n] Worker ()
+5
+src.out , wrk.out -> wrk.in , dst.in
+6 }
+The connection operator sequences all ports listed on the
+left- and right-hand side, and connects the 𝑛th port on the
+left hand side to the𝑛th port on the right-hand side. By offset-
+ting the left-hand side of the connection statement in line 5
+with a single source port and appending the sink port to the
+right hand side, we can effectively arrange the connections
+to form the cascade shown in Figure 4c.
+The connection operator also connects multiports within
+banks. In this case, the operator will implicitly unfold all
+port instances on both sides of the connection to form a flat
+list of ports. The unfolding happens such that we first list all
+ports of the first bank instance, then all ports of the second
+instance, and so on. Consider the following program:
+1
+reactor Node(w: int (3)) {
+2
+input[w] in: int
+3
+output[w] out: int
+4 }
+5 main reactor(w: int (3)) {
+6
+node = new[w] Node(w=w)
+7
+node.out -> node.in
+8 }
+This will create the pattern shown in Fig. 4d which is not
+very useful. Using the interleaved modifier on either side
+of the connection in line 7, we can modify the unfolding
+
+High-Performance Deterministic Concurrency using Lingua Franca
+Fork-Join
+dst
+src
+wrk[2]
+wrk[1]
+wrk[0]
+out[1]
+out[3]
+out[0]
+in[0]
+in[1]
+in[2]
+in
+out
+in
+out
+in
+out
+(a) Multiport connections
+Fork-Join-Broadcast
+wrk[2]
+wrk[1]
+wrk[0]
+dst
+src
+out
+in[0]
+in[1]
+in[2]
+in
+out
+in
+out
+in
+out
+(b) Broadcast connection
+Cascade
+wrk[1]
+in
+out
+wrk[0]
+in
+out
+dst
+in
+src
+out
+(c) A cascade composition
+Non-Interleaved
+node[2]
+in[2]
+in[1]
+in[0]
+out[0]
+out[1]
+out[2]
+node[1]
+in[2]
+in[1]
+in[0]
+out[0]
+out[1]
+out[2]
+node[0]
+in[2]
+in[1]
+in[0]
+out[0]
+out[1]
+out[2]
+(d) Multiports within a bank: direct connections
+Interleaved
+node[2]
+in[2]
+in[1]
+in[0]
+out[0]
+out[1]
+out[2]
+node[1]
+in[2]
+in[1]
+in[0]
+out[0]
+out[1]
+out[2]
+node[0]
+in[2]
+in[1]
+in[0]
+out[0]
+out[1]
+out[2]
+(e) Multiports within a bank: interleaved connections
+Figure 4. Various connection patterns in Lingua Franca
+strategy to first list all first port instances within all bank
+instances, then the second port instances within all bank
+instances, and so on. This creates the fully connected pattern
+shown in Figure 4e.
+All of the patterns discussed in this section are used ex-
+tensively in our benchmark implementations in Section 5.
+4.3
+Runtime Implementation
+The execution of each LF program is governed by a runtime.
+Most importantly, the runtime includes a scheduler which
+keeps track of all scheduled future events, controls the ad-
+vancement of logical time, and invokes any triggered reac-
+tions in the order specified by the dependency graph while
+aiming to exploit as much parallelism as possible. Lohstroh
+et al. have already sketched a simple scheduling algorithm
+for reactor programs [51]. In this section, we present a C++
+implementation of this scheduling algorithm that aims at ex-
+ploiting parallelism while keeping synchronization overhead
+to a minimum and avoiding contention on shared resources.
+Figure 5 gives an overview of the scheduling mechanism
+used in our runtime. The scheduler keeps track of future
+events in the event queue and processes them strictly in tag
+order. When processing an event, the scheduler first deter-
+mines all reactions that are triggered by the event and stores
+them in the reaction queue. Any reactions in the reaction
+queue for which all dependencies are met (as indicated by
+the APG) are forwarded to the ready queue and then picked
+up for execution by the worker threads. If the executed re-
+actions trigger any further reactions by setting ports, those
+reactions are inserted in the reaction queue. If a reaction
+Initial-
+Events
+Event Queue
+Reaction Queue
+Ready Queue
+Workers
+set()
+schedule()
+Figure 5. Scheduling mechanism in the LF runtime.
+schedules future events via an action, these new events are
+inserted into the event queue. Note that the scheduler always
+waits until all reactions at the current tag are processed be-
+fore advancing to the next tag and triggering new reactions.
+The most important task of the scheduler is to decide when
+any given reaction should be moved from the reaction queue
+to the ready queue. As the APG precisely defines the order-
+ing constraints of reactions, reaction scheduling is closely
+related to DAG-based scheduling strategies [2, 38]. However,
+the APG is not equivalent to a task graph as it may contain
+reactions that do not need to be executed. Most often only a
+fraction of the reactions is triggered at a particular tag. More-
+over, we do not know in advance precisely which reactions
+will be triggered for a given tag, as reactions may or may
+not send messages via their declared ports. In consequence
+an optimal schedule cannot be computed in advance.
+To decide whether a given triggered reaction is ready for
+execution, we need to check if it has a dependency on any
+other reaction that is triggered or currently executing. To
+avoid traversing the APG at runtime, we utilize a simple
+heuristic. Concretely, we assign a level (top level as defined
+in [38]) to each reaction. Any reactions with the same level
+do not depend on each other and hence can be executed in
+parallel. Our scheduler then processes reactions going from
+one level to the next. Once all reactions within a level are pro-
+cessed, all triggered reactions in the next level are moved to
+the ready queue. This approach avoids the need for analyzing
+the APG during execution, but also falls short on exploit-
+ing all opportunities for parallel execution. For instance, this
+approach does not execute reaction 2 of ProxyDelay in paral-
+lel with reaction 2 of Account. Nonetheless, our evaluation
+shows that this strategy is sufficient to efficiently exploit
+parallelism in most cases. Given the extensive research on
+DAG-scheduling, we are confident that we can apply more
+complex strategies in future work to also account for the
+missed opportunities for exploiting parallelism.
+Another limitation of our scheduling approach is that the
+scheduler only executes reactions that are triggered at the
+same tag. In particular, this may hinder exploiting pipeline
+parallelism in programs that do not use logical delays to
+create separate pipeline stages. However, this limitation can
+be overcome by using a federated execution strategy as dis-
+cussed by Bateni et al. [7].
+While the scheduling algorithm sketched in [51] and re-
+fined in this section is relatively straight forward to imple-
+ment, further optimizations where needed to achieve com-
+petitive performance. In the following, we detail the most
+important optimizations that we use in our C++ runtime.
+
+Menard & Lohstroh et al.
+Coordinating worker threads. In the above discussion
+we conceptually distinguished the scheduler from the work-
+ers. In an actual implementation, however, using a central
+scheduler and separate worker thread imposes a significant
+synchronization overhead. Instead, in our implementation,
+any of the worker threads can become the scheduler and
+move ready reactions to the ready queue or advance logical
+time to the next tag if all reactions have been processed. Fur-
+thermore, we exploit the fact that at any time we know the
+number of reactions to execute in parallel and use a counting
+semaphore to control the number of active workers.
+Lock-free data structures. The three queues and other
+data structures that are required for bookkeeping (e.g., a
+list of all set ports) are shared across workers. Using mu-
+texes for synchronization proved to be inefficient due to
+high contention on the shared resources, especially when
+many parallel reactions set ports or schedule actions. Instead,
+we utilize lock-free data structures where possible. For in-
+stance, the ready queue is implemented as a fixed size buffer
+paired with an atomic counter. Since we know precisely how
+many reactions can at most run in parallel (i.e. the maximum
+number of reactions in the APG that have the same level), we
+can fix the size of the queue. Every time new reactions are
+moved to the reaction queue, the atomic counter is set to the
+number of reactions in the queue. Each time a worker tries
+to execute a reaction it atomically decrements the counter.
+If the result is negative, then the queue is empty. Otherwise
+the result provides the index within the buffer to read from.
+We further exploit knowledge about the execution of re-
+actor programs. For instance, the scheduler advances logical
+time only once all reactions have been processed. This op-
+eration is safe without additional synchronization, as all of
+the workers are waiting for new reactions.
+Sparse multiports. In programs where reactions in mul-
+tiple reactors may trigger the same reaction (such as an
+account with an arbitrary number of users), the triggered
+reaction often needs to know which port(s) actually are set
+(contain data). If there are many upstream reactors and com-
+munication is sparse, simply checking all ports for presence
+can be inefficient. Instead, we expose an API for obtaining
+an iterator to only set ports. Note that this problem does not
+arise in actors, as no ports exist and messages are processed
+one by one, only considering those that are actually sent.
+5
+Performance evaluation
+The actor model is widely accepted for programming large
+concurrent applications, and implementations such as the
+C++ Actor Framework (CAF) [16] and Akka [66] are known
+to be fast and efficient in utilizing a larger number of threads.
+Compared to actors, LF imposes various restrictions that
+amount to a model of computation in which fewer behaviors
+are allowed. In this section, we show that these restrictions
+do not necessarily introduce overhead or higher execution
+times. In fact, LF is considerably faster for many benchmarks.
+5.1
+Methodology
+Our evaluation is based on the Savina benchmark suite [34]
+for actor languages and frameworks. While this suite has
+several issues, as Blessing et al. discuss in [13], Savina covers
+a wide range of patterns and, to the best of our knowledge,
+is the most comprehensive benchmark suite for actor frame-
+works that has been published. The Savina suite includes
+Akka implementations of all benchmarks. CAF implementa-
+tions of most Savina benchmarks are also available.2
+We ported 22 of the 30 Savina benchmarks to the C++
+target of LF.3 Due to the fundamental differences between
+the actor and reactor model, the process of porting bench-
+marks is not always straightforward. We aimed at closely
+resembling the original workloads and considered the inten-
+tion behind the individual benchmarks. We did not imple-
+ment the benchmarks Fork Join (actor creation), Fibonacci,
+Quicksort, Bitonic Sort, Sieve of Eratosthenes, Unbalanced
+Cobwebbed Tree, Online Facility Location, and Successive
+Over-Relaxation as they require the capability to dynamically
+create actors. In the reactor model, this can be achieved with
+mutations that may modify the reactor topology [50, 51].
+However, mutations are not yet fully implemented in LF,
+and a discussion of language-level constructs for supporting
+mutations is beyond the scope of this paper.
+We further omit the A*-Search and Logistic Map Series
+benchmarks from our presentation. A*-Search suffers from a
+severe race condition that results in wildly varying execution
+times [13]. Logistic Map Series is omitted as it violates actor
+semantics and requires explicit synchronization [13]. For this
+reason, the CAF implementation needs to use a blocking call,
+which makes it slower than the other implementations by
+at least two orders of magnitude. Since this is not a problem
+of CAF, but rather a problem in the benchmark design, we
+omit Logistic Map Series to avoid skewing the analysis.
+Figure 6 reports measured results for all supported bench-
+marks obtained with Akka, CAF, and the C++ target of LF.
+The plots show the mean execution times (including 99% con-
+fidence intervals) for a varying number of worker threads
+for each of the benchmarks. Not all benchmarks are imple-
+mented in CAF and hence it is missing in some plots.
+All measurements were performed on a workstation with
+an Intel Core i9-10900K processor with 32 GiB DDR4-2933
+RAM running Ubuntu 22.04 and using CAF version 17.6 and
+Akka version 2.6.17. Following the methodology of Savina,
+measurements exclude initialization and cleanup. Each mea-
+surement comprises 32 iterations. The first two iterations are
+2https://github.com/woelke/savina
+3Source code available at https://github.com/lf-lang/benchmarks-lingua-
+franca
+
+High-Performance Deterministic Concurrency using Lingua Franca
+Recursive Matrix Multiplication #20
+Radix Sort #22
+Filter Bank #23
+Trapezoidal Approximation #28
+Precise Pi Computation #29
+Cigarette Smokers #14
+Bank Transaction #16
+Producer Consumer (bounded) #11
+All−Pairs Shortest Path #17
+N Queens First K Solutions #19
+Big #8
+Concurrent Dictionary #9
+Concurrent Sorted Linked−List #10
+Dining Philosophers #12
+Sleeping Barber #13
+Ping Pong #1
+Thread Ring #2
+Counting Actor #3
+Fork Join (throughput) #4
+Chameneos #7
+1
+2
+4
+8
+12
+16
+20
+1
+2
+4
+8
+12
+16
+20
+1
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+12
+16
+20
+1
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+16
+20
+1
+2
+4
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+12
+16
+20
+1
+2
+4
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+12
+16
+20
+1
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+4
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+12
+16
+20
+1
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+12
+16
+20
+1
+2
+4
+8
+12
+16
+20
+1
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+4
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+12
+16
+20
+1
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+12
+16
+20
+1
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+4
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+12
+16
+20
+1
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+8
+12
+16
+20
+1
+2
+4
+8
+12
+16
+20
+1
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+4
+8
+12
+16
+20
+1
+2
+4
+8
+12
+16
+20
+1
+2
+4
+8
+12
+16
+20
+1
+2
+4
+8
+12
+16
+20
+1
+2
+4
+8
+12
+16
+20
+1
+2
+4
+8
+12
+16
+20
+0
+100
+200
+300
+0
+50
+100
+0
+100
+200
+300
+400
+500
+0
+100
+200
+300
+0
+50
+100
+150
+200
+0
+100
+200
+300
+400
+500
+0
+300
+600
+900
+0
+200
+400
+600
+800
+0
+50
+100
+150
+200
+0
+5000
+10000
+15000
+0
+500
+1000
+1500
+2000
+0
+500
+1000
+1500
+2000
+0
+200
+400
+600
+0
+50
+100
+150
+200
+0
+20
+40
+60
+80
+0
+200
+400
+600
+800
+0
+500
+1000
+1500
+0
+500
+1000
+1500
+0
+30
+60
+90
+120
+0
+250
+500
+750
+1000
+Number of Threads
+Mean Execution Time in ms
+Akka
+CAF
+LF C++ Target
+Figure 6. Mean execution times and 99% confidence intervals for various Savina benchmarks implemented in LF, CAF, and
+Akka, measured for a varying number of worker threads. The numbers prefixed with # are benchmark IDs as listed in [34].
+excluded from our analysis and are used to warm up caches
+and the JVM (in the case of Akka).
+5.2
+Discussion
+The first six plots in Figure 6 belong to the group of micro
+benchmarks in the Savina suite. These are designed to ex-
+pose overhead in the protocol used for exchanging messages
+and for scheduling. Overall, our C++ runtime shows com-
+parable performance to Akka and CAF. In Ping Pong and
+Thread Ring, our implementation is considerably faster than
+Akka but is still outperformed by CAF. For Counting Actor
+and Big, Akka performs better and the LF performance is
+slightly behind CAF. In Fork Join and Chameneos, the LF
+implementation outperforms both Akka and CAF, especially
+when using a larger number of worker threads.
+The next six plots (Concurrent Dictionary to Bank Trans-
+action) belong to the group of concurrency benchmarks. LF
+significantly outperforms CAF and Akka in all the concur-
+rency benchmarks (especially for a high number of worker
+threads). This highlights how concurrent behavior is ex-
+pressed naturally in LF and can be executed efficiently. Actor
+implementations of those benchmarks, on the other hand,
+have to synchronize explicitly and resort to potentially ex-
+pensive protocols (e.g., by sending acknowledge messages),
+or implement some other (blocking) synchronization strat-
+egy that violates actor semantics [13].
+The remaining plots belong to the group of parallelism
+benchmarks in the Savina suite4. Radix Sort and Filter Bank
+suffer somewhat from an inefficiency in our scheduler, as
+discussed in Section 4.3. In these particular benchmarks,
+our simple algorithm leads to a non-optimal execution as
+some reactions are executed later than they could. We will
+revise this algorithm in future work. However, the remain-
+ing parallelism benchmarks highlight that LF can efficiently
+implement parallel algorithms. Our LF implementations are
+on par with Akka and CAF and scale well with more threads.
+On average, LF outperforms both Akka and CAF. For 20
+threads, the C++ runtime achieves a speedup of 1.85𝑥 over
+Akka and a 1.42𝑥 speedup over CAF. These speedups were
+calculated using the geometric mean over the speedups of
+individual benchmarks. We conclude that LF can compete
+with and even outperform modern and highly optimized
+4Producer Consumer is actually listed as a concurrency benchmark, but we
+find it fits better to the group of parallelism benchmarks.
+
+Menard & Lohstroh et al.
+actor frameworks such as Akka and CAF. Particularly with
+workloads that require synchronization, LF significantly out-
+performs actor implementations. LF is as efficient as the
+actor frameworks in exploiting parallelism and scales well
+to a larger thread count. In summary, the deterministic con-
+currency provided by LF does not hinder performance but
+enables more efficient implementations. This is possible in
+part because the scheduler has insights into the program
+structure, and explicit synchronization can be avoided in LF,
+as opposed to many of Savina actor benchmarks.
+The performance comparison between C++ and Scala
+(Akka) needs to be taken with care, as other factors such
+as different library implementations and the behavior of
+the JVM may influence performance. For instance, the large
+discrepancy between Akka and our implementation in the
+Pi Precision benchmark is explained by a less efficient rep-
+resentation of large numbers in Scala/Java. However, the
+other benchmarks of the Savina suite do not depend on
+external libraries and are designed to be more portable be-
+tween languages. Also note that over all benchmarks CAF
+only achieves an average speedup of 1.09𝑥 over Akka for
+20 threads and is outperformed in 9 out of 16 benchmarks.
+For single threaded execution, Akka outperforms CAF in
+10 benchmarks and achieves an average speedup of 1.33x.
+This indicates that the implemented Scala workloads are
+comparable to the C++ implementations. Even considering a
+potential skew due to the JVM, our results clearly show that
+LF can compete with state-of-the-art actor frameworks.
+To better understand the impact of the optimizations dis-
+cussed in Section 4.3, Figure 7 also shows the speedup of our
+runtime for 20 worker threads compared to a less optimized
+runtime. This baseline is an older version of our runtime
+that is optimized in the sense that we used code profiling
+to identify obvious bottlenecks and eliminated them using
+common code optimization techniques, but that does not
+include the optimizations discussed in this paper. The av-
+erage overall speedup (geometric mean) achieved by our
+optimizations is 2.18𝑥. In particular, Big and Bank Transac-
+tion significantly benefit from our optimization for sparse
+communication patterns. The concurrency benchmarks (e.g.,
+Concurrent Dictionary and Dining Philosophers), are mostly
+improved by reducing the contention on shared resources.
+However, not all benchmarks benefit from our optimizations.
+The reduced performance in Ping Pong and Counting Actor
+shows that optimizing for efficient parallel execution also
+comes at a cost for simple sequential programs.
+6
+Related Work
+LF is closely related to the languages and frameworks evolved
+around Hewitt’s actor model [1, 33], including Akka [66],
+CAF [16], Ray [60], Erlang [3], Rebeca [70], P [22] and Pony
+[17]. Also reactive programming techniques [5], as used in
+frameworks like ReactiveX [56] and Reactors.IO [65] but
+1.0
+1.0
+1.0
+1.0
+1.0
+1.0
+1.0
+1.0
+1.0
+1.0
+1.0
+1.0
+1.0
+1.0
+1.0
+1.0
+1.0
+1.0
+1.0
+1.0
+0
+5
+10
+15
+20
+Ping Pong
+Thread Ring
+Counting Actor
+Fork Join (throughput)
+Chameneos
+Big
+Concurrent Dictionary
+Concurrent Sorted Linked−List
+Dining Philosophers
+Sleeping Barber
+Cigarette Smokers
+Bank Transaction
+Producer Consumer (bounded)
+All−Pairs Shortest Path
+N Queens First K Solutions
+Recursive Matrix Multiplication
+Radix Sort
+Filter Bank
+Trapezoidal Approximation
+Precise Pi Computation
+Benchmark
+Speedup
+Figure 7. Speedup achieved by our optimized C++ runtime
+for 20 worker threads compared to an unoptimized version.
+also in language-level constructs like event loops [74], are
+closely related to LF. While actors and reactive programming
+provide good resiliency and scalability, this comes at the
+cost of nondeterminism, which makes programs notoriously
+hard to test and debug [6, 75]. Even more problems arise
+if languages, frameworks, and libraries do not enforce the
+underlying model and invite the programmer to break its se-
+mantics [71]. Pony addresses the later problem by leveraging
+a strong type system similar to Rust to prevent data races at
+compile time. Rebeca provides a formalism and model check-
+ing techniques for analyzing and verifying actor networks.
+While this can improve confidence in a correct implemen-
+tation, the programmer is still responsible for finding this
+correct implementation. P goes a step further in that it also
+has an efficient runtime system and a compiler that generates
+correct-by-construction code with reasonable performance.
+Blessing et al. propose a strategy that maps actor commu-
+nication to a tree topology in order to guarantee a causal
+ordering of messages [12]. In a similar approach, Sang et al.
+utilize a DAG topology to achieve serializability in the pro-
+cessing of events. Orleans [14] is also based on an actor-like
+model and provides guarantees on atomicity on transactions.
+Finally, Reactors as defined by Field et al. [28] is a model that
+is closely related to actors but that supports both synchro-
+nous and asynchronous communication and also provides
+atomicity guarantees. All these strategies are most useful in
+distributed scenarios, in particular in presence of network
+failures. In this paper, however, we focus on the execution of
+a single host. Moreover, the determinism guarantees that LF
+makes are stronger. Nonetheless, such techniques are highly
+relevant to LF and could be deployed for ensuring fault-
+tolerant execution in distributed LF programs. We believe,
+however, that LF provides a more general solution, as the
+programmer can explicitly trade consistency for availability
+in distributed contexts [42], and hence the solution can be
+adjusted to the concrete application requirements.
+Dataflow models [11, 21, 44] and process networks [35,
+45] provide deterministic concurrency by creating statically
+connected networks of actors with deterministic semantics.
+
+High-Performance Deterministic Concurrency using Lingua Franca
+These models enable improved static analysis and optimiza-
+tion [30], but they limit the application’s flexibility and ca-
+pability to react to external events. Ohua [26] is another lan-
+guage providing parallelism through dataflow and is similar
+to LF in that it integrates with existing high-level languages.
+However, it falls short on exposing coordination facilities for
+individual nodes and does not provide a timed semantics.
+Deterministic concurrency is also found in synchronous
+languages such as Esterel [10], Lustre [31], and SIGNAL [9]
+as well as in Functional Reactive Programming (FRP) lan-
+guages, like Fran [25], FrTime [19], and Elm [20]. However,
+these languages are challenging to use for general purpose
+programming as they require pure functions and there is a
+lack of widely-applicable libraries. Only recently, side effects
+have been considered in a formal semantics for Esterel [29]
+and distributed dataflow [59]. In Lingua Franca, arbitrary
+code can be embedded in reactions and we can benefit from
+the available libraries for popular general purpose languages.
+Another interesting approach is taken by deterministic
+multithreading libraries such as DThreads [49] or Conse-
+quence [57], which enforce a total order for concurrent store
+operations. Recent work has made considerable progress in
+avoiding the bottlenecks of conventional DTM techniques [58].
+However, we argue that threads are not a convenient con-
+currency model for the reasons outlined in [40]. Moreover,
+threads do not allow for transparent distributed execution
+as is possible with (re)actors.
+7
+Conclusion
+Unlike actors and related models for asynchronous concur-
+rency, LF enforces determinism by default, and features asyn-
+chronous behavior only when introduced deliberately. Our
+evaluation, based on LF’s C++ target, shows that this de-
+terministic model does not impede performance. On the
+contrary, we achieve an average speedup of 1.85𝑥 over Akka
+and 1.42𝑥 over CAF. With LF, we manage to combine repro-
+ducible (and testable) behavior with good performance. Yet,
+our relatively simple scheduling strategy likely still leaves
+room for significant improvement. We leave it as future work
+to explore more advanced scheduling algorithms capable of
+exploiting more parallelism at runtime. We also aim to fur-
+nish full runtime support for mutations and implement the
+remaining Savina benchmarks that require them. Finally,
+we note that our implementation of the Savina benchmark
+suite is not only useful for comparing LF to actor-oriented
+frameworks; it also demonstrates that LF, which is still in its
+infancy, is already suitable for solving practical problems.
+Acknowledgments
+The work in this paper was supported in part by the German
+Federal Ministry of Education and Research (BMBF) as part
+of the Software Campus (01IS12051) and the program “Sou-
+verän. Digital. Vernetzt.”, joint project 6G-life (16KISK001K).
+This work was also supported in part by the National Sci-
+ence Foundation (NSF), award #CNS-1836601 (Reconciling
+Safety with the Internet), the iCyPhy Research Center (In-
+dustrial Cyber-Physical Systems), supported by Denso, Ford,
+Siemens, and Toyota, and the National Research Foundation
+(NRF) of Korea (No. NRF-2022R1F1A1065201).
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+page_content='de  TU Dresden, Germany  Marten Lohstroh  marten@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='edu  UC Berkeley, USA  Soroush Bateni  soroush@berkely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='edu  UC Berkely, USA  Matthew Chorlian  mattchorlian@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='edu  UC Berkeley, USA  Arthur Deng  langxing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='deng@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='edu  UC Berkeley, USA  Peter Donovan  peterdonovan@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='edu  UC Berkeley, USA  Clément Fournier  clement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='fournier@tu-dresden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='de  TU Dresden, Germany  Shaokai Lin  shaokai@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='edu  UC Berkeley, USA  Felix Suchert  felix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='suchert@tu-dresden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='de  TU Dresden, Germany  Tassilo Tanneberger  tassilo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='tanneberger@tu-dresden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='de  TU Dresden, Germany  Hokeun Kim  hokeun@hanyang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='kr  Hanyang University, South Korea  Jeronimo Castrillon  jeronimo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='castrillon@tu-dresden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='de  TU Dresden, Germany  Edward A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Lee  eal@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='edu  UC Berkeley, USA  Abstract  Actor frameworks and similar reactive programming tech-  niques are widely used for building concurrent systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' They  promise to be efficient and scale well to a large number of  cores or nodes in a distributed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' However, they also  expose programmers to nondeterminism, which often makes  implementations hard to understand, debug, and test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The  recently proposed reactor model is a promising alternative  that enables efficient deterministic concurrency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In this pa-  per, we show that determinacy does neither imply a loss in  expressivity nor in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' To show this, we evaluate  Lingua Franca (LF), a reactor-oriented coordination lan-  guage that equips mainstream programming languages with  a concurrency model that automatically takes advantage of  opportunities to exploit parallelism that do not introduce  nondeterminism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Our implementation of the Savina bench-  mark suite demonstrates that, in terms of execution time, the  runtime performance of LF programs even exceeds popular  and highly optimized actor frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' We compare against  Akka and CAF, which LF outperforms by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='86𝑥 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='42𝑥,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  CCS Concepts: • Computing methodologies → Concur-  rent programming languages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' • Software and its engi-  neering → Runtime environments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Source code gener-  ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Keywords: coordination, concurrency, determinism, perfor-  mance  1  Introduction  Theoreticians working on programming language semantics  have long understood the value of determinism as well as the  expressive power of nondeterminism in programming lan-  guages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In practice, however, today, nondeterminism creeps  into programming languages and frameworks not to benefit  from its expressiveness, but rather because of a widespread  perception that it is needed to get good performance on par-  allel hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' We show in this paper that it is not necessary  to sacrifice determinism to achieve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' We do this  by focusing on actor frameworks, which have proved pop-  ular and successful in many very demanding applications,  but admit nondeterminism that is often not actually needed  by their applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Exploiting parallel hardware such as multicore machines  to improve performance is only possible when programs  expose concurrency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Common abstractions for concurrency  include threads [18], remote procedure calls [63], publish-  subscribe [27], service-oriented architectures [64], and ac-  tors [1, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Each of these models has its own merits, but  they all introduce nondeterminism: situations where, for  a given state and input, the behavior of a program is not  uniquely defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' While nondeterminism can be useful in  some applications, most programming tasks benefit from  more repeatable behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Deterministic programs are easier  to understand, debug, and test (for each test vector, there is  one known-good response).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' For nondeterministic programs,  problematic behaviors might be harder to discover because  they may only occupy a small fraction of the state space [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  And reproducing failures can be extremely hard [40, 48, 61]  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='02444v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='PL]  6 Jan 2023  Menard & Lohstroh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  because they might occur only when the system is under a  specific amount of load [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Determinism is a subtle concept [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Here, we focus on a  particular form of determinism for programs, where a pro-  gram is deterministic if, given the same inputs, it always  produces the same outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' This definition does not require  that operations be performed in a particular order, and there-  fore is not at odds with concurrency and parallel execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  It is possible, but often not easy, to achieve this form of deter-  minism even when using nondeterministic abstractions such  as threads, actors, and asynchronous remote procedure calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  For simple enough programs, such as a chain of actors, if  communication is reliable, then execution will be determin-  istic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Some of the benchmarks we compare against in this  paper are deterministic in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' As we will show, how-  ever, even slightly more complex communication structures  result in nondeterminism that can be difficult to correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  In this paper, we evaluate a language-based coordination  approach to specifying concurrent software that preserves  determinism by default and only admits nondeterminism  when explicitly introduced by the programmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The coordi-  nation language Lingua Franca (LF) [53], which is based on  a concurrent model of computation called reactors [50, 51],  achieves this by analyzing program structure and ensuring  that data dependencies are observed correctly at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  An LF program defines reactive software components called  “reactors” and provides operators to compose them hierarchi-  cally (through containment) and bilaterally (via connections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Because the language supports both deterministic and non-  deterministic concurrency, it provides a fertile ground for  exploring the impact of determinism on performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  The semantics of the deterministic subset of LF can be  thought of as a deterministic variant of actors [1, 33, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' We  show in this paper that it delivers performance comparable  to popular nondeterministic realizations of actors on parallel  hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Like Akka [66] and CAF [16]—the frameworks we  compare against—LF orchestrates the execution of chunks  of code written in conventional programming languages,  allowing programmers to rely on the languages, libraries, and  tools that they are comfortable with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Unlike the frameworks  we compare against, LF is polyglot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' It currently supports C,  C++, Python, TypeScript, and Rust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' This paper focuses on  the runtime performance of the C++ target, which, as a core  contribution of this paper, has been optimized to efficiently  exploit concurrency on parallel hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Earlier work [53]  has only reported preliminary performance indications of  LF based on its C target, which is predominantly aimed at  running on embedded systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  At the core of LF’s concurrency model is a logical model of  time that gives a clear notion of simultaneity and avoids dead-  locks using dependency analysis based on causality inter-  faces [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' It is this timed semantics that enables efficient de-  terministic concurrency in LF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' However, the benchmarks we  compare against were created to evaluate actor frameworks,  User A  User B  Account  Deposit  Withdrawal  (a) Deposit and With-  drawal sent by different  users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  User  Account  Deposit  Withdrawal  (b)  Deposit  and  With-  drawal  sent  by  same  user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  User  Proxy  Account  Deposit  Withdrawal  Withdrawal  (c) Withdrawal sent via a proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Example actor programs that may expose nonde-  terministic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  which have no temporal semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' None of the benchmarks  take advantage of the time-related features of LF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' the tempo-  ral semantics is only used to deliver determinism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' We show that the reactor-oriented par-  adigm as implemented in Lingua Franca enables efficient  exploitation of parallel hardware without relinquishing deter-  minism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' For this, we explain the mechanisms through which  LF programs expose concurrency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' we present a language  extension that allows for the definition of scalable programs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  and we introduce an optimized C++ runtime for LF that  enables efficient parallel execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' We further present an ex-  tensive evaluation based on the Savina benchmark suite [34],  showing that our LF runtime outperforms Akka and CAF by  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='86𝑥 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='42𝑥, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' We first motivate our work (Section 2) and then  introduce LF (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' We then go into detail about the  concurrency in LF and introduce our optimized C++ runtime  (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Next, we report benchmark results (Section 5),  discuss related work (Section 6), and conclude (Section 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  2  Motivation  The actor model is widely accepted and deployed in produc-  tion for its promise to allow programmers to easily express  concurrency, provide high execution performance, and scale  well to large datasets and complex applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Moreover, in  contrast to thread-based programs, actor semantics prevents  low-level data races.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' However, like most message passing  paradigms, actors expose the programmer to nondetermin-  ism in the form of high-level data races [76], a problem that  becomes considerably challenging to manage as the com-  plexity of a program grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Consider the simple example in Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The Account  actor manages the balance of a bank account that two users  interact with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' User A sends a deposit message increasing  the account’s balance and User B sends a withdrawal mes-  sage decreasing the account’s balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' If we assume that the  High-Performance Deterministic Concurrency using Lingua Franca  account : Account  balance: float(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0)  1  2  userA : User  offset: time(1 sec)  value: float(20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0)  (1 sec)  userB : User  offset: time(2 sec)  value: float(-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0)  (2 sec)  (a) Simple LF implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Proxy  Account  2  1  userB : User  (2 sec)  userA : User  (1 sec)  (b) Adding a proxy reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  ProxyDelay  2  1  L  Account  2  1  userB : User  (2 sec)  userA : User  (1 sec)  (c) Adding a proxy introducing a logical delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Example LF implementation of the actor program shown in Figure 1a and variations thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  balance is initialized to 0 and the account only grants a with-  drawal if the resulting balance is not negative, then there are  two possible behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' If A’s message is processed first, then  the withdrawal is granted to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' If B’s message is processed  first, then the withdrawal is denied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The actor model assigns  no meaning to the ordering of messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Therefore, there is  no well-defined correct behavior for this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  The reader may object that for an application like that of  Figure 1a, the order of transactions is intrinsically nonde-  terministic, and any additional nondeterminism introduced  by the software framework is inconsequential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' However, if  we focus on testability, we see that even identical inputs can  yield different results, making testing more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' If we  focus on consistency, the problem that different observers of  the same events may see different behaviors becomes prob-  lematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In databases, it is common to assign time stamps  to external inputs and to then treat those timestamps as a  semantic property of the inputs and define the behavior of  the database relative to those time stamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' We adopt this  perspective in this paper, and rely on the definition of de-  terminism given by Lee [41]: “determinism is a property of  models, not of physical realizations,” and “A model is deter-  ministic if given all the inputs that are provided to the model,  the model defines exactly one possible behavior.” If we de-  fine “inputs” in Figure 1a to be time-stamped user queries  and “behavior” to be the sequence of actions taken by the  Account, then it is reasonable to demand determinism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Consider Figure 1b, which has only one user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Even if this  one user first sends a deposit and then a withdrawal mes-  sage, the actor model does not guarantee that the receiving  actor sees and processes the incoming messages in this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  While some actor frameworks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=', Akka and Erlang, guaran-  tee in-order message delivery, others, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=', AmbientTalk [77],  expressly do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Yet, even if the framework guarantees point-  to-point in-order message delivery, this property is not tran-  sitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' If we add a Proxy, as shown in Figure 1c, then we  cannot make any assumptions about the order in which  Account receives messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' This example further illustrates  that composing actors can have unexpected side effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Consequently, implementing solutions to practical concur-  rency problems with actors can be challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Even seem-  ingly simple concurrency problems like the one discussed  above require high programming discipline, and solutions are  typically difficult to maintain and tend to lack modularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In  addition, the inherent nondeterminism of actor frameworks  makes it hard to verify such solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Erroneous behavior  might only occur in a fraction of executions, and thus inte-  gration tests cannot reliably detect such “Heisenbugs” [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  In a recent study, Bagherzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' [4] analyzed bugs  in Akka programs that were discussed on StackOverflow or  GitHub and determined that 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='6% of the bugs are caused by  races.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' This makes high-level races the second most common  cause of bugs in Akka programs after errors in the program  logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In a similar study of 12 actor-based production systems,  Hedden and Zhao [32] determined that 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='2% of the reported  bugs were caused by bad message ordering, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='8% of bugs were  caused by incorrect coordination mechanisms, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='8% were  caused by erroneous coordination at shutdown, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='4%  of bugs were caused by erroneous coordination at startup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Note that these numbers only cover known bugs in their  studied projects and, as noted by the authors, the majority of  the reported message ordering bugs belonged to the Gatling  project because it already incorporated a debugging tool  called Bita [73] that is designed to detect such bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' We  suspect that there are more undetected bugs in projects that  do not use specialized debugging tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  The actor community has addressed the inherent nondeter-  minism of actors and the resulting bugs by introducing better  tools for analyzing and debugging actor programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' This in-  cludes TransPDOR [72], Bita [73], Actoverse [69], iDeA [55],  CauDEr [39], and Multiverse debugging [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' While these  are valuable solutions, we argue that a programming model  for expressing concurrent programs should provide deter-  ministic semantics by default and allow the programmer to  introduce nondeterminism only where it is desired and un-  derstood to do no harm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In such cases, the aforementioned  tools for nondeterministic behavior can still be utilized to  debug the implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  There are a number of ways to achieve deterministic con-  currency, including Kahn process networks [35, 36], many fla-  vors of dataflow models [21, 43, 62], physically asynchronous,  logically synchronous models [68], synchronous-reactive  languages [8, 24], and discrete-event systems [15, 23, 46, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Lohstroh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' [53] compare the reactor model to each of  these, showing that it has many of their best features and  Menard & Lohstroh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  fewer of their pitfalls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Lingua Franca builds on this reac-  tor model because it is more expressive than some of the  alternatives (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=', Kahn networks) and is stylistically close to  actors, which have proven effective in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In this paper,  we show that the resulting determinism does not incur a  performance penalty, but on the contrary, helps to achieve  improved performance in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  3  Introduction to Lingua Franca  Lingua Franca (LF) builds on the relatively new reactor-  oriented programming paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Intuitively, we can describe  reactors as deterministic actors with a discrete event execu-  tion semantics and explicitly declared ports and connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  A logical timeline is used to order events and ensure a deter-  ministic execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' As a polyglot language, LF incorporates  code in a target programming language to implement the  logic of each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' LF itself is only concerned with  the coordination aspect of a program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  In this section, we introduce the core concepts of reactors  and LF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Note however, that a full discussion of LF includ-  ing its syntax and tooling is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Instead, we base our discussion on LF’s diagrammatic repre-  sentation of programswhich gets synthesized from LF source  code automatically [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' A complete introduction to LF’s tex-  tual syntax is given by Lohstroh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='1  LF by Example  Figure 2a shows an LF implementation of the deposit/with-  drawal example in Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The program is assembled  from three reactor instances userA, userB and account,  shown as light gray boxes in the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Note that userA  and userB are instances of the same reactor class User and  hence share the same structure and functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In the di-  agram, black triangles denote ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In this example, both  users have an output port which is connected to a respective  input port at the account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' These ports and connections allow  the users to send requests to the account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  In LF, all computation is performed in reactive code seg-  ments called reactions that are implemented in an arbitrary  target language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In the diagram, reactions are represented  by dark gray chevrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' All reactions must explicitly declare  their triggers, other dependencies and potential effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='In  the example in Figure 2a, both users define a reaction that  is triggered by a timer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Timers are an LF construct used to  produce events in regular intervals or once at a specific time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  The timer of userA is configured to trigger an event one sec-  ond after program startup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' the timer of userB is configured  to trigger an event two seconds after program startup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The  corresponding reactions simply send a deposit or withdrawal  request by setting the user’s output port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  The Account reactor defines two reactions, one for each  of its inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Both reactions will simply try to apply the  requested change to the balance, which is stored in a state  variable local to the reactor instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Note that reactors may  define arbitrary state variables which are accessible by all  their own reactions (which does not include reactions of  contained reaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In addition to state variables, reactors  may also define parameters which can be set at instantiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  This mechanism allows the User reactor to be reusable, as  the precise time at which the timer triggers (offset) and  the amount to withdraw or deposit (value) are configurable  at instantiation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  The reader might notice that the separated reactions in  account duplicate logic and are not a practical solution, in  particular if there are many users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' We choose this represen-  tation to keep our exposition simple and avoid a detailed  discussion of LF syntax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Indeed, in LF a single reaction can  bind to an arbitrary number of upstream ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  When executed, the program will wait for 1 second before  triggering the timer of the userA reactor and invoking its  reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The event produced by this reaction will trigger  reaction 1 in account, which is invoked immediately af-  ter the first reaction completes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Two seconds after program  startup, userB will react and subsequently trigger reaction  2 in account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In this example, the deposit event (+20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0) oc-  curs earlier than the withdrawal event (-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0), and hence  our execution semantics ensures that the account processes  the deposit event before the withdrawal event, meaning the  balance will not become negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In a more realistic imple-  mentation, the two users would generate events sporadically  and have their reactions triggered not by a timer but a phys-  ical action (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' However, using a timer greatly  simplifies our exposition as we only have to consider a single  logical timeline along which events are ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Moreover,  such timers can be used to create regression tests that vali-  date program execution with specific input timings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Note that even when the two events occur logically si-  multaneously, meaning that both reactions in the account  reactor are triggered at the same logical time, the resulting  program will be deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' All reactions within the same  tag are executed according to a well-defined precedence re-  lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In particular, any reactions within the same reactor  are mutually exclusive and executed following the lexical  declaration order of the reactions in LF code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' This order is  also reflected by the numbers displayed on the reactions in  the diagrams in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' More details on the precedence  relation of reactions are given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  To deliberately change the order in which events occur, a  logical delay can be introduced in the program using a log-  ical action, as shown in Figure 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In the diagram, actions  are denoted by small white triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In contrast to ports,  which allow relaying events logically instantaneously from  one reactor to another, logical actions provide a mechanism  for scheduling new events at a later (logical) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Upon re-  ceiving an input, reaction 2 of the ProxyDelay reactor is trig-  gered, which schedules its logical action with a configurable  High-Performance Deterministic Concurrency using Lingua Franca  delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' This creates a new event which, when processed, trig-  gers reaction 1 of the ProxyDelay reactor, which retrieves  the original value and forwards it to its output port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  If we assume that a delay of 2 seconds is used for schedul-  ing the logical action, then the deposit message from userA  will only arrive at the account 3 seconds after startup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Hence  the deposit message will be processed after the withdrawal  message from userB, causing B’s request to be denied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  It is important to note that all of the discussed examples  are deterministic, regardless of the physical execution times  of reactions, as all events are unambiguously ordered along a  single logical timeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The physical timing of the events, on  the other hand, will be approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The contribution of this  paper is to show that such determinism does not necessarily  reduce performance and is also useful for applications that  have no need for explicit timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='2  Logical and physical time  All events have an associated tag, which is used to order  events on a logical timeline at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In time-sensitive  applications, tags are not purely used for logical ordering  but also relate to physical time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' By default, the runtime only  processes the events associated with a certain tag once the  current physical time 𝑇 is greater than the time value of  the tag 𝑡 (𝑇 > 𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' We say that logical time “chases” physical  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The relationship between physical and logical time in  the reactor model gives logical delays a useful semantics  and also permits the formulation of deadlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' This timed  semantics is particularly useful for software that operates in  cyber-physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' For a more in depth discussion of  LF’s timed-semantics, the interested reader may refer to [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  If an application has no need for any physical time prop-  erties, the concurrence of physical and logical time can be  turned off;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' in this case, the tags are used only to preserve  determinism, not to control timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Moreover, LF program-  mers are not required to explicitly control timing aspects of  their programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Delays can simply be omitted, for instance  when scheduling an action, in which case the runtime will  use the next available tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In consequence, also untimed gen-  eral purpose programs can benefit from the deterministic  concurrency enabled by LF’s timed-semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='3  Asynchrony and deliberate nondeterminism  The reactor model distinguishes logical actions and physical  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' A logical action is always scheduled synchronously  with a delay relative to the current tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' A physical action  may be scheduled from asynchronous contexts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' its event  is assigned a logical time based on the current reading of  physical time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Physical actions are the key mechanism for  handling sporadic inputs originating from physical processes  (such as users initiating withdrawal or deposit requests).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  The assignment of tags to physical actions is nondeter-  ministic in the sense that it is not defined by the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  However, once those tags are assigned, for example, to de-  posit or withdrawal requests by a user, the processing of the  events is deterministic and occurs in tag order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Hence, the  tags assigned to externally initiated events are considered as  part of the input, and given this input, the program remains  deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' This approach draws a clear perimeter around  the deterministic and therefore testable program logic while  allowing it to interact with sporadic external inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Physical actions can also be used within the program it-  self, for example, to nondeterministically assign a new tag  to a message received from another reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In this usage,  physical actions provide a means for deliberately introducing  actor-like nondeterminism into a program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  4  Efficient Deterministic Concurrency  LF programs are deterministic by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' This property is in-  herited from the reactor model that LF implements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Lohstroh  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' [53] explain why reactors behave deterministically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Their argument can be adapted to the concrete context of  the Lingua Franca language, but this is beyond the scope  of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Reactors are also concurrent, and, as we show  in this paper, the exposed concurrency is sufficient for the  runtime system to effectively exploit multi-core hardware to  where it matches or exceeds the performance of fundamen-  tally asynchronous and nondeterministic actor frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  In this section, we first show exactly how concurrency is ex-  posed and then describe in more depth how our C++ runtime  is implemented and how it utilizes parallel hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='1  Parallelism  The use of statically declared ports and connections as the  interfaces between reactors, as well as the declarations of  reaction dependencies, distinguish reactors from more dy-  namic models like actors or other asynchronous message  passing frameworks where communication is purely based  on addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' While the fixed topology of reactor programs  is less flexible and limits runtime adaptation, it also pro-  vides two key advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' First, it achieves a separation  of concerns between the functionality of components and  their composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Second, it makes explicit at the interface  level which dependencies exist between components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' As a  consequence, a dependency graph can be derived for any  composition of reactors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The dependency graph is an acyclic  precedence graph (APG) that organizes all reactions into  a partial order that captures all scheduling constraints that  must be observed to ensure that the execution of a reactor  program yields deterministic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Because this graph is  valid irrespective of the contents of the code that executes  when reactions are triggered, reactions can be treated as a  black box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' It is this property that enables the polyglot nature  of LF and exposes the concurrency in the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Figure 3 shows the dependency graph for the program  given in Figure 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The solid arrows represent dependencies  Menard & Lohstroh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  that arise because one reaction (possibly) sends data to the  other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The dashed arrows represent dependencies that arise  because the two reactions belong to the same reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Analo-  gous to the behaviors of actors, reactions of the same reactor  are mutually exclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The execution order is well-defined  and given by the lexical declaration order of the reactions in  LF code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' This order is also indicated by the numbers in the  reaction labels in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  The dependency graph precisely defines in which order  reactions need to be executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Independent reactions may  be executed in parallel without breaking determinism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' For  instance, the APG in Figure 3 tells us that reaction 1 of  ProxyDelay and the reactions of userA and userB can all  execute in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Note that the dependency graph is re-  quired to be acyclic as any cycle would violate causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The  LF compiler ensures that a valid program has an acyclic de-  pendency graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Any dependency cycles in LF programs can  be resolved by introducing a logical action and using it to  schedule a new event at a future tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Since in LF all dependencies are statically declared, there is  a lack of runtime agility compared to actors and similar mod-  els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The reactor model compensates this with mutations  that support runtime adaptations of the reactor topology and  the implied dependency graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' However, LF does not fully  implement mutations yet and a discussion of mutations is  beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='2  Scalable Connection Patterns  Creating individual reactor instances, ports and connections  becomes tedious for larger programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' To address this prob-  lem, we introduce an extension to the LF syntax that allows  to create multiple ports or reactor instances at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Further,  we introduce an overloading of LF’s connection operator  to create multiple connections over multiports and banks  at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' This mechanism allows realizing various complex  connection patterns in a single line of code and, as it is fully  parameterizable, allows LF programs to transparently scale  to a given problem size without recompiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Consider a simple fork-join program in LF:1  1  reactor Src(w: int (3)) {  2  output[w] out: int  3 }  4  reactor Worker {  5  input in: int  6  output out: int  7 }  8  reactor Sink(w: int (3)) {  9  input[w] in: int  10 }  11 main reactor(w: int (3)) {  12  src = new Src(w = w)  13  dst = new Sink(w = w)  14  wrk = new[w] Worker ()  15  src.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='out -> wrk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='in  16  wrk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='out -> dst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='in  17 }  The program defines a Src, a Worker, and a Sink reactor and  an unnamed main reactor that assembles the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Worker  defines two individual ports of type int called in and out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Src  1Implementation details are omitted for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Please refer to [53] for an  introduction to LF syntax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  userA  ProxyDelay  2  ProxyDelay  1  Account  1  Account  2  userB  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Reaction graph for the program in Figure 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  and Sink use our syntax extension to each define a multi-  port of width w, where w is a parameter and defaults to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The  main reactor creates all reactor instances and connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Concretely, it creates two individual instances of Src and Sink  and uses our syntax extension to create a bank of worker  reactors of width w (line 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The two connection statements  (line 15, 16) establish w connections each, one for each pair of  multiport and bank instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The resulting connection pat-  tern is illustrated in Figure 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Note that the precise number  of workers is configurable via the w parameter of the main  reactor, which can be specified when executing the program  without recompiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Hence the program can be configured  to an arbitrary number of workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  In this example, the source reactor produces three sepa-  rate values to be sent to the worker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Instead, if we want to  broadcast a single value to all workers, then we can use the  broadcast syntax (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=')+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Configuring the source reactor to  use a single output (by setting w=1 in line 12) and changing  line 15 to (src.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='out)+ -> wrk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='in creates the pattern in Figure 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Another common pattern that can be conveniently ex-  pressed using LF syntax is a cascade composition, illustrated  by the following program:  1 main reactor(n: int (2)) {  2  src = new Src(w = 1)  3  dst = new Sink(w = 1)  4  wrk = new[n] Worker ()  5  src.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='out , wrk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='out -> wrk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='in , dst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='in  6 }  The connection operator sequences all ports listed on the  left- and right-hand side, and connects the 𝑛th port on the  left hand side to the𝑛th port on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' By offset-  ting the left-hand side of the connection statement in line 5  with a single source port and appending the sink port to the  right hand side, we can effectively arrange the connections  to form the cascade shown in Figure 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  The connection operator also connects multiports within  banks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In this case, the operator will implicitly unfold all  port instances on both sides of the connection to form a flat  list of ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The unfolding happens such that we first list all  ports of the first bank instance, then all ports of the second  instance, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Consider the following program:  1  reactor Node(w: int (3)) {  2  input[w] in: int  3  output[w] out: int  4 }  5 main reactor(w: int (3)) {  6  node = new[w] Node(w=w)  7  node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='out -> node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='in  8 }  This will create the pattern shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' 4d which is not  very useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Using the interleaved modifier on either side  of the connection in line 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' we can modify the unfolding  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='High-Performance Deterministic Concurrency using Lingua Franca  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Fork-Join  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='dst  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='src  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='wrk[2]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='wrk[1]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='wrk[0]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='out[1]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='out[3]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='out[0]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='in[0]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='in[1]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='in[2]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='in  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='out  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='in  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='out  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='in  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='out  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='(a) Multiport connections  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Fork-Join-Broadcast  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='wrk[2]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
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+page_content='(d) Multiports within a bank: direct connections  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
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+page_content='(e) Multiports within a bank: interleaved connections  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Various connection patterns in Lingua Franca  strategy to first list all first port instances within all bank  instances, then the second port instances within all bank  instances, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' This creates the fully connected pattern  shown in Figure 4e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  All of the patterns discussed in this section are used ex-  tensively in our benchmark implementations in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='3  Runtime Implementation  The execution of each LF program is governed by a runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Most importantly, the runtime includes a scheduler which  keeps track of all scheduled future events, controls the ad-  vancement of logical time, and invokes any triggered reac-  tions in the order specified by the dependency graph while  aiming to exploit as much parallelism as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Lohstroh  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' have already sketched a simple scheduling algorithm  for reactor programs [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In this section, we present a C++  implementation of this scheduling algorithm that aims at ex-  ploiting parallelism while keeping synchronization overhead  to a minimum and avoiding contention on shared resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Figure 5 gives an overview of the scheduling mechanism  used in our runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The scheduler keeps track of future  events in the event queue and processes them strictly in tag  order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' When processing an event, the scheduler first deter-  mines all reactions that are triggered by the event and stores  them in the reaction queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Any reactions in the reaction  queue for which all dependencies are met (as indicated by  the APG) are forwarded to the ready queue and then picked  up for execution by the worker threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' If the executed re-  actions trigger any further reactions by setting ports, those  reactions are inserted in the reaction queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' If a reaction  Initial-  Events  Event Queue  Reaction Queue  Ready Queue  Workers  set()  schedule()  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Scheduling mechanism in the LF runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  schedules future events via an action, these new events are  inserted into the event queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Note that the scheduler always  waits until all reactions at the current tag are processed be-  fore advancing to the next tag and triggering new reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  The most important task of the scheduler is to decide when  any given reaction should be moved from the reaction queue  to the ready queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' As the APG precisely defines the order-  ing constraints of reactions, reaction scheduling is closely  related to DAG-based scheduling strategies [2, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' However,  the APG is not equivalent to a task graph as it may contain  reactions that do not need to be executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Most often only a  fraction of the reactions is triggered at a particular tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' More-  over, we do not know in advance precisely which reactions  will be triggered for a given tag, as reactions may or may  not send messages via their declared ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In consequence  an optimal schedule cannot be computed in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  To decide whether a given triggered reaction is ready for  execution, we need to check if it has a dependency on any  other reaction that is triggered or currently executing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' To  avoid traversing the APG at runtime, we utilize a simple  heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Concretely, we assign a level (top level as defined  in [38]) to each reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Any reactions with the same level  do not depend on each other and hence can be executed in  parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Our scheduler then processes reactions going from  one level to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Once all reactions within a level are pro-  cessed, all triggered reactions in the next level are moved to  the ready queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' This approach avoids the need for analyzing  the APG during execution, but also falls short on exploit-  ing all opportunities for parallel execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' For instance, this  approach does not execute reaction 2 of ProxyDelay in paral-  lel with reaction 2 of Account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Nonetheless, our evaluation  shows that this strategy is sufficient to efficiently exploit  parallelism in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Given the extensive research on  DAG-scheduling, we are confident that we can apply more  complex strategies in future work to also account for the  missed opportunities for exploiting parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Another limitation of our scheduling approach is that the  scheduler only executes reactions that are triggered at the  same tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In particular, this may hinder exploiting pipeline  parallelism in programs that do not use logical delays to  create separate pipeline stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' However, this limitation can  be overcome by using a federated execution strategy as dis-  cussed by Bateni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  While the scheduling algorithm sketched in [51] and re-  fined in this section is relatively straight forward to imple-  ment, further optimizations where needed to achieve com-  petitive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In the following, we detail the most  important optimizations that we use in our C++ runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Menard & Lohstroh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Coordinating worker threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In the above discussion  we conceptually distinguished the scheduler from the work-  ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In an actual implementation, however, using a central  scheduler and separate worker thread imposes a significant  synchronization overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Instead, in our implementation,  any of the worker threads can become the scheduler and  move ready reactions to the ready queue or advance logical  time to the next tag if all reactions have been processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Fur-  thermore, we exploit the fact that at any time we know the  number of reactions to execute in parallel and use a counting  semaphore to control the number of active workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Lock-free data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The three queues and other  data structures that are required for bookkeeping (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=', a  list of all set ports) are shared across workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Using mu-  texes for synchronization proved to be inefficient due to  high contention on the shared resources, especially when  many parallel reactions set ports or schedule actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Instead,  we utilize lock-free data structures where possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' For in-  stance, the ready queue is implemented as a fixed size buffer  paired with an atomic counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Since we know precisely how  many reactions can at most run in parallel (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' the maximum  number of reactions in the APG that have the same level), we  can fix the size of the queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Every time new reactions are  moved to the reaction queue, the atomic counter is set to the  number of reactions in the queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Each time a worker tries  to execute a reaction it atomically decrements the counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  If the result is negative, then the queue is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Otherwise  the result provides the index within the buffer to read from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  We further exploit knowledge about the execution of re-  actor programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' For instance, the scheduler advances logical  time only once all reactions have been processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' This op-  eration is safe without additional synchronization, as all of  the workers are waiting for new reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Sparse multiports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In programs where reactions in mul-  tiple reactors may trigger the same reaction (such as an  account with an arbitrary number of users), the triggered  reaction often needs to know which port(s) actually are set  (contain data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' If there are many upstream reactors and com-  munication is sparse, simply checking all ports for presence  can be inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Instead, we expose an API for obtaining  an iterator to only set ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Note that this problem does not  arise in actors, as no ports exist and messages are processed  one by one, only considering those that are actually sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  5  Performance evaluation  The actor model is widely accepted for programming large  concurrent applications, and implementations such as the  C++ Actor Framework (CAF) [16] and Akka [66] are known  to be fast and efficient in utilizing a larger number of threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Compared to actors, LF imposes various restrictions that  amount to a model of computation in which fewer behaviors  are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In this section, we show that these restrictions  do not necessarily introduce overhead or higher execution  times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In fact, LF is considerably faster for many benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='1  Methodology  Our evaluation is based on the Savina benchmark suite [34]  for actor languages and frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' While this suite has  several issues, as Blessing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' discuss in [13], Savina covers  a wide range of patterns and, to the best of our knowledge,  is the most comprehensive benchmark suite for actor frame-  works that has been published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The Savina suite includes  Akka implementations of all benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' CAF implementa-  tions of most Savina benchmarks are also available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='2  We ported 22 of the 30 Savina benchmarks to the C++  target of LF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='3 Due to the fundamental differences between  the actor and reactor model, the process of porting bench-  marks is not always straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' We aimed at closely  resembling the original workloads and considered the inten-  tion behind the individual benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' We did not imple-  ment the benchmarks Fork Join (actor creation), Fibonacci,  Quicksort, Bitonic Sort, Sieve of Eratosthenes, Unbalanced  Cobwebbed Tree, Online Facility Location, and Successive  Over-Relaxation as they require the capability to dynamically  create actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In the reactor model, this can be achieved with  mutations that may modify the reactor topology [50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  However, mutations are not yet fully implemented in LF,  and a discussion of language-level constructs for supporting  mutations is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  We further omit the A*-Search and Logistic Map Series  benchmarks from our presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' A*-Search suffers from a  severe race condition that results in wildly varying execution  times [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Logistic Map Series is omitted as it violates actor  semantics and requires explicit synchronization [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' For this  reason, the CAF implementation needs to use a blocking call,  which makes it slower than the other implementations by  at least two orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Since this is not a problem  of CAF, but rather a problem in the benchmark design, we  omit Logistic Map Series to avoid skewing the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Figure 6 reports measured results for all supported bench-  marks obtained with Akka, CAF, and the C++ target of LF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  The plots show the mean execution times (including 99% con-  fidence intervals) for a varying number of worker threads  for each of the benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Not all benchmarks are imple-  mented in CAF and hence it is missing in some plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  All measurements were performed on a workstation with  an Intel Core i9-10900K processor with 32 GiB DDR4-2933  RAM running Ubuntu 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='04 and using CAF version 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='6 and  Akka version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Following the methodology of Savina,  measurements exclude initialization and cleanup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Each mea-  surement comprises 32 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The first two iterations are  2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='com/woelke/savina  3Source code available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='com/lf-lang/benchmarks-lingua-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='franca  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='High-Performance Deterministic Concurrency using Lingua Franca  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Recursive Matrix Multiplication #20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Radix Sort #22  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Filter Bank #23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Trapezoidal Approximation #28  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Precise Pi Computation #29  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Cigarette Smokers #14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Bank Transaction #16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Producer Consumer (bounded) #11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='All−Pairs Shortest Path #17  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='N Queens First K Solutions #19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Big #8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Concurrent Dictionary #9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Concurrent Sorted Linked−List #10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Dining Philosophers #12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Sleeping Barber #13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Ping Pong #1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Thread Ring #2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Counting Actor #3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Fork Join (throughput) #4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Chameneos #7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
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+page_content='Number of Threads  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Mean Execution Time in ms  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Akka  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='CAF  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='LF C++ Target  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Mean execution times and 99% confidence intervals for various Savina benchmarks implemented in LF, CAF, and  Akka, measured for a varying number of worker threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The numbers prefixed with # are benchmark IDs as listed in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  excluded from our analysis and are used to warm up caches  and the JVM (in the case of Akka).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='2  Discussion  The first six plots in Figure 6 belong to the group of micro  benchmarks in the Savina suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' These are designed to ex-  pose overhead in the protocol used for exchanging messages  and for scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Overall, our C++ runtime shows com-  parable performance to Akka and CAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In Ping Pong and  Thread Ring, our implementation is considerably faster than  Akka but is still outperformed by CAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' For Counting Actor  and Big, Akka performs better and the LF performance is  slightly behind CAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In Fork Join and Chameneos, the LF  implementation outperforms both Akka and CAF, especially  when using a larger number of worker threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  The next six plots (Concurrent Dictionary to Bank Trans-  action) belong to the group of concurrency benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' LF  significantly outperforms CAF and Akka in all the concur-  rency benchmarks (especially for a high number of worker  threads).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' This highlights how concurrent behavior is ex-  pressed naturally in LF and can be executed efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Actor  implementations of those benchmarks, on the other hand,  have to synchronize explicitly and resort to potentially ex-  pensive protocols (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=', by sending acknowledge messages),  or implement some other (blocking) synchronization strat-  egy that violates actor semantics [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  The remaining plots belong to the group of parallelism  benchmarks in the Savina suite4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Radix Sort and Filter Bank  suffer somewhat from an inefficiency in our scheduler, as  discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In these particular benchmarks,  our simple algorithm leads to a non-optimal execution as  some reactions are executed later than they could.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' We will  revise this algorithm in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' However, the remain-  ing parallelism benchmarks highlight that LF can efficiently  implement parallel algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Our LF implementations are  on par with Akka and CAF and scale well with more threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  On average, LF outperforms both Akka and CAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' For 20  threads, the C++ runtime achieves a speedup of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='85𝑥 over  Akka and a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='42𝑥 speedup over CAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' These speedups were  calculated using the geometric mean over the speedups of  individual benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' We conclude that LF can compete  with and even outperform modern and highly optimized  4Producer Consumer is actually listed as a concurrency benchmark, but we  find it fits better to the group of parallelism benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Menard & Lohstroh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  actor frameworks such as Akka and CAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Particularly with  workloads that require synchronization, LF significantly out-  performs actor implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' LF is as efficient as the  actor frameworks in exploiting parallelism and scales well  to a larger thread count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In summary, the deterministic con-  currency provided by LF does not hinder performance but  enables more efficient implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' This is possible in  part because the scheduler has insights into the program  structure, and explicit synchronization can be avoided in LF,  as opposed to many of Savina actor benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  The performance comparison between C++ and Scala  (Akka) needs to be taken with care, as other factors such  as different library implementations and the behavior of  the JVM may influence performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' For instance, the large  discrepancy between Akka and our implementation in the  Pi Precision benchmark is explained by a less efficient rep-  resentation of large numbers in Scala/Java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' However, the  other benchmarks of the Savina suite do not depend on  external libraries and are designed to be more portable be-  tween languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Also note that over all benchmarks CAF  only achieves an average speedup of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='09𝑥 over Akka for  20 threads and is outperformed in 9 out of 16 benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  For single threaded execution, Akka outperforms CAF in  10 benchmarks and achieves an average speedup of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='33x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  This indicates that the implemented Scala workloads are  comparable to the C++ implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Even considering a  potential skew due to the JVM, our results clearly show that  LF can compete with state-of-the-art actor frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  To better understand the impact of the optimizations dis-  cussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='3, Figure 7 also shows the speedup of our  runtime for 20 worker threads compared to a less optimized  runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' This baseline is an older version of our runtime  that is optimized in the sense that we used code profiling  to identify obvious bottlenecks and eliminated them using  common code optimization techniques, but that does not  include the optimizations discussed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The av-  erage overall speedup (geometric mean) achieved by our  optimizations is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='18𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In particular, Big and Bank Transac-  tion significantly benefit from our optimization for sparse  communication patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' The concurrency benchmarks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=',  Concurrent Dictionary and Dining Philosophers), are mostly  improved by reducing the contention on shared resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  However, not all benchmarks benefit from our optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  The reduced performance in Ping Pong and Counting Actor  shows that optimizing for efficient parallel execution also  comes at a cost for simple sequential programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  6  Related Work  LF is closely related to the languages and frameworks evolved  around Hewitt’s actor model [1, 33], including Akka [66],  CAF [16], Ray [60], Erlang [3], Rebeca [70], P [22] and Pony  [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Also reactive programming techniques [5], as used in  frameworks like ReactiveX [56] and Reactors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='IO [65] but  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='0  0  5  10  15  20  Ping Pong  Thread Ring  Counting Actor  Fork Join (throughput)  Chameneos  Big  Concurrent Dictionary  Concurrent Sorted Linked−List  Dining Philosophers  Sleeping Barber  Cigarette Smokers  Bank Transaction  Producer Consumer (bounded)  All−Pairs Shortest Path  N Queens First K Solutions  Recursive Matrix Multiplication  Radix Sort  Filter Bank  Trapezoidal Approximation  Precise Pi Computation  Benchmark  Speedup  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Speedup achieved by our optimized C++ runtime  for 20 worker threads compared to an unoptimized version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  also in language-level constructs like event loops [74], are  closely related to LF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' While actors and reactive programming  provide good resiliency and scalability, this comes at the  cost of nondeterminism, which makes programs notoriously  hard to test and debug [6, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Even more problems arise  if languages, frameworks, and libraries do not enforce the  underlying model and invite the programmer to break its se-  mantics [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Pony addresses the later problem by leveraging  a strong type system similar to Rust to prevent data races at  compile time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Rebeca provides a formalism and model check-  ing techniques for analyzing and verifying actor networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  While this can improve confidence in a correct implemen-  tation, the programmer is still responsible for finding this  correct implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' P goes a step further in that it also  has an efficient runtime system and a compiler that generates  correct-by-construction code with reasonable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Blessing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' propose a strategy that maps actor commu-  nication to a tree topology in order to guarantee a causal  ordering of messages [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In a similar approach, Sang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  utilize a DAG topology to achieve serializability in the pro-  cessing of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Orleans [14] is also based on an actor-like  model and provides guarantees on atomicity on transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Finally, Reactors as defined by Field et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' [28] is a model that  is closely related to actors but that supports both synchro-  nous and asynchronous communication and also provides  atomicity guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' All these strategies are most useful in  distributed scenarios, in particular in presence of network  failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In this paper, however, we focus on the execution of  a single host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Moreover, the determinism guarantees that LF  makes are stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Nonetheless, such techniques are highly  relevant to LF and could be deployed for ensuring fault-  tolerant execution in distributed LF programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' We believe,  however, that LF provides a more general solution, as the  programmer can explicitly trade consistency for availability  in distributed contexts [42], and hence the solution can be  adjusted to the concrete application requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Dataflow models [11, 21, 44] and process networks [35,  45] provide deterministic concurrency by creating statically  connected networks of actors with deterministic semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  High-Performance Deterministic Concurrency using Lingua Franca  These models enable improved static analysis and optimiza-  tion [30], but they limit the application’s flexibility and ca-  pability to react to external events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Ohua [26] is another lan-  guage providing parallelism through dataflow and is similar  to LF in that it integrates with existing high-level languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  However, it falls short on exposing coordination facilities for  individual nodes and does not provide a timed semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Deterministic concurrency is also found in synchronous  languages such as Esterel [10], Lustre [31], and SIGNAL [9]  as well as in Functional Reactive Programming (FRP) lan-  guages, like Fran [25], FrTime [19], and Elm [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' However,  these languages are challenging to use for general purpose  programming as they require pure functions and there is a  lack of widely-applicable libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Only recently, side effects  have been considered in a formal semantics for Esterel [29]  and distributed dataflow [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In Lingua Franca, arbitrary  code can be embedded in reactions and we can benefit from  the available libraries for popular general purpose languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Another interesting approach is taken by deterministic  multithreading libraries such as DThreads [49] or Conse-  quence [57], which enforce a total order for concurrent store  operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Recent work has made considerable progress in  avoiding the bottlenecks of conventional DTM techniques [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  However, we argue that threads are not a convenient con-  currency model for the reasons outlined in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Moreover,  threads do not allow for transparent distributed execution  as is possible with (re)actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  7  Conclusion  Unlike actors and related models for asynchronous concur-  rency, LF enforces determinism by default, and features asyn-  chronous behavior only when introduced deliberately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Our  evaluation, based on LF’s C++ target, shows that this de-  terministic model does not impede performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' On the  contrary, we achieve an average speedup of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='85𝑥 over Akka  and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='42𝑥 over CAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' With LF, we manage to combine repro-  ducible (and testable) behavior with good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Yet,  our relatively simple scheduling strategy likely still leaves  room for significant improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' We leave it as future work  to explore more advanced scheduling algorithms capable of  exploiting more parallelism at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' We also aim to fur-  nish full runtime support for mutations and implement the  remaining Savina benchmarks that require them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Finally,  we note that our implementation of the Savina benchmark  suite is not only useful for comparing LF to actor-oriented  frameworks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' it also demonstrates that LF, which is still in its  infancy, is already suitable for solving practical problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  Acknowledgments  The work in this paper was supported in part by the German  Federal Ministry of Education and Research (BMBF) as part  of the Software Campus (01IS12051) and the program “Sou-  verän.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Digital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Vernetzt.”, joint project 6G-life (16KISK001K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  This work was also supported in part by the National Sci-  ence Foundation (NSF), award #CNS-1836601 (Reconciling  Safety with the Internet), the iCyPhy Research Center (In-  dustrial Cyber-Physical Systems), supported by Denso, Ford,  Siemens, and Toyota, and the National Research Foundation  (NRF) of Korea (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
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+page_content=' Why do  scala developers mix the actor model with other concurrency models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='.  In European Conference on Object-Oriented Programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
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+page_content=' Karmani, Steven Lauterburg, Axel Legay,  Darko Marinov, and Gul Agha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
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+page_content=' In  Formal Techniques for Distributed Systems, Holger Giese and Grigore  Rosu (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
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+page_content=' Springer Berlin Heidelberg, Berlin, Heidelberg, 219–234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
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+page_content='  Bita: Coverage-guided, automatic testing of actor programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In 2013  28th IEEE/ACM International Conference on Automated Software Engi-  neering (ASE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
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+page_content=' IEEE Internet Computing  14, 6 (2010), 80–83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
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+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Multiverse Debugging:  Non-Deterministic Debugging for Non-Deterministic Programs (Brave  New Idea Paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In 33rd European Conference on Object-Oriented Pro-  gramming (ECOOP 2019) (Leibniz International Proceedings in Infor-  matics (LIPIcs), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' 134), Alastair F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Donaldson (Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Schloss Dagstuhl–  Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 27:1–27:30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='4230/LIPIcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='ECOOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='27  [76] Carmen Torres Lopez, Stefan Marr, Elisa Gonzalez Boix, and Hanspeter  Mössenböck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Programming with Actors: State-of-the-Art and Re-  search Perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Springer International Publishing, Cham, Chapter  A Study of Concurrency Bugs and Advanced Development Support  for Actor-based Programs, 155–185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='1007/978-3-  030-00302-9_6  [77] Tom Van Cutsem, Elisa Gonzalez Boix, Christophe Scholliers, An-  doni Lombide Carreton, Dries Harnie, Kevin Pinte, and Wolfgang  De Meuter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' AmbientTalk: programming responsive mobile peer-  to-peer applications with actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Computer Languages, Systems &  Structures 40, 3-4 (2014), 112–136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  [78] Reinhard von Hanxleden, Edward A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Lee, Hauke Fuhrmann, Alexan-  der Schulz-Rosengarten, Sören Domrös, Marten Lohstroh, Soroush  Bateni, and Christian Menard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Pragmatics Twelve Years Later: A  Report on Lingua Franca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' In Leveraging Applications of Formal Methods,  Verification and Validation: Verification Principles: 11th International  Symposium on Leveraging Applications of Formal Methods, ISoLA 2022,  Tiziana Margaria and Bernhard Steffen (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Springer Nature Switzer-  land, 60–89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content='  [79] Bernard Zeigler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Theory of Modeling and Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
+page_content=' Wiley  Interscience, New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E0T4oBgHgl3EQfiQHX/content/2301.02444v1.pdf'}
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+Resonant noise amplification in a predator-prey model with quasi-discrete generations
+M. Giannakou1,3, B. Waclaw1,2
+1School of Physics and Astronomy, University of Edinburgh, James Clerk Maxwell Building,
+Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom
+2Dioscuri Centre for Physics and Chemistry of Bacteria,
+Institute of Physical Chemistry PAS, Kasprzaka 44/52, 01-224 Warsaw, Poland
+3Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudingerweg 9, 55128 Mainz, Germany
+Predator-prey models have been shown to exhibit resonance-like behaviour, in which random
+fluctuations in the number of organisms (demographic noise) are amplified when their frequency
+is close to the natural oscillatory frequency of the system. This behaviour has been traditionally
+studied in models with exponentially distributed replication and death times. Here we consider a
+biologically more realistic model, in which organisms replicate quasi-synchronously such that the
+distribution of replication times has a narrow maximum at some T > 0 corresponding to the mean
+doubling time. We show that when the frequency of replication f = 1/T is tuned to the natural
+oscillatory frequency of the predator-prey model, the system exhibits oscillations that are much
+stronger than in the model with Poissonian (non-synchronous) replication and death. The effect can
+be explained by resonant amplification of coloured noise generated by quasi-synchronous replication
+events. To show this, we consider a single-species model with quasi-synchronous replication. We
+calculate the spectrum and the amplitude of demographic noise in this model, and use these results
+to obtain these quantities for the two-species model.
+I.
+INTRODUCTION
+When a non-linear dynamical system capable of ex-
+hibiting damped oscillations is coupled to a source of
+random noise, it often begins to generate periodic oscilla-
+tions (a quasi-cycle) [1]. This resonance-like behaviour is
+caused by the amplification of noise frequencies that are
+in tune with the natural oscillatory frequency of the sys-
+tem. Importantly, the noise does not have to be external
+but it can be intrinsic to the system and arise from its
+microscopic stochastic dynamics.
+An important example is resonant amplification of
+demographic noise which has been found in stochas-
+tic models of biological populations [2–8]. However, all
+these models assume that reproduction is a Markov pro-
+cess: birth and death occur with certain (possibly state-
+dependent) rates. At any moment, the distribution of
+replication times is therefore exponential, with the max-
+imum at t = 0. However, biological organisms do not
+replicate in this way: all known organisms require a cer-
+tain minimum time to develop reproductive capability.
+Moreover, many organisms reproduce in quasi-discrete
+generations such that the time between consecutive repli-
+cation events has a narrow distribution that peaks around
+some characteristic time T called the generation time, or
+doubling time. For example, for the bacterium E. coli, T
+ranges between 20 min and a few hours and the coefficient
+of variation of the doubling time is 0.1 − 0.3, depending
+on growth conditions [9, 10].
+This leads to significant
+correlations between reproduction times of related indi-
+viduals. Modelling this process for a single species has a
+long history [11–17].
+The quasi-synchronous nature of replication suggest an
+interesting possibility: if a predator-prey system has a
+tendency to oscillate at a frequency similar to the inverse
+of the doubling time, synchronisation of the two oscilla-
+tions may lead to a substantial enhancement of resonant
+amplification of demographic noise.
+In this work, we investigate this scenario in a simple
+predator-prey model originally proposed in Ref. [2]. The
+model assumes two biological species interacting in a way
+that leads to damped predator-prey cycles in the limit of
+infinitely large populations. In the original model, demo-
+graphic noise due to stochastic replication of organisms
+led to persistent oscillations of small but non-zero ampli-
+tude and a Lorenz-like power spectrum. Here we show
+that when replication is no longer Poissonian but occurs
+in quasi-discrete generations, these oscillations increase
+dramatically in amplitude and can be as high as 50% of
+the total population size even when the number of organ-
+isms is very large (millions or more).
+II.
+MODEL
+Our model an extension of the Newman-McKane
+model [2]. We consider a well-mixed population of two
+types of organisms A, B. We shall call these organisms
+"cells" as if they were single-celled microorganisms, al-
+though the model is agnostic to the exact nature of these
+organisms.
+Let nA, nB be the number of cells of each
+type. Each cell has an internal state variable τ assigned
+at birth from a certain distribution R(τ), the same for
+both species. We shall call this variable a “timer”. The
+timer counts down from the assigned time interval; when
+it reaches zero, the cell produces an offspring and both
+cells are assigned new, randomly selected values of τ from
+R(τ). We shall assume that the distribution R(τ) is con-
+centrated around its mean value ⟨τ⟩ ≡ T > 0. Cells also
+die with per-capita rates dA = p2 − p1nB/K for type A,
+and dB = p4nA/K − p3(1 − nB/K) for type B. Here K
+plays a role similar to the carrying capacity in popula-
+arXiv:2301.13290v1  [cond-mat.stat-mech]  30 Jan 2023
+
+2
+tion dynamics models and sets the scale for the number
+of cells in the system: nA, nB ∼ K on average. We have
+used the same symbols for the parameters p1, p2, p3, p4
+as in Ref. [2], however their microscopic interpretation
+is slightly different. We will come back to this when we
+discuss the steady-state solution of the model.
+The dynamics of the model can be schematically rep-
+resented as a set of chemical-like equations:
+A → 2A
+(1)
+B → 2B
+(2)
+A → 0
+(3)
+B → 0
+(4)
+However, one must be careful with how these equations
+are interpreted in our model. Only the last two reactions
+have been used in the same way as in Ref. [2] to represent
+inhomogeneous Poisson processes occurring with state-
+dependent rates dA(nA, nB), dB(nA, nB). The first two
+reactions do not describe Poisson processes because the
+probability of replication depends on the internal state τ
+of each cell. Thus the model is non-Markovian.
+Let us briefly discuss some possible choices for the dis-
+tribution R(τ) of replication times.
+The case R(τ) =
+δ(τ − T) corresponds to all cells reproducing in per-
+fect synchrony; the generation time is T. The exponen-
+tial distribution R(τ) = (1/T) exp(−τ/T) represents the
+Poisson case: cells reproduce with rate 1/T per capita
+and the mean time to replication is T. In this case the
+model is Markovian and its behaviour is expected to be
+the same as the original model from Ref. [2]. Finally,
+R(τ) can be concentrated around τ = T but have a non-
+zero width. This represents quasi-synchronous replica-
+tion: all descendants of a given cell initially replicate in
+quasi-discrete generations, with progressive loss of syn-
+chronization over time.
+In this manuscript, we shall compare the behaviour
+of the model for two distributions R(τ): (i) exponential
+(the Poisson model) with mean time to division T, (ii)
+uniform on (T(1 − w), T(1 + w)) where w ≪ 1 controls
+the degree of correlation of replication times; synchrony
+is lost after ∼ 1/w generations.
+A.
+Biological interpretation
+We shall now provide a biological interpretation of the
+model. However, our analysis does not rely on this inter-
+pretation and the model is deliberately oversimplified and
+not intended to replicate any specific experiment. Our
+model could describe a population of micro-organisms
+of one species and two slightly different ecotypes. Both
+types replicate with the same time-independent rate.
+Type A’s basal death rate p2 decreases in the presence of
+B proportionally to the concentration of B times p1. This
+could be due to type B producing an essential chemical
+compound necessary for A to survive. This interpretation
+in consistent with p2 ≫ b that we will generally assume
+later. Therefore, a sufficient density of B is required for
+A to thrive.
+Type B, on the other hand, is killed by
+A (e.g., A releases a toxin that kills B) with rate equal
+to the abundance of A times p4. The additional term
+−p3(1 − xB) accounts for the increase in the death rate
+due to crowding. A similar term could be added to the
+equation for type A to make the model more symmetric
+but it would not qualitatively affect the dynamics of the
+model. All interactions described here have been demon-
+strated in microbial populations [18–20].
+III.
+ANALYSIS OF THE MODEL
+We first study the behaviour of the Poisson model
+in the infinite-population size limit (K → ∞) by ne-
+glecting fluctuations in the number of cells. We define
+xA = nA/K, xB = nB/K as the new state variables.
+The dynamics of the model can be approximated by two
+differential equations:
+dxA
+dt
+= xA (b − max[p2 − p1xB, 0]) ,
+(5)
+dxB
+dt
+= xB (b − max[p4xA − p3(1 − xB), 0]) ,
+(6)
+where b = ln(2)/T. In the above, we have used average
+rates of all processes represented by reactions (1-4), and
+assumed that all higher moments factorize into products
+of xA, xB. The max[. . . ] function ensures that the death
+terms contribute only if the corresponding rates are pos-
+itive.
+The non-zero steady-state solution of the model, with
+both species present, reads:
+x∗
+A = (p1 − p2)p3 + b(p1 + p3)
+p1p4
+,
+(7)
+x∗
+B = p2 − b
+p1
+.
+(8)
+To investigate the stability of this solution, we Taylor-
+expand equations (5-6) around the steady-state solution.
+This leads to the Jacobian matrix with the following
+eigenvalues:
+λ± = bp3 − p2p3 ±
+�
+(b − p2) (b(2p1 + p3)2 − p3 (−4p2
+1 + 4p1p2 + p2p3))
+2p1
+.
+(9)
+
+3
+0
+2
+4
+6
+8
+0
+3105
+6105
+9105
+t
+NA,NB
+0
+2
+4
+6
+8
+0
+3105
+6105
+9105
+t
+NA,NB
+Figure 1: Damped oscillations in the stochastic asynchronous
+model (points:
+blue for NA, yellow for NB) and its de-
+terministic counterpart (black lines), for {p1, p2, p3, p4, b} =
+{30, 7, 7, 10, 1}. Top: the deterministic solution differs from
+the stochastic simulation for the uniform initial distribu-
+tion of the timer variable.
+Both models assume the same
+initial condition xA(0) = 0.2, xB(0) = 0.2.
+Bottom:
+the
+agreement is very good when we compare the models af-
+ter the stochastic model reached a quasi-steady state timer
+distribution.
+The deterministic model uses the values of
+NA, NB = {797423, 293636} from the stochastic simulation
+for t0 = 1.2207 as its initial condition.
+In the regime that we are interested here (x∗
+A > 0, x∗
+B >
+0), the eigenvalues have a negative real part, meaning
+that the steady-state solution is stable to a small per-
+turbation.
+However, their imaginary part is generally
+non-zero, and hence the system will exhibit damped os-
+cillations while relaxing towards the steady state. This is
+illustrated in Figure 1, which shows plots of the determin-
+istic solution for {p1, p2, p3, p4, b} = {30, 7, 7, 10, 1}, for a
+short time (a few oscillations), starting from xA(0) =
+0.2, xB(0) = 0.2. On the same plot we show the results
+of a numerical simulation of the stochastic Poisson model
+with K = 106. Note the small discrepancy between both
+models (Fig. 1, top). This is due to the timer of all cells
+being initialized with uniformly distributed random num-
+bers in the stochastic simulation. This initial distribution
+is quite different than the quasi-steady state distribution
+obtained after a few cycles, which is implicitly assumed
+when deriving Eqs. (5, 6) from the microscopic rules (1-
+4). When we solve the deterministic model starting from
+a later point, using NA, NB from the stochastic model as
+the initial condition, the discrepancy vanishes (Fig. 1,
+bottom).
+A.
+Parameter selection
+The
+deterministic
+model
+has
+five
+parameters:
+p1, p2, p3, p4, b.
+We can put b = 1; this fixes the time
+scale.
+The remaining four parameters determine the
+frequency of small-amplitude oscillations, the damping
+coefficient, and steady-state occupations.
+Before we
+move on, we shall discuss how we select these parameters
+so that the model exhibits under-damped oscillations;
+this is required for the resonant amplification of noise.
+We have used Eqs. (9) together with Eqs. (7-8) to find
+a region in the parameter space {p1, p2, p3, p4} and b = 1
+of the deterministic model that corresponds to damped
+oscillations of frequency f0 = Im(λ)/(2π) ∈ (0.98, 1.02),
+damping coefficient |Re(λ)| < 0.5, and steady-state abun-
+dances 0.5 ± 0.1. We did this via Monte-Carlo sampling
+of the parameter space. We then used the selected val-
+ues as starting points for a root finding algorithm to find
+p1, p2, p3, p4 such that the frequency would be exactly
+f0 = 1, steady-state occupations xA = xB = 0.5, and the
+damping coefficient assumed one of three values: 0.5 (fast
+damping), 0.2 (slow damping) and 0.1 (minimal damp-
+ing).
+This
+procedure
+has
+produced
+three
+sets
+of
+pa-
+rameters:
+S0.5
+=
+{39.73, 20.86, 2., 4.},
+S0.2
+=
+{56.45, 29.23, 0.8, 2.8}, and S0.1 = {65.81, 33.91, 0.4, 2.4}.
+We shall use these parameters in the full, stochastic
+model.
+B.
+Numerical results
+We have simulated the stochastic model using a sim-
+ple tau-leaping algorithm with fixed-size time step dt =
+1/512 [21]. Figure 2 shows examples of time series ob-
+tained for the Poisson version of the model, and for
+the quasi-synchronous model with a narrow distribu-
+tion of doubling times. The parameters are S0.5, K =
+300000, w = 0.02, and the initial condition is nA(0) =
+0.2K, nB(0) = 0.2K. In both cases the natural oscilla-
+tory frequency of the model is f0 = 1 and the average
+doubling time is also T = 1.
+The quasi-synchronous model exhibits much larger os-
+cillations than the Poisson model.
+Accordingly, the
+Fourier spectrum of the quasi-synchronous model has a
+much more pronounced peak at frequency f0 = 1 (Fig.
+3).
+The observed increase in the amplitude of oscilla-
+tions occurs only when the doubling frequency is close to
+the natural oscillation frequency. Figure 4, top, shows
+that when T ̸= 1/f0 the amplitude is significantly re-
+duced; this resonance-like behaviour is not present in the
+Poisson model (black line in Fig. 4, top). Interestingly,
+the maximum amplitude is observed at a slightly lower
+b = 0.5 than expected (b = ln 2 = 0.69 which corresponds
+to T = 1). The resonance peak is also quite broad. This
+is a non-linear effect; for large amplitudes as observed
+here, the resonant frequency is slightly lower than f0 = 1.
+The resonance peak becomes sharper with increasing
+
+4
+2000
+4000
+6000
+8000
+10000
+100000
+110000
+120000
+130000
+140000
+150000
+160000
+t
+NA,NB
+2000
+4000
+6000
+8000
+10000
+100000
+110000
+120000
+130000
+140000
+150000
+160000
+t
+NA,NB
+Figure 2:
+Example time series NA(t), NB(t) in the Pois-
+son (top) and quasi-synchronous (bottom) models, for K =
+300000, w = 0.02. Individual oscillations cannot be seen due
+to the length of the time window shown here; the window
+contains a few thousand oscillations as those from Fig. 1.
+0.0
+0.5
+1.0
+1.5
+5
+6
+7
+8
+9
+10
+11
+12
+frequency
+ln(amplitude) [a.u]
+0.0
+0.5
+1.0
+1.5
+5
+6
+7
+8
+9
+10
+11
+12
+frequency
+ln(amplitude) [a.u]
+Figure 3: Fourier spectrum of NA(t) in the Poisson (left) and
+non-Poisson (right) models, for K = 300000, w = 0.02, the
+remaining parameters = S0.5. Moving average with a 20-point
+long window has been applied to smooth out the spectra.
+carrying capacity K (Fig. 4, middle). The amplitude of
+oscillations in the peak is independent of K for a wide
+range of K. This is very different to the scaling ∼ 1/
+√
+K
+observed in the Poisson case and also the scaling of CV
+in the quasi-synchronous model far away from the peak
+(Fig.
+5).
+The amplitude of oscillations in the quasi-
+synchronous model is more than 10% of the steady-state
+population abundance for K = 106, whereas in the Pois-
+son model with identical parameters it is less than 1%.
+Figure 4, bottom, shows that the height of the reso-
+nance peak decreases with increasing w. For w = 0.2,
+the peak is barely noticeable.
+On the other hand, all
+values w ≤ 0.1 produce a visible peak.
+These results suggest that quasi-synchronous replica-
+tion leads to a significant enhancement of demographic
+noise in the quasi-synchronous model.
+To understand
+this, let us first revisit what happens in the Poisson ver-
+0.5
+1.0
+1.5
+2.0
+2.5
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+b
+CV(NA)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+0.00
+0.05
+0.10
+0.15
+b
+CV(NA)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+b
+CV(NA)
+Figure 4: Top: Coefficient of variation (CV) of NA(t) for
+the Poisson (black) and quasi-synchronous (red) models, as a
+function of b. CV is a convenient measure of the amplitude
+of oscillations.
+A resonance peak can be seen at b ≈ 0.5.
+Middle: CV of NA(t) for the quasi-synchronous model with
+w = 0.02 and different K = 104, 3 × 104, 105, 3 × 105, 106
+(blue, yellow, green, red, violet). Bottom: CV versus b for
+different widths w = 0.02, 0.04, 0.08, 0.1, 0.2 of the doubling
+time distribution (colours from blue to violet) and K = 105.
+In all cases, parameters = S0.5.
+sion of the model [2]. In that model, demographic noise
+has a flat spectrum and contains a broad range of fre-
+quencies (white noise). Frequencies close to the frequency
+at which the system exhibits damped oscillations are am-
+plified; this leads to quasi-periodic oscillations with the
+
+5
+1×104
+5×104 1×105
+5×105 1×106
+0.005
+0.010
+0.020
+0.050
+K
+CV(NA) for b=2.5
+Figure 5:
+Coefficient of variation of NA(t) for the quasi-
+synchronous model as a function of K, calculated at b = 2.5
+(away from the resonance peak). Solid line represents the scal-
+ing ∼ 1/
+√
+K expected for the Poisson version of the model.
+amplitude ∼
+√
+K.
+However, since the average abun-
+dances NA, NB increase proportionally to K, the rela-
+tive magnitude of oscillations decreases as ∼ 1/
+√
+K with
+the increasing population size. Noise-induced oscillations
+are therefore significant only for relatively small systems
+K ≪ 106. In contrast, here we observe large, persistent
+oscillations even for K = 106. As we shall see, this can
+be explained by demographic noise being concentrated in
+a narrow range of frequencies in the non-Poisson model.
+IV.
+SINGLE-SPECIES MODEL
+To understand the spectrum of noise in the quasi-
+synchronous model, we consider a simpler one-species
+model. In this model, replication is non-Poissonian with
+mean doubling time ln(2)/b as in the two-species model,
+whereas death is a Poisson process with rate bN/K,
+where N is the total number of cells, K is the carrying
+capacity, and b is the replication rate.
+In the large-K limit, the average abundance x = N/K
+evolves according to the logistic equation,
+dx/dt = bx(1 − x).
+(10)
+The steady-state occupation is x∗ = 1. In the stochastic
+model (K < ∞), the number of cells is thus expected to
+fluctuate around the mean value N ∗ ∼= K.
+Figure 6, top, shows examples of N(t) for the model
+with Poisson and non-Poisson replication (w = 0.02), for
+K = 105. The quasi-synchronous model exhibits more
+regular oscillations. The standard deviation of N(t) is
+very similar to the Poisson model for w > 0.05 but
+rapidly increases for smaller w (Fig. 6, bottom).
+In what follows, we shall study this model analytically.
+In particular, we are interested in analytical expressions
+for (i) the correlation time of oscillations, (ii) the spec-
+trum of oscillations, (iii) the steady-state amplitude of
+oscillations. This will help us to better understand the
+behaviour of the two-species model from previous sec-
+tions.
+A.
+Preliminary considerations
+We begin by considering the behaviour of a large popu-
+lation of cells in which the cells can be assigned to groups
+depending on the phase φ of their cell cycle. The phase is
+not the same as the timer variable; instead, it should be
+interpreted as the difference between the timer variable
+and some arbitrary chosen reference timer. Let n(φ) be
+the number density of cells with phase φ. Let us further
+assume that, if all cells were synchronised (all φ being
+equal), the total number of cells would be described by
+a certain periodic function f(t). This function (besides
+a different amplitude) also describes number fluctuations
+in a group of cells that have the same phase φ. The total
+number of cells in the population is therefore
+N(t) =
+� ∞
+−∞
+n(φ)f(t − φ)dφ,
+(11)
+which is the convolution of f and n. The Fourier spec-
+trum of N is
+F[N](ω) = F[f](ω) F[n](ω).
+(12)
+Suppose n(φ) is Gaussian with variance σ2. We have
+F[N](ω) = F[f](ω)e−(1/2)σ2ω2.
+(13)
+If f is periodic with angular frequency ω0, then the
+lowest-frequency Fourier mode of N at ω = ω0 will be re-
+duced in comparison to f by e−(1/2)σ2ω2 due to the spread
+of the phases.
+All higher modes will be damped even
+more; we will neglect them for now. For ω0 = 2π/ ln 2
+assumed in our single-species model for b = 1 (dou-
+bling time ln 2) and σ2 ≪ 1, the reduction factor is
+e[2π2/(ln 2)2]σ2 ≈ e−41.1σ2.
+Suppose further than each generation causes the dis-
+tribution n to broaden, due to the finite width of the
+distribution of doubling times, so that σ2 = σ2
+0(t/ ln(2)),
+where σ2
+0 = (w2/3)(ln 2)2 is the variance of the uni-
+form distribution of doubling times used in the simu-
+lations. This corresponds to the variance of the phase
+distribution increasing by the variance of the doubling
+time distribution every generation. This will then lead
+to oscillations in N(t) (caused by quasi-synchronous
+replication) to decay exponentially with the rate γ =
+[2π2/(ln 2)2](ln 2)(w2/3) = [2π2/(3 ln 2)]w2 ≈ 9.5w2.
+Figure 7 shows that the predicted rate is in very good
+agreement with the decay rate observed in numerical sim-
+ulations.
+B.
+A more formal approach
+The above result can be derived more formally. We
+shall start by writing down the equation for the number
+
+6
+3750 3755 3760 3765
+99500
+100000
+100500
+101000
+t
+n
+3750 3755 3760 3765
+98500
+99000
+99500
+100000
+100500
+101000
+101500
+t
+n
+0.05 0.10 0.15 0.20
+0
+200
+400
+600
+800
+w
+std.dev.
+Figure 6: Fluctuations in the single-species model. Top: N(t)
+for K = 100000: Poisson model (left), and quasi-synchronous
+model with w = 0.02 (right). Bottom: standard deviation
+of fluctuations for different w. Points = simulation, line =
+equation (64) with γ = 9.5w2.
+No parameters have been
+fitted to data here.
+0
+50
+100
+150
+0
+20000
+40000
+60000
+80000
+100000
+t
+n
+0.02
+0.05 0.10 0.20
+0.005
+0.010
+0.050
+0.100
+0.500
+1
+ϵ
+γ
+Figure 7: Exponential decay of oscillation in the single-species
+model for K = 100000. Left: N(t) for w = 0.04. Right: the
+exponential decay rate γ as a function of w (points). Black
+line is the theoretical prediction γ ≈ 9.5w2.
+density of cells n(τ, t) at time t with the timer variable
+τ, for the time being neglecting stochastic noise:
+∂tn(τ, t) = ∂τn(τ, t) − ln 2
+T n(τ, t)
+� ∞
+0
+n(τ ′, t)
+K
+dτ ′
++2R(τ)n(0, t). (14)
+The first term corresponds to the timer counting back-
+ward. The second term represents death with rate pro-
+portional to the total size divided by K.
+The factor
+(ln 2)/T is required to have the correct behaviour in the
+limit of perfectly synchronous replication - we shall see
+this later. The third term represent replication that oc-
+curs when the timer reaches τ = 0 and is the product
+of the density of cells n(0, t) in that state and R(τ), the
+probability density function for the timer being reset to
+τ. We assume R(τ) to be normalized:
+� ∞
+0
+R(τ)dτ = 1,
+(15)
+and that R(τ) is concentrated around τ = T as in nu-
+merical simulations in previous sections.
+C.
+Stationary solution
+In the limit t → ∞, Eq. (14) becomes
+0 = ∂n(τ)
+∂τ
+− n(τ)J + 2n(0)R(τ),
+(16)
+where J = (ln 2)/T
+� ∞
+0 (n(τ)/K)dτ, and n(τ) does not
+depend on t. We can solve this equation for the steady-
+state distribution n∗(τ):
+n∗(τ) = n∗(0)eJτ
+�
+1 − 2
+� τ
+0
+e−Jτ ′R(τ ′)dτ ′
+�
+,
+(17)
+with the condition n∗(τ → ∞) = 0 implying that
+� ∞
+0
+e−JτR(τ)dτ = 1/2,
+(18)
+which fixes the value of J. If R is a uniform distribution
+with mean τ = T and width 2wT, we obtain from (18)
+that
+e−JT sinh(JTw)
+JTw
+= 1/2.
+(19)
+This equation must be solved for J numerically. In ad-
+dition, one must determine the value of n∗(0) from the
+relationship between J and n(τ):
+JK = n∗(0)
+� T (1+w)
+0
+eJτ
+�
+1 − 2
+� τ
+0
+e−Jτ ′R(τ ′)dτ ′
+�
+dτ
+(20)
+Figure 8 shows an example of n∗(τ) for w = 0.1, calcu-
+lated in this way. The cell number density is proportional
+to eJτ ≈ 2τ/T for τ < T, and rapidly falls down to zero
+for τ > T. The solution simplifies greatly in the limit
+w → 0, in which J tends to (ln 2)/T. The steady state
+number density becomes then
+n∗(τ) = K ln 2
+T 2τ/T
+(21)
+D.
+Evolution of a small perturbation
+We now consider the time evolution of a small pertur-
+bation to the steady state solution:
+n(τ, t) = n∗(τ)(1 + ϵ(τ, t)).
+(22)
+
+7
+0.0
+0.2
+0.4
+0.6
+0.8
+0.0
+0.5
+1.0
+1.5
+2.0
+τ
+n[τ]
+Figure 8: Stationary cell density n∗(τ) for T = ln 2, w =
+0.1. Red line = Eq. (20). Black dashed line = approximate
+solution (21).
+of the noise-less equation 14. Inserting this into Eq. 14
+gives:
+n∗∂tϵ = (∂τn∗)(1 + ϵ) + n∗∂τϵ
+− ln 2
+TK n∗(1 + ϵ)
+� ∞
+0
+n∗(τ ′)(1 + ϵ(τ ′, t))dτ ′
++ 2R(τ)n∗(1 + ϵ).
+(23)
+We note that ∂τn∗ = Jn∗ − 2n∗(0)R(τ), and keep only
+terms linear in ϵ:
+n∗∂tϵ = n∗∂τϵ − ln 2
+TK n∗
+� T
+0
+n∗(τ)ϵ(τ ′, t)dτ ′
++ 2R(τ)n∗(0)(ϵ(0, t) − ϵ(τ, t)).
+(24)
+We divide by n∗ and obtain
+∂tϵ = ∂τϵ − (ln 2)
+TK
+� ∞
+0
+n∗(τ ′)ϵ(τ ′, t)dτ ′
+− 2n∗(0)
+n∗(τ)R(τ)(ϵ(0, t) − ϵ(τ, t))
+(25)
+with the boundary condition ϵ(0, t) = ϵ(T, t). We now
+expand ϵ and n∗(0)
+n∗(τ)R(τ) as Fourier series (consistent with
+the b.c.):
+ϵ(τ, t) =
+∞
+�
+k=−∞
+Ak(t)e2πikτ/T ,
+(26)
+n∗(0)
+n∗(τ)R(τ) =
+∞
+�
+k=−∞
+Rk(t)e2πikτ/T .
+(27)
+where the coefficients {Rk} are given by
+Rk = 1
+T
+� T
+0
+e−2πikτ/T n∗(0)
+n∗(τ)R(τ)dτ.
+(28)
+The transformed equation reads
+�
+k
+∂tAke2πikτ/T =
+�
+k
+Ak(2πik/T)e2πikτ/T −
+−(ln 2)
+TK
+�
+k
+Ak
+� T
+0
+n∗(τ ′)e2πikτ ′/T dτ ′
++2
+�
+m
+Rme2πimτ/T
+��
+k
+Ak −
+�
+k
+Ake2πikτ/T
+�
+.(29)
+The sums in (29) can be compared term by term since
+they must be valid for any τ. This leads to the following
+equation for the Fourier coefficients Ak(t):
+∂tAk = (2πik/T)Ak
+− δk,0
+ln 2
+TK
+�
+m
+Am
+� T
+0
+n∗(τ ′)e2πimτ ′/T dτ ′
++ 2(Rk
+�
+m
+Am −
+�
+m
+RmAk−m).
+(30)
+In particular, for k > 0 we have
+∂tAk = (2πik/T)Ak + 2(Rk
+�
+m
+Am −
+�
+m
+RmAk−m).
+(31)
+Let us assume that the initial perturbation is a pure kth
+Fourier mode, i.e., Ak ̸= 0 only for a single value of k.
+The first term represents oscillations with period T/k
+of that mode, which essentially gives a travelling-wave
+type of solution ∼ exp(2πik(τ − t)/T). The second term
+represents damping with rate γk = −2(Rk −R0). We can
+calculate this rate using equation (28) in the limit w → 0,
+since then we have from Eq. (21) that n∗(0)/n∗(τ) =
+2−τ/T and hence
+Rk = 1
+T
+� T (1+w)
+T (1−w)
+e−2πikτ/T 2−τ/t
+1
+2Twdτ.
+(32)
+We obtain that
+γk = −2(Rk − R0) ∼= 2π2
+3T k2w2,
+(33)
+which, for T = ln 2 and k = 1 reproduces the decay rate
+γ ≈ 9.5w2 which we have already seen in Sec. IV A.
+E.
+Amplitude of oscillations for perfectly
+synchronous replication
+We shall now add noise to the model and see how it
+affects its behaviour. We shall first consider a fully syn-
+chronous replication with arbitrary period T, which leads
+to the following equation:
+∂tn = ∂τn − ln 2
+T n
+� T
+0
+n(τ ′, t)
+K
+dτ ′ +
+√
+n∗η,
+(34)
+with boundary conditions
+n(0, t) = (1/2)n(T, t).
+(35)
+The noise term
+√
+n∗η is due to death only, since replica-
+tion is perfectly synchronous. We assume η(τ, t) repre-
+sents uncorrelated white noise:
+⟨η(τ1, t1)η(τ2, t2)⟩ = Dδ(τ1 − τ2)δ(t1 − t2),
+(36)
+with some D > 0 to be specified later. It can be eas-
+ily verified that this form of noise arises from a master
+
+8
+equation for the model with no replication by perform-
+ing a van Kampen expansion [22] of the master equation.
+While it may be possible to derive the noise term also
+in the presence of non-Markovian replication, we find it
+easier to postulate that Eq.
+(36) generally holds, and
+justify it based on the agreement between the result of
+our calculation and the computer simulation (see below).
+In the absence of noise, equation (34) has the steady-
+state solution
+n∗(τ) = K ln 2
+T 2τ/T ,
+(37)
+� T
+0
+n∗(τ)dτ = K.
+(38)
+To solve the time-dependent equation with noise, we con-
+sider a small perturbation (similarly as in the previous
+section):
+n(τ, t) = n∗(τ)(1 + ϵ(τ, t)).
+(39)
+This gives
+n∗∂tϵ = (∂τn∗)(1 + ϵ) + n∗∂τϵ
+− ln 2
+TK n∗(1 + ϵ)
+� T
+0
+n∗(τ ′)(1 + ϵ(τ ′, t))dτ ′ +
+√
+n∗η.
+(40)
+We note that ∂τn∗ = ((ln 2)/T)n∗, and only keep terms
+linear in ϵ:
+n∗∂tϵ = n∗∂τϵ − (ln 2)2
+T 2
+n∗
+� T
+0
+2τ ′/T ϵ(τ ′, t)dτ ′ +
+√
+n∗η.
+(41)
+We divide by n∗ and obtain
+∂tϵ = ∂τϵ− (ln 2)2
+T 2
+� T
+0
+2τ ′/T ϵ(τ ′, t)dτ ′+(n∗)−1/2η, (42)
+with the following boundary and initial conditions:
+ϵ(0, t) = ϵ(T, t) and ϵ(τ, 0) = 0. Proceeding as in the
+previous section, we expand ϵ and (n∗)−1/2η as Fourier
+series (consistent with the b.c.):
+ϵ(τ, t) =
+∞
+�
+k=−∞
+Ak(t)e2πkiτ/T ,
+(43)
+(n∗(τ))−1/2η(τ, t) =
+∞
+�
+k=−∞
+ηk(t)e2πkiτ/T .
+(44)
+(45)
+The transformed equation reads
+�
+k
+∂tAke2πkiτ/T =
+�
+k
+Ak(2πik/T)e2πkiτ/T −
+−(ln 2)2
+T 2
+�
+k
+Ak
+� T
+0
+2τ ′/T e2πkiτ ′/T dτ ′
++
+�
+k
+ηk(t)e2πkiτ/T .
+The integral over dτ ′ gives
+� T
+0
+2τ ′/T e2πkiτ ′/T dτ ′ =
+iT
+i ln 2 − 2πk .
+(46)
+Comparing the sums in (46) term-by-term we notice that
+the (ln 2)2 term does not contain any factor e2πkiτ/T , so
+it only contributes to the constant term:
+δk,0
+(ln 2)2
+T
+�
+n
+An
+i
+i ln 2 − 2πn ≈ δk,0
+ln 2
+T A0,
+(47)
+where we have assumed that all An for n ̸= 0 are much
+smaller than A0 (we shall see later that this is the case).
+This leads to the following equation for the Fourier coef-
+ficients Ak(t):
+∂tAk = (2πik/T)Ak − ((ln 2)/T)δk,0A0 + ηk.
+(48)
+In particular, for k = 0 we have
+∂tA0 = −ln 2
+T A0 + η0,
+(49)
+which can be formally solved as
+A0(t) = 2−t/T
+� t
+0
+2t′/T η0(t′)dt′.
+(50)
+This gives
+�
+A0A†
+0
+�
+(t) = 2− 2t
+T
+� t
+0
+dt1
+� t
+0
+dt22
+t1+t2
+2
+�
+η0(t1)η†
+0(t2)
+�
+.
+(51)
+Equation (44) enables us to write
+ηk(t) = 1
+T
+� T
+0
+�K ln 2
+T
+�−1/2
+2− τ
+2T η(τ, t)e−2πikτ/T dτ.
+(52)
+The average of the noise term gives
+�
+ηk(t1)η†
+k(t2)
+�
+=
+1
+TK ln 2 ×
+×
+� T
+0
+dτ1
+� T
+0
+dτ22− τ1+τ2
+2T
+e− 2πik(τ1−τ2)
+T
+�
+η(τ1, t1)η†(τ2, t2)
+�
+= D
+K
+δ(t1 − t2)
+2(ln 2)2 .
+(53)
+We therefore have
+�
+A0A†
+0
+�
+(t) = 2−2t/T
+� t
+0
+22t′/T D
+K2(ln 2)2 dt′
+= DT
+K
+1 − 2−2t/T
+4(ln 2)3
+.
+(54)
+Proceeding similarly for k ̸= 0, we obtain:
+Ak(t) = e2πikt/T
+� t
+0
+e−2πikt′/T ηk(t′)dt′,
+(55)
+
+9
+from which we obtain that
+�
+AkA†
+k
+�
+(t) =
+� t
+0
+dt1
+� t
+0
+dt2e−2πik(t1−t2)/T �
+ηk(t1)η†
+k(t2)
+�
+= D
+K
+t
+2(ln 2)2 .
+(56)
+We can now calculate the standard deviation of ∆N, the
+difference between the total number of cells at time t and
+the average steady-state number:
+∆N(t) =
+� T
+0
+n∗(τ)ϵ(τ, t)dτ
+(57)
+=
+� T
+0
+K ln 2
+T
+2τ/T �
+k
+Ak(t)e2πikτ/T dτ
+=
+�
+k
+Ak(t)K ln 2
+T
+� T
+0
+2τ/T e2πikτ/T dτ
+= (K ln 2)
+�
+k
+Ak(t)
+i
+i ln 2 − 2πk .
+(58)
+This gives (we note that terms
+�
+AkA†
+n
+�
+with k ̸= n van-
+ish):
+�
+|∆N(t)|2�
+=
+(K ln 2)2
+∞
+�
+k=−∞
+�
+AkA†
+k
+�
+(t)
+(ln 2)2 + (2πk)2
+= KD
+�
+T 1 − 2−2t/T
+4(ln 2)3
++ 2
+∞
+�
+k=1
+t/2
+(ln 2)2 + (2πk)2
+�
+= KD
+�
+T 1 − 2−2t/T
+4(ln 2)3
++ t3(ln 2) − 2
+4(ln 2)2
+�
+.
+(59)
+In the limit t → ∞ this gives
+�
+|∆N(t)|2� ∼= KDt3(ln 2) − 2
+4(ln 2)2
+≈ 0.0413 KDt.
+(60)
+Equation (60) predicts that the amplitude of oscillations
+for perfectly synchronous replication increases linearly in
+time. However, recall that our result has been derived
+under the assumption of a small perturbation. In reality,
+the amplitude will be limited by non-linear effects.
+F.
+Amplitude of quasi-synchronous oscillations
+Let us now consider the case of quasi-synchronous
+replication. Rather than attempting to solve Eq. (34)
+with the extra term R(τ) as in Eq. (14), we observe (as
+argued in subsection IV D) that the k-th Fourier mode
+will be damped with rate k2γ. We thus consider the fol-
+lowing modification to Eq. (48) for k ̸= 0:
+∂tAk = (2πik/T − k2γ)Ak + ηk,
+(61)
+where γ is the damping coefficient derived previously.
+The equation for k = 0 remains unchanged. Proceeding
+as in Sec. IV E, we obtain
+�
+AkA†
+k
+�
+(t) = D
+K
+1 − e−2k2γt
+4(ln 2)2k2γ .
+(62)
+Inserting this into the equation for
+�
+|∆N(t)|2�
+we have
+in the limit t → ∞:
+�
+|∆N|2�
+=
+�
+|∆N(t → ∞)|2�
+=
+= KD
+�
+T
+4(ln 2)3 + 2
+∞
+�
+k=1
+1
+4k2γ
+1
+(ln 2)2 + (2πk)2
+�
+= KD
+�
+T
+4(ln 2)3 + π2 12 − 18 ln 2 + (ln 2)2
+12γ(ln 2)4
+�
+.
+(63)
+For T = ln 2 we obtain that
+�
+|∆N|2�
+≈ KD(0.5203 + 0.01355/γ).
+(64)
+It remains to relate D to the parameters of the model. We
+again assume that death is the main source of stochas-
+ticity, and that the contribution from quasi-synchronous
+replication is negligible. Consider a pure death process
+with the same total number of organisms K as the steady
+state total (38), and death rate d = (ln 2)/T as per Eq.
+(34). For short time intervals, the variance of the num-
+ber of organisms in the pure death process equals to
+�
+|∆N|2�
+= (Kd)t = (K(ln 2)/T)t (easy to derive from
+the general formula on p. 108-109 of Ref. [23]). On the
+other hand, from Eq. (59) we have that for small t and
+T = ln 2,
+�
+|∆N|2� ∼= KD
+3
+4 ln 2t.
+(65)
+Comparing the two formulas for
+�
+|∆N|2�
+, we obtain that
+D = 4 ln 2
+3
+≈ 0.924.
+(66)
+Figure 6 shows that equation (64) with the above value
+of D reproduces the variance from numerical simulations
+very well.
+G.
+Spectrum of fluctuations
+We can now obtain a very good analytic approxima-
+tion for the spectrum of normalized fluctuations y(t) =
+N(t)/K − 1 in the single-species model by Fourier-
+transforming the expression for ∆N:
+˜y(ω) = (ln 2)
+�
+k
+˜Ak(ω)
+i
+i ln 2 − 2πk ,
+(67)
+in which
+˜Ak(ω) = lim
+L→∞
+1
+√
+L
+� L/2
+−L/2
+Ak(t)eiωtdt.
+(68)
+
+10
+We have
+�
+| ˜A0(ω)|2�
+=
+�
+|˜η0|2�
+� ln 2
+T
+�2 + ω2 ,
+(69)
+�
+| ˜Ak(ω)|2�
+=
+�
+|˜ηk|2�
+(ωT − 2πk)2 + γ2T 2k4 .
+(70)
+in which
+�
+|˜ηk|2�
+is defined through the Fourier transform
+like in Eq. (68), and evaluates to
+�
+|˜η0|2�
+=
+�
+|˜ηk|2�
+=
+D
+K2(ln 2)2 = D2K−1,
+(71)
+with D2 =
+2
+3 ln 2 ≈ 0.962. This gives
+K
+�
+|˜y|2(ω)
+�
+= (ln 2)2 �
+k
+�
+| ˜Ak|2�
+(ω)
+(ln 2)2 + (2πk)2
+=
+D2
+� ln 2
+T
+�2 + ω2 +
++
+∞
+�
+k=1
+2D2T 2
+(ωT − 2πk)2 + γ2T 2k4
+(ln 2)2
+(ln 2)2 + (2πk)2
+≈
+D2
+� ln 2
+T
+�2 + ω2 +
++
+2D2T 2(ln 2)2
+[(ωT − 2π)2 + γ2T 2][(ln 2)2 + (2π)2] + . . . (72)
+where ‘. . . ’ stand for terms corresponding to higher har-
+monics which we neglect because they are strongly sup-
+pressed by the denominator increasing fast with k. For
+T = ln 2, we have
+K
+�
+|˜y|2(ω)
+� ∼=
+D2
+1 + ω2 +
+2(ln 2)2D2
+(ln 2)2 + (2π)2
+1
+(ω − 2π
+ln 2)2 + γ2 .
+(73)
+The formula as a function of frequency f = ω/(2π) reads:
+K
+�
+|˜y|2(f)
+� ∼=
+∼= D2/(2π)2
+1 + (2πf)2 +
+D2
+2(ln 2)2
+(2π)2
+(ln 2)2 + (2π)2
+1
+(2πf − 2π
+ln 2)2 + γ2 ,(74)
+where the factor 1/(2π)2 is required for correct normal-
+ization. Figure 9 shows that equation (74) agrees well
+with the numerically obtained spectrum for a broad range
+of w values.
+Now we have determined the spectrum of the single-
+species model, we can proceed to obtain the spectrum of
+the two-species model.
+V.
+APPROXIMATE ANALYTIC SOLUTION OF
+THE TWO-SPECIES MODEL
+A.
+Linear approximation
+We shall assume that near the steady state the dy-
+namics of the two-species model can be described by lin-
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+100
+200
+300
+400
+f
+|ΔN
+ 2
+1/2=|Ky2
+1/2
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+100
+200
+300
+400
+f
+|ΔN
+ 2
+1/2=|Ky2
+1/2
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+500
+1000
+1500
+f
+|ΔN
+ 2
+1/2=|Ky2
+1/2
+Figure 9: Fourier spectrum |K˜y2|1/2 of N(t) in the single-
+species model for K = 105. Blue points = simulations, red
+lines = theoretical prediction (no fitting) obtained from Eq.
+(74). From top to bottom: w = 0.2, 0.1, 0.05.
+earized equations
+dyA/dt = aAByB + ηA,
+(75)
+dyB/dt = aBAyA + aBByB + ηB,
+(76)
+where yA = xA − x∗
+A, yB = xB − x∗
+B (x∗
+A, x∗
+B are steady-
+state concentrations), and aAB, aBA, aBB are given by
+the following expressions
+aAB = p3(b − p2) + p1(b + p3)
+p4
+,
+(77)
+aBB = p3(b − p2)
+p1
+,
+(78)
+aBA = p4(b − p2)
+p1
+.
+(79)
+Here ηA, ηB represent noise (not necessarily white noise)
+due to replication and death of both species.
+Fourier-transforming Eqs. (75, 76) leads to the follow-
+ing expression for the spectrum of yA:
+
+11
+�
+|˜yA(ω)|2�
+= (a2
+BB + ω2)
+�
+|ηA|2�
+− 2aABaBB
+�
+|ηAηB|2�
++ a2
+AB
+�
+|ηB|2�
+a2
+ABa2
+BA + 2aABaBAω2 + a2
+BBω2 + ω4
+.
+(80)
+The formula for the spectrum of yB (not shown) is very
+similar. Figure 10 shows that Eq. (80) works well for
+the Poisson case (asynchronous replication), for which we
+assume
+�
+|ηA|2�
+=
+�
+|ηB|2�
+= (2b)(K/2) = bK (recall that
+K/2 is the average number of organisms for our choice
+of the parameters, and the factor 2b is due to both birth
+and death contributing equally near the steady state),
+and
+�
+|ηAB|2�
+= 0.
+We now want to establish whether Eq. (80) also works
+for the non-Poissonian case, with an appropriate choise
+of the coloured noise ηA, ηB, based on the single-species
+calculation presented in the previous Sec. IV G. We as-
+sume that fluctuations around the steady state in the
+single species model can effectively be described by the
+following equation:
+dy/dt = −by + coloured noise.
+(81)
+Since we know the spectrum of y, we can calculate the
+spectrum of the coloured noise as dy/dt or, in Fourier
+space, by multiplying Eq. (73) by ω2:
+�
+|˜ηA,B|2(ω)
+�
+= K−1
+A,Bω2
+�
+D2
+b2 + ω2 +
++
+2(ln 2)2D2
+(ln 2)2 + (2π)2
+1
+(ω − 2π
+T )2 + γ2
+�
+,
+(82)
+with KA = Kx∗
+A, KB = Kx∗
+B. We then insert Eq. (82)
+into Eq. (80), assuming again that
+�
+|ηAB|2�
+= 0 (justi-
+fied since both species replicate independently with rates
+unaffected by the other species).
+Figure 11, shows that this simple approach works quite
+well for different frequencies of replication (controlled
+by b).
+If the replication frequency is slightly detuned
+from the natural frequency of the system (Fig. 11, top),
+two peaks are visible in the spectrum:
+a sharp peak
+coming from the quasi-synchronous birth events, and a
+much wider but lower peak corresponding to white noise-
+induced oscillations at the natural frequency f ≈ 1.
+Figure 11, bottom, shows that when b = 0.7 is tuned in
+to the resonant frequency of the system, only one peak is
+visible, with a slight broadening towards lower frequen-
+cies.
+B.
+Amplitude of oscillations
+We can now calculate the variance of ∆NA(t) - the dif-
+ference between the actual NA(t) and the average number
+KA = Kx∗
+A of organisms. We have:
+�
+|∆NA(t)|2�
+= K2
+A
+2π
+� ∞
+−∞
+�
+|˜yA|2(ω)
+�
+dω = K2
+A
+2π
+� ∞
+−∞
+(a2
+BB + ω2)
+�
+|ηA|2(ω)
+�
++ a2
+AB
+�
+|ηB|2(ω)
+�
+a2
+ABa2
+BA + 2aABaBAω2 + a2
+BBω2 + ω4 dω,
+(83)
+in which we used Eq. (80) and assumed no correlation
+between the noise ηA and ηB (
+�
+|ηAηB|2�
+= 0).
+In the case of Poisson replication, we put
+�
+|ηA|2�
+=
+�
+|ηB|2�
+= (2b)(K/2) and evaluate the integral (83) nu-
+merically. For our usual choice of the parameters S0.5,
+we obtain
+�
+|∆NA(t)|2�
+theor ≈ 5.48 × 105 which is very
+close to the numerical estimate from the simulation,
+�
+|∆NA(t)|2�
+sim ≈ 5.56 × 105.
+In the case of quasi-synchronous replication, we insert
+Eqs. (82) into Eq. (83), and again integrate numerically
+over ω. Figure 12 shows the plot of
+�
+|∆NA(t)|2�
+obtained
+in this way, compared to the simulation results. We no-
+tice that the analytic formula correctly reproduces the
+trend but the theoretically predicted values are generally
+larger than the ones from the simulation. However, the
+agreement is still quite good, given that our formula has
+been derived using many approximations.
+Note that we used equation (83), which is the same as
+the formula for the Poisson case [2], but with coloured
+noise given by Eq.
+(82) instead of white noise.
+The
+fact that this approach works means that oscillations in
+the system with quasi-synchronous replication can be un-
+derstood as being caused by resonant amplification of
+coloured, non-Poissonian noise.
+To get some qualitative insight into the behaviour of
+Eq. (83), we consider the case γ → 0. For T = 1, b = ln 2,
+we can expand the formula under the integral in (83)
+around ω = 2π, which enables us to carry out the integral
+analytically. We obtain
+
+12
+�
+|∆NA(t)|2� ∼=
+KD2π2(a2
+BB + a2
+AB + 4π2)(ln 2)2
+γ(a2
+ABa2
+BA + 4(2aABaBA + a2
+BB)π2 + 16π4)(4π2 + (ln 2)2).
+(84)
+We see that, since γ ∼ w2, the variance of ∆NA in-
+creases as 1/w2 as reproduction becomes more and more
+synchronous for w → 0.
+This is similar to the effect
+of a long delay in reaction kinetics [24]. The relation-
+ship
+�
+|∆NA(t)|2�
+∝ K/γ can be interpreted as an ef-
+fective reduction in the number of replicating entities;
+cells originating from a common ancestor replicate quasi-
+synchronously when their sub-population is much less
+than 1/γ. The system thus consists of Kγ ≪ K of such
+groups of cells, which increases demographic noise by a
+factor 1/√γ, and the variance of N by 1/γ.
+Let us now consider how small w needs to be for the
+variance to start deviating from the Poisson case, i.e.,
+how synchronous replication must be to make difference
+to random, asynchronous replication. Equation (83) can
+be rewritten as follows:
+�
+|∆NA(t)|2�
+= KAD2
+2π
+� ∞
+−∞
+F(ω)(f0(ω) + fγ(ω))dω,
+(85)
+where
+F(ω) = a2
+BB + a2
+AB(KB/KA) + ω2
+(ω2 − Ω2)2 + c
+(86)
+is
+the
+resonance
+response
+function,
+with
+c
+=
+−aABaBAa2
+BB − a4
+BB/4, and Ω2 = −aABaBA − a2
+BB/2 =
+4π2 (squared resonant frequency for our parameters
+S0.5). The function
+f0(ω) =
+ω2
+ω2 + (ln 2)2
+(87)
+is the γ-independent contribution from stochastic repli-
+cation, and the function
+fγ(ω) =
+2(ln 2)2ω2
+(γ2 + (ω − 2π)2)(4π2 + (ln 2)2)
+(88)
+is the γ-dependent contribution.
+F(ω) has full width at maximum height (FWHM) ap-
+proximately equal to √c, whereas fγ(ω) has FWHM
+equal to ≈ 2γ ≈ 13.16w2.
+We expect that when the
+contribution from fγ(ω) near the peak of F(ω) is larger
+than the contribution from f0(ω), the variance of NA will
+be dominated by synchronous replication. For γ < √c,
+these contributions can be crudely estimated as follows:
+� 2π+√c/2
+2π−√c/2
+f0(ω)dω ≈ √c,
+(89)
+and
+� 2π+γ
+2π−γ
+fγ(ω)dω ≈ 2/γ,
+(90)
+0.0
+0.5
+1.0
+1.5
+0
+200
+400
+600
+800
+1000
+1200
+f
+|ΔN
+
+A
+2
+1/2=|KyA
+2
+
+1/2
+Figure 10: Plot of the spectrum of yA versus the frequency
+f = ω/(2π), for the Poisson model. Blue = simulation with
+K = 105, b = 1, and the remaining parameters as in S0.5. Red
+= analytic expression (80) with
+�
+|ηA|2�
+=
+�
+|ηB|2�
+= bK, and
+�
+|ηAB|2�
+= 0.
+so that the contribution from synchronous replication be-
+comes comparable to death-induced noise for γ < 2/√c,
+or when w < 0.55c−1/4. As the expression is rather in-
+sensitive to the value of c, we can conclude that devia-
+tions from the Poisson, asynchronous replication should
+already be visible even for relatively large values of w.
+VI.
+CONCLUSION
+We have revisited a stochastic two-species model of
+the predator-prey type [2], which exhibits oscillations
+for a wide range of parameters of the model. We have
+modified the model so that both species replicate quasi-
+synchronously, with doubling times drawn from a narrow
+distribution. We have shown that coloured demographic
+noise generated by this process leads to much stronger os-
+cillations than the Poisson process of replication assumed
+in earlier works. Coloured noise has been shown to affect
+population dynamics in single-species models [25]; here,
+we not only derive its spectrum from the underlying mi-
+croscopic dynamics, but also show how it affects more
+complex models.
+Our result, while obtained for an abstract mathemati-
+cal model, may be relevant for real biological populations,
+in particular for microorganism which often replicate in
+quasi-discrete generations. We expect to see the same
+behaviour in other models that exhibit quasi-cycles [26–
+31].
+The phenomenon of coloured noise amplification may
+
+13
+0.0
+0.5
+1.0
+1.5
+0
+500
+1000
+1500
+2000
+2500
+f
+|ΔN
+
+A
+2
+1/2=|KyA
+2
+
+1/2
+0.0
+0.5
+1.0
+1.5
+0
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+f
+|ΔN
+
+A
+2
+1/2=|KyA
+2
+
+1/2
+Figure 11:
+Plot of the spectrum of yA versus frequency
+f = ω/(2π), for the non-Poisson model with the replication
+frequency slightly detuned (b = 0.9, top) and in resonance
+(b = 0.7, bottom). Blue = simulation with K = 105, w =
+0.08, and the remaining parameters as in S0.5. Red = ana-
+lytic expression 80 with the noise terms from Eq. (82), and
+γ = 6.579w2 (here T = 1).
+0.05
+0.10
+0.15
+0.20
+0
+2 × 106
+4 × 106
+6 × 106
+8 × 106
+1 × 107
+w
+<|ΔNA
+2|>
+Figure 12: Plot of
+�
+|∆NA(t)|2�
+versus w, for the non-Poisson
+model with replication frequency b = 0.7. Blue = simulation
+with K = 105, and the remaining parameters as in S0.5. Red
+= analytic expression 83.
+be further augmented in situations in which oscillations
+in the population abundance become synchronized with
+reproductive cycles. We leave this interesting problem
+for future studies.
+Acknowledgments
+B.W. acknowledges funding under Dioscuri, a pro-
+gramme initiated by the Max Planck Society, jointly
+managed with the National Science Centre in Poland,
+and mutually funded by Polish Ministry of Science and
+Higher Education and German Federal Ministry of Edu-
+cation and Research (UMO-2019/02/H/NZ6/00003).
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+
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+page_content=' EH9 3FD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' United Kingdom  2Dioscuri Centre for Physics and Chemistry of Bacteria,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Institute of Physical Chemistry PAS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Kasprzaka 44/52,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' 01-224 Warsaw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Poland  3Institut für Physik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Johannes Gutenberg-Universität Mainz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Staudingerweg 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' 55128 Mainz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Germany  Predator-prey models have been shown to exhibit resonance-like behaviour,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' in which random  fluctuations in the number of organisms (demographic noise) are amplified when their frequency  is close to the natural oscillatory frequency of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' This behaviour has been traditionally  studied in models with exponentially distributed replication and death times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Here we consider a  biologically more realistic model, in which organisms replicate quasi-synchronously such that the  distribution of replication times has a narrow maximum at some T > 0 corresponding to the mean  doubling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We show that when the frequency of replication f = 1/T is tuned to the natural  oscillatory frequency of the predator-prey model, the system exhibits oscillations that are much  stronger than in the model with Poissonian (non-synchronous) replication and death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The effect can  be explained by resonant amplification of coloured noise generated by quasi-synchronous replication  events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' To show this, we consider a single-species model with quasi-synchronous replication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We  calculate the spectrum and the amplitude of demographic noise in this model, and use these results  to obtain these quantities for the two-species model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  INTRODUCTION  When a non-linear dynamical system capable of ex-  hibiting damped oscillations is coupled to a source of  random noise, it often begins to generate periodic oscilla-  tions (a quasi-cycle) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' This resonance-like behaviour is  caused by the amplification of noise frequencies that are  in tune with the natural oscillatory frequency of the sys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Importantly, the noise does not have to be external  but it can be intrinsic to the system and arise from its  microscopic stochastic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  An important example is resonant amplification of  demographic noise which has been found in stochas-  tic models of biological populations [2–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' However, all  these models assume that reproduction is a Markov pro-  cess: birth and death occur with certain (possibly state-  dependent) rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' At any moment, the distribution of  replication times is therefore exponential, with the max-  imum at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' However, biological organisms do not  replicate in this way: all known organisms require a cer-  tain minimum time to develop reproductive capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Moreover, many organisms reproduce in quasi-discrete  generations such that the time between consecutive repli-  cation events has a narrow distribution that peaks around  some characteristic time T called the generation time, or  doubling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' For example, for the bacterium E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' coli, T  ranges between 20 min and a few hours and the coefficient  of variation of the doubling time is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='3, depending  on growth conditions [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  This leads to significant  correlations between reproduction times of related indi-  viduals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Modelling this process for a single species has a  long history [11–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  The quasi-synchronous nature of replication suggest an  interesting possibility: if a predator-prey system has a  tendency to oscillate at a frequency similar to the inverse  of the doubling time, synchronisation of the two oscilla-  tions may lead to a substantial enhancement of resonant  amplification of demographic noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  In this work, we investigate this scenario in a simple  predator-prey model originally proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The  model assumes two biological species interacting in a way  that leads to damped predator-prey cycles in the limit of  infinitely large populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' In the original model, demo-  graphic noise due to stochastic replication of organisms  led to persistent oscillations of small but non-zero ampli-  tude and a Lorenz-like power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Here we show  that when replication is no longer Poissonian but occurs  in quasi-discrete generations, these oscillations increase  dramatically in amplitude and can be as high as 50% of  the total population size even when the number of organ-  isms is very large (millions or more).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  MODEL  Our model an extension of the Newman-McKane  model [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We consider a well-mixed population of two  types of organisms A, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We shall call these organisms  "cells" as if they were single-celled microorganisms, al-  though the model is agnostic to the exact nature of these  organisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Let nA, nB be the number of cells of each  type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Each cell has an internal state variable τ assigned  at birth from a certain distribution R(τ), the same for  both species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We shall call this variable a “timer”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The  timer counts down from the assigned time interval;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' when  it reaches zero, the cell produces an offspring and both  cells are assigned new, randomly selected values of τ from  R(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We shall assume that the distribution R(τ) is con-  centrated around its mean value ⟨τ⟩ ≡ T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Cells also  die with per-capita rates dA = p2 − p1nB/K for type A,  and dB = p4nA/K − p3(1 − nB/K) for type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Here K  plays a role similar to the carrying capacity in popula-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='13290v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='stat-mech]  30 Jan 2023  2  tion dynamics models and sets the scale for the number  of cells in the system: nA, nB ∼ K on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We have  used the same symbols for the parameters p1, p2, p3, p4  as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' [2], however their microscopic interpretation  is slightly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We will come back to this when we  discuss the steady-state solution of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  The dynamics of the model can be schematically rep-  resented as a set of chemical-like equations:  A → 2A  (1)  B → 2B  (2)  A → 0  (3)  B → 0  (4)  However, one must be careful with how these equations  are interpreted in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Only the last two reactions  have been used in the same way as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' [2] to represent  inhomogeneous Poisson processes occurring with state-  dependent rates dA(nA, nB), dB(nA, nB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The first two  reactions do not describe Poisson processes because the  probability of replication depends on the internal state τ  of each cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Thus the model is non-Markovian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Let us briefly discuss some possible choices for the dis-  tribution R(τ) of replication times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  The case R(τ) =  δ(τ − T) corresponds to all cells reproducing in per-  fect synchrony;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' the generation time is T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The exponen-  tial distribution R(τ) = (1/T) exp(−τ/T) represents the  Poisson case: cells reproduce with rate 1/T per capita  and the mean time to replication is T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' In this case the  model is Markovian and its behaviour is expected to be  the same as the original model from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Finally,  R(τ) can be concentrated around τ = T but have a non-  zero width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' This represents quasi-synchronous replica-  tion: all descendants of a given cell initially replicate in  quasi-discrete generations, with progressive loss of syn-  chronization over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  In this manuscript, we shall compare the behaviour  of the model for two distributions R(τ): (i) exponential  (the Poisson model) with mean time to division T, (ii)  uniform on (T(1 − w), T(1 + w)) where w ≪ 1 controls  the degree of correlation of replication times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' synchrony  is lost after ∼ 1/w generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Biological interpretation  We shall now provide a biological interpretation of the  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' However, our analysis does not rely on this inter-  pretation and the model is deliberately oversimplified and  not intended to replicate any specific experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Our  model could describe a population of micro-organisms  of one species and two slightly different ecotypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Both  types replicate with the same time-independent rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Type A’s basal death rate p2 decreases in the presence of  B proportionally to the concentration of B times p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' This  could be due to type B producing an essential chemical  compound necessary for A to survive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' This interpretation  in consistent with p2 ≫ b that we will generally assume  later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Therefore, a sufficient density of B is required for  A to thrive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Type B, on the other hand, is killed by  A (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=', A releases a toxin that kills B) with rate equal  to the abundance of A times p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The additional term  −p3(1 − xB) accounts for the increase in the death rate  due to crowding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' A similar term could be added to the  equation for type A to make the model more symmetric  but it would not qualitatively affect the dynamics of the  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' All interactions described here have been demon-  strated in microbial populations [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  ANALYSIS OF THE MODEL  We first study the behaviour of the Poisson model  in the infinite-population size limit (K → ∞) by ne-  glecting fluctuations in the number of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We define  xA = nA/K, xB = nB/K as the new state variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  The dynamics of the model can be approximated by two  differential equations:  dxA  dt  = xA (b − max[p2 − p1xB, 0]) ,  (5)  dxB  dt  = xB (b − max[p4xA − p3(1 − xB), 0]) ,  (6)  where b = ln(2)/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' In the above, we have used average  rates of all processes represented by reactions (1-4), and  assumed that all higher moments factorize into products  of xA, xB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The max[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' ] function ensures that the death  terms contribute only if the corresponding rates are pos-  itive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  The non-zero steady-state solution of the model, with  both species present, reads:  x∗  A = (p1 − p2)p3 + b(p1 + p3)  p1p4  ,  (7)  x∗  B = p2 − b  p1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (8)  To investigate the stability of this solution, we Taylor-  expand equations (5-6) around the steady-state solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  This leads to the Jacobian matrix with the following  eigenvalues:  λ± = bp3 − p2p3 ±  �  (b − p2) (b(2p1 + p3)2 − p3 (−4p2  1 + 4p1p2 + p2p3))  2p1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (9)  3  0  2  4  6  8  0  3\uf4a0105  6\uf4a0105  9\uf4a0105  t  NA,NB  0  2  4  6  8  0  3\uf4a0105  6\uf4a0105  9\uf4a0105  t  NA,NB  Figure 1: Damped oscillations in the stochastic asynchronous  model (points:  blue for NA, yellow for NB) and its de-  terministic counterpart (black lines), for {p1, p2, p3, p4, b} =  {30, 7, 7, 10, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Top: the deterministic solution differs from  the stochastic simulation for the uniform initial distribu-  tion of the timer variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Both models assume the same  initial condition xA(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='2, xB(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Bottom:  the  agreement is very good when we compare the models af-  ter the stochastic model reached a quasi-steady state timer  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  The deterministic model uses the values of  NA, NB = {797423, 293636} from the stochastic simulation  for t0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='2207 as its initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  In the regime that we are interested here (x∗  A > 0, x∗  B >  0), the eigenvalues have a negative real part, meaning  that the steady-state solution is stable to a small per-  turbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  However, their imaginary part is generally  non-zero, and hence the system will exhibit damped os-  cillations while relaxing towards the steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' This is  illustrated in Figure 1, which shows plots of the determin-  istic solution for {p1, p2, p3, p4, b} = {30, 7, 7, 10, 1}, for a  short time (a few oscillations), starting from xA(0) =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='2, xB(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' On the same plot we show the results  of a numerical simulation of the stochastic Poisson model  with K = 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Note the small discrepancy between both  models (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' 1, top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' This is due to the timer of all cells  being initialized with uniformly distributed random num-  bers in the stochastic simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' This initial distribution  is quite different than the quasi-steady state distribution  obtained after a few cycles, which is implicitly assumed  when deriving Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (5, 6) from the microscopic rules (1-  4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' When we solve the deterministic model starting from  a later point, using NA, NB from the stochastic model as  the initial condition, the discrepancy vanishes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' 1,  bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Parameter selection  The  deterministic  model  has  five  parameters:  p1, p2, p3, p4, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  We can put b = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' this fixes the time  scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  The remaining four parameters determine the  frequency of small-amplitude oscillations, the damping  coefficient, and steady-state occupations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Before we  move on, we shall discuss how we select these parameters  so that the model exhibits under-damped oscillations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  this is required for the resonant amplification of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  We have used Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (9) together with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (7-8) to find  a region in the parameter space {p1, p2, p3, p4} and b = 1  of the deterministic model that corresponds to damped  oscillations of frequency f0 = Im(λ)/(2π) ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='98, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='02),  damping coefficient |Re(λ)| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5, and steady-state abun-  dances 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We did this via Monte-Carlo sampling  of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We then used the selected val-  ues as starting points for a root finding algorithm to find  p1, p2, p3, p4 such that the frequency would be exactly  f0 = 1, steady-state occupations xA = xB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5, and the  damping coefficient assumed one of three values: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5 (fast  damping), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='2 (slow damping) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='1 (minimal damp-  ing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  This  procedure  has  produced  three  sets  of  pa-  rameters:  S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  =  {39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='73, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='86, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=', 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' },  S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='2  =  {56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='45, 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='23, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='8}, and S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='1 = {65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='81, 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='91, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  We shall use these parameters in the full, stochastic  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Numerical results  We have simulated the stochastic model using a sim-  ple tau-leaping algorithm with fixed-size time step dt =  1/512 [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Figure 2 shows examples of time series ob-  tained for the Poisson version of the model, and for  the quasi-synchronous model with a narrow distribu-  tion of doubling times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The parameters are S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5, K =  300000, w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='02, and the initial condition is nA(0) =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='2K, nB(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='2K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' In both cases the natural oscilla-  tory frequency of the model is f0 = 1 and the average  doubling time is also T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  The quasi-synchronous model exhibits much larger os-  cillations than the Poisson model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Accordingly, the  Fourier spectrum of the quasi-synchronous model has a  much more pronounced peak at frequency f0 = 1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  The observed increase in the amplitude of oscilla-  tions occurs only when the doubling frequency is close to  the natural oscillation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Figure 4, top, shows  that when T ̸= 1/f0 the amplitude is significantly re-  duced;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' this resonance-like behaviour is not present in the  Poisson model (black line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' 4, top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Interestingly,  the maximum amplitude is observed at a slightly lower  b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5 than expected (b = ln 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='69 which corresponds  to T = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The resonance peak is also quite broad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' This  is a non-linear effect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' for large amplitudes as observed  here, the resonant frequency is slightly lower than f0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  The resonance peak becomes sharper with increasing  4  2000  4000  6000  8000  10000  100000  110000  120000  130000  140000  150000  160000  t  NA,NB  2000  4000  6000  8000  10000  100000  110000  120000  130000  140000  150000  160000  t  NA,NB  Figure 2:  Example time series NA(t), NB(t) in the Pois-  son (top) and quasi-synchronous (bottom) models, for K =  300000, w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Individual oscillations cannot be seen due  to the length of the time window shown here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' the window  contains a few thousand oscillations as those from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  5  6  7  8  9  10  11  12  frequency  ln(amplitude) [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='u]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  5  6  7  8  9  10  11  12  frequency  ln(amplitude) [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='u]  Figure 3: Fourier spectrum of NA(t) in the Poisson (left) and  non-Poisson (right) models, for K = 300000, w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='02, the  remaining parameters = S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Moving average with a 20-point  long window has been applied to smooth out the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  carrying capacity K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' 4, middle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The amplitude of  oscillations in the peak is independent of K for a wide  range of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' This is very different to the scaling ∼ 1/  √  K  observed in the Poisson case and also the scaling of CV  in the quasi-synchronous model far away from the peak  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  The amplitude of oscillations in the quasi-  synchronous model is more than 10% of the steady-state  population abundance for K = 106, whereas in the Pois-  son model with identical parameters it is less than 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Figure 4, bottom, shows that the height of the reso-  nance peak decreases with increasing w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' For w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='2,  the peak is barely noticeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  On the other hand, all  values w ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='1 produce a visible peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  These results suggest that quasi-synchronous replica-  tion leads to a significant enhancement of demographic  noise in the quasi-synchronous model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  To understand  this, let us first revisit what happens in the Poisson ver-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='12  b  CV(NA)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='15  b  CV(NA)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='12  b  CV(NA)  Figure 4: Top: Coefficient of variation (CV) of NA(t) for  the Poisson (black) and quasi-synchronous (red) models, as a  function of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' CV is a convenient measure of the amplitude  of oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  A resonance peak can be seen at b ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Middle: CV of NA(t) for the quasi-synchronous model with  w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='02 and different K = 104, 3 × 104, 105, 3 × 105, 106  (blue, yellow, green, red, violet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Bottom: CV versus b for  different widths w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='08, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='2 of the doubling  time distribution (colours from blue to violet) and K = 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  In all cases, parameters = S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  sion of the model [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' In that model, demographic noise  has a flat spectrum and contains a broad range of fre-  quencies (white noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Frequencies close to the frequency  at which the system exhibits damped oscillations are am-  plified;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' this leads to quasi-periodic oscillations with the  5  1×104  5×104 1×105  5×105 1×106  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='050  K  CV(NA) for b=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  Figure 5:  Coefficient of variation of NA(t) for the quasi-  synchronous model as a function of K, calculated at b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  (away from the resonance peak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Solid line represents the scal-  ing ∼ 1/  √  K expected for the Poisson version of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  amplitude ∼  √  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  However, since the average abun-  dances NA, NB increase proportionally to K, the rela-  tive magnitude of oscillations decreases as ∼ 1/  √  K with  the increasing population size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Noise-induced oscillations  are therefore significant only for relatively small systems  K ≪ 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' In contrast, here we observe large, persistent  oscillations even for K = 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' As we shall see, this can  be explained by demographic noise being concentrated in  a narrow range of frequencies in the non-Poisson model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  SINGLE-SPECIES MODEL  To understand the spectrum of noise in the quasi-  synchronous model, we consider a simpler one-species  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' In this model, replication is non-Poissonian with  mean doubling time ln(2)/b as in the two-species model,  whereas death is a Poisson process with rate bN/K,  where N is the total number of cells, K is the carrying  capacity, and b is the replication rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  In the large-K limit, the average abundance x = N/K  evolves according to the logistic equation,  dx/dt = bx(1 − x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (10)  The steady-state occupation is x∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' In the stochastic  model (K < ∞), the number of cells is thus expected to  fluctuate around the mean value N ∗ ∼= K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Figure 6, top, shows examples of N(t) for the model  with Poisson and non-Poisson replication (w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='02), for  K = 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The quasi-synchronous model exhibits more  regular oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The standard deviation of N(t) is  very similar to the Poisson model for w > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='05 but  rapidly increases for smaller w (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' 6, bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  In what follows, we shall study this model analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  In particular, we are interested in analytical expressions  for (i) the correlation time of oscillations, (ii) the spec-  trum of oscillations, (iii) the steady-state amplitude of  oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' This will help us to better understand the  behaviour of the two-species model from previous sec-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Preliminary considerations  We begin by considering the behaviour of a large popu-  lation of cells in which the cells can be assigned to groups  depending on the phase φ of their cell cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The phase is  not the same as the timer variable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' instead, it should be  interpreted as the difference between the timer variable  and some arbitrary chosen reference timer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Let n(φ) be  the number density of cells with phase φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Let us further  assume that, if all cells were synchronised (all φ being  equal), the total number of cells would be described by  a certain periodic function f(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' This function (besides  a different amplitude) also describes number fluctuations  in a group of cells that have the same phase φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The total  number of cells in the population is therefore  N(t) =  � ∞  −∞  n(φ)f(t − φ)dφ,  (11)  which is the convolution of f and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The Fourier spec-  trum of N is  F[N](ω) = F[f](ω) F[n](ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (12)  Suppose n(φ) is Gaussian with variance σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We have  F[N](ω) = F[f](ω)e−(1/2)σ2ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (13)  If f is periodic with angular frequency ω0, then the  lowest-frequency Fourier mode of N at ω = ω0 will be re-  duced in comparison to f by e−(1/2)σ2ω2 due to the spread  of the phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  All higher modes will be damped even  more;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' we will neglect them for now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' For ω0 = 2π/ ln 2  assumed in our single-species model for b = 1 (dou-  bling time ln 2) and σ2 ≪ 1, the reduction factor is  e[2π2/(ln 2)2]σ2 ≈ e−41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='1σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Suppose further than each generation causes the dis-  tribution n to broaden, due to the finite width of the  distribution of doubling times, so that σ2 = σ2  0(t/ ln(2)),  where σ2  0 = (w2/3)(ln 2)2 is the variance of the uni-  form distribution of doubling times used in the simu-  lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' This corresponds to the variance of the phase  distribution increasing by the variance of the doubling  time distribution every generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' This will then lead  to oscillations in N(t) (caused by quasi-synchronous  replication) to decay exponentially with the rate γ =  [2π2/(ln 2)2](ln 2)(w2/3) = [2π2/(3 ln 2)]w2 ≈ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Figure 7 shows that the predicted rate is in very good  agreement with the decay rate observed in numerical sim-  ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  A more formal approach  The above result can be derived more formally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We  shall start by writing down the equation for the number  6  3750 3755 3760 3765  99500  100000  100500  101000  t  n  3750 3755 3760 3765  98500  99000  99500  100000  100500  101000  101500  t  n  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='20  0  200  400  600  800  w  std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Figure 6: Fluctuations in the single-species model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Top: N(t)  for K = 100000: Poisson model (left), and quasi-synchronous  model with w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='02 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Bottom: standard deviation  of fluctuations for different w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Points = simulation, line =  equation (64) with γ = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  No parameters have been  fitted to data here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  0  50  100  150  0  20000  40000  60000  80000  100000  t  n  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='500  1  ϵ  γ  Figure 7: Exponential decay of oscillation in the single-species  model for K = 100000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Left: N(t) for w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Right: the  exponential decay rate γ as a function of w (points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Black  line is the theoretical prediction γ ≈ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  density of cells n(τ, t) at time t with the timer variable  τ, for the time being neglecting stochastic noise:  ∂tn(τ, t) = ∂τn(τ, t) − ln 2  T n(τ, t)  � ∞  0  n(τ ′, t)  K  dτ ′  +2R(τ)n(0, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (14)  The first term corresponds to the timer counting back-  ward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The second term represents death with rate pro-  portional to the total size divided by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  The factor  (ln 2)/T is required to have the correct behaviour in the  limit of perfectly synchronous replication - we shall see  this later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The third term represent replication that oc-  curs when the timer reaches τ = 0 and is the product  of the density of cells n(0, t) in that state and R(τ), the  probability density function for the timer being reset to  τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We assume R(τ) to be normalized:  � ∞  0  R(τ)dτ = 1,  (15)  and that R(τ) is concentrated around τ = T as in nu-  merical simulations in previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Stationary solution  In the limit t → ∞, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (14) becomes  0 = ∂n(τ)  ∂τ  − n(τ)J + 2n(0)R(τ),  (16)  where J = (ln 2)/T  � ∞  0 (n(τ)/K)dτ, and n(τ) does not  depend on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We can solve this equation for the steady-  state distribution n∗(τ):  n∗(τ) = n∗(0)eJτ  �  1 − 2  � τ  0  e−Jτ ′R(τ ′)dτ ′  �  ,  (17)  with the condition n∗(τ → ∞) = 0 implying that  � ∞  0  e−JτR(τ)dτ = 1/2,  (18)  which fixes the value of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' If R is a uniform distribution  with mean τ = T and width 2wT, we obtain from (18)  that  e−JT sinh(JTw)  JTw  = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (19)  This equation must be solved for J numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' In ad-  dition, one must determine the value of n∗(0) from the  relationship between J and n(τ):  JK = n∗(0)  � T (1+w)  0  eJτ  �  1 − 2  � τ  0  e−Jτ ′R(τ ′)dτ ′  �  dτ  (20)  Figure 8 shows an example of n∗(τ) for w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='1, calcu-  lated in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The cell number density is proportional  to eJτ ≈ 2τ/T for τ < T, and rapidly falls down to zero  for τ > T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The solution simplifies greatly in the limit  w → 0, in which J tends to (ln 2)/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The steady state  number density becomes then  n∗(τ) = K ln 2  T 2τ/T  (21)  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Evolution of a small perturbation  We now consider the time evolution of a small pertur-  bation to the steady state solution:  n(τ, t) = n∗(τ)(1 + ϵ(τ, t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (22)  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  τ  n[τ]  Figure 8: Stationary cell density n∗(τ) for T = ln 2, w =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Red line = Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Black dashed line = approximate  solution (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  of the noise-less equation 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Inserting this into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' 14  gives:  n∗∂tϵ = (∂τn∗)(1 + ϵ) + n∗∂τϵ  − ln 2  TK n∗(1 + ϵ)  � ∞  0  n∗(τ ′)(1 + ϵ(τ ′, t))dτ ′  + 2R(τ)n∗(1 + ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (23)  We note that ∂τn∗ = Jn∗ − 2n∗(0)R(τ), and keep only  terms linear in ϵ:  n∗∂tϵ = n∗∂τϵ − ln 2  TK n∗  � T  0  n∗(τ)ϵ(τ ′, t)dτ ′  + 2R(τ)n∗(0)(ϵ(0, t) − ϵ(τ, t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (24)  We divide by n∗ and obtain  ∂tϵ = ∂τϵ − (ln 2)  TK  � ∞  0  n∗(τ ′)ϵ(τ ′, t)dτ ′  − 2n∗(0)  n∗(τ)R(τ)(ϵ(0, t) − ϵ(τ, t))  (25)  with the boundary condition ϵ(0, t) = ϵ(T, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We now  expand ϵ and n∗(0)  n∗(τ)R(τ) as Fourier series (consistent with  the b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' ):  ϵ(τ, t) =  ∞  �  k=−∞  Ak(t)e2πikτ/T ,  (26)  n∗(0)  n∗(τ)R(τ) =  ∞  �  k=−∞  Rk(t)e2πikτ/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (27)  where the coefficients {Rk} are given by  Rk = 1  T  � T  0  e−2πikτ/T n∗(0)  n∗(τ)R(τ)dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (28)  The transformed equation reads  �  k  ∂tAke2πikτ/T =  �  k  Ak(2πik/T)e2πikτ/T −  −(ln 2)  TK  �  k  Ak  � T  0  n∗(τ ′)e2πikτ ′/T dτ ′  +2  �  m  Rme2πimτ/T  ��  k  Ak −  �  k  Ake2πikτ/T  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (29)  The sums in (29) can be compared term by term since  they must be valid for any τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' This leads to the following  equation for the Fourier coefficients Ak(t):  ∂tAk = (2πik/T)Ak  − δk,0  ln 2  TK  �  m  Am  � T  0  n∗(τ ′)e2πimτ ′/T dτ ′  + 2(Rk  �  m  Am −  �  m  RmAk−m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (30)  In particular, for k > 0 we have  ∂tAk = (2πik/T)Ak + 2(Rk  �  m  Am −  �  m  RmAk−m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (31)  Let us assume that the initial perturbation is a pure kth  Fourier mode, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=', Ak ̸= 0 only for a single value of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  The first term represents oscillations with period T/k  of that mode, which essentially gives a travelling-wave  type of solution ∼ exp(2πik(τ − t)/T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The second term  represents damping with rate γk = −2(Rk −R0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We can  calculate this rate using equation (28) in the limit w → 0,  since then we have from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (21) that n∗(0)/n∗(τ) =  2−τ/T and hence  Rk = 1  T  � T (1+w)  T (1−w)  e−2πikτ/T 2−τ/t  1  2Twdτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (32)  We obtain that  γk = −2(Rk − R0) ∼= 2π2  3T k2w2,  (33)  which, for T = ln 2 and k = 1 reproduces the decay rate  γ ≈ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5w2 which we have already seen in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Amplitude of oscillations for perfectly  synchronous replication  We shall now add noise to the model and see how it  affects its behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We shall first consider a fully syn-  chronous replication with arbitrary period T, which leads  to the following equation:  ∂tn = ∂τn − ln 2  T n  � T  0  n(τ ′, t)  K  dτ ′ +  √  n∗η,  (34)  with boundary conditions  n(0, t) = (1/2)n(T, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (35)  The noise term  √  n∗η is due to death only, since replica-  tion is perfectly synchronous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We assume η(τ, t) repre-  sents uncorrelated white noise:  ⟨η(τ1, t1)η(τ2, t2)⟩ = Dδ(τ1 − τ2)δ(t1 − t2),  (36)  with some D > 0 to be specified later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' It can be eas-  ily verified that this form of noise arises from a master  8  equation for the model with no replication by perform-  ing a van Kampen expansion [22] of the master equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  While it may be possible to derive the noise term also  in the presence of non-Markovian replication, we find it  easier to postulate that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (36) generally holds, and  justify it based on the agreement between the result of  our calculation and the computer simulation (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  In the absence of noise, equation (34) has the steady-  state solution  n∗(τ) = K ln 2  T 2τ/T ,  (37)  � T  0  n∗(τ)dτ = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (38)  To solve the time-dependent equation with noise, we con-  sider a small perturbation (similarly as in the previous  section):  n(τ, t) = n∗(τ)(1 + ϵ(τ, t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (39)  This gives  n∗∂tϵ = (∂τn∗)(1 + ϵ) + n∗∂τϵ  − ln 2  TK n∗(1 + ϵ)  � T  0  n∗(τ ′)(1 + ϵ(τ ′, t))dτ ′ +  √  n∗η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (40)  We note that ∂τn∗ = ((ln 2)/T)n∗, and only keep terms  linear in ϵ:  n∗∂tϵ = n∗∂τϵ − (ln 2)2  T 2  n∗  � T  0  2τ ′/T ϵ(τ ′, t)dτ ′ +  √  n∗η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (41)  We divide by n∗ and obtain  ∂tϵ = ∂τϵ− (ln 2)2  T 2  � T  0  2τ ′/T ϵ(τ ′, t)dτ ′+(n∗)−1/2η, (42)  with the following boundary and initial conditions:  ϵ(0, t) = ϵ(T, t) and ϵ(τ, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Proceeding as in the  previous section, we expand ϵ and (n∗)−1/2η as Fourier  series (consistent with the b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' ):  ϵ(τ, t) =  ∞  �  k=−∞  Ak(t)e2πkiτ/T ,  (43)  (n∗(τ))−1/2η(τ, t) =  ∞  �  k=−∞  ηk(t)e2πkiτ/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (44)  (45)  The transformed equation reads  �  k  ∂tAke2πkiτ/T =  �  k  Ak(2πik/T)e2πkiτ/T −  −(ln 2)2  T 2  �  k  Ak  � T  0  2τ ′/T e2πkiτ ′/T dτ ′  +  �  k  ηk(t)e2πkiτ/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  The integral over dτ ′ gives  � T  0  2τ ′/T e2πkiτ ′/T dτ ′ =  iT  i ln 2 − 2πk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (46)  Comparing the sums in (46) term-by-term we notice that  the (ln 2)2 term does not contain any factor e2πkiτ/T , so  it only contributes to the constant term:  δk,0  (ln 2)2  T  �  n  An  i  i ln 2 − 2πn ≈ δk,0  ln 2  T A0,  (47)  where we have assumed that all An for n ̸= 0 are much  smaller than A0 (we shall see later that this is the case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  This leads to the following equation for the Fourier coef-  ficients Ak(t):  ∂tAk = (2πik/T)Ak − ((ln 2)/T)δk,0A0 + ηk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (48)  In particular, for k = 0 we have  ∂tA0 = −ln 2  T A0 + η0,  (49)  which can be formally solved as  A0(t) = 2−t/T  � t  0  2t′/T η0(t′)dt′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (50)  This gives  �  A0A†  0  �  (t) = 2− 2t  T  � t  0  dt1  � t  0  dt22  t1+t2  2  �  η0(t1)η†  0(t2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (51)  Equation (44) enables us to write  ηk(t) = 1  T  � T  0  �K ln 2  T  �−1/2  2− τ  2T η(τ, t)e−2πikτ/T dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (52)  The average of the noise term gives  �  ηk(t1)η†  k(t2)  �  =  1  TK ln 2 ×  ×  � T  0  dτ1  � T  0  dτ22− τ1+τ2  2T  e− 2πik(τ1−τ2)  T  �  η(τ1, t1)η†(τ2, t2)  �  = D  K  δ(t1 − t2)  2(ln 2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (53)  We therefore have  �  A0A†  0  �  (t) = 2−2t/T  � t  0  22t′/T D  K2(ln 2)2 dt′  = DT  K  1 − 2−2t/T  4(ln 2)3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (54)  Proceeding similarly for k ̸= 0, we obtain:  Ak(t) = e2πikt/T  � t  0  e−2πikt′/T ηk(t′)dt′,  (55)  9  from which we obtain that  �  AkA†  k  �  (t) =  � t  0  dt1  � t  0  dt2e−2πik(t1−t2)/T �  ηk(t1)η†  k(t2)  �  = D  K  t  2(ln 2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (56)  We can now calculate the standard deviation of ∆N, the  difference between the total number of cells at time t and  the average steady-state number:  ∆N(t) =  � T  0  n∗(τ)ϵ(τ, t)dτ  (57)  =  � T  0  K ln 2  T  2τ/T �  k  Ak(t)e2πikτ/T dτ  =  �  k  Ak(t)K ln 2  T  � T  0  2τ/T e2πikτ/T dτ  = (K ln 2)  �  k  Ak(t)  i  i ln 2 − 2πk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (58)  This gives (we note that terms  �  AkA†  n  �  with k ̸= n van-  ish):  �  |∆N(t)|2�  =  (K ln 2)2  ∞  �  k=−∞  �  AkA†  k  �  (t)  (ln 2)2 + (2πk)2  = KD  �  T 1 − 2−2t/T  4(ln 2)3  + 2  ∞  �  k=1  t/2  (ln 2)2 + (2πk)2  �  = KD  �  T 1 − 2−2t/T  4(ln 2)3  + t3(ln 2) − 2  4(ln 2)2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (59)  In the limit t → ∞ this gives  �  |∆N(t)|2� ∼= KDt3(ln 2) − 2  4(ln 2)2  ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0413 KDt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (60)  Equation (60) predicts that the amplitude of oscillations  for perfectly synchronous replication increases linearly in  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' However, recall that our result has been derived  under the assumption of a small perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' In reality,  the amplitude will be limited by non-linear effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Amplitude of quasi-synchronous oscillations  Let us now consider the case of quasi-synchronous  replication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Rather than attempting to solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (34)  with the extra term R(τ) as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (14), we observe (as  argued in subsection IV D) that the k-th Fourier mode  will be damped with rate k2γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We thus consider the fol-  lowing modification to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (48) for k ̸= 0:  ∂tAk = (2πik/T − k2γ)Ak + ηk,  (61)  where γ is the damping coefficient derived previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  The equation for k = 0 remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Proceeding  as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' IV E, we obtain  �  AkA†  k  �  (t) = D  K  1 − e−2k2γt  4(ln 2)2k2γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (62)  Inserting this into the equation for  �  |∆N(t)|2�  we have  in the limit t → ∞:  �  |∆N|2�  =  �  |∆N(t → ∞)|2�  =  = KD  �  T  4(ln 2)3 + 2  ∞  �  k=1  1  4k2γ  1  (ln 2)2 + (2πk)2  �  = KD  �  T  4(ln 2)3 + π2 12 − 18 ln 2 + (ln 2)2  12γ(ln 2)4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (63)  For T = ln 2 we obtain that  �  |∆N|2�  ≈ KD(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5203 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='01355/γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (64)  It remains to relate D to the parameters of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We  again assume that death is the main source of stochas-  ticity, and that the contribution from quasi-synchronous  replication is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Consider a pure death process  with the same total number of organisms K as the steady  state total (38), and death rate d = (ln 2)/T as per Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' For short time intervals, the variance of the num-  ber of organisms in the pure death process equals to  �  |∆N|2�  = (Kd)t = (K(ln 2)/T)t (easy to derive from  the general formula on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' 108-109 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' On the  other hand, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (59) we have that for small t and  T = ln 2,  �  |∆N|2� ∼= KD  3  4 ln 2t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (65)  Comparing the two formulas for  �  |∆N|2�  , we obtain that  D = 4 ln 2  3  ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='924.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (66)  Figure 6 shows that equation (64) with the above value  of D reproduces the variance from numerical simulations  very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Spectrum of fluctuations  We can now obtain a very good analytic approxima-  tion for the spectrum of normalized fluctuations y(t) =  N(t)/K − 1 in the single-species model by Fourier-  transforming the expression for ∆N:  ˜y(ω) = (ln 2)  �  k  ˜Ak(ω)  i  i ln 2 − 2πk ,  (67)  in which  ˜Ak(ω) = lim  L→∞  1  √  L  � L/2  −L/2  Ak(t)eiωtdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (68)  10  We have  �  | ˜A0(ω)|2�  =  �  |˜η0|2�  � ln 2  T  �2 + ω2 ,  (69)  �  | ˜Ak(ω)|2�  =  �  |˜ηk|2�  (ωT − 2πk)2 + γ2T 2k4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (70)  in which  �  |˜ηk|2�  is defined through the Fourier transform  like in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (68), and evaluates to  �  |˜η0|2�  =  �  |˜ηk|2�  =  D  K2(ln 2)2 = D2K−1,  (71)  with D2 =  2  3 ln 2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' This gives  K  �  |˜y|2(ω)  �  = (ln 2)2 �  k  �  | ˜Ak|2�  (ω)  (ln 2)2 + (2πk)2  =  D2  � ln 2  T  �2 + ω2 +  +  ∞  �  k=1  2D2T 2  (ωT − 2πk)2 + γ2T 2k4  (ln 2)2  (ln 2)2 + (2πk)2  ≈  D2  � ln 2  T  �2 + ω2 +  +  2D2T 2(ln 2)2  [(ωT − 2π)2 + γ2T 2][(ln 2)2 + (2π)2] + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (72)  where ‘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' ’ stand for terms corresponding to higher har-  monics which we neglect because they are strongly sup-  pressed by the denominator increasing fast with k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' For  T = ln 2, we have  K  �  |˜y|2(ω)  � ∼=  D2  1 + ω2 +  2(ln 2)2D2  (ln 2)2 + (2π)2  1  (ω − 2π  ln 2)2 + γ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (73)  The formula as a function of frequency f = ω/(2π) reads:  K  �  |˜y|2(f)  � ∼=  ∼= D2/(2π)2  1 + (2πf)2 +  D2  2(ln 2)2  (2π)2  (ln 2)2 + (2π)2  1  (2πf − 2π  ln 2)2 + γ2 ,(74)  where the factor 1/(2π)2 is required for correct normal-  ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Figure 9 shows that equation (74) agrees well  with the numerically obtained spectrum for a broad range  of w values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Now we have determined the spectrum of the single-  species model, we can proceed to obtain the spectrum of  the two-species model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  APPROXIMATE ANALYTIC SOLUTION OF  THE TWO-SPECIES MODEL  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Linear approximation  We shall assume that near the steady state the dy-  namics of the two-species model can be described by lin-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  0  100  200  300  400  f  |ΔN  \uf02d 2  1/2=|Ky2\uf02d  1/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  0  100  200  300  400  f  |ΔN  \uf02d 2  1/2=|Ky2\uf02d  1/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  0  500  1000  1500  f  |ΔN  \uf02d 2  1/2=|Ky2\uf02d  1/2  Figure 9: Fourier spectrum |K˜y2|1/2 of N(t) in the single-  species model for K = 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Blue points = simulations, red  lines = theoretical prediction (no fitting) obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' From top to bottom: w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  earized equations  dyA/dt = aAByB + ηA,  (75)  dyB/dt = aBAyA + aBByB + ηB,  (76)  where yA = xA − x∗  A, yB = xB − x∗  B (x∗  A, x∗  B are steady-  state concentrations), and aAB, aBA, aBB are given by  the following expressions  aAB = p3(b − p2) + p1(b + p3)  p4  ,  (77)  aBB = p3(b − p2)  p1  ,  (78)  aBA = p4(b − p2)  p1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (79)  Here ηA, ηB represent noise (not necessarily white noise)  due to replication and death of both species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Fourier-transforming Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (75, 76) leads to the follow-  ing expression for the spectrum of yA:  11  �  |˜yA(ω)|2�  = (a2  BB + ω2)  �  |ηA|2�  − 2aABaBB  �  |ηAηB|2�  + a2  AB  �  |ηB|2�  a2  ABa2  BA + 2aABaBAω2 + a2  BBω2 + ω4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (80)  The formula for the spectrum of yB (not shown) is very  similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Figure 10 shows that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (80) works well for  the Poisson case (asynchronous replication), for which we  assume  �  |ηA|2�  =  �  |ηB|2�  = (2b)(K/2) = bK (recall that  K/2 is the average number of organisms for our choice  of the parameters, and the factor 2b is due to both birth  and death contributing equally near the steady state),  and  �  |ηAB|2�  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  We now want to establish whether Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (80) also works  for the non-Poissonian case, with an appropriate choise  of the coloured noise ηA, ηB, based on the single-species  calculation presented in the previous Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' IV G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We as-  sume that fluctuations around the steady state in the  single species model can effectively be described by the  following equation:  dy/dt = −by + coloured noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (81)  Since we know the spectrum of y, we can calculate the  spectrum of the coloured noise as dy/dt or, in Fourier  space, by multiplying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (73) by ω2:  �  |˜ηA,B|2(ω)  �  = K−1  A,Bω2  �  D2  b2 + ω2 +  +  2(ln 2)2D2  (ln 2)2 + (2π)2  1  (ω − 2π  T )2 + γ2  �  ,  (82)  with KA = Kx∗  A, KB = Kx∗  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We then insert Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (82)  into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (80), assuming again that  �  |ηAB|2�  = 0 (justi-  fied since both species replicate independently with rates  unaffected by the other species).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Figure 11, shows that this simple approach works quite  well for different frequencies of replication (controlled  by b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  If the replication frequency is slightly detuned  from the natural frequency of the system (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' 11, top),  two peaks are visible in the spectrum:  a sharp peak  coming from the quasi-synchronous birth events, and a  much wider but lower peak corresponding to white noise-  induced oscillations at the natural frequency f ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Figure 11, bottom, shows that when b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='7 is tuned in  to the resonant frequency of the system, only one peak is  visible, with a slight broadening towards lower frequen-  cies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Amplitude of oscillations  We can now calculate the variance of ∆NA(t) - the dif-  ference between the actual NA(t) and the average number  KA = Kx∗  A of organisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We have:  �  |∆NA(t)|2�  = K2  A  2π  � ∞  −∞  �  |˜yA|2(ω)  �  dω = K2  A  2π  � ∞  −∞  (a2  BB + ω2)  �  |ηA|2(ω)  �  + a2  AB  �  |ηB|2(ω)  �  a2  ABa2  BA + 2aABaBAω2 + a2  BBω2 + ω4 dω,  (83)  in which we used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (80) and assumed no correlation  between the noise ηA and ηB (  �  |ηAηB|2�  = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  In the case of Poisson replication, we put  �  |ηA|2�  =  �  |ηB|2�  = (2b)(K/2) and evaluate the integral (83) nu-  merically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' For our usual choice of the parameters S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5,  we obtain  �  |∆NA(t)|2�  theor ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='48 × 105 which is very  close to the numerical estimate from the simulation,  �  |∆NA(t)|2�  sim ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='56 × 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  In the case of quasi-synchronous replication, we insert  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (82) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (83), and again integrate numerically  over ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Figure 12 shows the plot of  �  |∆NA(t)|2�  obtained  in this way, compared to the simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We no-  tice that the analytic formula correctly reproduces the  trend but the theoretically predicted values are generally  larger than the ones from the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' However, the  agreement is still quite good, given that our formula has  been derived using many approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Note that we used equation (83), which is the same as  the formula for the Poisson case [2], but with coloured  noise given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (82) instead of white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  The  fact that this approach works means that oscillations in  the system with quasi-synchronous replication can be un-  derstood as being caused by resonant amplification of  coloured, non-Poissonian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  To get some qualitative insight into the behaviour of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (83), we consider the case γ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' For T = 1, b = ln 2,  we can expand the formula under the integral in (83)  around ω = 2π, which enables us to carry out the integral  analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We obtain  12  �  |∆NA(t)|2� ∼=  KD2π2(a2  BB + a2  AB + 4π2)(ln 2)2  γ(a2  ABa2  BA + 4(2aABaBA + a2  BB)π2 + 16π4)(4π2 + (ln 2)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  (84)  We see that, since γ ∼ w2, the variance of ∆NA in-  creases as 1/w2 as reproduction becomes more and more  synchronous for w → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  This is similar to the effect  of a long delay in reaction kinetics [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The relation-  ship  �  |∆NA(t)|2�  ∝ K/γ can be interpreted as an ef-  fective reduction in the number of replicating entities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  cells originating from a common ancestor replicate quasi-  synchronously when their sub-population is much less  than 1/γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The system thus consists of Kγ ≪ K of such  groups of cells, which increases demographic noise by a  factor 1/√γ, and the variance of N by 1/γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Let us now consider how small w needs to be for the  variance to start deviating from the Poisson case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=',  how synchronous replication must be to make difference  to random, asynchronous replication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Equation (83) can  be rewritten as follows:  �  |∆NA(t)|2�  = KAD2  2π  � ∞  −∞  F(ω)(f0(ω) + fγ(ω))dω,  (85)  where  F(ω) = a2  BB + a2  AB(KB/KA) + ω2  (ω2 − Ω2)2 + c  (86)  is  the  resonance  response  function,  with  c  =  −aABaBAa2  BB − a4  BB/4, and Ω2 = −aABaBA − a2  BB/2 =  4π2 (squared resonant frequency for our parameters  S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' The function  f0(ω) =  ω2  ω2 + (ln 2)2  (87)  is the γ-independent contribution from stochastic repli-  cation, and the function  fγ(ω) =  2(ln 2)2ω2  (γ2 + (ω − 2π)2)(4π2 + (ln 2)2)  (88)  is the γ-dependent contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  F(ω) has full width at maximum height (FWHM) ap-  proximately equal to √c, whereas fγ(ω) has FWHM  equal to ≈ 2γ ≈ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='16w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  We expect that when the  contribution from fγ(ω) near the peak of F(ω) is larger  than the contribution from f0(ω), the variance of NA will  be dominated by synchronous replication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' For γ < √c,  these contributions can be crudely estimated as follows:  � 2π+√c/2  2π−√c/2  f0(ω)dω ≈ √c,  (89)  and  � 2π+γ  2π−γ  fγ(ω)dω ≈ 2/γ,  (90)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  0  200  400  600  800  1000  1200  f  |ΔN  \uf02d  A  2  1/2=|KyA  2  \uf02d  1/2  Figure 10: Plot of the spectrum of yA versus the frequency  f = ω/(2π), for the Poisson model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Blue = simulation with  K = 105, b = 1, and the remaining parameters as in S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Red  = analytic expression (80) with  �  |ηA|2�  =  �  |ηB|2�  = bK, and  �  |ηAB|2�  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  so that the contribution from synchronous replication be-  comes comparable to death-induced noise for γ < 2/√c,  or when w < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='55c−1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' As the expression is rather in-  sensitive to the value of c, we can conclude that devia-  tions from the Poisson, asynchronous replication should  already be visible even for relatively large values of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  CONCLUSION  We have revisited a stochastic two-species model of  the predator-prey type [2], which exhibits oscillations  for a wide range of parameters of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We have  modified the model so that both species replicate quasi-  synchronously, with doubling times drawn from a narrow  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We have shown that coloured demographic  noise generated by this process leads to much stronger os-  cillations than the Poisson process of replication assumed  in earlier works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Coloured noise has been shown to affect  population dynamics in single-species models [25];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' here,  we not only derive its spectrum from the underlying mi-  croscopic dynamics, but also show how it affects more  complex models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Our result, while obtained for an abstract mathemati-  cal model, may be relevant for real biological populations,  in particular for microorganism which often replicate in  quasi-discrete generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We expect to see the same  behaviour in other models that exhibit quasi-cycles [26–  31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  The phenomenon of coloured noise amplification may  13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  0  500  1000  1500  2000  2500  f  |ΔN  \uf02d  A  2  1/2=|KyA  2  \uf02d  1/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5  0  1000  2000  3000  4000  5000  6000  7000  f  |ΔN  \uf02d  A  2  1/2=|KyA  2  \uf02d  1/2  Figure 11:  Plot of the spectrum of yA versus frequency  f = ω/(2π), for the non-Poisson model with the replication  frequency slightly detuned (b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='9, top) and in resonance  (b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='7, bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Blue = simulation with K = 105, w =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='08, and the remaining parameters as in S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Red = ana-  lytic expression 80 with the noise terms from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' (82), and  γ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='579w2 (here T = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='20  0  2 × 106  4 × 106  6 × 106  8 × 106  1 × 107  w  <|ΔNA  2|>  Figure 12: Plot of  �  |∆NA(t)|2�  versus w, for the non-Poisson  model with replication frequency b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Blue = simulation  with K = 105, and the remaining parameters as in S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Red  = analytic expression 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  be further augmented in situations in which oscillations  in the population abundance become synchronized with  reproductive cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' We leave this interesting problem  for future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='  Acknowledgments  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' acknowledges funding under Dioscuri, a pro-  gramme initiated by the Max Planck Society, jointly  managed with the National Science Centre in Poland,  and mutually funded by Polish Ministry of Science and  Higher Education and German Federal Ministry of Edu-  cation and Research (UMO-2019/02/H/NZ6/00003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Martini, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' Lu, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FQT4oBgHgl3EQfSjYz/content/2301.13290v1.pdf'}
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+Molecular Language Model as Multi-task Generator
+Yin Fang 1 2 3 Ningyu Zhang 1 4 2 Zhuo Chen 1 2 Xiaohui Fan 3 5 Huajun Chen 1 2
+Abstract
+Molecule generation with desired properties has
+grown immensely in popularity by disruptively
+changing the way scientists design molecular
+structures and providing support for chemical and
+materials design. However, despite the promis-
+ing outcome, previous machine learning-based
+deep generative models suffer from a reliance on
+complex, task-specific fine-tuning, limited dimen-
+sional latent spaces, or the quality of expert rules.
+In this work, we propose MOLGEN, a pre-trained
+molecular language model that effectively learns
+and shares knowledge across multiple generation
+tasks and domains.
+Specifically, we pre-train
+MOLGEN with the chemical language SELFIES
+on more than 100 million unlabelled molecules.
+We further propose multi-task molecular prefix
+tuning across several molecular generation tasks
+and different molecular domains (synthetic &
+natural products) with a self-feedback mecha-
+nism. Extensive experiments show that MOLGEN
+can obtain superior performances on well-known
+molecular generation benchmark datasets. The
+further analysis illustrates that MOLGEN can ac-
+curately capture the distribution of molecules, im-
+plicitly learn their structural characteristics, and
+efficiently explore the chemical space with the
+guidance of multi-task molecular prefix tuning1.
+1College of Computer Science and Technology, Zhejiang
+University 2Alibaba-Zhejiang University Joint Research Insti-
+tute of Frontier Technologies 3Pharmaceutical Informatics Insti-
+tute, College of Pharmaceutical Sciences, Zhejiang University
+4School of Software Technology, Zhejiang University 5Future
+Health Laboratory, Innovation Center of Yangtze River Delta,
+Zhejiang University. Correspondence to: Ningyu Zhang <zhangn-
+ingyu@zju.edu.cn>, Huajun Chen <huajunsir@zju.edu.cn>.
+Work in progress.
+1Codes, datasets, and the pre-trained model will be available in
+https://github.com/zjunlp/MolGen.
+1. Introduction
+Molecular generation – synthesizing and designing novel
+molecules with desirable properties2 – holds an important
+place in chemical science, with numerous applications in
+drug discovery (Wang et al., 2022a). The task of molec-
+ular generation is rather challenging, whereas the chemi-
+cal space of molecules, being both discrete and vast with
+an estimated size of 1033, renders exhaustive searches in-
+feasible (Polishchuk et al., 2013). Early, deep generative
+models (Jin et al., 2020; Zang & Wang, 2020; Luo et al.,
+2021; Shi et al., 2020b) have been proposed as one of the
+most promising tools for exploring the broader synthetically
+accessible chemical space. Their ability to automatically
+generate chemically valid and similar molecular structures
+has proven invaluable for tasks such as the inverse design of
+functional compounds (Flam-Shepherd et al., 2022).
+Current deep generative models typically involve initial
+training of an unconditional generative model through a
+large dataset of existing molecules, followed by the employ-
+ment of additional reward functions (Cao & Kipf, 2018;
+Popova et al., 2018; You et al., 2018; Popova et al., 2019;
+Shi et al., 2020b; Zang & Wang, 2020) or property pre-
+dictors (Simonovsky & Komodakis, 2018; Jin et al., 2019;
+G´omez-Bombarelli et al., 2018; Liu et al., 2018) to guide
+the synthesis of new molecules with desired properties. De-
+spite their widespread use, the performance of these ap-
+proaches is limited by their reliance on heavy task-specific
+fine-tuning (Xie et al., 2021), fixed-dimensional latent gen-
+eration space (Wang et al., 2022b), and expert-provided
+generation rules (Sun et al., 2022), which impede efficient
+exploration of the broader chemical space.
+Recent advancements in language models have demon-
+strated their potential for understanding complex molec-
+ular distributions (Flam-Shepherd et al., 2022). Researchers
+have begun integrating SMILES (Weininger, 1988), a linear
+string notation for describing molecular structures, with pre-
+trained language models to gain a more comprehensive un-
+derstanding of the underlying molecular structures and their
+representations (Irwin et al., 2022). However, the brittle-
+ness of SMILES may lead to a high proportion of generated
+SMILES strings being chemically invalid. These strings can
+2This paper focuses on generating 2D molecules based on given
+molecules, also known as molecular optimization.
+arXiv:2301.11259v1  [cs.LG]  26 Jan 2023
+
+Molecular Language Model as Multi-task Generator
+either be syntactically incorrect (e.g., not corresponding to
+molecular graphs) or violate fundamental chemical princi-
+ples (e.g., exceeding the maximum number of inter-atomic
+valence bonds). Another defect is that current language
+model-based approaches rely on task-specific fine-tuning
+with different optimization objectives, making each progress
+incompatible and hindering knowledge sharing among mul-
+tiple downstream tasks. Besides, almost all previous studies
+have primarily focused on one single domain of synthetic
+molecules, neglecting natural products. Notably, natural
+products, characterized by enormous scaffold diversity and
+structural complexity, exhibit a distinct distribution com-
+pared to synthetic molecules and confer additional oppor-
+tunities and challenges for numerous molecular generation
+applications such as drug discovery (Atanasov et al., 2021).
+In this paper, we propose MOLGEN, a novel pre-trained
+molecular language model for molecular generation. Specif-
+ically, to effectively explore the chemical universe, Firstly,
+we pre-train a molecular language model based on the chem-
+ical language SELFIES (Krenn et al., 2020), where each
+combination of symbols in its alphabet maps to a chemically
+valid graph. By combining bidirectional and auto-regressive
+Transformers (Vaswani et al., 2017), the pre-trained molecu-
+lar language model can learn intrinsic structural regularities
+by mapping corrupted SELFIES to their original forms.
+Then, to break down the boundaries between diverse molec-
+ular tasks and domains (synthetic & natural products), we
+introduce Multi-task molecular Prefix Tuning (MPT). It al-
+lows the model to acquire general features that can be trans-
+ferred to various domains, thus simplifying the process of
+adapting to new tasks. Moreover, by incorporating a simple
+self-feedback paradigm that assigns probabilities to candi-
+date molecules based on their property scores, MOLGEN is
+incentivized to generate molecules with desired properties.
+We conduct comprehensive experiments on both synthetic
+and natural product molecular datasets. The considerable
+performance gains on molecular generation benchmarks
+demonstrate the capability of MOLGEN to generate chemi-
+cally valid molecules and effectively navigate the chemical
+space. We further investigate MOLGEN’s understanding
+of molecular distributions, recognition of crucial substruc-
+tures, and the effect of MPT. Our findings confirm that
+MOLGEN preserves generation diversity while mastering
+the distribution and structural characteristics of molecules.
+Additionally, MPT improves the performance on various
+tasks, owing to its ability of knowledge sharing.
+2. Related Work
+Molecular generation.
+The past decade has seen signifi-
+cant progress in molecule generation, particularly in deep
+generative models (G´omez-Bombarelli et al., 2018). Prior
+works typically formulate this task as a sequence genera-
+tion problem. Specifically, (Kusner et al., 2017; G´omez-
+Bombarelli et al., 2018; Segler et al., 2018; Kwon et al.,
+2021) start by generating SMILES strings due to the abun-
+dance of sequence modeling tools such as recurrent neural
+networks, dilated convolutions, and attention mechanisms.
+Additionally, as molecules can be naturally viewed as two-
+dimensional graphs, various molecular graph-based genera-
+tive methods have emerged (Ma et al., 2018; Simonovsky &
+Komodakis, 2018; Jin et al., 2020; Zang & Wang, 2020; Luo
+et al., 2021). Based on these two forms of representation,
+existing approaches can be categorized into four venues.
+Bayesian Optimization (BO) (G´omez-Bombarelli et al.,
+2018; Jin et al., 2018; Winter et al., 2019) learns a continu-
+ous latent space of molecules and optimizes the target prop-
+erties by navigating through this space. However, BO often
+requires a long evaluation time to optimize the objective
+function (Du et al., 2022). Reinforcement Learning (RL)-
+based approaches aim to train an agent to select actions,
+such as adding substructures, in an explicit chemical space
+to optimize properties (Cao & Kipf, 2018; Popova et al.,
+2018; You et al., 2018; Popova et al., 2019; Shi et al., 2020b;
+Zang & Wang, 2020), but these methods are often difficult to
+optimize due to high variance (Xie et al., 2021). Variational
+Auto-Encoder (VAE) can be another solution (Simonovsky
+& Komodakis, 2018; Jin et al., 2019; G´omez-Bombarelli
+et al., 2018; Liu et al., 2018), but their performance is highly
+dependent on the quality of the fixed-dimensional latent
+space. Lastly, Genetic Algorithms (GA) (Jensen, 2019; Ahn
+et al., 2020; Nigam et al., 2020) use predefined mutation
+and crossover rules from reference compound libraries or
+domain expertise to generate molecules. While flexible, it
+can be challenging to acquire the necessary prior knowledge
+and rules, limiting the search process’s efficiency.
+Pre-trained language models.
+Pre-trained language
+models (PLMs) have waved the community as fundamen-
+tal infrastructure (Kenton & Toutanova, 2019; Lewis et al.,
+2020; Brown et al., 2020). Intuitively, PLMs are a promis-
+ing solution for molecular generation, and several pioneers
+have started to leverage SMILES-based language models
+with promising performance (Bagal et al., 2021; Irwin et al.,
+2022; Flam-Shepherd et al., 2022; Ross et al., 2022; Chilin-
+garyan et al., 2022; Pan, 2023). To date, Chemformer
+(Irwin et al., 2022) is the only publicly available PLM that
+can address molecular generation tasks. Note that similar
+to human language, molecular language also has syntax,
+and just as the syntax of natural languages imposes a gram-
+matical structure that allows words to relate to each other
+in specific ways, biological symbols also combine in spe-
+cific structural manners. However, SMILES, as a traditional
+molecular language, contains complex and restrictive gram-
+mars, which results in most sequences over the appropriate
+character set not belonging to well-defined molecules. Com-
+
+Molecular Language Model as Multi-task Generator
+pared with Chemformer and other previous approaches,
+MOLGEN has two main differences: (1) We pre-train the
+language model with SELFIES, a chemical language repre-
+sentation of molecules that is 100% robust to both syntactic
+and semantic issues. (2) We formulate downstream tasks
+with multi-task molecular prefix tuning to share knowledge
+across tasks and domains (synthetic & natural products).
+3. Methodology
+As shown in Figure 2, we illustrate the general framework
+of MOLGEN, which is based on pre-training a molecular
+language model using SELFIES. We introduce the technical
+details of problem formulation (§3.1), generative molecular
+pre-training (§3.2), and multi-task molecular prefix tuning
+(§3.3) in the following sections.
+3.1. Problem Formulation and Background
+We focus on three standard molecular generation setting in
+this paper. Concretely, define M = {Mi}n
+i=1 as a set of
+molecules, A(M) : M → R as a property score function,
+and sim(M, M ′) ∈ [0, 1] as a similarity function measures
+the similarity between the molecules M and M ′. Then we
+formulate the molecular generation tasks as follows:
+• Distribution learning: learning a generative model
+pθ(·) from M, such that the model can sample a valid
+molecule M with a high probability pθ(M).
+• Targeted molecule discovery:
+learning a targeted
+molecule discovery model pθ(·) to maximize the ex-
+pected property score EM∼pθ[A(M)].
+• Constrained molecular property optimization: learn-
+ing a constrained optimization model pθ(·|M), to
+maximize EM ′|M∼pθ[A(M ′)] while ensuring that
+sim(M, M ′) > δ is satisfied. Here M ′ is the optimized
+molecule based on M, and δ is a similarity threshold.
+To address the task of molecular generation with PLMs,
+the current study (Irwin et al., 2022) follows BART (Lewis
+et al., 2020), a large pre-trained Seq2seq language model
+standard in the literature, to corrupt SMILES and then opti-
+mizing a reconstruction loss for pre-training. Specifically,
+the pre-training includes random sampling tokens within
+the original SMILES string S = {s1, · · · , sj, · · · , sl} and
+replacing them with a special token [MASK], encoding the
+corrupted SMILES using a bidirectional model, and then
+calculating the likelihood of S with a left-to-right autoregres-
+sive decoder. Formally, we have the cross-entropy between
+the decoder’s output, and the original input constitutes the
+(a) SMILES maps to a wide range of structures, including
+molecules, non-molecular graphs, and invalid (non-graph) struc-
+tures. On the other hand, SELFIES is a surjective representation
+that maps exclusively to molecular graphs.
+(b) Each token in SELFIES points to a specific structure or refer-
+ence, avoiding the presence of meaningless substrings caused by
+unreasonable division in SMILES (e.g., “)” and “/”).
+(c) Reconstruct the SELFIES from the corrupted one by optimizing
+the negative log-likelihood object.
+Figure 1. Generative molecular pre-training with SELFIES.
+reconstruction loss as:
+Lce(S) = −
+l
+�
+j=1
+�
+s
+ptrue (s|S, S<j) log pθ (s|S, S<j; θ) ,
+where
+S<j
+denotes
+the
+partitial
+original
+sequence
+{s0, · · · , sj−1} and s0 is a pre-defined start token <s>.
+ptrue is a one-hot distribution under the standard maximum
+likelihood estimation:
+ptrue (s|S, S<j) =
+�
+1,
+s = sj
+0,
+s ̸= sj
+.
+3.2. Generative Molecular Pre-training with SELFIES
+SMILES and SELFIES are two molecular languages that
+map a string of tokens to a molecular graph. As illustrated
+in Fig 1(a), in SMILES, molecules are represented as chains
+of atoms (main string in green) with branches (red) de-
+fined within parentheses and ring closures (blue) indicated
+by two matching numbers. Although SMILES has been a
+workhorse in cheminformatics for the past few decades, it is
+
+Invalid
+Structure
+Graphs
+Molecules
+CC(C)CC1=CC=C2C(C1)CCO2
+[C][C][Branch1 ][C][C]C][C][=C][C][=C][C][Branch1][Ring2][C][Ring1][=Branch1][C][C][O][Ring1][=Branch1]C
+N
+[=Ring1]
+[/C]]
+[=O]
+[C]
+[C@@H1][C][=C] [C][=C][C] [=C][Ring]][=Branch1]
+Bidirectional
+Autoregressive
+Encoder
+Decoder
+[C] [=C] 
+<mask>
+[C] 
+<mask>
+[=Branch1]
+<s>
+[C][=C][C][=C]  [C] [=C][Ring]Molecular Language Model as Multi-task Generator
+Figure 2. Multi-task molecular Prefix Tuning (MPT), which can share knowledge across tasks and domains (synthetic & natural products).
+inherently fragile as there is no mechanism to guarantee that
+molecular strings can be valid regarding syntax and physical
+principles. This means that SMILES is a surjective repre-
+sentation from strings to structures that include molecular
+graphs but also non-molecular (semantically invalid) graphs
+and other structures which cannot be interpreted as graphs
+(syntactically invalid) (Krenn et al., 2020).
+This weakness has been addressed by a 100% robust molec-
+ular language SELFIES, which is a surjective mapping from
+strings to molecular graphs. It means that SELFIES can-
+not produce invalid molecules since every combination of
+symbols in the SELFIES alphabet maps to a chemically
+valid graph (Krenn et al., 2022). Both rings and branches
+in SELFIES are defined at a single location, eliminating
+syntactic invalidity caused by unbalanced parentheses or
+ring identifiers. Special symbols start a brand ([Branch1])
+or ring ([Ring1]), and the subsequent token in the string de-
+fines the length of the branch or ring according to its specific
+derivation rules (Krenn et al., 2022).
+To pre-train the molecular language model, we collect more
+than 100 million unlabelled molecules as our pre-training
+corpus by randomly selecting from the publicly accessible
+ZINC-15 dataset (Sterling & Irwin, 2015) (the same pre-
+training corpus of Chemformer (Irwin et al., 2022) for a
+fair comparison). The selection process strictly follows the
+constraints outlined in Irwin et al. (2022), ensuring that the
+molecules are reactive, available for purchase, have a molec-
+ular weight of ≤ 500 Daltons, and a LogP (octanol-water
+partition coefficient) of ≤ 5. The molecules are then con-
+verted to SELFIES strings and segmented into tokens. The
+use of SELFIES eliminates the need for training a specific
+tokenizer to identify frequently occurring substring patterns.
+Instead, we construct an alphabet from the dataset of SELF-
+IES strings, as depicted in Fig 1(b), effectively avoiding
+the generation of meaningless tokens caused by imprecise
+tokenization. We then conduct token masking on the input
+SELFIES strings S∗ = {s∗
+1, · · · , s∗
+l } and train a Seq2Seq
+model to reconstruct the original SELFIES by optimizing
+the negative log-likelihood Lce(S∗), as shown in Fig 1(c).
+3.3. Multi-task Molecular Prefix Tuning
+In current task-specific fine-tuning strategies to leverage
+PLMs, the inputs for molecular generation tasks contain
+diverse and structured knowledge from various molecular
+domains, making it difficult to effectively learn shared pa-
+rameters. To address this challenge, we propose to break
+down the boundaries between tasks and domains through
+Multi-task molecular Prefix Tuning (MPT). Specifically, we
+prepend l tunable prefix vectors, shared between all gener-
+ation tasks across domains, to the keys and values of the
+multi-head attention at every layer. Let T = {t}n
+t=1 denote
+the task type on one molecular domain; the attention score
+output on each head for the task t can be formulated as:
+headt = Attn
+�
+xtW t
+q, [P k, XtW t
+k], [P v, XtW t
+v]
+�
+,
+where the sequence of m vectors Xt ∈ Rm×d denotes the
+input to a Transformer layer, x ∈ Rd is a query vector,
+and W q, W k, W v ∈ Rd×dh are project matrics that map
+inputs to queries, keys, and values. Two sets of l prefix
+vectors P k, P v ∈ Rl×d, are concatenated with the original
+attention key and value in order to capture shared knowledge
+across all generation tasks. From an alternative view, the
+
+Task 1 : Distribution Learning
+x L
+Synthetic molecules
+Add & Layer Norm
+Feed Forward
+Add & Layer Norm 
+Task 2 : Targeted Molecule Discovery
+Attention
+Qt
+PhKt
+Natural product molecules
+Low
+Property Score
+High
+Pk
+wa
+w
+P
+Task 3 : Constrained Molecular Property Optimization
+Hidden States
+Prefix shared between
+tasks and domains
+sim=0.81
+Multi-Head
+Task t = 1
+Task t = 2
+-8.21
+Property Score
+-2.92
+Task t = 3Molecular Language Model as Multi-task Generator
+computation of headt can be equivalent becomes:
+headt = softmax
+�
+xtW t
+q[P k, XtW t
+k]⊤� �
+P v
+XtW t
+v
+�
+= softmax
+�
+xtW t
+q
+�
+P ⊤
+k
+(W t
+k)⊤(Xt)⊤
+�� �
+P v
+XtW t
+v
+�
+= λ(x)softmax
+�
+xtW t
+qP ⊤
+k
+�
+P v
++ (1 − λ(x))softmax
+�
+xW t
+q(W t
+k)⊤(Xt)⊤�
+XtW t
+v
+= λ(x)
+Attn
+�
+xtW t
+q, P k, P v
+�
+�
+��
+�
+attention of prefix shared between tasks
++ (1 − λ(x)) Attn
+�
+xtW t
+q, XtW t
+k, XtW t
+v
+�
+�
+��
+�
+original attention
+,
+where λ(x) is a scalar whose two terms in the denominator
+represent the sum of prefixes attention weights and original
+sequence normalized attention weights, respectively:
+λ(x) =
+�
+i exp(xtW t
+qP ⊤
+k )i
+�
+i exp(xtW t
+qP ⊤
+k )i + �
+j exp(xtW t
+q(W t
+k)⊤(Xt)⊤)j
+.
+This modification to the original head attention through
+linear interpolation with shared prefixes enables knowledge
+sharing across multiple domains and tasks.
+Coordinating generation candidate with self-feedback.
+For molecular generation tasks such as targeted molecule
+discovery and constrained optimization, the target is to as-
+sign higher probabilities to molecules with desired prop-
+erties during inference, which is non-trivial for vanilla au-
+toregressive generation. To this end, we introduce a simple
+mechanism dubbed coordinating generation candidate with
+self-feedback inspired by fine-tuning with human feedback
+(Ouyang et al., 2022) to bridge this gap. We hypothesize that
+the probability of a generated candidate molecule should be
+well-correlated with its property score as evaluated by an
+automatic metric A. That is, for non-reference candidate
+SELFIES ˆ
+S∗, we assume that:
+ptrue(S∗
+i |S∗) > ptrue(S∗
+j |S∗),
+∀S∗
+i , S∗
+j ∈ ˆ
+S∗
+A(S∗
+i ) > A(S∗
+j ),
+where S∗
+i and S∗
+j are two different candidate SELFIES, and
+S∗ = {s∗
+1, · · · , s∗
+l } is the reference SELFIES. We utilize
+our model to generate various candidate SELFIES ˆ
+S∗, each
+with distinct property scores, and then encourage the model
+to assign higher estimated probabilities to better candidates
+by MPT with a rank loss (Liu et al., 2022):
+Lrank(S∗) =
+�
+i
+�
+j>i
+max
+�
+0, f
+�
+S∗
+j
+�
+− f (S∗
+i ) + λij
+�
+,
+where A(S∗
+i ) > A(S∗
+j ), ∀i, j, i < j. λij = (j − i) ∗ λ is
+the margin multiplied by the difference in rank between the
+candidates, and f(S∗
+i ) is the estimated log-probability:
+f(S∗) =
+l
+�
+t=1
+log pθ (s∗
+t | S∗, S∗
+<t; θ) .
+Intuitively, rank loss occurs when candidates with low prop-
+erty scores have higher estimated probabilities than those
+with high property scores. To balance the trade-off between
+desired properties and similarity to original molecules, we
+combine the rank and cross-entropy loss as:
+L = Lce + γLrank,
+where γ is the weight of the rank loss. Here we utilize
+label smoothing (Szegedy et al., 2016) to modify the target
+distribution of ptrue in Lrank to a “soft” label by assigning
+probability mass β to other tokens:
+ptrue
+�
+s∗|S∗, S∗
+<j
+�
+=
+�
+1 − β,
+s∗ = s∗
+j
+β
+N−1,
+s∗ ̸= s∗
+j
+,
+where N is the length of the dictionary. This allows for
+both sequence-level coordination and token-level prediction
+accuracy to be taken into account, enabling the model to
+navigate the orientation of generated molecules effectively.
+4. Experiments
+4.1. Experimental Setup
+Evaluation Tasks.
+We conduct experiments by compar-
+ing the proposed model, MOLGEN, with state-of-the-art
+approaches on three tasks. Distribution learning evaluates
+the model’s capacity to learn the data distribution and gen-
+erate diverse and realistic molecules. Targeted molecule
+discovery focuses on generating novel molecules with opti-
+mized chemical properties. Constrained molecular prop-
+erty optimization aims at modifying the given molecule
+to improve desired properties while satisfying a similarity
+constraint. In addition to the three major tasks mentioned
+above, MOLGEN can be applied to other tasks, as detailed
+in Appendix A.
+Dataset.
+For downstream generation tasks, the molecular
+data we use mainly falls into two categories: synthetic and
+natural product datasets. For synthetic datasets, we use the
+benchmark ZINC250k dataset (Irwin et al., 2012) containing
+about 250,000 drug-like molecules for the optimization task,
+and the ZINC-based MOSES dataset (Polykovskiy et al.,
+2018) containing 1,936,962 molecules for the generation
+task. Moreover, we collect 30,926 natural product com-
+pounds from the NPASS database (Zhao et al., 2022) and
+conduct all tasks on this dataset, detailed in Appendix B.
+
+Molecular Language Model as Multi-task Generator
+Table 1. Generation performance on the Synthetic MOSES dataset.
+METHOD
+VALIDITY(↑)
+FRAG(↑)
+SCAF(↑)
+SNN(↑)
+INTDIV(↑)
+FCD(↓)
+NOVELTY(↑)
+AAE
+0.9368
+0.9910
+0.9022
+0.6081
+0.8557
+0.5555
+0.7931
+LATENTGAN
+0.8966
+0.9986
+0.8867
+0.5132
+0.8565
+0.2968
+0.9498
+CHARRNN
+0.9748
+0.9998
+0.9242
+0.6015
+0.8562
+0.0732
+0.8419
+VAE
+0.9767
+0.9994
+0.9386
+0.6257
+0.8558
+0.0990
+0.6949
+JTN-VAE
+1.0000
+0.9965
+0.8964
+0.5477
+0.8551
+0.3954
+0.9143
+LIMO
+1.0000
+0.9562
+0.1073
+0.6125
+0.8544
+0.1532
+0.8956
+CHEMFORMER-LARGE
+0.9843
+0.9889
+0.9248
+0.5622
+0.8553
+0.0061
+0.9581
+MOLGEN
+1.0000
+0.9999
+0.9999
+0.9996
+0.8567
+0.0015
+1.0000
+Table 2. Generation performance on the Natural Product dataset.
+METHOD
+VALIDITY(↑)
+FRAG(↑)
+SCAF(↑)
+SNN(↑)
+INTDIV(↑)
+FCD(↓)
+NOVELTY(↑)
+AAE
+0.0082
+0.9687
+0.2638
+0.3680
+0.8704
+4.1089
+0.9943
+LATENTGAN
+0.9225
+0.2771
+0.0884
+0.5321
+0.6009
+45.5327
+0.9949
+CHARRNN
+0.7351
+0.8816
+0.5212
+0.4179
+0.8756
+2.2115
+0.9792
+VAE
+0.2627
+0.8840
+0.4563
+0.3950
+0.8719
+4.3175
+0.9912
+LIMO
+1.0000
+0.7242
+0.0005
+0.3416
+0.7726
+31.8366
+0.9962
+CHEMFORMER-LARGE
+0.9825
+0.9826
+0.4126
+0.5875
+0.8650
+0.8346
+0.9947
+MOLGEN
+1.0000
+0.9994
+0.8404
+0.8148
+0.8878
+0.6519
+0.9987
+4.2. Main Results
+Distribution Learning.
+Distribution learning models are
+mainly used for building virtual libraries for computer-
+assisted drug discovery (van Hilten et al., 2019). Relying on
+a set of manually or automatically selected compounds, dis-
+tribution learning models can generate a larger dataset that
+preserves implicit rules from the reference dataset. In this
+section, we adopt seven widely used metrics, detailed in Ap-
+pendix E, to evaluate the ability of MOLGEN to model real
+molecules. We generate 10,000 molecules on the synthetic
+dataset following the setting in Polykovskiy et al. (2018)
+and 80,000 molecules on the natural product dataset.
+Table 1 and 2 offer the following observations: (1) MOL-
+GEN holds a perfect ability to produce valid molecules with-
+out additional valency check like JTN-VAE (Jin et al., 2018).
+Since LIMO (Eckmann et al., 2022) is also modeled based
+on SELFIES, the molecules it generates also have 100%
+validity. However, due to the complexity of internal struc-
+tures and scaffolds in natural products, most models fail to
+produce valid molecules on the natural product dataset. The
+performance of Chemformer (Irwin et al., 2022) is better
+than others, largely because the model learns the SMILES
+grammar during large-scale pre-training. (2) On synthetic
+datasets, almost all models generate molecules with similar
+fragments and scaffolds to the reference molecules. How-
+ever, MOLGEN excels at capturing substructure distribution
+in natural products. (3) MOLGEN achieves the highest SNN
+and lowest FCD scores, indicating that it effectively masters
+the statistics of the dataset in terms of biological properties
+and topology structure. Additionally, the good performance
+Table 3. Comparison of QED and p-logP maximization methods
+on the synthetic ZINC dataset. As the p-logP score can increase
+linearly with molecule length, we divide baselines into upper and
+lower groups for a fair comparison. The first group of models
+has a limit on the output length (The maximum molecule size of
+ZINC250k), while the second has no limit.
+P-LOGP
+QED
+METHOD
+1ST
+2ND
+3RD
+1ST
+2ND
+3RD
+ZINC
+4.52
+4.30
+4.23
+0.948
+0.948
+0.948
+GCPN
+7.98
+7.85
+7.80
+0.948
+0.947
+0.946
+MOLDQN
+11.80
+11.80
+11.80
+0.948
+0.943
+0.943
+LIMO
+10.50
+9.69
+9.60
+0.947
+0.946
+0.945
+MOLGEN
+30.51
+28.98
+28.95
+0.948
+0.948
+0.948
+JT-VAE
+5.30
+4.93
+4.49
+0.925
+0.911
+0.910
+GRAPHAF
+12.23
+11.29
+11.05
+0.948
+0.948
+0.947
+GRAPHDF
+13.70
+13.18
+13.17
+0.948
+0.948
+0.948
+MARS
+44.99
+44.32
+43.81
+0.948
+0.948
+0.948
+MOLGEN
+80.30
+74.70
+69.85
+0.948
+0.948
+0.948
+of IntDiv and Novelty metrics suggests that MOLGEN is
+favorable for discovering new chemical structures and ex-
+ploring unknown chemical space without overfitting. We
+provide a visual comparison of the training set and generated
+molecules in Appendix F.1.
+Targeted Molecule Discovery.
+In this task, we aim to
+generate molecules with high property scores using penal-
+ized logP (p-logP) (Jin et al., 2018) and QED (Bickerton
+et al., 2012) as the optimization targets. P-logP refers to
+the logP score penalized by ring size and synthetic acces-
+sibility, while QED refers to the quantitative estimation of
+drug-likeness. Both scores are calculated by empirical pre-
+
+Molecular Language Model as Multi-task Generator
+diction models, and we utilize the script based on the official
+implementation of Shi et al. (2020b) to ensure comparabil-
+ity. Unlike the distribution learning task that only employs
+the basic MPT strategy, for this task, we enhance the MPT
+method by incorporating the self-feedback paradigm to more
+effectively guide the generation direction of the molecules.
+In Table 3, we present the top-3 property scores of molecules
+generated by each model on the synthetic ZINC dataset, fol-
+lowing the convention of previous works (Shi et al., 2020b;
+Eckmann et al., 2022). The first line provides a summary
+of the top-3 property scores of the ZINC250K dataset. For
+a fair comparison, we divide the baselines into two groups
+based on the output length. The models in the first group
+have a maximum output length limit, while the second group
+has no such restriction. As MOLGEN supports variable-
+length output, we compare it under these two settings.
+From Table 3, we notice that MOLGEN achieves better per-
+formance than all baselines for p-logP score and obtains
+comparable results for QED, suggesting that the combi-
+nation of multi-task molecular prefix tuning and the self-
+feedback paradigm effectively encourages the molecular
+PLM to assign higher probabilities to candidate molecules
+with desired properties, providing a significant advantage
+for optimizing molecules. However, it should be noted that
+the p-logP metric heavily depend on molecule length (Xie
+et al., 2021; Eckmann et al., 2022), and while we obtain
+the highest property score in the unlimited length setting
+for a fair comparison, large molecules such as these are
+unrealistic for practical drug discovery. Moreover, QED is
+known to have boundary effects around its maximum score
+of 0.948 (Zhou et al., 2019), and drugs with a QED score
+above 0.9 are usually very rare. Despite these limitations,
+the good performance of MOLGEN demonstrates its strong
+ability to explore the complex chemical space and generate
+diverse molecular graphs.
+For natural products, we use shared prefixes with synthetic
+molecules to facilitate knowledge transfer across different
+domains. The results, shown in Appendix F.2 and Figure 10,
+demonstrate MOLGEN’s ability to generate molecules with
+high QED and p-logP scores. As per our reference nat-
+ural product dataset, only 0.701% of molecules have a
+QED score greater than 0.9, with the highest score being
+0.9439 (as seen in Appendix B Table 5). Thus, achiev-
+ing the maximum score of 0.9478 is sufficient for practi-
+cal drug discovery purposes. Additionally, MOLGEN can
+generate molecules with a p-logP score of 54.33, which is
+significantly higher than the maximum score of 17.69 in the
+reference set, further demonstrating MOLGEN’s strong per-
+formance in generating molecules with desired properties.
+Constrained Molecular Property Optimization.
+For
+this task, it aims to improve the p-logP score of a given
+Table 4. The average p-logP improvements of generated molecules
+over inputs under similarity constraint δ = {0.6, 0.4}.
+IMPROVEMENT
+METHOD
+δ = 0.6
+δ = 0.4
+JT-VAE
+0.28±0.79
+1.03±1.39
+GCPN
+0.79±0.63
+2.49±1.30
+MOLDQN
+1.86±1.21
+3.37±1.62
+VSEQ2SEQ
+2.33±1.17
+3.37±1.75
+VJTNN
+2.33±1.24
+3.55±1.67
+GA
+3.44±1.09
+5.93±1.41
+GRAPHAF
+4.98±6.49
+8.21±6.51
+GRAPHDF
+4.51±5.80
+9.19±6.43
+LIMO
+1.80±2.00
+3.60±2.30
+CHEMFORMER-LARGE
+2.48±0.89
+3.56±1.32
+MOLGEN
+12.08±0.82
+12.35±1.21
+molecule while maintaining a similarity of at least δ to the
+original molecule. Following previous works (Jin et al.,
+2018; Shi et al., 2020b; Luo et al., 2021; Eckmann et al.,
+2022), we optimize the p-logP scores of 800 molecules
+from the synthetic ZINC250K dataset with the lowest
+scores and use the Tanimoto similarity with Morgan fin-
+gerprints (Rogers & Hahn, 2010) to measure the similarity
+between the optimized molecules and input.
+We report the mean and standard deviation of the success
+rate under each similarity constraint, which is the percent-
+age of molecules that can be successfully optimized over
+800 molecules.
+As summarized in Table 4, MOLGEN
+yields better results under the two similarity constraints
+compared to baseline models, validating its ability to search
+for molecules with higher property scores in the nearby
+chemical space. Surprisingly, MOLGEN outperforms mod-
+els that rely on additional reward functions (GCPN (You
+et al., 2018), MolDQN (Zhou et al., 2019), GraphAF (Shi
+et al., 2020b), GraphDF (Luo et al., 2021)) or property pre-
+dictors (JT-VAE (Jin et al., 2018), VJTNN (Jin et al., 2019),
+LIMO (Eckmann et al., 2022)), further emphasizing the
+power of the self-feedback paradigm in MPT.
+Following the same setting, we conduct experiments on the
+QED score for synthetic molecules and on two molecular
+properties for natural products using shared prefixes. Fig-
+ure 11 and 12 in Appendix F.3 visually demonstrate the
+results of constrained optimization examples. These exam-
+ples show that despite the complex molecular structure and
+long length of natural products, MOLGEN can improve the
+property score while maintaining a certain level of similarity
+between the input and the modified molecule. Additionally,
+MOLGEN is able to maintain the diversity of generated
+molecules while exploring the chemical space in close prox-
+imity to the input molecule.
+
+Molecular Language Model as Multi-task Generator
+Figure 3. Comparison of molecules generated by different models
+on properties, atom counts, ring counts, molecular weights, and
+Bertz complexity (S-PLM stands for SMILES-based PLM).
+4.3. A Closer Look at MOLGEN
+MOLGEN
+learns
+complex
+molecular
+distributions.
+First of all, we evaluate the performance of the pre-trained
+MOLGEN by comparing it with the most popular deep gen-
+erative Graph-based (Jin et al., 2018), VAE-based (Blaschke
+et al., 2018), and SMILES-based language models (Irwin
+et al., 2022) in terms of the distribution of properties and
+structures of the generated molecules. The training and gen-
+eration settings of all models are the same as the distribution
+learning task on the synthetic MOSES dataset.
+As shown in the 2D histograms of p-logP and QED scores in
+Figure 3, VAE-based and SMILES-based PLMs tend to pro-
+duce molecules with larger p-logP and QED scores than the
+training data. In comparison, the Graph-based model learns
+the main mode of p-logP in the training data, while MOL-
+GEN performs a bit better - similar results are observed for
+QED. Furthermore, in terms of molecular topology, PLMs
+outperform the others in perceiving atom numbers, ring
+numbers, and molecular weights, with MOLGEN producing
+a slightly closer match to the training distribution. All the
+models are capable of picking up on molecular Bertz com-
+plexity. Overall, PLMs, especially MOLGEN, can capture
+the properties and structural characteristics of the training
+molecules while preserving the diversity of generation.
+MOLGEN captures molecular substructures implicitly.
+In this section, we investigate the capability of PLMs to
+capture essential substructures when using different molec-
+ular languages (i.e., SMILES and SELFIES). To provide an
+intuitive understanding, we visualize the attention weights
+of each token in the same molecule. Specifically, we ex-
+tract and normalize the attention weights from the last self-
+attention layer, as illustrated in Figure 4.
+Figure 4. Visualization of learned attention maps for the same
+molecule by different PLM models. Note that the models we
+compared are of a similar parameter scale.
+The attention map produced by MOLGEN shows that the
+fluoro group has higher attention weights, followed by the
+phenyl and hydroxyl groups. This is because the fluoro
+group has a strong ability to acquire electrons, which greatly
+affects the polarity of the molecule. The phenyl group is
+a frequently occurring organic functional group, while the
+hydroxyl group largely affects the interaction force between
+the molecule and water. However, SMILES-based PLMs
+may allocate more attention to symbols or numbers that
+are meaningless when they are independent, which may
+negatively impact the model’s ability to perceive frequently
+occurring substructures. For further illustration, additional
+results are presented in Figure 13 in Appendix F.4. It is
+clear that, by using a precise vocabulary without any mean-
+ingless tokens, MOLGEN is able to focus on tokens without
+distractions and capture molecular substructures implicitly.
+Figure 5. Property dynamics across MOLGEN stages (Figure 8 in
+Appendix F.2 provides a more detailed look at these changes).
+Multi-task molecular prefix tuning facilitates property
+optimization.
+Finally, we conduct an ablation study to
+investigate the impact of MPT combined with the self-
+feedback paradigm on the optimization of molecular proper-
+ties. Starting from a batch of molecules to be optimized in
+
+0.75
+0.75
+0.75
+0.75
+0.75
+0.50
+0.50
+0.50
+0.50
+0.50
+0.25
+0.25
+0.25
+0.25
+0.25
+-10
+0
+-10
+0
+-10
+0
+-10
+0
+-10
+0
+10
+20
+3010
+20
+30
+0
+5
+0
+5
+Density
+D
+200
+300
+400 200
+300
+400 0
+1000
+0
+10000.6
+0.25
+OH
+0.5
+0.20
+0.4
+Branch
+Branch2
+0.15
+0.3
+[Branch]
+C
+[F]
+0.10
+0.2
+Branch1
+IC]
+[F]
+[F]
+0.05
+0.
+[=C]
+[Ring]]
+#Branch2
+B.B2.5
+1.762
+0.860
+0.9
+0.883
+0.0
+-6.050-7.031
+-12.265 -13.152
+0.811
+0.8
+2.5
+0.7530.7h
+-5.0
+QED
+0.7
+7.5
+0.6
+10.0
+0.5
+0.488
+0.461
+12.5
+0.4
+Natural Product
+Synthetic
+Natural Product
+SyntheticMolecular Language Model as Multi-task Generator
+natural products and synthetic compounds, Figure 5 illus-
+trates the changes in property scores of molecules generated
+by different model architectures.
+Without MPT, the pre-trained model can learn similar prop-
+erty scores to the starting molecules. Moreover, we observe
+an exponential increase in scores from the first two stages
+to the last stage when MPT is implemented. This confirms
+that MPT enables MOLGEN to generate molecules with de-
+sired properties by sharing knowledge across different tasks
+and domains and encouraging the model to assign higher
+estimated probabilities to better candidate molecules.
+5. Conclusion and Future Work
+In this paper, we propose MOLGEN, a pre-trained molec-
+ular language model that utilizes the chemical language
+SELFIES and multi-task molecular prefix tuning to navi-
+gate knowledge sharing across diverse tasks and domains.
+Our in-depth study on MOLGEN demonstrates its ability to
+effectively identify essential molecular substructures and
+generate molecules with targeted properties. Interesting
+future directions include: 1) applying MOLGEN to other
+tasks such as retrosynthesis and reaction prediction (Shi
+et al., 2020a), 2) exploring multimodal pre-training like Ed-
+wards et al. (2022); Su et al. (2022), 3) incorporating addi-
+tional sources of knowledge. We will release the pre-trained
+model, code and data, hoping our work has the potential
+to pave the way for more efficient and accurate molecular
+generation in various fields.
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+
+Molecular Language Model as Multi-task Generator
+A. Availability of MOLGEN
+We will release MOLGEN on Hugging Face to support the wider scientific community soon. Note that in addition to the
+three major tasks discussed in this paper, MOLGEN can also be simply applied to other tasks such as reaction prediction and
+retrosynthetic analysis. However, due to limitations in computing resources, we have only conducted experiments on the
+three generation tasks in this study. We leave the other tasks as our future work.
+B. Data Information
+In this section, we present additional information about the dataset used in our study. For the natural product dataset, we
+sourced 30,926 compounds from the Natural Product Activity & Species Source Database (NPASS) 3 (Zhao et al., 2022).
+From this, we randomly selected 30,126 molecules for training and 800 molecules for testing and used the same sets for all
+downstream molecule generation tasks.
+The statistics of our datasets are shown in Table 5. It can be observed that the natural product dataset has a distinct
+distribution compared to the synthetic dataset, with a wider range of p-logP scores and lower QED scores. This highlights
+the added complexity of optimizing the properties of natural products.
+Table 5. Data statistics.
+LENGTH
+PENALIZED LOGP
+QED
+DATASET
+MIN
+MAX
+MEAN
+MIN
+MAX
+MEAN
+MIN
+MAX
+MEAN
+MOSES
+13
+55
+35
+-10.241
+3.329
+-0.027
+0.191
+0.948
+0.807
+ZINC250K
+8
+72
+37
+-22.189
+5.073
+-0.622
+0.117
+0.948
+0.732
+NATURAL PRODUCT
+2
+436
+55
+-51.083
+17.691
+-2.186
+0.005
+0.944
+0.438
+C. Comparison with SMILES-based PLM
+In this section, we compare the differences between two molecular language models, Chemformer (Irwin et al., 2022)
+and MOLGEN. For fairness, we select the large version of Chemformer for comparison in our paper, as it has a similar
+architecture to MOLGEN. We utilize the same number of pre-training data of 100M from the ZINC-15 dataset (Sterling &
+Irwin, 2015). MOLGEN utilizes a smaller and more specific vocabulary size of 185, while Chemformer uses a vocabulary
+size of 523. This allows MOLGEN to more effectively capture important molecular substructure information.
+D. Compared Baselines
+We illustrate the baselines we compare within our experiments. We reproduce these baselines using their open-source codes
+and under the same experimental settings. The baselines include:
+• JT-VAE (Jin et al., 2018), a VAE-based generative model which generates a scaffold junction tree and assembles nodes
+in the tree into a molecular graph.
+• GCPN (You et al., 2018), a reinforcement learning-based method that can construct a molecule by optimizing a reward
+composed of adversarial loss and molecular property objectives.
+• MolGQN (Zhou et al., 2019), a reinforcement learning-based method that utilizes double Q-learning and chemical
+domain knowledge.
+• MARS (Xie et al., 2021), a sampling approach based on Markov chain Monte Carlo using an adaptive fragment-editing
+proposal distribution with GNN.
+• GraphAF (Shi et al., 2020b), an autoregressive flow model that generates molecular graphs by adding edges and nodes
+sequentially.
+• GraphDF (Luo et al., 2021), a normalizing flow model using a discrete latent variable model and is fine-tuned with
+reinforcement learning.
+3https://bidd.group/NPASS/
+
+Molecular Language Model as Multi-task Generator
+• LIMO (Eckmann et al., 2022), a VAE-based model that leverages a variational autoencoder-generated latent space.
+• Chemformer (Irwin et al., 2022), a pre-trained molecular language model that conducts on SMILES representations.
+E. Experiment Metrics and Details
+In this section, we describe the evaluation metrics, training procedures, and hyperparameters used for each task and dataset
+in our experiments. MOLGEN is implemented using Pytorch and trained on 6 Nvidia V100 GPUs. For pre-training, we use
+the LAMB optimizer with a linear warm-up of the learning rate for the first 180,000 gradient updates, followed by a linear
+decay for the remaining training steps. The pre-training process consisted of 600 million steps with a batch size of 256
+molecules per GPU. We present the specific experimental settings and parameters in Table 6.
+Table 6. Hyper-parameter settings.
+HYPER-PARAMETERS
+VALUE
+MAXIMUM SEQUENCE LENGTH
+{55, 148, 436}
+LEARNING RATE
+{1E-5, 3E-5, 1E-4}
+BATCH SIZE
+{8, 32, 64, 200, 256}
+WEIGHT OF RANK LOSS γ
+{1,3,5}
+PREFIX LENGTH
+5
+Multi-task Molecular Prefix Tuning
+To efficiently share parameters and knowledge, we first pre-train a prefix on all
+tasks and datasets. Next, we initialize the prefix for each task with this pre-trained prefix and then optimize it for the specific
+task. It is important to note that during our experiments, we empirically find that using Multi-task molecular Prefix Tuning
+(MPT) across various molecular tasks and data sources does not result in negative transfer. Therefore, we implement it
+on all tasks and datasets to achieve the best performance.
+Distribution Learning
+We outline the metrics used to evaluate the performance of the generative models in our experi-
+ments, including:
+• Validity measures the proportion of generated molecules conforming to valence rules.
+• Fragment similarity (Frag), which compares the distribution of BRICS fragments in the generated and reference sets.
+For example, the Frag metric will be large if the molecules in both sets have similar fragments. Conversely, if some
+fragments are over- or under-represented (or never present) in the generated set, the metric will be low.
+• Scaffold similarity (Scaff), which compares the frequencies of Bemis–Murcko scaffolds (contains all molecule’s linker
+fragments connecting rings and ring structures) in the generated and reference sets. That is, if the model rarely produces
+a specific chemotype from a reference set, the metric will be low.
+• Similarity to the nearest neighbor (SNN), which measures the average Tanimoto similarity between a molecule from
+the generated set and its nearest neighbor molecule in the reference dataset. If generated molecules are far from the
+manifold of the reference set, then the similarity to the nearest neighbor will be low.
+• Internal diversity (IntDiv), which assesses the chemical diversity of generated molecules by calculating the average
+Tanimoto coefficient within the generated set.
+• Fr´echet ChemNet Distance (FCD), which considers chemically and biologically relevant information about molecules.
+It can detect if generated molecules have similar biological and chemical properties as real molecules.
+• Novelty, which measures the percentage of the generated molecules that do not appear in the training set and assesses
+the ability to explore unknown chemical space.
+To obtain the results presented in Table 1 and 2, MOLGEN is trained using the AdamW optimizer with a batch size of 200
+for the MOSES dataset and 32 for the natural product dataset on 6 Nvidia V100 GPUs for 100 epochs. A linear warm-up of
+20000 steps was also employed.
+
+Molecular Language Model as Multi-task Generator
+Targeted Molecule Discovery / Constrained Molecular Property Optimization
+For both targeted molecule discovery
+and constrained property optimization, we first use the pre-trained MOLGEN to generate 30 candidates for each data sample
+in synthetic compounds and 8 candidates for natural products and then train the model on 6 Nvidia V100 GPUs for 100
+epochs. The batch size is set to 6 for the synthetic dataset and 6 for the natural product dataset. We use AdamW as the
+optimizer, with a linear warm-up of 20000 steps.
+F. Additional Visualization of Molecules Generated in Generation Tasks
+F.1. Distribution Learning.
+In this section, we provide additional visualizations of the molecules generated by MOLGEN in the distribution learning task.
+Figures 6 and 7 show a comparison of molecules from the training set and those generated by MOLGEN for both synthetic
+and natural product datasets. The visualizations demonstrate that MOLGEN effectively captures the structural characteristics
+of molecules from different domains, despite their distinct differences in structure.
+Figure 6. Visualizations of training and generated molecules of Synthetic compounds.
+Figure 7. Visualizations of training and generated molecules of Natural Products.
+
+Molecular Language Model as Multi-task Generator
+F.2. Targeted Molecule Discovery.
+In this section, we provide additional visualizations to further support our claims in the main text. To show in more detail
+the molecular property changes in Figure 5, we visualize the distribution dynamics of p-logP and QED scores in Figure 8.
+This helps to confirm that our pre-trained MOLGEN model is able to effectively learn the property distribution of molecules
+and that the use of MPT improves the scores of generated molecules, moving them closer to the desired properties.
+Figure 8. Property distribution dynamics across MOLGEN stages.
+Moreover, the generated molecules that achieve the largest scores on both data sources are shown in Figure 9 and 10. We
+can conclude that although ultra-long carbon chains are typical structures of high p-logP molecules, MOLGEN can still
+maintain high scores without losing diversity. Besides, MOLGEN is capable of generating molecules with high QED scores
+while maintaining the structural properties of different domains.
+F.3. Constrained Molecular Property Optimization.
+We further illustrate several constrained optimization examples of both QED and p-logP in Figure 11 and 12. These examples
+further demonstrate MOLGEN’s ability to optimize molecules while preserving their general structure. Furthermore,
+MOLGEN can also achieve good performance on more difficult natural product generation tasks, which illustrates its ability
+to explore broader chemical space.
+
+0.5
+2.5
+12.5
+0.15
+0.4
+2.0
+I logP
+10.0
+1.5
+0.10
+QED
+QED
+7.5
+1.0
+5.0 
+0.05
+0.1
+0.5
+2.5
+0.00 Ermm
+0.0 H-ttttt- tm
+0.0 Ell
+0.0 ETEF TL1
+20
+-10
+0
+10
+20
+30
+-20
+-10
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.6
+0.7
+0.8
+0.9
+Natural Product
+Synthetic
+Natural Product
+Synthetic
+0.30
+12
+0.15
+0.25
+6
+10 
+logP
+8
+penlized 1
+QED
+0.10
+0.15
+6
+.0.10
+4
+0.05
+2
+0.05
+2
+0.00 EFEHH
+0.00 E-++++---mrii
+0E
+-20
+-10
+0
+10
+20
+-20
+-10
+0
+10
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.6
+0.7
+0.8
+0.9
+1.0
+Natural Product
+Synthetic
+Natural Product
+SyntheticMolecular Language Model as Multi-task Generator
+Figure 9. Molecule samples with high p-logP and QED score generated by MOLGEN from Synthetic compounds.
+Figure 10. Molecule samples with high p-logP and QED score generated by MOLGEN from Natural Products.
+
+OH
+OH
+CMolecular Language Model as Multi-task Generator
+Figure 11. Illustrations of Synthetic molecules in constrained optimization of QED and p-logP.
+Figure 12. Illustrations of Natural Products in constrained optimization of QED and p-logP.
+
+sim=0.581
+sim=0.615
+sim=0.679
+sim=0.824
+sim=0.698
+ sim=0.651
+sim=0.721
+sim=0.790
+ sim=0.788 
+sim=0.818
+sim=0.807
+sim=0.850
+sim=0.829
+sim=0.815
+sim=0.711
+sim=0.836sim=0.605
+sim=0.476
+ sim=0.577
+sim=0.539
+sim=0.698
+sim=0.651
+sim=0.721
+sim=0.790
+HO
+sim=0.525
+sim=0.566
+sim=0.4154
+sim=0.525
+sim=0.581
+sim=0.6860
+sim=0.487
+sim=0.5182Molecular Language Model as Multi-task Generator
+F.4. More molecule samples of attention visualization.
+Lastly, we evaluate the representation capabilities of different PLMs by visualizing the attention weights of each token in the
+same molecule, using the same setting as shown in Figure 4.
+Figure 13. Visualization of the learned attention map.
+As depicted in Figure 13, MOLGEN assigns more attention to important functional groups, such as carboxamide, which
+requires higher extra energy to break, and carboxyl, which is highly polar. In contrast, the performance of SMILES-based
+PLMs is not as effective, as they tend to be distracted by irrelevant tokens. This highlights the advantage of MOLGEN’s
+precise vocabulary in effectively identifying and focusing on important structural elements of molecules.
+
+0.40
+IC
+C
+0.18
+0.200
+[/C]
+[/C]
+0.16
+0.175
+0.30
+[=C]
+=C
+0.14
+0.150
+0.25
+[/C]
+[/C]
+[C]
+0.125
+[C
+0.12
+0.20
+[Branch]]
+[Branchl
+0.100
+0.10
+[C]
+[C
+0.10
+0.075
+0.08
+[N]
+TN
+0.05
+0.050
+0.06
+=0
+=0
+NH.
+de
+[=Branc
+[=Branch
+OH
+0.150
+0.125
+Ringl
+0.025
+0.025
+[=Branch
+[=Branck
\ No newline at end of file
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf,len=1526
+page_content='Molecular Language Model as Multi-task Generator  Yin Fang 1 2 3 Ningyu Zhang 1 4 2 Zhuo Chen 1 2 Xiaohui Fan 3 5 Huajun Chen 1 2  Abstract  Molecule generation with desired properties has  grown immensely in popularity by disruptively  changing the way scientists design molecular  structures and providing support for chemical and  materials design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' However, despite the promis-  ing outcome, previous machine learning-based  deep generative models suffer from a reliance on  complex, task-specific fine-tuning, limited dimen-  sional latent spaces, or the quality of expert rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  In this work, we propose MOLGEN, a pre-trained  molecular language model that effectively learns  and shares knowledge across multiple generation  tasks and domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Specifically, we pre-train  MOLGEN with the chemical language SELFIES  on more than 100 million unlabelled molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  We further propose multi-task molecular prefix  tuning across several molecular generation tasks  and different molecular domains (synthetic &  natural products) with a self-feedback mecha-  nism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Extensive experiments show that MOLGEN  can obtain superior performances on well-known  molecular generation benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' The  further analysis illustrates that MOLGEN can ac-  curately capture the distribution of molecules, im-  plicitly learn their structural characteristics, and  efficiently explore the chemical space with the  guidance of multi-task molecular prefix tuning1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  1College of Computer Science and Technology, Zhejiang  University 2Alibaba-Zhejiang University Joint Research Insti-  tute of Frontier Technologies 3Pharmaceutical Informatics Insti-  tute, College of Pharmaceutical Sciences, Zhejiang University  4School of Software Technology, Zhejiang University 5Future  Health Laboratory, Innovation Center of Yangtze River Delta,  Zhejiang University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Correspondence to: Ningyu Zhang <zhangn-  ingyu@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='cn>, Huajun Chen <huajunsir@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='cn>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Work in progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  1Codes, datasets, and the pre-trained model will be available in  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='com/zjunlp/MolGen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Introduction  Molecular generation – synthesizing and designing novel  molecules with desirable properties2 – holds an important  place in chemical science, with numerous applications in  drug discovery (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' The task of molec-  ular generation is rather challenging, whereas the chemi-  cal space of molecules, being both discrete and vast with  an estimated size of 1033, renders exhaustive searches in-  feasible (Polishchuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Early, deep generative  models (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Zang & Wang, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=',  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2020b) have been proposed as one of the  most promising tools for exploring the broader synthetically  accessible chemical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Their ability to automatically  generate chemically valid and similar molecular structures  has proven invaluable for tasks such as the inverse design of  functional compounds (Flam-Shepherd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Current deep generative models typically involve initial  training of an unconditional generative model through a  large dataset of existing molecules, followed by the employ-  ment of additional reward functions (Cao & Kipf, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Popova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' You et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Popova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Zang & Wang, 2020) or property pre-  dictors (Simonovsky & Komodakis, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  G´omez-Bombarelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2018) to guide  the synthesis of new molecules with desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' De-  spite their widespread use, the performance of these ap-  proaches is limited by their reliance on heavy task-specific  fine-tuning (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2021), fixed-dimensional latent gen-  eration space (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022b), and expert-provided  generation rules (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022), which impede efficient  exploration of the broader chemical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Recent advancements in language models have demon-  strated their potential for understanding complex molec-  ular distributions (Flam-Shepherd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Researchers  have begun integrating SMILES (Weininger, 1988), a linear  string notation for describing molecular structures, with pre-  trained language models to gain a more comprehensive un-  derstanding of the underlying molecular structures and their  representations (Irwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' However, the brittle-  ness of SMILES may lead to a high proportion of generated  SMILES strings being chemically invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' These strings can  2This paper focuses on generating 2D molecules based on given  molecules, also known as molecular optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='11259v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='LG]  26 Jan 2023  Molecular Language Model as Multi-task Generator  either be syntactically incorrect (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', not corresponding to  molecular graphs) or violate fundamental chemical princi-  ples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', exceeding the maximum number of inter-atomic  valence bonds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Another defect is that current language  model-based approaches rely on task-specific fine-tuning  with different optimization objectives, making each progress  incompatible and hindering knowledge sharing among mul-  tiple downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Besides, almost all previous studies  have primarily focused on one single domain of synthetic  molecules, neglecting natural products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Notably, natural  products, characterized by enormous scaffold diversity and  structural complexity, exhibit a distinct distribution com-  pared to synthetic molecules and confer additional oppor-  tunities and challenges for numerous molecular generation  applications such as drug discovery (Atanasov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  In this paper, we propose MOLGEN, a novel pre-trained  molecular language model for molecular generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Specif-  ically, to effectively explore the chemical universe, Firstly,  we pre-train a molecular language model based on the chem-  ical language SELFIES (Krenn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2020), where each  combination of symbols in its alphabet maps to a chemically  valid graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' By combining bidirectional and auto-regressive  Transformers (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2017), the pre-trained molecu-  lar language model can learn intrinsic structural regularities  by mapping corrupted SELFIES to their original forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Then, to break down the boundaries between diverse molec-  ular tasks and domains (synthetic & natural products), we  introduce Multi-task molecular Prefix Tuning (MPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' It al-  lows the model to acquire general features that can be trans-  ferred to various domains, thus simplifying the process of  adapting to new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Moreover, by incorporating a simple  self-feedback paradigm that assigns probabilities to candi-  date molecules based on their property scores, MOLGEN is  incentivized to generate molecules with desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  We conduct comprehensive experiments on both synthetic  and natural product molecular datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' The considerable  performance gains on molecular generation benchmarks  demonstrate the capability of MOLGEN to generate chemi-  cally valid molecules and effectively navigate the chemical  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' We further investigate MOLGEN’s understanding  of molecular distributions, recognition of crucial substruc-  tures, and the effect of MPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Our findings confirm that  MOLGEN preserves generation diversity while mastering  the distribution and structural characteristics of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Additionally, MPT improves the performance on various  tasks, owing to its ability of knowledge sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Related Work  Molecular generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  The past decade has seen signifi-  cant progress in molecule generation, particularly in deep  generative models (G´omez-Bombarelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Prior  works typically formulate this task as a sequence genera-  tion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Specifically, (Kusner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' G´omez-  Bombarelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Segler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Kwon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=',  2021) start by generating SMILES strings due to the abun-  dance of sequence modeling tools such as recurrent neural  networks, dilated convolutions, and attention mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Additionally, as molecules can be naturally viewed as two-  dimensional graphs, various molecular graph-based genera-  tive methods have emerged (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Simonovsky &  Komodakis, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Zang & Wang, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Luo  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Based on these two forms of representation,  existing approaches can be categorized into four venues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Bayesian Optimization (BO) (G´omez-Bombarelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=',  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Winter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2019) learns a continu-  ous latent space of molecules and optimizes the target prop-  erties by navigating through this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' However, BO often  requires a long evaluation time to optimize the objective  function (Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Reinforcement Learning (RL)-  based approaches aim to train an agent to select actions,  such as adding substructures, in an explicit chemical space  to optimize properties (Cao & Kipf, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Popova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=',  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' You et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Popova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Zang & Wang, 2020), but these methods are often difficult to  optimize due to high variance (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Variational  Auto-Encoder (VAE) can be another solution (Simonovsky  & Komodakis, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' G´omez-Bombarelli  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2018), but their performance is highly  dependent on the quality of the fixed-dimensional latent  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Lastly, Genetic Algorithms (GA) (Jensen, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Ahn  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Nigam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2020) use predefined mutation  and crossover rules from reference compound libraries or  domain expertise to generate molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' While flexible, it  can be challenging to acquire the necessary prior knowledge  and rules, limiting the search process’s efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Pre-trained language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Pre-trained language  models (PLMs) have waved the community as fundamen-  tal infrastructure (Kenton & Toutanova, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=',  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Intuitively, PLMs are a promis-  ing solution for molecular generation, and several pioneers  have started to leverage SMILES-based language models  with promising performance (Bagal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Irwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=',  2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Flam-Shepherd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Chilin-  garyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Pan, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' To date, Chemformer  (Irwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022) is the only publicly available PLM that  can address molecular generation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Note that similar  to human language, molecular language also has syntax,  and just as the syntax of natural languages imposes a gram-  matical structure that allows words to relate to each other  in specific ways, biological symbols also combine in spe-  cific structural manners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' However, SMILES, as a traditional  molecular language, contains complex and restrictive gram-  mars, which results in most sequences over the appropriate  character set not belonging to well-defined molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Com-  Molecular Language Model as Multi-task Generator  pared with Chemformer and other previous approaches,  MOLGEN has two main differences: (1) We pre-train the  language model with SELFIES, a chemical language repre-  sentation of molecules that is 100% robust to both syntactic  and semantic issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' (2) We formulate downstream tasks  with multi-task molecular prefix tuning to share knowledge  across tasks and domains (synthetic & natural products).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Methodology  As shown in Figure 2, we illustrate the general framework  of MOLGEN, which is based on pre-training a molecular  language model using SELFIES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' We introduce the technical  details of problem formulation (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='1), generative molecular  pre-training (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='2), and multi-task molecular prefix tuning  (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='3) in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Problem Formulation and Background  We focus on three standard molecular generation setting in  this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Concretely, define M = {Mi}n  i=1 as a set of  molecules, A(M) : M → R as a property score function,  and sim(M, M ′) ∈ [0, 1] as a similarity function measures  the similarity between the molecules M and M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Then we  formulate the molecular generation tasks as follows:  Distribution learning: learning a generative model  pθ(·) from M, such that the model can sample a valid  molecule M with a high probability pθ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Targeted molecule discovery:  learning a targeted  molecule discovery model pθ(·) to maximize the ex-  pected property score EM∼pθ[A(M)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Constrained molecular property optimization: learn-  ing a constrained optimization model pθ(·|M), to  maximize EM ′|M∼pθ[A(M ′)] while ensuring that  sim(M, M ′) > δ is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Here M ′ is the optimized  molecule based on M, and δ is a similarity threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  To address the task of molecular generation with PLMs,  the current study (Irwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022) follows BART (Lewis  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2020), a large pre-trained Seq2seq language model  standard in the literature, to corrupt SMILES and then opti-  mizing a reconstruction loss for pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Specifically,  the pre-training includes random sampling tokens within  the original SMILES string S = {s1, · · · , sj, · · · , sl} and  replacing them with a special token [MASK], encoding the  corrupted SMILES using a bidirectional model, and then  calculating the likelihood of S with a left-to-right autoregres-  sive decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Formally, we have the cross-entropy between  the decoder’s output, and the original input constitutes the  (a) SMILES maps to a wide range of structures, including  molecules, non-molecular graphs, and invalid (non-graph) struc-  tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' On the other hand, SELFIES is a surjective representation  that maps exclusively to molecular graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  (b) Each token in SELFIES points to a specific structure or refer-  ence, avoiding the presence of meaningless substrings caused by  unreasonable division in SMILES (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', “)” and “/”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  (c) Reconstruct the SELFIES from the corrupted one by optimizing  the negative log-likelihood object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Generative molecular pre-training with SELFIES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  reconstruction loss as:  Lce(S) = −  l  �  j=1  �  s  ptrue (s|S, S<j) log pθ (s|S, S<j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' θ) ,  where  S<j  denotes  the  partitial  original  sequence  {s0, · · · , sj−1} and s0 is a pre-defined start token <s>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  ptrue is a one-hot distribution under the standard maximum  likelihood estimation:  ptrue (s|S, S<j) =  �  1,  s = sj  0,  s ̸= sj  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Generative Molecular Pre-training with SELFIES  SMILES and SELFIES are two molecular languages that  map a string of tokens to a molecular graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' As illustrated  in Fig 1(a), in SMILES, molecules are represented as chains  of atoms (main string in green) with branches (red) de-  fined within parentheses and ring closures (blue) indicated  by two matching numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Although SMILES has been a  workhorse in cheminformatics for the past few decades,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' it is  Invalid  Structure  Graphs  Molecules  CC(C)CC1=CC=C2C(C1)CCO2  [C][C][Branch1 ][C][C]C][C][=C][C][=C][C][Branch1][Ring2][C][Ring1][=Branch1][C][C][O][Ring1][=Branch1]C  N  [=Ring1]  [/C]]  [=O]  [C]  [C@@H1][C][=C] [C][=C][C] [=C][Ring]][=Branch1]  Bidirectional  Autoregressive  Encoder  Decoder  [C] [=C]  <mask>  [C]  <mask>  [=Branch1]  <s>  [C][=C][C][=C]  [C] [=C][Ring]Molecular Language Model as Multi-task Generator  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Multi-task molecular Prefix Tuning (MPT), which can share knowledge across tasks and domains (synthetic & natural products).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  inherently fragile as there is no mechanism to guarantee that  molecular strings can be valid regarding syntax and physical  principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' This means that SMILES is a surjective repre-  sentation from strings to structures that include molecular  graphs but also non-molecular (semantically invalid) graphs  and other structures which cannot be interpreted as graphs  (syntactically invalid) (Krenn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  This weakness has been addressed by a 100% robust molec-  ular language SELFIES, which is a surjective mapping from  strings to molecular graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' It means that SELFIES can-  not produce invalid molecules since every combination of  symbols in the SELFIES alphabet maps to a chemically  valid graph (Krenn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Both rings and branches  in SELFIES are defined at a single location, eliminating  syntactic invalidity caused by unbalanced parentheses or  ring identifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Special symbols start a brand ([Branch1])  or ring ([Ring1]), and the subsequent token in the string de-  fines the length of the branch or ring according to its specific  derivation rules (Krenn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  To pre-train the molecular language model, we collect more  than 100 million unlabelled molecules as our pre-training  corpus by randomly selecting from the publicly accessible  ZINC-15 dataset (Sterling & Irwin, 2015) (the same pre-  training corpus of Chemformer (Irwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022) for a  fair comparison).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' The selection process strictly follows the  constraints outlined in Irwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' (2022), ensuring that the  molecules are reactive, available for purchase, have a molec-  ular weight of ≤ 500 Daltons, and a LogP (octanol-water  partition coefficient) of ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' The molecules are then con-  verted to SELFIES strings and segmented into tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' The  use of SELFIES eliminates the need for training a specific  tokenizer to identify frequently occurring substring patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Instead, we construct an alphabet from the dataset of SELF-  IES strings, as depicted in Fig 1(b), effectively avoiding  the generation of meaningless tokens caused by imprecise  tokenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' We then conduct token masking on the input  SELFIES strings S∗ = {s∗  1, · · · , s∗  l } and train a Seq2Seq  model to reconstruct the original SELFIES by optimizing  the negative log-likelihood Lce(S∗), as shown in Fig 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Multi-task Molecular Prefix Tuning  In current task-specific fine-tuning strategies to leverage  PLMs, the inputs for molecular generation tasks contain  diverse and structured knowledge from various molecular  domains, making it difficult to effectively learn shared pa-  rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' To address this challenge, we propose to break  down the boundaries between tasks and domains through  Multi-task molecular Prefix Tuning (MPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Specifically, we  prepend l tunable prefix vectors, shared between all gener-  ation tasks across domains, to the keys and values of the  multi-head attention at every layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Let T = {t}n  t=1 denote  the task type on one molecular domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' the attention score  output on each head for the task t can be formulated as:  headt = Attn  �  xtW t  q, [P k, XtW t  k], [P v, XtW t  v]  �  ,  where the sequence of m vectors Xt ∈ Rm×d denotes the  input to a Transformer layer, x ∈ Rd is a query vector,  and W q, W k, W v ∈ Rd×dh are project matrics that map  inputs to queries, keys, and values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Two sets of l prefix  vectors P k, P v ∈ Rl×d, are concatenated with the original  attention key and value in order to capture shared knowledge  across all generation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' From an alternative view, the  Task 1 : Distribution Learning  x L  Synthetic molecules  Add & Layer Norm  Feed Forward  Add & Layer Norm  Task 2 : Targeted Molecule Discovery  Attention  Qt  PhKt  Natural product molecules  Low  Property Score  High  Pk  wa  w  P  Task 3 : Constrained Molecular Property Optimization  Hidden States  Prefix shared between  tasks and domains  sim=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='81  Multi-Head  Task t = 1  Task t = 2  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='21  Property Score  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='92  Task t = 3Molecular Language Model as Multi-task Generator  computation of headt can be equivalent becomes:  headt = softmax  �  xtW t  q[P k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' XtW t  k]⊤� �  P v  XtW t  v  �  = softmax  �  xtW t  q  �  P ⊤  k  (W t  k)⊤(Xt)⊤  �� �  P v  XtW t  v  �  = λ(x)softmax  �  xtW t  qP ⊤  k  �  P v  + (1 − λ(x))softmax  �  xW t  q(W t  k)⊤(Xt)⊤�  XtW t  v  = λ(x)  Attn  �  xtW t  q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' P k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' P v  �  �  ��  �  attention of prefix shared between tasks  + (1 − λ(x)) Attn  �  xtW t  q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' XtW t  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' XtW t  v  �  �  ��  �  original attention  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  where λ(x) is a scalar whose two terms in the denominator  represent the sum of prefixes attention weights and original  sequence normalized attention weights,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' respectively:  λ(x) =  �  i exp(xtW t  qP ⊤  k )i  �  i exp(xtW t  qP ⊤  k )i + �  j exp(xtW t  q(W t  k)⊤(Xt)⊤)j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  This modification to the original head attention through  linear interpolation with shared prefixes enables knowledge  sharing across multiple domains and tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Coordinating generation candidate with self-feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  For molecular generation tasks such as targeted molecule  discovery and constrained optimization, the target is to as-  sign higher probabilities to molecules with desired prop-  erties during inference, which is non-trivial for vanilla au-  toregressive generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' To this end, we introduce a simple  mechanism dubbed coordinating generation candidate with  self-feedback inspired by fine-tuning with human feedback  (Ouyang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022) to bridge this gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' We hypothesize that  the probability of a generated candidate molecule should be  well-correlated with its property score as evaluated by an  automatic metric A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' That is, for non-reference candidate  SELFIES ˆ  S∗, we assume that:  ptrue(S∗  i |S∗) > ptrue(S∗  j |S∗),  ∀S∗  i , S∗  j ∈ ˆ  S∗  A(S∗  i ) > A(S∗  j ),  where S∗  i and S∗  j are two different candidate SELFIES, and  S∗ = {s∗  1, · · · , s∗  l } is the reference SELFIES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' We utilize  our model to generate various candidate SELFIES ˆ  S∗, each  with distinct property scores, and then encourage the model  to assign higher estimated probabilities to better candidates  by MPT with a rank loss (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022):  Lrank(S∗) =  �  i  �  j>i  max  �  0, f  �  S∗  j  �  − f (S∗  i ) + λij  �  ,  where A(S∗  i ) > A(S∗  j ), ∀i, j, i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' λij = (j − i) ∗ λ is  the margin multiplied by the difference in rank between the  candidates, and f(S∗  i ) is the estimated log-probability:  f(S∗) =  l  �  t=1  log pθ (s∗  t | S∗, S∗  <t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Intuitively, rank loss occurs when candidates with low prop-  erty scores have higher estimated probabilities than those  with high property scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' To balance the trade-off between  desired properties and similarity to original molecules, we  combine the rank and cross-entropy loss as:  L = Lce + γLrank,  where γ is the weight of the rank loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Here we utilize  label smoothing (Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2016) to modify the target  distribution of ptrue in Lrank to a “soft” label by assigning  probability mass β to other tokens:  ptrue  �  s∗|S∗, S∗  <j  �  =  �  1 − β,  s∗ = s∗  j  β  N−1,  s∗ ̸= s∗  j  ,  where N is the length of the dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' This allows for  both sequence-level coordination and token-level prediction  accuracy to be taken into account, enabling the model to  navigate the orientation of generated molecules effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Experiments  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Experimental Setup  Evaluation Tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  We conduct experiments by compar-  ing the proposed model, MOLGEN, with state-of-the-art  approaches on three tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Distribution learning evaluates  the model’s capacity to learn the data distribution and gen-  erate diverse and realistic molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Targeted molecule  discovery focuses on generating novel molecules with opti-  mized chemical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Constrained molecular prop-  erty optimization aims at modifying the given molecule  to improve desired properties while satisfying a similarity  constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' In addition to the three major tasks mentioned  above, MOLGEN can be applied to other tasks, as detailed  in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  For downstream generation tasks, the molecular  data we use mainly falls into two categories: synthetic and  natural product datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' For synthetic datasets, we use the  benchmark ZINC250k dataset (Irwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2012) containing  about 250,000 drug-like molecules for the optimization task,  and the ZINC-based MOSES dataset (Polykovskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=',  2018) containing 1,936,962 molecules for the generation  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Moreover, we collect 30,926 natural product com-  pounds from the NPASS database (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022) and  conduct all tasks on this dataset, detailed in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Molecular Language Model as Multi-task Generator  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Generation performance on the Synthetic MOSES dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  METHOD  VALIDITY(↑)  FRAG(↑)  SCAF(↑)  SNN(↑)  INTDIV(↑)  FCD(↓)  NOVELTY(↑)  AAE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content=' Generation performance on the Natural Product dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  METHOD  VALIDITY(↑)  FRAG(↑)  SCAF(↑)  SNN(↑)  INTDIV(↑)  FCD(↓)  NOVELTY(↑)  AAE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content='9912  LIMO  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content='9962  CHEMFORMER-LARGE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='9825  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content='4126  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content='9947  MOLGEN  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content='9987  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Main Results  Distribution Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Distribution learning models are  mainly used for building virtual libraries for computer-  assisted drug discovery (van Hilten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Relying on  a set of manually or automatically selected compounds, dis-  tribution learning models can generate a larger dataset that  preserves implicit rules from the reference dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' In this  section, we adopt seven widely used metrics, detailed in Ap-  pendix E, to evaluate the ability of MOLGEN to model real  molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' We generate 10,000 molecules on the synthetic  dataset following the setting in Polykovskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' (2018)  and 80,000 molecules on the natural product dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Table 1 and 2 offer the following observations: (1) MOL-  GEN holds a perfect ability to produce valid molecules with-  out additional valency check like JTN-VAE (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Since LIMO (Eckmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022) is also modeled based  on SELFIES, the molecules it generates also have 100%  validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' However, due to the complexity of internal struc-  tures and scaffolds in natural products, most models fail to  produce valid molecules on the natural product dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' The  performance of Chemformer (Irwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022) is better  than others, largely because the model learns the SMILES  grammar during large-scale pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' (2) On synthetic  datasets, almost all models generate molecules with similar  fragments and scaffolds to the reference molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' How-  ever, MOLGEN excels at capturing substructure distribution  in natural products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' (3) MOLGEN achieves the highest SNN  and lowest FCD scores, indicating that it effectively masters  the statistics of the dataset in terms of biological properties  and topology structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Additionally, the good performance  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Comparison of QED and p-logP maximization methods  on the synthetic ZINC dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' As the p-logP score can increase  linearly with molecule length, we divide baselines into upper and  lower groups for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' The first group of models  has a limit on the output length (The maximum molecule size of  ZINC250k), while the second has no limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  P-LOGP  QED  METHOD  1ST  2ND  3RD  1ST  2ND  3RD  ZINC  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content='50  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content='945  MOLGEN  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='51  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content='948  JT-VAE  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='30  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content='948  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='948  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='947  GRAPHDF  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='70  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='18  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='948  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='948  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='948  MARS  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='99  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='32  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='81  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='948  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='948  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='948  MOLGEN  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='30  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='70  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='948  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='948  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='948  of IntDiv and Novelty metrics suggests that MOLGEN is  favorable for discovering new chemical structures and ex-  ploring unknown chemical space without overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' We  provide a visual comparison of the training set and generated  molecules in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Targeted Molecule Discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  In this task, we aim to  generate molecules with high property scores using penal-  ized logP (p-logP) (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2018) and QED (Bickerton  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2012) as the optimization targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' P-logP refers to  the logP score penalized by ring size and synthetic acces-  sibility, while QED refers to the quantitative estimation of  drug-likeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Both scores are calculated by empirical pre-  Molecular Language Model as Multi-task Generator  diction models, and we utilize the script based on the official  implementation of Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' (2020b) to ensure comparabil-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Unlike the distribution learning task that only employs  the basic MPT strategy, for this task, we enhance the MPT  method by incorporating the self-feedback paradigm to more  effectively guide the generation direction of the molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  In Table 3, we present the top-3 property scores of molecules  generated by each model on the synthetic ZINC dataset, fol-  lowing the convention of previous works (Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Eckmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' The first line provides a summary  of the top-3 property scores of the ZINC250K dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' For  a fair comparison, we divide the baselines into two groups  based on the output length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' The models in the first group  have a maximum output length limit, while the second group  has no such restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' As MOLGEN supports variable-  length output, we compare it under these two settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  From Table 3, we notice that MOLGEN achieves better per-  formance than all baselines for p-logP score and obtains  comparable results for QED, suggesting that the combi-  nation of multi-task molecular prefix tuning and the self-  feedback paradigm effectively encourages the molecular  PLM to assign higher probabilities to candidate molecules  with desired properties, providing a significant advantage  for optimizing molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' However, it should be noted that  the p-logP metric heavily depend on molecule length (Xie  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Eckmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022), and while we obtain  the highest property score in the unlimited length setting  for a fair comparison, large molecules such as these are  unrealistic for practical drug discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Moreover, QED is  known to have boundary effects around its maximum score  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='948 (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2019), and drugs with a QED score  above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='9 are usually very rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Despite these limitations,  the good performance of MOLGEN demonstrates its strong  ability to explore the complex chemical space and generate  diverse molecular graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  For natural products, we use shared prefixes with synthetic  molecules to facilitate knowledge transfer across different  domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' The results, shown in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='2 and Figure 10,  demonstrate MOLGEN’s ability to generate molecules with  high QED and p-logP scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' As per our reference nat-  ural product dataset, only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='701% of molecules have a  QED score greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='9, with the highest score being  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='9439 (as seen in Appendix B Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Thus, achiev-  ing the maximum score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='9478 is sufficient for practi-  cal drug discovery purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Additionally, MOLGEN can  generate molecules with a p-logP score of 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='33, which is  significantly higher than the maximum score of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='69 in the  reference set, further demonstrating MOLGEN’s strong per-  formance in generating molecules with desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Constrained Molecular Property Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  For  this task, it aims to improve the p-logP score of a given  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' The average p-logP improvements of generated molecules  over inputs under similarity constraint δ = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  IMPROVEMENT  METHOD  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='6  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='4  JT-VAE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='28±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='79  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='03±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='39  GCPN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='79±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='63  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='49±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='30  MOLDQN  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='86±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='21  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='37±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='62  VSEQ2SEQ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='33±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='17  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='37±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='75  VJTNN  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='33±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='24  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='55±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='67  GA  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='44±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='09  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='93±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='41  GRAPHAF  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='98±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='49  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='21±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='51  GRAPHDF  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='51±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='80  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='19±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='43  LIMO  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='80±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='00  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='60±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='30  CHEMFORMER-LARGE  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='48±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='89  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='56±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='32  MOLGEN  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='08±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='82  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='35±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='21  molecule while maintaining a similarity of at least δ to the  original molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Following previous works (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=',  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Eckmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=',  2022), we optimize the p-logP scores of 800 molecules  from the synthetic ZINC250K dataset with the lowest  scores and use the Tanimoto similarity with Morgan fin-  gerprints (Rogers & Hahn, 2010) to measure the similarity  between the optimized molecules and input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  We report the mean and standard deviation of the success  rate under each similarity constraint, which is the percent-  age of molecules that can be successfully optimized over  800 molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  As summarized in Table 4, MOLGEN  yields better results under the two similarity constraints  compared to baseline models, validating its ability to search  for molecules with higher property scores in the nearby  chemical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Surprisingly, MOLGEN outperforms mod-  els that rely on additional reward functions (GCPN (You  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2018), MolDQN (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2019), GraphAF (Shi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2020b), GraphDF (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2021)) or property pre-  dictors (JT-VAE (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2018), VJTNN (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2019),  LIMO (Eckmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022)), further emphasizing the  power of the self-feedback paradigm in MPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Following the same setting, we conduct experiments on the  QED score for synthetic molecules and on two molecular  properties for natural products using shared prefixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Fig-  ure 11 and 12 in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='3 visually demonstrate the  results of constrained optimization examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' These exam-  ples show that despite the complex molecular structure and  long length of natural products, MOLGEN can improve the  property score while maintaining a certain level of similarity  between the input and the modified molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Additionally,  MOLGEN is able to maintain the diversity of generated  molecules while exploring the chemical space in close prox-  imity to the input molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Molecular Language Model as Multi-task Generator  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Comparison of molecules generated by different models  on properties, atom counts, ring counts, molecular weights, and  Bertz complexity (S-PLM stands for SMILES-based PLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' A Closer Look at MOLGEN  MOLGEN  learns  complex  molecular  distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  First of all, we evaluate the performance of the pre-trained  MOLGEN by comparing it with the most popular deep gen-  erative Graph-based (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2018), VAE-based (Blaschke  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2018), and SMILES-based language models (Irwin  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022) in terms of the distribution of properties and  structures of the generated molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' The training and gen-  eration settings of all models are the same as the distribution  learning task on the synthetic MOSES dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  As shown in the 2D histograms of p-logP and QED scores in  Figure 3, VAE-based and SMILES-based PLMs tend to pro-  duce molecules with larger p-logP and QED scores than the  training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' In comparison, the Graph-based model learns  the main mode of p-logP in the training data, while MOL-  GEN performs a bit better - similar results are observed for  QED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Furthermore, in terms of molecular topology, PLMs  outperform the others in perceiving atom numbers, ring  numbers, and molecular weights, with MOLGEN producing  a slightly closer match to the training distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' All the  models are capable of picking up on molecular Bertz com-  plexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Overall, PLMs, especially MOLGEN, can capture  the properties and structural characteristics of the training  molecules while preserving the diversity of generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  MOLGEN captures molecular substructures implicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  In this section, we investigate the capability of PLMs to  capture essential substructures when using different molec-  ular languages (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', SMILES and SELFIES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' To provide an  intuitive understanding, we visualize the attention weights  of each token in the same molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Specifically, we ex-  tract and normalize the attention weights from the last self-  attention layer, as illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Visualization of learned attention maps for the same  molecule by different PLM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Note that the models we  compared are of a similar parameter scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  The attention map produced by MOLGEN shows that the  fluoro group has higher attention weights, followed by the  phenyl and hydroxyl groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' This is because the fluoro  group has a strong ability to acquire electrons, which greatly  affects the polarity of the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' The phenyl group is  a frequently occurring organic functional group, while the  hydroxyl group largely affects the interaction force between  the molecule and water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' However, SMILES-based PLMs  may allocate more attention to symbols or numbers that  are meaningless when they are independent, which may  negatively impact the model’s ability to perceive frequently  occurring substructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' For further illustration, additional  results are presented in Figure 13 in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' It is  clear that, by using a precise vocabulary without any mean-  ingless tokens, MOLGEN is able to focus on tokens without  distractions and capture molecular substructures implicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Property dynamics across MOLGEN stages (Figure 8 in  Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='2 provides a more detailed look at these changes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Multi-task molecular prefix tuning facilitates property  optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Finally, we conduct an ablation study to  investigate the impact of MPT combined with the self-  feedback paradigm on the optimization of molecular proper-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content='4  Natural Product  Synthetic  Natural Product  SyntheticMolecular Language Model as Multi-task Generator  natural products and synthetic compounds, Figure 5 illus-  trates the changes in property scores of molecules generated  by different model architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Without MPT, the pre-trained model can learn similar prop-  erty scores to the starting molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Moreover, we observe  an exponential increase in scores from the first two stages  to the last stage when MPT is implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' This confirms  that MPT enables MOLGEN to generate molecules with de-  sired properties by sharing knowledge across different tasks  and domains and encouraging the model to assign higher  estimated probabilities to better candidate molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Conclusion and Future Work  In this paper, we propose MOLGEN, a pre-trained molec-  ular language model that utilizes the chemical language  SELFIES and multi-task molecular prefix tuning to navi-  gate knowledge sharing across diverse tasks and domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Our in-depth study on MOLGEN demonstrates its ability to  effectively identify essential molecular substructures and  generate molecules with targeted properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Interesting  future directions include: 1) applying MOLGEN to other  tasks such as retrosynthesis and reaction prediction (Shi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2020a), 2) exploring multimodal pre-training like Ed-  wards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' (2022), 3) incorporating addi-  tional sources of knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' We will release the pre-trained  model, code and data, hoping our work has the potential  to pave the way for more efficient and accurate molecular  generation in various fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content=', Neelakantan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Shyam, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Sastry, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content=' URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content=' Bartsmiles: Gener-  ative masked language models for molecular representa-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' CoRR, abs/2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 1(4):45024, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Krenn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Ai, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Barthel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Carson, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Frei, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Frey,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Friederich, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Gaudin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Gayle, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content=', Lo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Moosavi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content=', Pollice, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content=', Schatzschneider, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Schwaller, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Skreta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Smit,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Strieth-Kalthoff, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Sun, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Tom, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content=' SELFIES and the future of molecular  string representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Patterns, 3(10):100588, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Kusner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content=' PMLR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content=' Evolutionary  design of molecules based on deep learning and a genetic  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Scientific reports, 11(1):1–11, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Lewis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content=' Association for Computational Linguistics,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content=' In ICML, volume 139  of Proceedings of Machine Learning Research, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content=' Constrained generation  of semantically valid graphs via regularizing variational  autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content='  Augmenting genetic algorithms with deep neural  networks for exploring the chemical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content=', Almeida, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content=', Agarwal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content=', Gray, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content=' Moflow: An invertible flow model  for generating molecular graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' In KDD, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' 617–626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  ACM, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Zhao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Zhou, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Xu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=',  Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Fan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Hou, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Npass database  update 2023: quantitative natural product activity and  species source database for biomedical research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Nucleic  Acids Research, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Zhou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Kearnes, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', Zare, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', and Riley, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Op-  timization of molecules via deep reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Scientific reports, 9(1):1–10, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Molecular Language Model as Multi-task Generator  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Availability of MOLGEN  We will release MOLGEN on Hugging Face to support the wider scientific community soon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Note that in addition to the  three major tasks discussed in this paper, MOLGEN can also be simply applied to other tasks such as reaction prediction and  retrosynthetic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' However, due to limitations in computing resources, we have only conducted experiments on the  three generation tasks in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' We leave the other tasks as our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Data Information  In this section, we present additional information about the dataset used in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' For the natural product dataset, we  sourced 30,926 compounds from the Natural Product Activity & Species Source Database (NPASS) 3 (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  From this, we randomly selected 30,126 molecules for training and 800 molecules for testing and used the same sets for all  downstream molecule generation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  The statistics of our datasets are shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' It can be observed that the natural product dataset has a distinct  distribution compared to the synthetic dataset, with a wider range of p-logP scores and lower QED scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' This highlights  the added complexity of optimizing the properties of natural products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Data statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  LENGTH  PENALIZED LOGP  QED  DATASET  MIN  MAX  MEAN  MIN  MAX  MEAN  MIN  MAX  MEAN  MOSES  13  55  35  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='241  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='329  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='027  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='191  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='948  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='807  ZINC250K  8  72  37  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='189  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='073  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='622  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='117  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='948  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='732  NATURAL PRODUCT  2  436  55  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='083  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='691  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='186  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='944  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='438  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Comparison with SMILES-based PLM  In this section, we compare the differences between two molecular language models, Chemformer (Irwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022)  and MOLGEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' For fairness, we select the large version of Chemformer for comparison in our paper, as it has a similar  architecture to MOLGEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' We utilize the same number of pre-training data of 100M from the ZINC-15 dataset (Sterling &  Irwin, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' MOLGEN utilizes a smaller and more specific vocabulary size of 185, while Chemformer uses a vocabulary  size of 523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' This allows MOLGEN to more effectively capture important molecular substructure information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Compared Baselines  We illustrate the baselines we compare within our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' We reproduce these baselines using their open-source codes  and under the same experimental settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' The baselines include:  JT-VAE (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2018), a VAE-based generative model which generates a scaffold junction tree and assembles nodes  in the tree into a molecular graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  GCPN (You et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2018), a reinforcement learning-based method that can construct a molecule by optimizing a reward  composed of adversarial loss and molecular property objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  MolGQN (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2019), a reinforcement learning-based method that utilizes double Q-learning and chemical  domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  MARS (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2021), a sampling approach based on Markov chain Monte Carlo using an adaptive fragment-editing  proposal distribution with GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  GraphAF (Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2020b), an autoregressive flow model that generates molecular graphs by adding edges and nodes  sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  GraphDF (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2021), a normalizing flow model using a discrete latent variable model and is fine-tuned with  reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  3https://bidd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='group/NPASS/  Molecular Language Model as Multi-task Generator  LIMO (Eckmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022), a VAE-based model that leverages a variational autoencoder-generated latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Chemformer (Irwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=', 2022), a pre-trained molecular language model that conducts on SMILES representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Experiment Metrics and Details  In this section, we describe the evaluation metrics, training procedures, and hyperparameters used for each task and dataset  in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' MOLGEN is implemented using Pytorch and trained on 6 Nvidia V100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' For pre-training, we use  the LAMB optimizer with a linear warm-up of the learning rate for the first 180,000 gradient updates, followed by a linear  decay for the remaining training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' The pre-training process consisted of 600 million steps with a batch size of 256  molecules per GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' We present the specific experimental settings and parameters in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Hyper-parameter settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  HYPER-PARAMETERS  VALUE  MAXIMUM SEQUENCE LENGTH  {55, 148, 436}  LEARNING RATE  {1E-5, 3E-5, 1E-4}  BATCH SIZE  {8, 32, 64, 200, 256}  WEIGHT OF RANK LOSS γ  {1,3,5}  PREFIX LENGTH  5  Multi-task Molecular Prefix Tuning  To efficiently share parameters and knowledge, we first pre-train a prefix on all  tasks and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Next, we initialize the prefix for each task with this pre-trained prefix and then optimize it for the specific  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' It is important to note that during our experiments, we empirically find that using Multi-task molecular Prefix Tuning  (MPT) across various molecular tasks and data sources does not result in negative transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Therefore, we implement it  on all tasks and datasets to achieve the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Distribution Learning  We outline the metrics used to evaluate the performance of the generative models in our experi-  ments, including:  Validity measures the proportion of generated molecules conforming to valence rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Fragment similarity (Frag), which compares the distribution of BRICS fragments in the generated and reference sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  For example, the Frag metric will be large if the molecules in both sets have similar fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Conversely, if some  fragments are over- or under-represented (or never present) in the generated set, the metric will be low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Scaffold similarity (Scaff), which compares the frequencies of Bemis–Murcko scaffolds (contains all molecule’s linker  fragments connecting rings and ring structures) in the generated and reference sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' That is, if the model rarely produces  a specific chemotype from a reference set, the metric will be low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Similarity to the nearest neighbor (SNN), which measures the average Tanimoto similarity between a molecule from  the generated set and its nearest neighbor molecule in the reference dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' If generated molecules are far from the  manifold of the reference set, then the similarity to the nearest neighbor will be low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Internal diversity (IntDiv), which assesses the chemical diversity of generated molecules by calculating the average  Tanimoto coefficient within the generated set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Fr´echet ChemNet Distance (FCD), which considers chemically and biologically relevant information about molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  It can detect if generated molecules have similar biological and chemical properties as real molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Novelty, which measures the percentage of the generated molecules that do not appear in the training set and assesses  the ability to explore unknown chemical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  To obtain the results presented in Table 1 and 2, MOLGEN is trained using the AdamW optimizer with a batch size of 200  for the MOSES dataset and 32 for the natural product dataset on 6 Nvidia V100 GPUs for 100 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' A linear warm-up of  20000 steps was also employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Molecular Language Model as Multi-task Generator  Targeted Molecule Discovery / Constrained Molecular Property Optimization  For both targeted molecule discovery  and constrained property optimization, we first use the pre-trained MOLGEN to generate 30 candidates for each data sample  in synthetic compounds and 8 candidates for natural products and then train the model on 6 Nvidia V100 GPUs for 100  epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' The batch size is set to 6 for the synthetic dataset and 6 for the natural product dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' We use AdamW as the  optimizer, with a linear warm-up of 20000 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Additional Visualization of Molecules Generated in Generation Tasks  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Distribution Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  In this section, we provide additional visualizations of the molecules generated by MOLGEN in the distribution learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Figures 6 and 7 show a comparison of molecules from the training set and those generated by MOLGEN for both synthetic  and natural product datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' The visualizations demonstrate that MOLGEN effectively captures the structural characteristics  of molecules from different domains, despite their distinct differences in structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Visualizations of training and generated molecules of Synthetic compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Visualizations of training and generated molecules of Natural Products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Molecular Language Model as Multi-task Generator  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Targeted Molecule Discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  In this section, we provide additional visualizations to further support our claims in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' To show in more detail  the molecular property changes in Figure 5, we visualize the distribution dynamics of p-logP and QED scores in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  This helps to confirm that our pre-trained MOLGEN model is able to effectively learn the property distribution of molecules  and that the use of MPT improves the scores of generated molecules, moving them closer to the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Property distribution dynamics across MOLGEN stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Moreover, the generated molecules that achieve the largest scores on both data sources are shown in Figure 9 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' We  can conclude that although ultra-long carbon chains are typical structures of high p-logP molecules, MOLGEN can still  maintain high scores without losing diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Besides, MOLGEN is capable of generating molecules with high QED scores  while maintaining the structural properties of different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Constrained Molecular Property Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  We further illustrate several constrained optimization examples of both QED and p-logP in Figure 11 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' These examples  further demonstrate MOLGEN’s ability to optimize molecules while preserving their general structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Furthermore,  MOLGEN can also achieve good performance on more difficult natural product generation tasks, which illustrates its ability  to explore broader chemical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content=' Molecule samples with high p-logP and QED score generated by MOLGEN from Synthetic compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Molecule samples with high p-logP and QED score generated by MOLGEN from Natural Products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  OH  OH  CMolecular Language Model as Multi-task Generator  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Illustrations of Synthetic molecules in constrained optimization of QED and p-logP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+page_content='5182Molecular Language Model as Multi-task Generator  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' More molecule samples of attention visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Lastly, we evaluate the representation capabilities of different PLMs by visualizing the attention weights of each token in the  same molecule, using the same setting as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' Visualization of the learned attention map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content='  As depicted in Figure 13, MOLGEN assigns more attention to important functional groups, such as carboxamide, which  requires higher extra energy to break, and carboxyl, which is highly polar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' In contrast, the performance of SMILES-based  PLMs is not as effective, as they tend to be distracted by irrelevant tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
+page_content=' This highlights the advantage of MOLGEN’s  precise vocabulary in effectively identifying and focusing on important structural elements of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFIT4oBgHgl3EQfayvO/content/2301.11259v1.pdf'}
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+arXiv:2301.11260v1  [cs.LG]  26 Jan 2023
+Maximum Optimality Margin: A Unified Approach for
+Contextual Linear Programming and Inverse Linear
+Programming
+Chunlin Sun∗, Shang Liu∗, Xiaocheng Li
+Institute for Computational and Mathematical Engineering, Stanford University, chunlin@stanford.edu
+Imperial College Business School, Imperial College London, (s.liu21, xiaocheng.li)@imperial.ac.uk
+Abstract
+In this paper, we study the predict-then-optimize problem where the output of a machine learning
+prediction task is used as the input of some downstream optimization problem, say, the objective
+coefficient vector of a linear program. The problem is also known as predictive analytics or contextual
+linear programming. The existing approaches largely suffer from either (i) optimization intractability
+(a non-convex objective function)/statistical inefficiency (a suboptimal generalization bound) or (ii)
+requiring strong condition(s) such as no constraint or loss calibration. We develop a new approach
+to the problem called maximum optimality margin which designs the machine learning loss function
+by the optimality condition of the downstream optimization. The max-margin formulation enjoys
+both computational efficiency and good theoretical properties for the learning procedure.
+More
+importantly, our new approach only needs the observations of the optimal solution in the training
+data rather than the objective function, which makes it a new and natural approach to the inverse
+linear programming problem under both contextual and context-free settings; we also analyze the
+proposed method under both offline and online settings, and demonstrate its performance using
+numerical experiments.
+1
+Introduction
+The predict-then-optimize problem considers a learning problem under a decision making context where
+the output of a machine learning model serves as the input of a downstream optimization problem
+(e.g. a linear program). The ultimate goal of the learner is to prescribe a decision/solution for the
+downstream optimization problem using directly the input (variables) of the machine learning model
+but without full observation of the input of the optimization problem. A similar problem formulation
+was also studied as prescriptive analytics (Bertsimas and Kallus, 2020) and contextual linear program-
+ming (Hu et al., 2022). While Elmachtoub and Grigas (2022) justify the importance of leveraging the
+optimization problem structure when building the machine learning model and the existing efforts on
+exploiting the optimization structure have been largely inadequate. In this paper, we delve deeper into
+the structural properties of the optimization problem and propose a new approach called maximum
+optimality margin which builds a max-margin learning model based on the optimality condition of the
+downstream optimization problem. More importantly, our approach only needs the observations of the
+optimal solution in the training data rather than the objective function, thus it draws an interesting
+connection to the inverse optimization problem. The connection gives a new shared perspective on both
+the predict-then-optimize problem and the inverse optimization problem, and our analysis reveals a scale
+inconsistency issue that arises practically and theoretically for many existing methods.
+∗ Equal contribution.
+1
+
+Now we present the problem formulation and provide an overview of the existing techniques and
+related literature. Consider a linear program (LP) that takes the following standard form
+LP(c, A, b) := min c⊤x,
+(1)
+s.t. Ax = b, x ≥ 0.
+where c ∈ Rn, A ∈ Rm×n, and b ∈ Rm are the inputs of the LP. In addition, there is an available feature
+vector z ∈ Rd that encodes the useful covariates (side information) associated with the LP.
+Predict-then-optimize/Contextual linear programming
+The problem of predict-the-optimize or contextual LP is stated as follows. A set of training data
+DML(T ) := {(ct, At, bt, zt)}T
+t=1
+consists of i.i.d. samples from an unknown distribution P, where ct ∈ Rn, At ∈ Rm×n, bt ∈ Rm, and
+zt ∈ Rd. Throughout the paper, we assume the constraints have a fixed dimensionality for notational
+simplicity, while all the results can be easily extended to the case of variable dimensionality. The goal
+of a conventional machine learning (ML) model is to identify a function g(·; Θ) : Rd → Rn that best
+predicts the objective coefficient vector ct using the covariates zt. However, the predict-then-optimize
+problem has a slightly different pipeline for the testing phase. It aims to map from the observation of
+the context and the knowledge of the constraints to a decision (from data to decision):
+(Anew, bnew, znew) → xnew
+without the observation of cnew, where the tuple (cnew, Anew, bnew, znew) is a new test sample from P.
+That is, for this new test sample, the decision maker knows the constraints (Anew, bnew), and the aim is
+to predict the unknown objective cnew from znew and prescribe the decision variable xnew.
+There are commonly three performance measures for the problem (as noted by Chen and Kılınç-Karzan
+(2020))
+• Prediction loss: lpre(ˆΘ) = E
+�
+∥cnew − ˆcnew∥2
+2
+�
+• Estimate loss: lest(ˆΘ) = E
+�
+c⊤
+newxnew − c⊤
+newx∗
+new
+�
+• Suboptimality loss: lsub(ˆΘ) = E
+�
+ˆc⊤
+newx∗
+new − ˆc⊤
+newˆxnew
+�
+where ˆcnew = g(znew; ˆΘ) is the predicted output of the ML model. Here ˆxnew and x∗
+new are the optimal
+solutions of LP(ˆcnew, Anew, bnew) and LP(cnew, Anew, bnew), respectively. The expectations are taken with
+respect to this new sample (following the distribution P).
+The prediction loss is aligned with the standard ML problems, where the L2 loss can also be re-
+placed by other proper metrics. The estimate loss captures the quality of the recommended decision
+xnew under the true (unobserved) cnew and it is a more natural loss given the interests in the down-
+stream optimization task. The suboptimality loss measures how well the predicted ˆcnew explains the
+realized optimal solution x∗
+new. It is more commonly adopted in the inverse linear programming litera-
+ture (Mohajerin Esfahani et al., 2018; Bärmann et al., 2018; Chen and Kılınç-Karzan, 2020).
+Inverse linear programming
+Our proposed approach to the predict-then-optimize problem also solves a seemingly unrelated prob-
+lem – inverse linear programming. In the language of predict-then-optimize, the problem of inverse linear
+2
+
+programming considers a set of training data
+Dinv(T ) := {(x∗
+t , At, bt, zt)}T
+t=1 .
+Similarly as the previous case, the tuples (ct, At, bt, zt)’s are generated from an unknown distribution P.
+Differently, for inverse linear programming, the optimal solution x∗
+t instead of the objective coefficient
+vectors ct is given in the training data. While the classic setting of inverse linear programming does
+not consider the context, i.e., zt ≡ 1 for all t, several recent works (Mohajerin Esfahani et al., 2018;
+Besbes et al., 2021) study the contextual case. The goal of the inverse linear programming is similar as
+that of the predict-then-optimize problem, to learn a function g(·; Θ) that maps from the context zt to
+the objective ct. We emphasize that the observation of x∗
+t gives much less information than that of ct
+which makes the inverse problem more challenging than the predict-then-optimize problem.
+In the following, we briefly review the representative existing approaches to tackle the two problems.
+Predicting the objective.
+The first class of approaches treats the predict-then-optimize problem as a two-step procedure.
+The learner first learns a function g(·; ˆΘ) : z → c from the training data, and then prescribes the
+final output xnew by the optimal solution of LP(g(znew; ˆΘ), Anew, bnew). There are mainly two exist-
+ing routes for how the parameter ˆΘ should be estimated from the training data.
+The first route is
+to ignore the constraint (At, bt) (in the training data) and treat the training problem as a pure ML
+problem (Ho-Nguyen and Kılınç-Karzan, 2022).
+The issue of this route is that there can be a mis-
+alignment between the ML loss lpre(·) and the downstream optimization loss lest(·) or lsub(·). Empiri-
+cally, it causes sample inefficiency if one is interested in lest(·) (noted by Elmachtoub and Grigas (2022)
+and also observed in our numerical experiments).
+Theoretically, to establish a performance guaran-
+tee for the optimization loss, it requires a calibration condition which can be hard to satisfy/verify
+(Ho-Nguyen and Kılınç-Karzan, 2022). The second route is to employ the optimization loss lest(·) for
+the estimation of ˆΘ (Elmachtoub and Grigas, 2022; Elmachtoub et al., 2020). However, the loss function
+is generally non-convex in the parameter Θ. A convex surrogate loss called SPO+ is proposed, but the
+sample complexity yielded under the surrogate loss can be of a suboptimal rate (El Balghiti et al., 2019).
+In the literature of inverse LP, the (convex) suboptimality loss lsub(·) is often used instead of lest(·).
+Our proposed approach falls in this category of first predicting the objective and then solving the LP.
+Different from all the existing works, we derive a new formulation based on the optimality condition of
+the optimization problem that features for both statistical and computational efficiency.
+Predicting the optimal solution.
+The second class of approaches treats the predict-then-optimize problem as a one-step procedure, and
+aims to learn an end-to-end function that maps from the context zt directly to the optimal solution x∗
+t
+(Bertsimas and Kallus, 2020; Hu et al., 2022). This end-to-end treatment works well in the unconstrained
+setting. But for the constrained setting, the optimal solution of an LP generally stays at the corner of
+the feasible simplex, and the end-to-end mapping can hardly predict such a corner solution. In the
+domain of inverse linear programming, a recent work (Tan et al., 2020) also proposes a loss function
+minimizing the gap between the predicted optimal solution and the observed optimal solution. The idea is
+more aligned with the early formulation of inverse optimization (Zhang and Liu, 1996; Ahuja and Orlin,
+2001). However, the formulation in (Tan et al., 2020) is more of a conceptual framework that is hardly
+computationally tractable.
+Consistency/Feasibility check.
+3
+
+For the inverse linear programming problem, a common approach in literature (Zhang and Liu, 1996;
+Ahuja and Orlin, 2001; Boyd and Vandenberghe, 2004; Bertsimas et al., 2015; Bärmann et al., 2018;
+Besbes et al., 2021) is to transform the knowledge of the optimal solution x∗
+t equivalently to constraints
+on the objective vector ct ∈ Ct for some polyhedron Ct (through the optimality condition), and then uti-
+lize these Ct to make inference about the objective vector. There can be two issues with this approach.
+First, many existing results study the context-free case and require a non-empty intersection of the poly-
+hedrons, i.e., ∩T
+t=1Ct ̸= ∅. This requirement may fail in a noisy or contextual setting. Second, from a
+methodology perspective, it is difficult to relate the structure of Ct with the covariates zt. To this end,
+our paper complements the results of the existing literature by providing a new approach for the inverse
+linear programming problem for both context-free and contextual setting and with less assumption.
+2
+LP Optimality Structure and Main Algorithm
+Now we first present some preliminaries on linear programming and then describe our main algorithm.
+Let x∗ = (x1, ..., xn)⊤ be the optimal solution of the LP(c, A, b). The optimal basis B∗ and its complement
+N ∗ of an LP are defined as follows
+B∗ := {i : x∗
+i > 0},
+N ∗ := {i : x∗
+i = 0}.
+For a set B ⊂ [n], we use AB to denote the submatrix of A with column indices corresponding to B
+and cB to denote the subvector with corresponding dimensions. We make the following assumptions on
+the LP’s nondegeneracy.
+Assumption 1 (Nondegeneracy). All the linear programs in this paper are nondegenerate, i.e., satisfying
+the following two conditions:
+(a) The LP is feasible and has a unique optimal solution.
+(b) All the submatrices AB are invertible for |B| = m. The optimal basis satisfies |B∗| = m.
+The assumption is standard in the literature of LP. It is a mild one in that any LP can satisfy the
+assumption under an arbitrarily small perturbation (Megiddo and Chandrasekaran, 1989). We also note
+that our focus on the standard-form LP (1) is without loss of generality because all the LPs can be
+written in the standard form and the results in this paper can be easily adapted to an LP of other forms.
+The following lemma describes the optimality condition for a linear program in terms of its input.
+Lemma 1 (Luenberger et al. (1984)). Let B ⊂ [n] be a feasible basis of LP(c, A, b), i.e., A−1
+B b ≥ 0, and let
+N = [n]\B be its complement. Then, under Assumption 1, the following inequality holds (element-wise)
+c⊤
+N − c⊤
+BA−1
+B AN ≥ 0
+(2)
+if and only if B = B∗.
+The result provides a foundation for the simplex method to solve LPs. When B = B∗, the vector
+p∗ := c⊤
+B∗A−1
+B∗ ∈ Rm will be the optimal solution of the dual LP of (1), also known as the dual price. It
+assigns a shadow price for per unit consumption of each constraint. For a decision variable index i ∈ [n],
+when ci > A⊤
+i p∗ (where Ai is the i-th column of A), it means the choice of the i-th variable is not cost
+effective, and thus x∗
+i = 0.
+4
+
+2.1
+Maximum optimality margin
+Now we present our approach of maximum optimality margin. Consider the following training data
+Dinv(T ) = {(x∗
+t , At, bt, zt)}T
+t=1 .
+Note that the training data only consists of the observations of the optimal solution x∗
+t , so all the algo-
+rithms and analyses apply to both the predict-then-optimize problem and the inverse linear programming
+problem. Denote the sets of basic variables and non-basic variables of the t-th training sample as B∗
+t ∈ [n]
+and N ∗
+t ∈ [n], respectively. Then the training data can be equivalently re-written as
+Dinv(T ) = {(x∗
+t , At, bt, zt, B∗
+t , N ∗
+t )}T
+t=1 .
+Specifically, we start with a linear mapping from the covariates zt to the objective vector ct, i.e.,
+ct = g(zt; Θ) := Θzt.
+Our maximum optimality margin method solves the following optimization problem.
+ˆΘ := arg min
+Θ∈K
+λ
+2 ∥Θ∥2
+2 + 1
+T
+T
+�
+t=1
+∥st∥1
+s.t.
+ˆct = Θzt,
+t = 1, ..., T,
+(3)
+ˆc⊤
+t,N ∗
+t − ˆc⊤
+t,B∗
+t A−1
+t,B∗
+t At,N ∗
+t ≥ 1|N ∗
+t | − st,
+t = 1, ..., T,
+where the decision variables are Θ ∈ Rn×d, ˆct ∈ Rn, and st ∈ R|Nt|. K is a convex set to be determined.
+To interpret the optimization problem (3):
+• The equality constraints encode the linear prediction function, and ˆct’s are the predicted value of
+ct under parameter Θ.
+• For the inequality constraints, the vector 1|N ∗
+t | ∈ R|N ∗
+t | is an all-one vector of dimension |N ∗
+t |. The
+left-hand-side of the inequality comes from the optimality condition (2) in Lemma 1, while the
+right-hand-side represents the slackness or margin of the optimality condition. From the objective
+function, it is easy to see that the optimal solution will always render st ≥ 0. Then, when st ∈ [0, 1]
+element-wise for all t, the inequality constraints fulfill the optimality condition (2) perfectly, which
+means that the optimal solution ˆΘ is consistent with all the training samples. In other words, when
+st ∈ [0, 1] element-wise for all t, if we predict the objective coefficient ct with ˆct = ˆΘzt, the two
+LPs, LP(ct, At, bt) and LP(ˆct, At, bt) share the same optimal solution for all t (by Lemma 1), and
+it thus results in a zero estimate loss lest and a zero suboptimality loss lsub on the training data.
+When some coordinate of st is greater than 1, it means the optimal solution under the predicted
+objective ˆct = ˆΘzt no longer satisfies the optimality condition of the original LP(ct, At, bt), and
+consequently, this results in a mismatch between the two optimal solutions of LP(ct, At, bt) and
+LP(ˆct, At, bt), and thus a non-zero loss of lest and lsub.
+• For the objective function, the second part is justified by the above discussion on the inequality
+constraints. We desire to have a smaller value of st as this leads to a better satisfaction of the
+optimality condition. The first part of the objective function regularizes the parameter Θ. The
+rationale is that the left-hand-side of the inequality constraints that represents the realized margin
+of the optimal condition, scales linearly with Θ. We hope the margin to be large but do not want
+the large margin is due to the scaling up of Θ.
+5
+
+We note that the optimization problem (3) has linear constraints and a quadratic objective, and
+thus it is convex. Algorithm 1 describes the procedure of maximum optimality margin for solving the
+predict-then-optimize problem and the inverse LP problem. It is a two-step procedure in that it first
+estimates the parameter and then solves the optimization problem based on the estimate ˆΘ.
+Algorithm 1 Maximum Optimality Margin (MOM) Algorithm
+1: Input: Dataset D(T ) = {(x∗
+t , At, bt, zt, B∗
+t , N ∗
+t )}T
+t=1, test sample (Anew, bnew, znew), λ, K
+2: Solve the optimization problem (3) and obtain the estimate ˆΘ
+3: Predict ˆcnew = ˆΘznew and solve the following LP
+min ˆc⊤
+newx,
+s.t. Anewx = bnew, x ≥ 0.
+4: Denote its optimal solution as ˆxnew
+5: Output: ˆxnew and ˆΘ
+2.2
+Interpreting the formulation
+The key idea of the MOM formulation is that it does not explicitly minimize the error of predicting
+the objective ct’s as a stand-alone machine learning problem, but it aims to minimize the violation –
+equivalently, maximize the margin – of the optimality conditions of the training samples (the inequality
+constraint in (3)). Intuitively, as long as we find a prediction ˆct that shares the same optimal solution
+x∗
+t with ct for most t (hopefully), we should not bother with the error between ˆct and ct. This distin-
+guishes from all the existing methods for predict-then-optimize and inverse LP in that it integrates the
+optimization problem’s structure naturally into the machine learning training procedure. We make the
+following remarks.
+First, our method does not require the knowledge of ct’s in the training data. As far as we concern,
+it is the first method that works for both predict-then-optimize and inverse LP problems. Beyond that,
+this special feature of our method has a modeling advantage of scale consistency or scale invariance.
+Specifically, we note that for the LP (1), both the objective vector c and αc for any α > 0 lead to
+the same optimal solution. Hence the loss of training a machine learning model should ideally bear a
+scale invariance in terms of the objective vector. That is, a prediction of ˆc and a prediction of αˆc for
+any α > 0 should incur the same training loss as both of them lead to the same prescribed solution.
+Our method enjoys this scale invariance property, since it only utilizes the optimal solution x∗
+t in the
+training phase but does not involve ct. In comparison, a method that minimizes the prediction loss lpre
+apparently does not have this scale invariance, so do some inverse LP algorithms (see Section 3.3). This
+property can be critical for some application contexts such as revealed preference or stated preference
+citepbeigman2006learning,zadimoghaddam2012efficiently where one aims to predict the utility ct of a
+customer based on observed covariates zt. In such contexts, even if we have observations of the ct’s
+through surveying the customers’ utilities, these observations might suffer from some scale contamination
+because each customer may have their own scale for measuring utilities. And therefore, to strive for an
+accurate prediction of the observed ct can be misleading.
+Second, the inequality constraint in (3) is critical.
+Two tempting alternatives are to replace the
+inequality constraints of (3) with the following:
+ˆc⊤
+t,N ∗
+t − ˆc⊤
+t,B∗
+t A−1
+t,B∗
+t At,N ∗
+t ≥ 0,
+(4)
+ˆc⊤
+t,N ∗
+t − ˆc⊤
+t,B∗
+t A−1
+t,B∗
+t At,N ∗
+t ≥ −st.
+(5)
+6
+
+For (4), it directly employs the optimality condition (2) in Lemma 1. The issue with this formulation is
+that there may exist no Θ that is consistent all the training data, and thus (4) will lead to an infeasible
+problem. In comparison, the inequality constraints of (3) can be viewed as a softer version of (4). For
+(5), it suffers a similar scale invariance problem as mentioned above. Specifically, the constraint (5) will
+result in ˆΘ → 0 as the left-hand-side of (5) scales linearly with ˆΘ. To prevent this, one can impose an
+additional unit sphere constraint by ∥Θ∥2 = 1 but this will lead to a non-convexity issue.
+Lastly, we note a few additional points for the formulation. First, the formulation does not involve bt in
+the optimization problem (3) either. This is due to that bt’s information is encoded by (At, B∗
+t , N ∗
+t ) under
+the standard-form LP (1). Second, the formulation is presented under a linear mapping of zt to ct; non-
+linearity dependency can be introduced by kernelizing the original covariates zt (Hofmann et al., 2008).
+Third, the MOM formulation aims to find a parameter ˆΘ that best satisfies the optimality condition, and
+it measures the quality of satisfaction by the total margin of optimality condition violation. It does not
+strive for a structured prediction of the optimal bases Bt’s which will lead to a less tractable formulation
+such as structured support vector machine (See Appendix C.1).
+3
+Theoretical Analysis
+Now we analyze the maximum optimality margin method under both offline and online settings. We
+make the following boundedness assumptions mainly for notation simplicity. Specifically, we note that the
+parameter ¯σ captures some “stability” of the LP’s optimal solution under perturbation of the constraint
+matrix. While our algorithm utilizes the optimal solutions in the training data, a better stability of the
+optimal solutions will lead to a more reliable learning of the parameter.
+Assumption 2 (Boundedness). Let the tuple (c, A, b, z, B∗, N ∗) be a sample drawn from the distribution
+P. The following assumptions hold with probability 1.
+(a) There exists a constant ¯σ such that σmax(A−1
+B∗ ) ≤ ¯σ, where σmax(·) denotes the largest singular
+value function of a matrix.
+(b) The covariates vector z is bounded by 1, i.e., ∥z∥2 ≤ 1.
+(c) All entries of the LP’s input (A, b, c) are within [−1, 1].
+(d) The feasible region {x ≥ 0 : Ax = b} is bounded by the 2-norm unit ball.
+3.1
+Separable Case
+We begin our discussion with the separable case where there exists a parameter that meets the optimality
+conditions on all the training samples with a margin.
+Assumption 3. There exists Θ∗ such that the following inequality holds element-wise almost surely,
+ˆc⊤
+N ∗ − ˆc⊤
+B∗A−1
+B∗ AN ∗ ≥ 1
+(6)
+where ˆc = Θ∗z and (c, A, b, z, B∗, N ∗) is sampled from P. Suppose ∥Θ∗∥F ≤ ¯Θ for some constant ¯Θ > 0.
+The assumption is weaker than to assume c = Θ∗z almost surely. It only requires the existence
+of a linear mapping ˆc = Θ∗z that meets the optimality condition almost surely. It can happen that
+Assumption 3 holds but c ̸= ˆc almost surely. The right-hand-side of (6) changes from that of (2) in
+Lemma 1 from 0 to 1. The change is not essential when the LP is nondegenerate almost surely; one can
+always scale up the parameter Θ∗ to meet this margin requirement.
+7
+
+Proposition 1. Under Assumptions 1, 2 and 3, let ˆΘ denote the output of Algorithm 1 with T training
+samples, λ =
+1
+√
+T and K = {Θ ∈ Rn×d : ∥Θ∥F ≤ ¯Θ}. Suppose (cnew, Anew, bnew, znew) is a new sample
+from P and denote ˆcnew = ˆΘznew. Let x∗
+new and ˆxnew denote the optimal solutions of LP(cnew, Anew, bnew)
+and LP(ˆcnew, Anew, bnew). Then we have
+E [P (x∗
+new = ˆxnew)] ≥ 1 − 28 + 8 ¯Θ2 + 7m¯σ2
+√
+T
+,
+(7)
+where m is the number of constraints in each LP, the expectation is taken with respect to the training
+samples, and the probability is with respect to the new test sample.
+The proposition states that under the separable case, the algorithm can predict the optimal solution
+accurately with high probability. We note that the result does not concern the prediction error in terms
+of the objective vector, and this fact is aligned with the design of MOM that focuses on the optimality
+condition instead of the objective vector. Intuitively, even when one makes some error in predicting the
+objective cnew, the prediction of the optimal solution can still be accurate due to the simplex structure
+of LP. The proof of the proposition mimics the algorithm stability analysis (Bousquet and Elisseeff,
+2002; Shalev-Shwartz et al., 2010) where the regularization term in (3) plays a role of stabilizing the
+estimation. Here the performance bound is presented in the sense of on expectation. We remark that
+a high probability bound (removing the expectation in (7)) can also be achieved following the recent
+analysis of (Feldman and Vondrak, 2019).
+3.2
+Inseparable case
+Now we move on to the inseparable case where Assumption 3 no longer holds. Let
+Dnew = (c, A, b, z, B∗, N ∗)
+denote a new sample from P. Define the margin-violation loss function
+l(Dnew; Θ) := ∥s∥1 where s := 1|N ∗| − ˆc⊤
+N ∗ − ˆc⊤
+B∗A−1
+B∗
+t AN ∗ and ˆc = Θz.
+The loss function is inherited from the objective function of the MOM formulation (3). For the inseparable
+case, we can derive the following generalization bound as Proposition 1.
+Proposition 2. Under Assumptions 1 and 2, let ˆΘ denote the output of Algorithm 1 with T training
+samples, under the choice of λ =
+1
+√
+T and K = {Θ ∈ Rn×d : ∥Θ∥F ≤ ¯Θ} for some ¯Θ > 0. Then we have
+E[l(Dnew; ˆΘ)] ≤ min
+Θ∈K E[l(Dnew; Θ)] + 28 + 8 ¯Θ2 + 7m¯σ2
+√
+T + 1
+,
+where the expectation is taken with respect to both the new sample Dnew and the training samples.
+The proposition follows the same analysis as Proposition 1 and the right-hand-side involves an addi-
+tional term which will become zero under the separability condition of Assumption 3. In the previous
+section, we motivate the optimization formulation (3) of MOM in its own right. The following corollary
+reveals an interesting connection between its objective function and the suboptimality loss. Specifically,
+the margin violation loss optimized in MOM can be viewed as an upper bound of the suboptimality loss.
+Proposition 3. The following inequality holds for any Θ ∈ Rn×d,
+lsub(Θ) ≤ E[l(Dnew; Θ)].
+8
+
+Therefore, under Assumptions 1 and 2, and the same setup as Proposition 2,
+lsub(ˆΘ) ≤ min
+Θ∈K E[l(Dnew; Θ)] + 28 + 8 ¯Θ2 + 7m¯σ2
+√
+T
+.
+The additional term on the right-hand-side in both Proposition 2 and Proposition 3 can be viewed as
+a measure of separability in terms of the LP’s optimality condition. Specifically, if one can predict the
+objective c very well with the covariates z, then this term will become close to zero; otherwise, this term
+will become large. While the standard measurement of the predictive power of z concerns the minimum
+error in predicting c, it does not account for the structure of the downstream LP optimization problem.
+3.3
+Online maximum optimality margin
+The MOM formulation also enables an online algorithm which reduces the quadratic program of (3) to
+a subgradient descent step per sample. Specifically, we can view the margin violation of the t-th sample
+as a piecewise-linear function of Θ, i.e.,
+l(Dt; Θ) = ∥st∥1 =
+���1|N ∗
+t | − ˆc⊤
+t,N ∗
+t − ˆc⊤
+t,B∗
+t A−1
+t,B∗
+t At,N ∗
+t
+���
+1
+where Dt represents the t-th sample in the dataset D(T ). The function l(Dt; Θ) is a convex function with
+respect to Θ and the piecewise linearity enables an efficient calculation of the subgradients. Algorithm
+2 arises from an online subgradient descent algorithm with respect to the cumulative loss
+T
+�
+t=1
+l(Dt; Θ).
+Algorithm 2 Online MOM Algorithm
+1: Input: Dataset D(T ) = {(x∗
+t , At, bt, zt, B∗
+t , N ∗
+t )}T
+t=1, step size η, K
+2: Initialize Θ1 = 0 ∈ Rn×d
+3: for t = 1, ..., T do
+4:
+Predict the objective by ˆct = Θtzt
+5:
+Solve the following LP and denote its optimal solution by xt
+min ˆc⊤
+t x,
+s.t. Atx = bt, x ≥ 0.
+6:
+Update
+Θt+1 = ProjK (Θt − η · ∂Θl(Dt; Θt))
+7: end for
+8: Output: {xt}T
+t=1, {ΘT }T +1
+t=1
+Proposition 4. Under Assumptions 1 and 2, with the choice of η =
+2 ¯Θ
+(√n+¯σ·mn)
+√
+T and K = {Θ ∈ Rn×d :
+∥Θ∥F ≤ ¯Θ} for some ¯Θ > 0, the outputs of Algorithm 2 satisfy
+1
+T E
+� T
+�
+t=1
+ˆc⊤
+t (x∗
+t − xt)
+�
+≤ E
+�
+min
+Θ∈K
+T
+�
+t=1
+l(Dt; Θ)
+�
++ 3 ¯Θ√n + 3¯σ ¯Θ · mn
+√
+T
+(8)
+where Dnew = (cnew, Anew, bnew, znew) denotes a new sample.
+9
+
+Moreover, under Assumption 3, let ˆcnew = ˆΘznew where ˆΘ is uniformly randomly sampled from
+{Θt}T +1
+t=1 . Let x∗
+new and ˆxnew denote the optimal solutions of LP(cnew, Anew, bnew) and LP(ˆcnew, Anew, bnew).
+Then we have
+E [P (x∗
+new = ˆxnew)] ≥ 1 − 3 ¯Θ√n + 3¯σ ¯Θ · mn
+√
+T + 1
+(9)
+where the expectation is taken with respect to both training data samples and the new sample.
+The proposition states that the results in Proposition 1 and Proposition 2 can also be achieved by a
+subgradient descent algorithm from an online perspective. The bound (8) also recovers the regret bounds
+in the online inverse optimization literature (Bärmann et al., 2018; Chen and Kılınç-Karzan, 2020) under
+a contextual setting. However, the algorithms therein require the convex set K not containing the origin
+(which will significantly restrict the predictive power of zt → ct). Otherwise, we find numerically that
+these existing algorithms will drive the parameter Θt → 0 and ˆct → 0, and consequently provide no
+meaningful prediction of the optimal solution. This reinforces the point of scale invariance raised earlier.
+As mentioned earlier, the margin-based formulation of MOM resolves the issue of scale invariance, and
+as a result, Algorithm 2 can provide an additional bound (9) and also perform stably well in numerical
+experiments, which we refer to Appendix A.4.
+3.4
+Tighter bound under separability – optimality-driven perceptron
+We conclude this section with Algorithm 3 that achieves a faster rate under the separability condition
+(Assumption 3). Specifically, the algorithm is an online algorithm that utilizes the knowledge of sepa-
+rability. The idea is to treat each inequality constraint in (3) as an individual binary classification task
+of distinguishing non-basic variables from basic variables, and view the margin of optimality condition
+as the margin for classification. Then when the margin (the reduced cost in Algorithm 3) drops below a
+threshold of 1/2, we perform an update of the parameter.
+Proposition 5. Under Assumptions 1, 2 and 3, the number of mistakes made by Algorithm 3 is inde-
+pendent of T ,
+# {t ∈ [T ] : x∗
+t ̸= xt} ≤ ¯Θ2 + ¯σ2 ¯Θ2m2n.
+In addition, suppose (cnew, Anew, bnew, znew) is a new sample from P and denote ˆcnew = ˆΘznew where
+ˆΘ is uniformly randomly sampled from {Θt}T +1
+t=1 . Let x∗
+new and ˆxnew denote the optimal solutions of
+LP(cnew, Anew, bnew) and LP(ˆcnew, Anew, bnew),
+E [P(x∗
+new = ˆxnew)] ≥ 1 −
+¯Θ2 + ¯σ2 ¯Θ2m2n
+T + 1
+,
+where the expectation is taken with respect to both the training data and the new data.
+Proposition 5 provides an upper bound on the number of mistakes made by Algorithm 3.
+The
+algorithm improves the dependency on T compared with the previous bounds in Proposition 1 and
+Proposition 4.
+However, we note this improvement sacrifices the dependency on either the problem
+size m and n or the constants of ¯Θ and ¯σ. In this light, the result is more of theoretical interests
+that completes our discussion.
+In the numerical experiments in Appendix A.4, we observe that the
+performance of Algorithm 3 is indeed worse than Algorithm 1 and Algorithm 2.
+4
+Numerical Experiments and Discussions
+We conduct numerical experiments for two underlying LP problems – the shortest path problem and the
+fractional knapsack problem. We present one experiment here and defer the remaining ones to Appendix
+10
+
+Algorithm 3 Optimality-driven Perceptron Algorithm
+1: Input: Dataset D(T ) = {(x∗
+t , At, bt, zt, B∗
+t , N ∗
+t )}T
+t=1
+2: Let Θ1 = 0 ∈ Rn×d
+3: %% We use (M)i to denote the i-th row of the matrix M
+4: for t = 1, ..., T do
+5:
+Predict the objective by ˆct = Θtzt
+6:
+Solve the following LP and denote its optimal solution by xt
+min ˆc⊤
+t x,
+s.t. Atx = bt, x ≥ 0.
+7:
+Let Θtmp = Θt
+8:
+for i = 1, ..., n do
+9:
+if i /∈ N ∗
+t then
+10:
+Continue
+11:
+end if
+12:
+Predict the objective by ˆc(i)
+t
+= Θtmpzt
+13:
+Compute the reduced cost
+r(i)⊤
+t
+= ˆc(i)⊤
+t
+− ˆc(i)⊤
+t,B∗
+t A−1
+t,B∗
+t At
+14:
+if r(i)
+t,i ≤ 1
+2 then
+15:
+Update
+(Θtmp)i ← (Θtmp)i + z⊤
+t
+(Θtmp)B∗
+t ← (Θtmp)B∗
+t − A−1
+B∗
+t Atzt
+16:
+end if
+17:
+end for
+18:
+Let Θt+1 = Θtmp
+19: end for
+20: Output: {xt}T
+t=1, {Θt}T +1
+t=1
+A. The experiments compare the proposed MOM algorithms against several benchmarks and illustrate
+different aspects, such as loss performance, computational time, scale consistency/invariance, sample
+complexity, kernelized version of MOM, and online setting.
+Specifically, we consider a shortest path (SP) problem here following the setup of Elmachtoub and Grigas
+(2022). The SP problem is defined on a 5 × 5 grid network with n = 40 directed edges associated with
+a cost vector c ∈ Rn, and the covariates z ∈ Rd with d = 6. For the training data, the covariates
+are generated from the Gaussian distribution, and the objective vector is generated from a polynomial
+mapping from the covariates with additional white noise. For this numerical experiment, we consider
+the performance measure of relative loss defined by
+Relative Loss := c⊤
+new(xnew − x∗
+new)
+c⊤x∗new
+where cnew is the objective vector of a new test sample, xnew is the predicted optimal solution, and x∗
+new
+is the true optimal solution. Indeed, this relative loss normalizes the estimate loss lest with the optimal
+objective value. We refer more details on the experiment setup to Appendix A.
+Figure 1 presents the experiment results for two MOM algorithms and a few benchmarks, and each
+panel represents a different degree of the polynomial that governs the true mapping from the covariates
+to the objective vector; a higher degree indicates a stronger non-linearity between the covariates and the
+objective. We make the following observations:
+11
+
+RF
+OLS
+Ridge
+SPO+
+MOM
+MOM-OGD
+0.00
+0.05
+0.10
+0.15
+Relative Loss
+Degree 1
+RF
+OLS
+Ridge
+SPO+
+MOM
+MOM-OGD
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Degree 2
+RF
+OLS
+Ridge
+SPO+
+MOM
+MOM-OGD
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Degree 4
+RF
+OLS
+Ridge
+SPO+
+MOM
+MOM-OGD
+0
+10
+20
+30
+40
+Degree 6
+Random Forest
+Ordinary Least Squares
+Ridge Regression
+SPO+
+Maximum Optimality Margin
+MOM-Online Gradient Descent
+Figure 1: Relative loss. The plot (means and confidence intervals) is generated based on 30 random
+trials each with 1000 training samples and 1000 test samples (20% of training samples are used for tuning
+hyper-parameters). The degree indicates the degree of the true polynomial function that generates the
+objective c from the covariates z. RF denotes random forests, OLS denotes ordinary least squares, Ridge
+denotes the ridge linear regression, SPO+ implements the algorithm of Elmachtoub and Grigas (2022),
+MOM implements Algorithm 1, and MOM-OGD implements Algorithm 2. For all these methods, they
+first predict the objective vector for a test sample, and solve an LP with the predicted objective for the
+optimal solution as the predicted solution for this test sample.
+First, from an information viewpoint, all four benchmark algorithms utilize the observations of ct’s on
+the training data, while the two MOM algorithms only observe x∗
+t ’s on the training data. In this sense,
+our algorithms give a better predictive performance with less amount of information, and therefore our
+algorithms are the only ones applicable to the inverse LP problem.
+Second, other than the two MOM algorithms, the SPO+ performs the best among the four benchmark
+algorithms. However, the SPO+ takes significantly longer training time than all the other algorithms.
+Although SPO+ is a stochastic gradient descent-based algorithm, the calculation of the stochastic gra-
+dient at each step requires solving k LPs with k being the number of training samples in the mini-batch.
+Comparatively, our MOM Algorithm 1 only solves a quadratic program for once, and its online version
+of Algorithm 2 allows a direct calculation of the online sub-gradient which makes it even faster than
+Algorithm 1.
+Third, the three ML-based methods (RF, OLS, Ridge) perform generally worse than the other three
+methods (SPO+, MOM, MOM-OGD) because they do not take into account the optimization structure.
+To see this, for the shortest path problem, there may be some large entries for the cost vector c. The ML
+models treat all the dimensions of c equally and spend too much effort in fitting those large entries (as
+they cause large L2 errors). However, from the optimization perspective, an accurate estimation of those
+large entries is not useful in that the optimal path will avoid those edges. That highlights the intuition
+behind the SPO+ and our MOM algorithms – one needs to incorporate the optimization structure into
+the learning model.
+Concluding remarks. In this paper, we develop a new approach of maximum optimality margin
+for the problems of predict-then-optimize and inverse LP. The MOM framework leads to new charac-
+terizations of the difficulty of the problem through the separability condition of Assumption 3 and the
+violation loss l(Dnew; Θ) which appears in Propositions 2, 3, and 4. With the MOM perspective, we
+derive bounds on correctly predicting the optimal solution,
+P(x∗
+new = ˆxnew)
+12
+
+in Propositions 1, 4 and 5, which is rarely seen in the existing literature even under strong conditions.
+Without the separability condition of Assumption 3, we draw a connection of our MOM objective value
+with the suboptimality loss lsub in the inverse LP literature and derive performance bounds in terms
+of lsub. Numerically, we observe the MOM algorithms also perform well under the estimate loss lest.
+To theoretically quantify the performance of MOM under lest, we conjecture that one needs to impose
+stronger structures on the dual LPs. Besides, another interesting future direction to pursue is to generalize
+the algorithms and analyses of MOM to non-linear optimization problems.
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+Mohajerin Esfahani, Peyman, Soroosh Shafieezadeh-Abadeh, Grani A Hanasusanto, Daniel Kuhn. 2018.
+Data-driven inverse optimization with imperfect information. Mathematical Programming 167(1) 191–
+234.
+Shalev-Shwartz, Shai, Ohad Shamir, Nathan Srebro, Karthik Sridharan. 2010. Learnability, stability and
+uniform convergence. The Journal of Machine Learning Research 11 2635–2670.
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+of Computational and Applied Mathematics 72(2) 261–273.
+14
+
+A
+Numerical Experiments
+In this section, we provide the details of the experiment setup and more numerical results.
+A.1
+Basic Setup
+For the numerical experiments, we compare the performance of our proposed algorithms against several
+benchmark methods, and we test the performance under both online and offline settings. Specifically,
+we consider the following benchmark methods where all of them can only be used to solve the predict-
+then-optimize problem (but not the contextual linear programming) as they all require the observations
+of ct on the training data.
+1. Linear regression models where we consider ordinary least squares (OLS) and ridge regression (RR).
+Accordingly, the parameter estimation follows the following loss functions:
+lOLS = 1
+2∥Θz − c∥2
+2,
+lRidge = 1
+2∥Θz − c∥2
+2 + λ
+2 ∥Θ∥2
+F.
+2. Random Forest (RF). We apply RF algorithm with squared error criterion
+lRF = ∥ˆc − c∥2
+2.
+3. Predict-then-optimize method. We take SPO+ algorithm (Elmachtoub and Grigas, 2022) as an
+example, where the loss function is as follows
+lSPO+ = (2Θz − c)⊤x∗(c) −
+min
+Ax=b,x≥0{(2Θz − c)⊤x}.
+Following the previous implementations, we optimize the SPO+ loss under stochastic gradient
+descent (SGD) approach and Frobenius regularization.
+Underlying LP problems.
+We study two canonical LP problems for the numerical experiments, namely the shortest path (SP)
+problem and the fractional knapsack (FK) problem, and we present their results in the following two
+sections A.2 and A.3, respectively. The experiment setup follows that of Elmachtoub and Grigas (2022);
+Ho-Nguyen and Kılınç-Karzan (2022); Besbes et al. (2021).
+Performance Measurements.
+We compare different algorithms by the performance measure of relative loss for the offline setting
+and cumulative Regret in the online setting.
+Under the offline setting, the relative loss normalizes the estimate loss lest by the optimal value,
+Rel-LossSP := c⊤(x − x∗)
+c⊤x∗
+.
+Specifically, for the shortest path (SP) problem, the optimal value is always non-zero (unless for trivial
+c) so the loss is well defined.
+For the fractional knapsack (FK) problem, due to the random noise, the optimal value may be zero
+with positive probability. So we consider another normalization which leads to
+Rel-LossFK := c⊤(x∗ − x)
+∥c∥2
+.
+15
+
+Under the online setting, we define the cumulative regret by the cumulative sum of the relative loss
+of each time step.
+Overview of the results.
+In Appendix A.2, we demonstrate the algorithm performance under the effect of degree (see Figure 1)
+and also investigate numerically the issue of scale inconsistency/invariance (see Figure 2). In Appendix
+A.3, we examine the sample complexity under the offline setting (see Figure 3) and compare the perfor-
+mance of several online algorithms (see Figure 5). In addition, we consider the kernelized version of the
+MOM formulation and present its performance in Figure 4.
+A.2
+Shortest Path Problem
+For the shortest path probelm, we follow the experiment setup of (Elmachtoub and Grigas, 2022). Con-
+sider a 5 × 5 grid network with n = 40 directed edges associated with a cost vector c ∈ Rn. Each edge
+is either from south to north or west to east, where the goal is to minimize the cost to traverse from
+the southwest corner to the northeast corner. For each node of the network, there is a flow balance
+constraint, encoded by Ax = b. The problem can then be formulated as the standard-form LP (1).
+For the methods of ordinary least square, ridge regression, and our maximum optimality margin,
+we solve the corresponding optimization problems using cvxpy package with GUROBI solver. For the
+random forest algorithm, we implement the sklearn.ensemble.RandomForestRegressor package. For
+the SPO+ algorithm and the online version of our MOM approach, we implement gradient descent
+approach. For each simulation trial, the training sets and test sets both consist of N = 1000 sample
+sizes. All regularization parameters are chosen from 10−6 to 102, and all step sizes are chosen from
+10−3 to 101.
+All projection radius (the diameter of K) are chosen from 0.8∥B∥F to 2.0∥B∥F.
+The
+hyper-parameters are chosen via a validation subset of N/4 samples from the training data.
+Data generation.
+Now we describe the data generation mechanism for Figure 1. For each trial of our experiments,
+we generate a gound-truth coefficient matrix Θ∗ ∈ Rn×d independently, where each component of Θ∗ is
+drawn from Bernoulli(0.5). Here n = 40 and d = 6.
+1. For the feature vector zt: the first d − 1 components are generated independently from a Unif[0, 1]
+distribution, and the last component of each zt is set to be 1 (to model the intercept term).
+2. The true cost vectors are generated as follows:
+ct,j =
+� 1
+√
+d
+(Θ∗zt)j + 3
+�deg
++ 1
+where deg is a fixed integer to be chosen to represent the non-linearity of the problem.
+Scale inconsistency.
+Figure 2 presents an experiment on scale attack that aims to illustrate the robustness of the underlying
+methods in terms of scale invariance. Specifically, in generating the training data, we follow the same
+mechanism as Figure 1 except that we multiply some objective vectors ct’s with a factor αt. Specifically,
+we have
+ct,j =
+�� 1
+√
+d
+(Θ∗zt)j + 3
+�deg
++ 1
+�
+· αt,
+16
+
+where the scaling factor
+αt =
+
+
+
+1 + ¯α,
+if zt,1 > 0.5,
+1,
+else.
+RF
+OLS
+Ridge
+SPO+
+MOM
+MOM-OGD
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+0.200
+Relative Loss
+Attack Power 0.1
+RF
+OLS
+Ridge
+SPO+
+MOM
+MOM-OGD
+0.00
+0.05
+0.10
+0.15
+0.20
+Attack Power 1.0
+RF
+OLS
+Ridge
+SPO+
+MOM
+MOM-OGD
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+Attack Power 3.0
+Random Forest
+Ordinary Least Squares
+Ridge Regression
+SPO+
+Maximum Optimality Margin
+MOM-Online Gradient Descent
+Figure 2: Relative loss under different scale attack.
+In Figure 2, we note that the performances of three ML models are more or less affected by this
+scale attack. However, the scale attack will not change the optimal solutions on the training data, so
+the method of SPO+ and our MOM algorithms which all rely on the optimal solutions, give a stable
+performance under this scale attack. Interestingly, the scale noise αt here is dependent on the covariates
+zt. When these two are independent, we establish that the linear regression model also has the property
+of scale invariance, which we refer to Appendix C.2.
+A.3
+Fractional Knapsack Problem
+For this experiment, we follow the problem setup of (Ho-Nguyen and Kılınç-Karzan, 2022). Specifically,
+each decision maker is presented with a bunch of items, and their aim is to maximize the utility (or
+equivalently, to minimize the cost) under a knapsack constraint. Different from the discrete case, frac-
+tional knapsack problem allows the decision maker to select a fraction of some certain item, making it a
+linear program. Mathematically, the decision maker solves the following LP
+min
+x,s1,s2 c⊤x,
+s.t. p⊤x + s1 = B, x + s2 = 1, x, s1, s2 ≥ 0.
+Here p ∈ Rn can be interpreted as the price vector, and B ∈ R is the budget of the decision maker. The
+corresponding utility vector should be interpreted as the opposite of the objective vector c since the LP
+considers a minimization problem. The slack variables s1 and s2 are introduced simply to convert the
+problem into a standard form.
+Data generation.
+For each simulation trial, we generate a true coefficient matrix Θ∗ ∈ Rn×d, where each component
+is a Bernoulli random variable with expectation 1/2. Here n = 10 and d = 5. We then generate the
+price vector, where each component is selected to be a random integer between 1 and 1000 in the offline
+setting. To ease the pain of large constants, we re-scale the price vector in the online setting of the next
+17
+
+subsection, where each component is uniformly distributed between 0 and 1. We generate two auxiliary
+variables low and high, where low = maxj pj and high = 1⊤p − u · low, and u ∼ Unif[0, 1]. Then the
+budget is chosen as Unif[low, high].
+1. For the feature vector zt: the first d − 1 components are generated independently from a Unif[0, 1]
+distribution, and the last component of each zt is set to be 1 (to model the intercept term).
+2. The true cost vectors are generated as follows:
+ct,j = (Θ∗zt)deg
+j
+· ǫt,j + ¯ηηt,j,
+where deg is a fixed integer that represents the intensity of the non-linearity dependency, ǫt,j
+represents the intrinsic diverse noise, and ηt,j is the additive noise. The constants ¯ǫ and ¯η are
+non-negative parameters chosen to control the power of each noise. Specifically,
+ǫt,j ∼ Unif[1 − ¯ǫ, 1 + ¯ǫ],
+2ηt,j + 1 ∼ Exponential(1).
+Sample complexity.
+100
+200
+500
+1000
+2000
+Number of Training Samples
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Relative Loss
+RF
+OLS
+Ridge
+SPO+
+MOM
+MOM-OGD
+Figure 3: Sample complexity of different algorithms.
+Figure 3 illustrates the sample efficiency of different algorithms, and the plot is based on an average
+over 30 random trails. The parameters are chosen by deg = 1, ¯ǫ = 0.1, ¯η = 1.0, and ¯α = 0.0. For
+training sets, we generate different numbers of training samples N = 100, 200, 500, 1000, 5000, and test
+the performance against 1000 test samples. As the underlying true model is a linear model, the two
+linear ML models unsurprisingly achieve the best performance. The performance of the random forest
+algorithm is similar to that of offline MOM algorithm (Algorithm 1), and both are slightly worse than
+that of linear regression models. As for SPO+ method, their averaged performance is undermined by the
+noise to a fairly large extent, compared with its performance under noiseless environment (see Figure 1
+and Figure 2). The performance difference between SPO+ and MOM can be partially explained by the
+different ways of dealing with noisy data. The SPO+ method directly utilizes the optimal solution of
+the noisy data to the gradient, while the optimal solution could probably vary drastically due to some
+18
+
+noise to the cost vector. In comparison, the margin-based MOM performs more steadily: for those noisy
+cost vectors, their optimal solutions could be far from the noiseless cost vectors, but the change with
+respect to optimal basis will usually be only a few components which only affects a few corresponding
+lines in estimated Θ. Compared to these ML models which utilize the observations of ct, although the
+MOM algorithm (Algorithm 1) consumes less information in the training phase, it does not exhibit a
+higher requirement of the number of training samples. The online version of MOM (Algorithm 2) does
+see a decreasing error with more training samples, and this is because the nature of the gradient-based
+algorithm does not take the full advantage of observed samples.
+Kernelization and non-linear MOM.
+In Figure 1, the performances of all the methods deteriorate significantly as there is a stronger non-
+linearity between the objective c and the covariates z. Here we explore the kernelized version of MOM
+and find that it effectively improves the performance of MOM when the underlying dependency is non-
+linear. Specifically, we consider three kernelized versions of MOM by different ways to kernelize the
+covariates: Linear MOM (Linear), Polynomial Kernelized MOM (PolyKer), and Radial Basis Function
+Kernelized MOM (RbfKer). The degree of the polynomial kernelized MOM is chosen from {1, 2, 3, 4}
+and the scale parameters γ’s of both kernelized MOM methods are chosen from {0.1, 0.5, 1, 2, 3, 4, 5}.
+All regularization parameters are chosen from 10−6 to 102. The hyper-parameters are determined via a
+validation set of 25% training samples.
+Linear
+PolyKer
+RbfKer
+0.0
+0.1
+0.2
+0.3
+Relative Loss
+Degree 1
+Linear
+PolyKer
+RbfKer
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+Degree 2
+Linear
+PolyKer
+RbfKer
+0.00
+0.05
+0.10
+0.15
+0.20
+Degree 4
+Linear
+PolyKer
+RbfKer
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+Degree 6
+Linear
+Polynomial Kernelized
+Radial Basis Function Kernelized
+Figure 4: Relative loss for kernelized MOMs.
+Figure 4 presents the performance of the three kernelized MOM models. The performance of linear
+MOM (Algorithm 1) is worsened as the degree of the model grows (just as Figure 1), since the linear
+mapping induces a stronger model misspecificaion. But kernelized MOM methods give good performance
+under high degrees. In comparison of Figure 1, we note that the kernelized versions of MOM can be
+quite effective in handling nonlinear dependency, and the loss does not increase much with the increase
+of the degree of non-linearity.
+A.4
+Online Setting
+Next we analyze the online setting for the MOM algorithms. We compare the performance of the OGD
+version of MOM (Algorithm 2), the perceptron version of MOM (Algorithm 3), and also a follow-the-
+regularized-leader (FTRL) version of the MOM Algorithm 1 as follows. Basically, Algorithm 4 solves the
+19
+
+MOM optimization formulation 3 at each time t and use the estimated parameter to predict the optimal
+solution of the next time step.
+Algorithm 4 Follow-the-Regularized-Leader MOM
+1: Input: Dataset D(T ) = {(x∗
+t , At, bt, zt, B∗
+t , N ∗
+t )}T
+t=1, the set K
+2: Initialize Θ1 = 0 ∈ Rn×d
+3: for t = 1, ..., T do
+4:
+Predict the objective by ˆct = Θtzt
+5:
+Solve the following LP and denote its optimal solution by xt
+min ˆc⊤
+t x,
+s.t. Atx = bt, x ≥ 0.
+6:
+Solve the optimization problem (3) with {(x∗
+s, As, bs, zs, B∗
+s, N ∗
+s )}t
+s=1, λ = 1/
+√
+t and K
+7:
+Let Θt+1 be the optimal solution
+8: end for
+9: Output: {xt}T
+t=1
+Algorithm 4 gives another interpretation of the MOM formulation. Under an online setting, if we
+solve the MOM optimization problem (3) at each time, this is equivalent to a follow-the-regularized
+online algorithm to solve the problem. The theoretical analysis of Algorithm 4 can be extended from
+Proposition 1 and Proposition 2.
+0
+200
+400
+600
+800
+1000
+Time Step
+0
+20
+40
+60
+80
+100
+120
+Averaged Regret
+Alg3: Perceptron
+Alg1/4: MOM-FTRL
+Alg2: MOM-OGD
+Figure 5: Cumulative regret curve (averaged over 30 trials) for online algorithms.
+Figure 5 presents the cumulative regret of several algorithms for a fractional knapsack problem. In
+this numerical experiment, we let the objective vector c = Θ∗z almost surely for some fixed Θ∗ so as
+to achieve the separability condition. We emphasize that it only ensures the existence of a parameter
+Θ∗ to meet the optimality condition (2) but not with a margin of 1 as Assumption 3. From the plot,
+we observe that Algorithm 3 has the worst performance though it features for the best theoretical
+dependency on T . We remark that this regret curve does not contradict the bounded number of mistakes
+in Proposition 5; we find that when we increase the horizon to T = 100, 000, the regret curve of Algorithm
+3 becomes flattened. For Algorithm 2 and Algorithm 4, both achieve significantly better performance.
+Comparatively, 4 performs better than Algorithm 2, with the price of more computation cost (to solve
+a quadratic program at each time period).
+Importantly, we also implement the online algorithm of (Bärmann et al., 2018; Chen and Kılınç-Karzan,
+2020) under the same setting, and it incurs a regret linearly increasing with T ; so we do not include its
+regret curve as it will clap all the regret curves of Figure 5 to x-axis. A closer investigation shows that
+20
+
+the online algorithm of (Bärmann et al., 2018; Chen and Kılınç-Karzan, 2020) will render the estimate
+Θt → 0, as noted by the scale inconsistency issue in earlier sections. In contrast, our margin-based
+formulation plays an important rule to ensure that a small optimality condition violation is caused by
+discovering the correct mapping from z to c but not by the scale of the predicted ˆc.
+B
+Proofs
+B.1
+Proof of Lemma 1
+Here we show a stronger version of Lemma 1 as follows.
+Lemma 2. Consider an LP of the standard form (1). For any feasible basis B ⊂ [n] satisfying |B| = m
+and its complement N = [n]\B, denote x = (x1, ..., xn)⊤ and r = (r1, ..., rn)⊤ as the solution and reduced
+cost vector corresponding the basis B, respectively, i.e.,
+xB = A−1
+B b, xN = 0, r = c − A⊤(A−1
+B )⊤cB.
+Denote x∗ = (x∗
+1, ..., x∗
+n)⊤ as one optimal solution of LP (1). Then, we have that rB = 0, and
+c⊤x − c⊤x∗ ≤ max
+i∈[n] x∗
+i ·
+�
+i∈N
+(−ri)+.
+(10)
+Furthermore, (10) implies that x is an optimal solution if r ≥ 0.
+Proof. First, it is easy to verify that rB = 0. Then, we only need to show that the inequality (10) holds.
+For any feasible solution x′ = (x′
+1, ..., x′
+n) ≥ 0 satisfying Ax′ = b, we have
+x′
+B = A−1
+B b − A−1
+B AN x′
+N .
+It implies
+c⊤x′ = c⊤
+BA−1
+B b − c⊤
+BA−1
+B AN x′
+N + c⊤
+N x′
+N = c⊤
+BA−1
+B b + r⊤
+N x′
+N .
+(11)
+Next, from (11), we have
+c⊤x − c⊤x′ = −r⊤
+N x′
+N
+≤
+�
+i∈N
+x′
+i(−ri)+
+≤ max
+i∈[n] x′
+i ·
+�
+i∈N
+(−ri)+,
+where the first line comes from equality (11) directly, and the last two lines come from the non-negativity
+of x′ and (ri)+ for all i. Note that the above inequality holds for any feasible solution x′; by plugging in
+x′ = x∗, we obtain (10).
+B.2
+Proof of Proposition 2
+We first show Proposition 2 and then utilize the result for the proof of 1. The key is to utilize the algorithm
+stability analysis where we cite the result from (Shalev-Shwartz et al., 2010) as the following proposition.
+We refer to (Bousquet and Elisseeff, 2002; Shalev-Shwartz et al., 2010; Feldman and Vondrak, 2019) for
+more related analysis such as the high probability bound.
+21
+
+Theorem 1 (Theorem 3 in Shalev-Shwartz et al. (2010)). Let f : H× Z → R be such that H is bounded
+by B and f(h, z) is convex and L-Lipschitz with respect to h. Let , z0, z1, ..., zT be i.i.d. samples and let
+ˆhλ = arg min
+h∈H
+� T
+�
+t=1
+f(h, zi) + λ
+2 ∥h∥2
+2
+�
+.
+Then, we have
+E
+�
+f(ˆhλ, z0)
+�
+≤ inf
+h∈H E [f(h, z0)] + λ
+2 B2 + 4(L + λB)2
+λT
+.
+Proof. We refer to Theorem 2 and Theorem 3 in Shalev-Shwartz et al. (2010).
+Now we show Proposition 2 with Theorem 1.
+Proof. Recall that
+Dnew = (c, A, b, z, B∗, N ∗)
+denotes a new sample from P, and the loss function
+l(Dnew; Θ) := ∥s∥1 where s := (1|N ∗| − ˆc⊤
+N ∗ − ˆc⊤
+B∗A−1
+B∗ AN ∗)+.
+Here, (·)+ denotes the entry-wise positive part function, and B∗ and N ∗ are defined as in Section 2.
+Here, with a slight abuse of notation, we drop the sample tuple Dnew and let l(Θ) = l (Dnew; Θ). Then,
+under Assumption 2, l(Θ) is (2 + √m¯σ)-Lipschitz with respect to Θ in the Frobenius norm for a fixed
+Dnew. To see this, for any two parameters Θ = (Θ1, ..., Θn)⊤, ˆΘ = (ˆΘ1, ..., ˆΘn)⊤ ∈ Rn×d
+|l(Θ) − l(Θ′)| =
+�����
+�
+i∈N
+�
+(1 − Θ⊤
+i z + A⊤
+i (A−1
+B∗ )⊤ΘB∗z)+ − (1 − ˆΘ⊤
+i z + A⊤
+i (A−1
+B∗ )⊤ ˆΘB∗z)+
+������
+≤
+�
+i∈N
+���(1 − Θ⊤
+i z + A⊤
+i (A−1
+B∗ )⊤ΘB∗z)+ − (1 − ˆΘ⊤
+i z + A⊤
+i (A−1
+B∗ )⊤ ˆΘB∗z)+
+���
+≤
+�
+i∈N
+���−Θ⊤
+i z + A⊤
+i (A−1
+B∗ )⊤ΘB∗z + ˆΘ⊤
+i z − A⊤
+i (A−1
+B∗ )⊤ ˆΘB∗z
+���
+≤
+�
+i∈N
+����(Θi − ˆΘi)⊤z
+��� +
+���A⊤
+i (A−1
+B∗ )⊤(ΘB∗ − ˆΘB∗)z
+���
+�
+≤
+�
+i∈N
+����Θi − ˆΘi
+���
+2 ∥z∥2 + σmax
+�
+(A−1
+B∗ )⊤(ΘB∗ − ˆΘB∗)
+�
+∥Ai∥2 ∥z∥2
+�
+≤ 2∥Θ − ˆΘ∥F + √mσmax
+�
+(A−1
+B∗ )⊤�
+σmax
+�
+ΘB∗ − ˆΘB∗
+�
+≤ (2 + √m¯σ)
+���Θ − ˆΘ
+���
+F .
+Here the first line comes from the definition of l(·), the second and fourth lines are obtained by the
+convexity of the absolute value function, the third line comes from a direct computation of the positive
+part function, the fifth line is obtained by Cauchy’s inequality and the definition of σmax, the sixth line
+comes from the inequality that
+σmax(XY ) ≤ σmax(X)σmax(Y )
+for any two matrices X, Y , and the last line comes from Assumption 2 and the inequality σmax(X) ≤
+∥X∥F for any matrix X.
+By Theorem 1 and Assumption 2, we have
+E[l(ˆΘ)] ≤ min
+Θ∈K E[l(Θ)] + λ
+2
+¯Θ2 + 4(2 + √m¯σ + λ¯Θ)2
+λT
+,
+(12)
+22
+
+where
+ˆΘ = arg min
+Θ∈K
+� T
+�
+t=1
+l(Dnew, Θ) + λ
+2 ∥Θ∥F
+�
+.
+Finally, plugging λ =
+1
+√
+T into (12), we have
+E[l(ˆΘ)] ≤ min
+Θ∈K E[l(Θ)] + 28 + 8 ¯Θ2 + 7m¯σ2
+√
+T
+.
+B.3
+Proof of Proposition 1
+Proof. Under Assumption 3, there exists a Θ∗ ∈ K such that the following inequality holds almost surely
+Θ∗
+N ∗z − A⊤
+N ∗(A−1
+B∗ )⊤Θ∗
+B∗z ≥ 1.
+Thus we have
+min
+Θ∈K E[l(Θ)] = 0,
+where l(Θ) is defined following the previous proof of Proposition 2. Then, from Proposition 2,
+E[l(ˆΘ)] ≤ 28 + 8 ¯Θ2 + 7m¯σ2
+√
+T
+.
+(13)
+For a new sample (cnew, Anew, bnew, znew) with ˆcnew = ˆΘznew, we can utilize Lemma 2 and bound the
+suboptimality loss as follows
+E
+�
+ˆc⊤
+newx∗
+new − ˆc⊤
+newˆxnew
+�
+≤ E
+
+max
+i∈[n](ˆxnew)i ·
+�
+i∈N ∗
+new
+(−ri)+
+
+
+≤ E
+
+ �
+i∈N ∗
+new
+(−ri)+
+
+
+≤ E
+
+ �
+i∈N ∗
+new
+(1 − ri)+
+
+
+≤ 28 + 8 ¯Θ2 + 7m¯σ2
+√
+T
+.
+(14)
+Here the first line comes from Lemma 2, the second line comes from Assumption 2, the third line comes
+from the monotonicity of the positive part function, and the last line comes from (13).
+Now, we note that if ˆΘ renders any non-basic variables in N ∗ as basic variables, i.e.,
+ˆc⊤
+new,N ∗
+new − ˆc⊤
+new,B∗newA−1
+new,B∗newAnew,N ∗
+new ≥ 0
+does not hold, we have l(Dnew; ˆΘ) ≥ 1. Thus, by applying Markov’s inequality to , we have that with
+probability no less than 1− 28+8 ¯Θ2+7m¯σ2
+√
+T
+, ˆΘ can identify both the true optimal basis and the true optimal
+solution correctly.
+B.4
+Proof of Proposition 4
+The proof in this part is basically an application of the following lemma.
+23
+
+Lemma 3 (Theorem 3.1 in Hazan (2016)). Let {ft(x)}T
+t=1 be a sequence of convex functions defined
+on {x : ∥x∥2 ≤ K}. Suppose ∥∇ft(x)∥2 ≤ G for all x such that ∥x∥2 ≤ K and all t = 1, ..., T . Let
+xt+1 = xt −
+2K
+G
+√
+t∇ft(xt). Then, the following inequality holds
+T
+�
+t=1
+ft(xt) −
+min
+x:∥x∥2≤K
+T
+�
+t=1
+ft(x) ≤ 3KG
+√
+T.
+Proof. We refer to Theorem 3.1 in Hazan (2016).
+Next, we show Proposition 4.
+Proof. Recall the definition of lt(Dt, Θ) as follows
+l(Dt; Θ) =
+���(1|N ∗
+t | − ˆc⊤
+t,N ∗
+t − ˆc⊤
+t,B∗
+t A−1
+t,B∗
+t At,N ∗
+t )+
+���
+1 ,
+where ˆc = Θzt, and (·)+ denotes the entry-wise positive part function. For the sake of simplicity, we use
+lt(Θ) to denote l(Dt; Θ). By calculating the derivative of lt(Θ), we have
+∇ΘNtlt(Θ) = −gtz⊤
+t , ∇ΘBt lt(Θ) = (A−1
+t,Bt)⊤At,Ntgtz⊤.
+(15)
+Here, gt ∈ Rn−m is defined as follows. For the i-th element in Nt for i = 1, ..., m, correspondingly, we
+define the i-th element of gt as
+gt,i =
+
+
+
+1,
+if
+��
+eNt − ΘNtzt + A⊤
+t,Nt(A−1
+t,Bt)⊤ΘBtzt
+�
++
+�
+i
+> 0
+0,
+otherwise.
+Then, we have
+∥∇lt(Θ)∥F ≤ ∥gtz⊤
+t ∥F + ∥(A−1
+t,Bt)⊤At,Ntgtz⊤∥F
+≤ ∥gt∥2∥zt∥2 + ∥A−1
+t,Bt∥F∥At,Nt∥F ∥gt∥2∥zt∥2
+≤ √n + ∥A−1
+t,Bt∥F ∥At,Nt∥F · √n
+≤ √n + ¯σmn,
+where the first line comes from the equalities in (15) and the triangle inequality inequality ∥A + B∥F ≤
+∥A∥F + ∥B∥F for any two matrices A, B with the same size, the second line comes from the inequality
+∥AB∥F ≤ ∥A∥F ∥B∥F for any two matrices A, B and the fact that ∥g∥F = ∥g∥2 for any vector g, the
+third line comes from the definition of gt and Assumption 2 that ∥zt∥2 ≤ 1, and the last line comes from
+Assumption 2 that all entries of At are in [−1, 1] and the inequality ∥A∥F ≤ √mσmax(A) for any matrix
+A ∈ Rm×m. Next, by Lemma 3, with η =
+2 ¯Θ
+(√n+¯σmn)
+√
+T ,
+T
+�
+t=1
+lt(Θt) −
+min
+Θ:∥Θ∥F ≤ ˆΘ
+T
+�
+t=1
+lt(Θ) ≤
+�
+3 ¯Θ√n + 3¯σ ¯Θmn
+� √
+T.
+(16)
+24
+
+Then, we show inequality (8) by
+1
+T E[z⊤
+t Θ⊤
+t (x∗
+t − xt)] ≤ E
+� T
+�
+t=1
+lt(Θt)
+�
+≤ E
+�
+min
+Θ:∥Θ∥F ≤ ˆΘ
+T
+�
+t=1
+lt(Θ)
+�
++ 3 ¯Θ√n + 3¯σ ¯Θ · mn
+√
+T
+.
+Here, we can obtain the first line by Lemma 2 and a similar proof as in the proof of Proposition 1, and
+obtain the second line by inequality (16).
+Furthermore, under Assumption 3, we have with probability 1,
+min
+Θ:∥Θ∥F ≤ ˆΘ
+T
+�
+t=1
+lt(Θ) = 0,
+which implies
+E
+� T
+�
+t=1
+lt(Θt)
+�
+≤ 3 ¯Θ√n + 3¯σ ¯Θ · mn
+√
+T
+.
+(17)
+Recall that ˆΘ denotes the matrix sampled uniformly from {Θt}T
+t=1, x∗
+new denotes the optimal solution
+of LP(cnew, Anew, bnew), ˆxnew denotes the optimal solution of LP(ˆcnew, Anew, bnew), and ˆcnew = ˆΘznew.
+Similar to the last paragraph of the proof of Proposition 1, we have if ˆΘ misclassifies any non-basic
+variable in Nnew, we have l(Dnew; ˆΘ) ≥ 1. Thus,
+E[P(x∗
+new ̸= ˆxnew)] ≤ E[P(l(Dnew; ˆΘ) ≥ 1)]
+≤ E[l(Dnew; ˆΘ)]
+=
+1
+T + 1E
+�T +1
+�
+t=1
+l(Dnew; Θt)
+�
+=
+1
+T + 1E
+�T +1
+�
+t=1
+l(Dt; Θt)
+�
+≤ 3 ¯Θ√n + 3¯σ ¯Θ · mn
+√
+T + 1
+,
+by which we have shown that, with probability no less than 1 − 3 ¯Θ√n+3¯σ ¯Θ·mn
+√T +1
+, Algorithm 2 can identify
+both the true optimal basis and the true optimal solution correctly. Here, the first line comes from the
+fact that l(Dnew; ˆΘ) ≥ 1 if x∗
+new ̸= ˆxnew, the second line comes from Markov’s inequality, the third
+line comes from the definition of ˆΘ, the forth line comes from the fact that Dt and Dnew are two i.i.d.
+samples that are also independent of Θt, and the last line comes from inequality (17).
+B.5
+Proof of Proposition 5
+The proof integrates the standard analysis of the perceptron algorithm with the structure of the linear
+program.
+Proof. In this part, we denote Θt,i as the value of matrix Θtmp at the beginning on the t-th iteration of
+the outer loop and the i-th iteration of the inner loop for t ∈ [T ] and i ∈ [n]. Also, we view Θt,n+1 and
+25
+
+Θt+1,1 as the same to avoid undefined boundary cases. Recall the definition of the reduced cost vector
+rt(Θ) = Θzt − A⊤
+t (A−1
+t,Bt)⊤ΘBtzt, for t = 1, ..., T,
+which is a linear function of Θ entry-wisely.
+Thus, we have that for each t = 1, ..., T and i ∈ [n],
+there exists a matrix Wt,i ∈ Rn∗d such that r(i)
+t,i (Θ) = Trace(W ⊤
+t,iΘ) and ∥Wt,i∥F ≤ 1 + ¯σ√mn, where
+Trace(W) =
+n�
+i=1
+wii for any square matrix W = (wij)n
+i,j=1 ∈ Rn×n. Then, we define
+ht,i(Θ) = sign(Trace(W ⊤
+t,iΘ) − .5),
+where sign(·) denotes the sign function.
+Moreover, if there is an i ∈ Nt at some time t such that
+ht,i(Θt,i) = −1, we have Θt,i+1 = Θt,i+Wt,i. The updating rule in Algorithm 3 is obtained as mentioned
+above. Specifically, as in Algorithm 3, for any i ∈ Nt and t ∈ [T ], Wt,i is defined as follows
+(Wt,i)i = z⊤
+t , (Wt,i)B∗
+t = A−1
+B∗
+t Atzt,
+and all other entries are 0. Then, we have
+∥Wt,i∥2
+F = ∥zt∥2
+F + ∥A−1
+B∗
+t Atzt∥2
+F
+≤ ∥zt∥2
+F + ∥AB∗
+t ∥2
+F ∥At∥2
+F ∥zt∥2
+F ,
+(18)
+≤ 1 + ¯σ2m2n
+where the first line comes directly from the definition of Wt,i, the second line comes from the inequality
+that ∥AB∥F ≤ ∥A∥F ∥B∥F for any two matrices A, B, and the last line comes from Assumption 2.
+We say that Algorithm 3 misclassifies one non-basic variable i ∈ Nt if ht,i(Θt,i) ≤ 0, and it misclas-
+sifies one basic variable i ∈ Bt if ht,i(ΘT ) > 0. Since the values of entries of the reduced cost vector
+corresponding to the true basis Bt are 0 for all t and i, Algorithm 3 makes a mistake only if it misclassi-
+fies one non-basic variable. Denote the number of identification mistakes at the t-th iteration as Kt for
+t = 1, ..., T . Let K =
+T�
+t=1
+Kt be the number of all mistakes. From the updating rule, inequality (18) and
+the triangle inequality of the Frobenius norm, we have ∥Θt,i+1∥2
+F ≤ Kt(1 + ¯σ2m2n), and then,
+∥ΘT +1∥2
+F ≤ K(1 + ¯σ2m2n).
+(19)
+Moreover, under Assumption 3, there exists a matrix Θ∗ such that
+Trace(W ⊤
+t,iΘ∗) ≥ 1,
+if i ∈ Nt,
+Trace(W ⊤
+t,iΘ∗) ≤ 0,
+if i ∈ Bt,
+for all t = 1, ..., T . Then, we have once a mistake is made for some i ∈ Nt at time t
+Trace((Θt,i+1 − Θt,i)⊤Θ∗) = Trace(W ⊤
+t,iΘ∗) ≥ 1,
+which implies
+trace(Θ⊤
+T,n+1Θ∗) ≤ K.
+(20)
+26
+
+Then, combining inequalities (19) and (20), we have
+K ≤ trace(Θ⊤
+T,n+1Θ∗)
+≤ ¯Θ∥ΘT,n+1∥F
+≤ ¯Θ
+�
+K(1 + ¯σ2m2n),
+where the first line comes from (20), the second line comes from Assumption 3 that ∥Θ∗∥F ≤ ¯Θ and the
+Cauchy-Schwartz inequality, and the last line comes from (19). Dividing both sides by
+√
+K and taking
+square,
+K ≤ ¯Θ2 + ¯σ2 ¯Θ2m2n.
+Moreover, Lemma 1 tells that one can recover the optimal solution if the optimal basis is identified.
+This statement implies that K is an upper bound of times that we cannot identify the true optimal
+solutions by Algorithm 3. Thus, we have
+|{t ∈ [T ] : x∗
+t ̸= xt}| ≤ ¯Θ2 + ¯σ2 ¯Θ2m2n.
+For the generalization bound, we apply the symmetry of samples. Following a similar argument as
+in the last paragraph of the proof of Proposition 4, we obtain
+E [P(x∗
+new ̸= ˆxnew)] ≤
+¯Θ2 + ¯σ2 ¯Θ2m2n
+T
+.
+C
+Additional Discussions
+C.1
+Why Structured Prediction Not Work
+We argued earlier that the inequality constraints in (3) is critical in that alternative formulations such
+as (4) and (5) will fail. Here we explore another alternative – structured support vector machine (SVM)
+(Joachims et al., 2009) and show that it leads to an computationally intractable formulation of the
+problem.
+The goal of a structured SVM classifier under our context to estimate the optimal basis (or equiv-
+alently, the non-basic variables) by observing {(z, A)} and a predictor trained on {(zt, At)}’s and their
+corresponding labels {yt}’s, where we define (yt)i = +1 for i ∈ Nt and (yt)i = −1 for i ∈ Bt. The
+Maximum Optimality Margin approach is a principle to maximize the estimated reduced cost vectors for
+non-basic variables. To express this principle more explicitly, the margin term one wants to maximize in
+structured SVM can be written as
+max
+Θ∈K
+�
+j∈N
+ˆrj,
+where ˆrj’s are to be specified later.
+For the simplicity of notations, we define some auxiliary vectors and matrices named as 1, JBt, JNt,
+and Φt,yt, where
+1 ∈ Rn−m, 1j = 1, ∀j ∈ [n − m],
+JBtv = vBt, ∀v ∈ Rn,
+JNtv = vNt, ∀v ∈ Rn,
+27
+
+Φt,yt := JNt − A⊤
+t,NtA−⊤
+t,BtJBt.
+Specifically, the estimated reduced cost with respect to yt can be written as
+ˆrt,yt = Φt,ytΘzt.
+Hence the margin term (where the subscript t is sometimes omitted for simplicity) is
+�
+j∈N
+ˆrj = 1⊤ˆr = ⟨Θ, Φ⊤
+y 1z⊤⟩,
+which implies the feature map
+φ((zt, At), yt) = Φ⊤
+t,yt1z⊤
+t .
+Note that the corresponding label space
+Y = {y ∈ {−1, +1}n, #{i, yi = +1} = m}
+is exponentially large in n in general cases. We also define some measurement of difference ∆(·, ·) ∈
+Y × Y → R, where ∆(y, y′) ≥ 0 and ∆(y, y) = 0. For example, the ∆(y, y′) function can be
+1{y ̸= y′}
+and we retrieve the multiclass Hinge loss. Such ∆(y, y′) can also be defined to be the Hamming distance
+and so on.
+Equipped with those notations, the structured SVM problem can now be formulated as:
+min
+Θ∈K,s∈RN,y∈Y
+λ
+2 ∥Θ∥2 + 1
+N
+N
+�
+i=1
+si,
+s.t. si ≥ ∆(yi, y) − ⟨Θ, φ((zi, Ai), yi)⟩ + ⟨Θ, φ((zi, Ai), y)⟩,
+∀i ∈ [N], ∀y ∈ Y,
+si ≥ 0,
+∀i ∈ [N].
+We thereby note that solving the above problem requires solving another sub-problem where
+gi(Θ) := s∗
+i = max
+y∈Y {∆(yi, y) − ⟨Θ, φ((zi, Ai), yi)⟩ + ⟨Θ, φ((zi, Ai), y)⟩} .
+But the above sub-problem is computationally intractable, since the precise evaluation of gi(Θ)
+requires solving a discrete optimization problem with exponentially large feasible set Y. Such an obstacle
+makes the training hard to implement. Besides, even if we get the training result ˜Θ, the following inference
+problem
+˜f(z, A) = arg max
+y∈Y
+⟨˜Θ, φ((z, A), y)⟩
+is still highly intractable.
+C.2
+Scale Consistency of Least Squares Linear Regression
+In the previous sections, we raise the point of scale consistency several times. Specifically, the key point
+made is that the objective vectors of c and αc for α > 0 produce the same optimal solution. Therefore,
+the algorithm should account for this scale invariance as there might be some scale contamination of the
+training data in application contexts such as revealed preference/stated preference. Here we present a
+self-contained result on the scale consistency of the linear regression model that might be of independent
+interests.
+Consider the linear regression model where one wants to estimate the true coefficient matrix using
+28
+
+cost vectors ci ∈ Rn and feature vectors zi ∈ Rd.
+Instead of observing true ci’s, we only observe
+their perturbed/contaminated versions with scale noises (1 + αi)ci’s, where αi’s here are some random
+variables. We claim that if αi’s are i.i.d. generated and independent of zi and ci, then the ordinary least
+squares method is still consistent if E[α] = 0.
+Proposition 6 (Scale Consistency of Ordinary Least Squares). Assume c = Θ∗z + ǫ, where E[ǫ|z] = 0,
+Θ∗ ∈ Rn×d is the true underlying coefficient matrix. Assume α is independent of z and c. Assume we
+observe N i.i.d. samples (zi, (1 + αi)ci)’s. Assume zi, ci, and αi have finite second order moments,
+which implies that they follow the strong law of large numbers. Further assume that ci and αi have finite
+fourth order moment, which implies the strong law of large numbers for c2
+i and α2
+i . Assume E[ziz⊤
+i ] = Σ,
+and Σ is non-singular. We estimate the underlying coefficient matrix by ordinary least squares method:
+ˆΘN = arg min
+Θ
+�
+fN(Θ) :=
+N
+�
+i=1
+1
+N ∥(1 + αi)ci − Θzi∥2
+2
+�
+.
+Then
+ˆΘN
+a.s.
+−−→ (1 + E[α])Θ∗,
+as N → ∞.
+Proof. W.l.o.g. we only prove the one-dimensional case c ∈ R, since multi-dimensional cases can be
+proved similarly by breaking Θ = (Θ1, . . . , Θn)⊤ into n independent Θi’s. For one-dimensional case, the
+estimator is now minimizing
+fN(Θ) = 1
+N
+N
+�
+i=1
+∥z⊤
+i Θ − (1 + αi)ci∥2
+2
+= Θ⊤
+�
+1
+N
+N
+�
+i=1
+ziz⊤
+i
+�
+Θ −
+�
+1
+N
+N
+�
+i=1
+2(1 + αi)ciz⊤
+i
+�
+Θ + 1
+N
+N
+�
+i=1
+(1 + αi)2c2
+i .
+Note that we assume that zi’s, ci’s, c2
+i ’s, αi’s, α2
+i ’s all have finite second order moments, which implies
+1
+N
+N
+�
+i=1
+ziz⊤
+i
+a.s.
+−−→ Σ,
+as N → ∞,
+1
+N
+N
+�
+i=1
+2(1 + αi)ciz⊤
+i
+a.s.
+−−→ 2E[(1 + α)cz⊤],
+as N → ∞,
+1
+N
+N
+�
+i=1
+(1 + αi)2c2
+i
+a.s.
+−−→ E[(1 + α)2c2] = (1 + 2E[α] + E[α2])E[c2] < ∞,
+as N → ∞.
+Combining the boundedness of the last component with the fact that Σ is non-singular and positive
+semidefinite and the fact that 2E[(1 + α)cz⊤] is bounded (which will be shown later), we have
+fN
+a.s.
+−−→ f,
+as N → ∞,
+where f is a positive definite quadratic function of Θ.
+Therefore, the unique minimizer of f must be its first order stationary point. We now compute the
+29
+
+partial derivatives. We have
+E[(1 + α)cz⊤] = E[1 + α]E[cz⊤]
+= (1 + E[α])E[E[cz⊤|z]]
+= (1 + E[α])E[E[Θ∗⊤zz⊤ + ǫz⊤|z]]
+= (1 + E[α])E[Θ∗⊤zz⊤],
+= (1 + E[α])Θ∗⊤Σ.
+where the first equality is from the fact that αi’s are independent of zi’s and ci’s, the second equality
+follows the tower law of conditional expectation, the third equality comes from the linear assumption,
+and the fourth equality comes from E[ǫ|z] = 0. It follows immediately that
+∂f
+∂Θ = 2ΣΘ − 2(1 + E[α])ΣΘ∗.
+Since Σ is non-singular, we have proved that the unique minimizer of f is exactly (1 + E[α])Θ∗.
+Then from the fact that fN
+a.s.
+−−→ f and f is positive definite quadratic function, we have
+ˆΘN = arg min fN
+a.s.
+−−→ arg min f = (1 + E[α])Θ∗,
+as N → ∞.
+Specifically, if E[α] = 0, we retrieve the true Θ∗.
+30
+
diff --git a/LdFIT4oBgHgl3EQfbCu_/content/tmp_files/load_file.txt b/LdFIT4oBgHgl3EQfbCu_/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..9027cd1352fd8579ed85ed71ef85ee712a7ea447
--- /dev/null
+++ b/LdFIT4oBgHgl3EQfbCu_/content/tmp_files/load_file.txt
@@ -0,0 +1,998 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf,len=997
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='11260v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='LG]  26 Jan 2023  Maximum Optimality Margin: A Unified Approach for  Contextual Linear Programming and Inverse Linear  Programming  Chunlin Sun∗, Shang Liu∗, Xiaocheng Li  Institute for Computational and Mathematical Engineering, Stanford University, chunlin@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='edu  Imperial College Business School, Imperial College London, (s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='liu21, xiaocheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='li)@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='uk  Abstract  In this paper, we study the predict-then-optimize problem where the output of a machine learning  prediction task is used as the input of some downstream optimization problem, say, the objective  coefficient vector of a linear program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The problem is also known as predictive analytics or contextual  linear programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The existing approaches largely suffer from either (i) optimization intractability  (a non-convex objective function)/statistical inefficiency (a suboptimal generalization bound) or (ii)  requiring strong condition(s) such as no constraint or loss calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We develop a new approach  to the problem called maximum optimality margin which designs the machine learning loss function  by the optimality condition of the downstream optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The max-margin formulation enjoys  both computational efficiency and good theoretical properties for the learning procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  More  importantly, our new approach only needs the observations of the optimal solution in the training  data rather than the objective function, which makes it a new and natural approach to the inverse  linear programming problem under both contextual and context-free settings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' we also analyze the  proposed method under both offline and online settings, and demonstrate its performance using  numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  1  Introduction  The predict-then-optimize problem considers a learning problem under a decision making context where  the output of a machine learning model serves as the input of a downstream optimization problem  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' a linear program).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The ultimate goal of the learner is to prescribe a decision/solution for the  downstream optimization problem using directly the input (variables) of the machine learning model  but without full observation of the input of the optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' A similar problem formulation  was also studied as prescriptive analytics (Bertsimas and Kallus, 2020) and contextual linear program-  ming (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' While Elmachtoub and Grigas (2022) justify the importance of leveraging the  optimization problem structure when building the machine learning model and the existing efforts on  exploiting the optimization structure have been largely inadequate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' In this paper, we delve deeper into  the structural properties of the optimization problem and propose a new approach called maximum  optimality margin which builds a max-margin learning model based on the optimality condition of the  downstream optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' More importantly, our approach only needs the observations of the  optimal solution in the training data rather than the objective function, thus it draws an interesting  connection to the inverse optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The connection gives a new shared perspective on both  the predict-then-optimize problem and the inverse optimization problem, and our analysis reveals a scale  inconsistency issue that arises practically and theoretically for many existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  ∗ Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  1  Now we present the problem formulation and provide an overview of the existing techniques and  related literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Consider a linear program (LP) that takes the following standard form  LP(c, A, b) := min c⊤x,  (1)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Ax = b, x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  where c ∈ Rn, A ∈ Rm×n, and b ∈ Rm are the inputs of the LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' In addition, there is an available feature  vector z ∈ Rd that encodes the useful covariates (side information) associated with the LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Predict-then-optimize/Contextual linear programming  The problem of predict-the-optimize or contextual LP is stated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' A set of training data  DML(T ) := {(ct, At, bt, zt)}T  t=1  consists of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' samples from an unknown distribution P, where ct ∈ Rn, At ∈ Rm×n, bt ∈ Rm, and  zt ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Throughout the paper, we assume the constraints have a fixed dimensionality for notational  simplicity, while all the results can be easily extended to the case of variable dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The goal  of a conventional machine learning (ML) model is to identify a function g(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Θ) : Rd → Rn that best  predicts the objective coefficient vector ct using the covariates zt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' However, the predict-then-optimize  problem has a slightly different pipeline for the testing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' It aims to map from the observation of  the context and the knowledge of the constraints to a decision (from data to decision):  (Anew, bnew, znew) → xnew  without the observation of cnew, where the tuple (cnew, Anew, bnew, znew) is a new test sample from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  That is, for this new test sample, the decision maker knows the constraints (Anew, bnew), and the aim is  to predict the unknown objective cnew from znew and prescribe the decision variable xnew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  There are commonly three performance measures for the problem (as noted by Chen and Kılınç-Karzan  (2020))  Prediction loss: lpre(ˆΘ) = E  �  ∥cnew − ˆcnew∥2  2  �  Estimate loss: lest(ˆΘ) = E  �  c⊤  newxnew − c⊤  newx∗  new  �  Suboptimality loss: lsub(ˆΘ) = E  �  ˆc⊤  newx∗  new − ˆc⊤  newˆxnew  �  where ˆcnew = g(znew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' ˆΘ) is the predicted output of the ML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Here ˆxnew and x∗  new are the optimal  solutions of LP(ˆcnew, Anew, bnew) and LP(cnew, Anew, bnew), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The expectations are taken with  respect to this new sample (following the distribution P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  The prediction loss is aligned with the standard ML problems, where the L2 loss can also be re-  placed by other proper metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The estimate loss captures the quality of the recommended decision  xnew under the true (unobserved) cnew and it is a more natural loss given the interests in the down-  stream optimization task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The suboptimality loss measures how well the predicted ˆcnew explains the  realized optimal solution x∗  new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' It is more commonly adopted in the inverse linear programming litera-  ture (Mohajerin Esfahani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Bärmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Chen and Kılınç-Karzan, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Inverse linear programming  Our proposed approach to the predict-then-optimize problem also solves a seemingly unrelated prob-  lem – inverse linear programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' In the language of predict-then-optimize, the problem of inverse linear  2  programming considers a set of training data  Dinv(T ) := {(x∗  t , At, bt, zt)}T  t=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Similarly as the previous case, the tuples (ct, At, bt, zt)’s are generated from an unknown distribution P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Differently, for inverse linear programming, the optimal solution x∗  t instead of the objective coefficient  vectors ct is given in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' While the classic setting of inverse linear programming does  not consider the context, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', zt ≡ 1 for all t, several recent works (Mohajerin Esfahani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Besbes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', 2021) study the contextual case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The goal of the inverse linear programming is similar as  that of the predict-then-optimize problem, to learn a function g(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Θ) that maps from the context zt to  the objective ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We emphasize that the observation of x∗  t gives much less information than that of ct  which makes the inverse problem more challenging than the predict-then-optimize problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  In the following, we briefly review the representative existing approaches to tackle the two problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Predicting the objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  The first class of approaches treats the predict-then-optimize problem as a two-step procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  The learner first learns a function g(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' ˆΘ) : z → c from the training data, and then prescribes the  final output xnew by the optimal solution of LP(g(znew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' ˆΘ), Anew, bnew).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' There are mainly two exist-  ing routes for how the parameter ˆΘ should be estimated from the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  The first route is  to ignore the constraint (At, bt) (in the training data) and treat the training problem as a pure ML  problem (Ho-Nguyen and Kılınç-Karzan, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  The issue of this route is that there can be a mis-  alignment between the ML loss lpre(·) and the downstream optimization loss lest(·) or lsub(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Empiri-  cally, it causes sample inefficiency if one is interested in lest(·) (noted by Elmachtoub and Grigas (2022)  and also observed in our numerical experiments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Theoretically, to establish a performance guaran-  tee for the optimization loss, it requires a calibration condition which can be hard to satisfy/verify  (Ho-Nguyen and Kılınç-Karzan, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The second route is to employ the optimization loss lest(·) for  the estimation of ˆΘ (Elmachtoub and Grigas, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Elmachtoub et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' However, the loss function  is generally non-convex in the parameter Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' A convex surrogate loss called SPO+ is proposed, but the  sample complexity yielded under the surrogate loss can be of a suboptimal rate (El Balghiti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  In the literature of inverse LP, the (convex) suboptimality loss lsub(·) is often used instead of lest(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Our proposed approach falls in this category of first predicting the objective and then solving the LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Different from all the existing works, we derive a new formulation based on the optimality condition of  the optimization problem that features for both statistical and computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Predicting the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  The second class of approaches treats the predict-then-optimize problem as a one-step procedure, and  aims to learn an end-to-end function that maps from the context zt directly to the optimal solution x∗  t  (Bertsimas and Kallus, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' This end-to-end treatment works well in the unconstrained  setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' But for the constrained setting, the optimal solution of an LP generally stays at the corner of  the feasible simplex, and the end-to-end mapping can hardly predict such a corner solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' In the  domain of inverse linear programming, a recent work (Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', 2020) also proposes a loss function  minimizing the gap between the predicted optimal solution and the observed optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The idea is  more aligned with the early formulation of inverse optimization (Zhang and Liu, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Ahuja and Orlin,  2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' However, the formulation in (Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', 2020) is more of a conceptual framework that is hardly  computationally tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Consistency/Feasibility check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  3  For the inverse linear programming problem, a common approach in literature (Zhang and Liu, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Ahuja and Orlin, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Boyd and Vandenberghe, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Bertsimas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Bärmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Besbes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', 2021) is to transform the knowledge of the optimal solution x∗  t equivalently to constraints  on the objective vector ct ∈ Ct for some polyhedron Ct (through the optimality condition), and then uti-  lize these Ct to make inference about the objective vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' There can be two issues with this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  First, many existing results study the context-free case and require a non-empty intersection of the poly-  hedrons, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', ∩T  t=1Ct ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' This requirement may fail in a noisy or contextual setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Second, from a  methodology perspective, it is difficult to relate the structure of Ct with the covariates zt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' To this end,  our paper complements the results of the existing literature by providing a new approach for the inverse  linear programming problem for both context-free and contextual setting and with less assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  2  LP Optimality Structure and Main Algorithm  Now we first present some preliminaries on linear programming and then describe our main algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Let x∗ = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', xn)⊤ be the optimal solution of the LP(c, A, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The optimal basis B∗ and its complement  N ∗ of an LP are defined as follows  B∗ := {i : x∗  i > 0},  N ∗ := {i : x∗  i = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  For a set B ⊂ [n], we use AB to denote the submatrix of A with column indices corresponding to B  and cB to denote the subvector with corresponding dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We make the following assumptions on  the LP’s nondegeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Assumption 1 (Nondegeneracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' All the linear programs in this paper are nondegenerate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', satisfying  the following two conditions:  (a) The LP is feasible and has a unique optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  (b) All the submatrices AB are invertible for |B| = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The optimal basis satisfies |B∗| = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  The assumption is standard in the literature of LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' It is a mild one in that any LP can satisfy the  assumption under an arbitrarily small perturbation (Megiddo and Chandrasekaran, 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We also note  that our focus on the standard-form LP (1) is without loss of generality because all the LPs can be  written in the standard form and the results in this paper can be easily adapted to an LP of other forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  The following lemma describes the optimality condition for a linear program in terms of its input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Lemma 1 (Luenberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' (1984)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Let B ⊂ [n] be a feasible basis of LP(c, A, b), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', A−1  B b ≥ 0, and let  N = [n]\\B be its complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Then, under Assumption 1, the following inequality holds (element-wise)  c⊤  N − c⊤  BA−1  B AN ≥ 0  (2)  if and only if B = B∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  The result provides a foundation for the simplex method to solve LPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' When B = B∗, the vector  p∗ := c⊤  B∗A−1  B∗ ∈ Rm will be the optimal solution of the dual LP of (1), also known as the dual price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' It  assigns a shadow price for per unit consumption of each constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' For a decision variable index i ∈ [n],  when ci > A⊤  i p∗ (where Ai is the i-th column of A), it means the choice of the i-th variable is not cost  effective, and thus x∗  i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='1  Maximum optimality margin  Now we present our approach of maximum optimality margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Consider the following training data  Dinv(T ) = {(x∗  t , At, bt, zt)}T  t=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Note that the training data only consists of the observations of the optimal solution x∗  t , so all the algo-  rithms and analyses apply to both the predict-then-optimize problem and the inverse linear programming  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Denote the sets of basic variables and non-basic variables of the t-th training sample as B∗  t ∈ [n]  and N ∗  t ∈ [n], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Then the training data can be equivalently re-written as  Dinv(T ) = {(x∗  t , At, bt, zt, B∗  t , N ∗  t )}T  t=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Specifically, we start with a linear mapping from the covariates zt to the objective vector ct, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=',  ct = g(zt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Θ) := Θzt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Our maximum optimality margin method solves the following optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  ˆΘ := arg min  Θ∈K  λ  2 ∥Θ∥2  2 + 1  T  T  �  t=1  ∥st∥1  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  ˆct = Θzt,  t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', T,  (3)  ˆc⊤  t,N ∗  t − ˆc⊤  t,B∗  t A−1  t,B∗  t At,N ∗  t ≥ 1|N ∗  t | − st,  t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', T,  where the decision variables are Θ ∈ Rn×d, ˆct ∈ Rn, and st ∈ R|Nt|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' K is a convex set to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  To interpret the optimization problem (3):  The equality constraints encode the linear prediction function, and ˆct’s are the predicted value of  ct under parameter Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  For the inequality constraints, the vector 1|N ∗  t | ∈ R|N ∗  t | is an all-one vector of dimension |N ∗  t |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The  left-hand-side of the inequality comes from the optimality condition (2) in Lemma 1, while the  right-hand-side represents the slackness or margin of the optimality condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' From the objective  function, it is easy to see that the optimal solution will always render st ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Then, when st ∈ [0, 1]  element-wise for all t, the inequality constraints fulfill the optimality condition (2) perfectly, which  means that the optimal solution ˆΘ is consistent with all the training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' In other words, when  st ∈ [0, 1] element-wise for all t, if we predict the objective coefficient ct with ˆct = ˆΘzt, the two  LPs, LP(ct, At, bt) and LP(ˆct, At, bt) share the same optimal solution for all t (by Lemma 1), and  it thus results in a zero estimate loss lest and a zero suboptimality loss lsub on the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  When some coordinate of st is greater than 1, it means the optimal solution under the predicted  objective ˆct = ˆΘzt no longer satisfies the optimality condition of the original LP(ct, At, bt), and  consequently, this results in a mismatch between the two optimal solutions of LP(ct, At, bt) and  LP(ˆct, At, bt), and thus a non-zero loss of lest and lsub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  For the objective function, the second part is justified by the above discussion on the inequality  constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We desire to have a smaller value of st as this leads to a better satisfaction of the  optimality condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The first part of the objective function regularizes the parameter Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The  rationale is that the left-hand-side of the inequality constraints that represents the realized margin  of the optimal condition, scales linearly with Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We hope the margin to be large but do not want  the large margin is due to the scaling up of Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  5  We note that the optimization problem (3) has linear constraints and a quadratic objective, and  thus it is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Algorithm 1 describes the procedure of maximum optimality margin for solving the  predict-then-optimize problem and the inverse LP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' It is a two-step procedure in that it first  estimates the parameter and then solves the optimization problem based on the estimate ˆΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Algorithm 1 Maximum Optimality Margin (MOM) Algorithm  1: Input: Dataset D(T ) = {(x∗  t , At, bt, zt, B∗  t , N ∗  t )}T  t=1, test sample (Anew, bnew, znew), λ, K  2: Solve the optimization problem (3) and obtain the estimate ˆΘ  3: Predict ˆcnew = ˆΘznew and solve the following LP  min ˆc⊤  newx,  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Anewx = bnew, x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  4: Denote its optimal solution as ˆxnew  5: Output: ˆxnew and ˆΘ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='2  Interpreting the formulation  The key idea of the MOM formulation is that it does not explicitly minimize the error of predicting  the objective ct’s as a stand-alone machine learning problem, but it aims to minimize the violation –  equivalently, maximize the margin – of the optimality conditions of the training samples (the inequality  constraint in (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Intuitively, as long as we find a prediction ˆct that shares the same optimal solution  x∗  t with ct for most t (hopefully), we should not bother with the error between ˆct and ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' This distin-  guishes from all the existing methods for predict-then-optimize and inverse LP in that it integrates the  optimization problem’s structure naturally into the machine learning training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We make the  following remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  First, our method does not require the knowledge of ct’s in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' As far as we concern,  it is the first method that works for both predict-then-optimize and inverse LP problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Beyond that,  this special feature of our method has a modeling advantage of scale consistency or scale invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Specifically, we note that for the LP (1), both the objective vector c and αc for any α > 0 lead to  the same optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Hence the loss of training a machine learning model should ideally bear a  scale invariance in terms of the objective vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' That is, a prediction of ˆc and a prediction of αˆc for  any α > 0 should incur the same training loss as both of them lead to the same prescribed solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Our method enjoys this scale invariance property, since it only utilizes the optimal solution x∗  t in the  training phase but does not involve ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' In comparison, a method that minimizes the prediction loss lpre  apparently does not have this scale invariance, so do some inverse LP algorithms (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' This  property can be critical for some application contexts such as revealed preference or stated preference  citepbeigman2006learning,zadimoghaddam2012efficiently where one aims to predict the utility ct of a  customer based on observed covariates zt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' In such contexts, even if we have observations of the ct’s  through surveying the customers’ utilities, these observations might suffer from some scale contamination  because each customer may have their own scale for measuring utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' And therefore, to strive for an  accurate prediction of the observed ct can be misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Second, the inequality constraint in (3) is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Two tempting alternatives are to replace the  inequality constraints of (3) with the following:  ˆc⊤  t,N ∗  t − ˆc⊤  t,B∗  t A−1  t,B∗  t At,N ∗  t ≥ 0,  (4)  ˆc⊤  t,N ∗  t − ˆc⊤  t,B∗  t A−1  t,B∗  t At,N ∗  t ≥ −st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  (5)  6  For (4), it directly employs the optimality condition (2) in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The issue with this formulation is  that there may exist no Θ that is consistent all the training data, and thus (4) will lead to an infeasible  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' In comparison, the inequality constraints of (3) can be viewed as a softer version of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' For  (5), it suffers a similar scale invariance problem as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Specifically, the constraint (5) will  result in ˆΘ → 0 as the left-hand-side of (5) scales linearly with ˆΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' To prevent this, one can impose an  additional unit sphere constraint by ∥Θ∥2 = 1 but this will lead to a non-convexity issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Lastly, we note a few additional points for the formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' First, the formulation does not involve bt in  the optimization problem (3) either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' This is due to that bt’s information is encoded by (At, B∗  t , N ∗  t ) under  the standard-form LP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Second, the formulation is presented under a linear mapping of zt to ct;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' non-  linearity dependency can be introduced by kernelizing the original covariates zt (Hofmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Third, the MOM formulation aims to find a parameter ˆΘ that best satisfies the optimality condition, and  it measures the quality of satisfaction by the total margin of optimality condition violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' It does not  strive for a structured prediction of the optimal bases Bt’s which will lead to a less tractable formulation  such as structured support vector machine (See Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  3  Theoretical Analysis  Now we analyze the maximum optimality margin method under both offline and online settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We  make the following boundedness assumptions mainly for notation simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Specifically, we note that the  parameter ¯σ captures some “stability” of the LP’s optimal solution under perturbation of the constraint  matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' While our algorithm utilizes the optimal solutions in the training data, a better stability of the  optimal solutions will lead to a more reliable learning of the parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Assumption 2 (Boundedness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Let the tuple (c, A, b, z, B∗, N ∗) be a sample drawn from the distribution  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The following assumptions hold with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  (a) There exists a constant ¯σ such that σmax(A−1  B∗ ) ≤ ¯σ, where σmax(·) denotes the largest singular  value function of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  (b) The covariates vector z is bounded by 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', ∥z∥2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  (c) All entries of the LP’s input (A, b, c) are within [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  (d) The feasible region {x ≥ 0 : Ax = b} is bounded by the 2-norm unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='1  Separable Case  We begin our discussion with the separable case where there exists a parameter that meets the optimality  conditions on all the training samples with a margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' There exists Θ∗ such that the following inequality holds element-wise almost surely,  ˆc⊤  N ∗ − ˆc⊤  B∗A−1  B∗ AN ∗ ≥ 1  (6)  where ˆc = Θ∗z and (c, A, b, z, B∗, N ∗) is sampled from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Suppose ∥Θ∗∥F ≤ ¯Θ for some constant ¯Θ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  The assumption is weaker than to assume c = Θ∗z almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' It only requires the existence  of a linear mapping ˆc = Θ∗z that meets the optimality condition almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' It can happen that  Assumption 3 holds but c ̸= ˆc almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The right-hand-side of (6) changes from that of (2) in  Lemma 1 from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The change is not essential when the LP is nondegenerate almost surely;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' one can  always scale up the parameter Θ∗ to meet this margin requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  7  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Under Assumptions 1, 2 and 3, let ˆΘ denote the output of Algorithm 1 with T training  samples, λ =  1  √  T and K = {Θ ∈ Rn×d : ∥Θ∥F ≤ ¯Θ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Suppose (cnew, Anew, bnew, znew) is a new sample  from P and denote ˆcnew = ˆΘznew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Let x∗  new and ˆxnew denote the optimal solutions of LP(cnew, Anew, bnew)  and LP(ˆcnew, Anew, bnew).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Then we have  E [P (x∗  new = ˆxnew)] ≥ 1 − 28 + 8 ¯Θ2 + 7m¯σ2  √  T  ,  (7)  where m is the number of constraints in each LP, the expectation is taken with respect to the training  samples, and the probability is with respect to the new test sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  The proposition states that under the separable case, the algorithm can predict the optimal solution  accurately with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We note that the result does not concern the prediction error in terms  of the objective vector, and this fact is aligned with the design of MOM that focuses on the optimality  condition instead of the objective vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Intuitively, even when one makes some error in predicting the  objective cnew, the prediction of the optimal solution can still be accurate due to the simplex structure  of LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The proof of the proposition mimics the algorithm stability analysis (Bousquet and Elisseeff,  2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Shalev-Shwartz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', 2010) where the regularization term in (3) plays a role of stabilizing the  estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Here the performance bound is presented in the sense of on expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We remark that  a high probability bound (removing the expectation in (7)) can also be achieved following the recent  analysis of (Feldman and Vondrak, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='2  Inseparable case  Now we move on to the inseparable case where Assumption 3 no longer holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Let  Dnew = (c, A, b, z, B∗, N ∗)  denote a new sample from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Define the margin-violation loss function  l(Dnew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Θ) := ∥s∥1 where s := 1|N ∗| − ˆc⊤  N ∗ − ˆc⊤  B∗A−1  B∗  t AN ∗ and ˆc = Θz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  The loss function is inherited from the objective function of the MOM formulation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' For the inseparable  case, we can derive the following generalization bound as Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Under Assumptions 1 and 2, let ˆΘ denote the output of Algorithm 1 with T training  samples, under the choice of λ =  1  √  T and K = {Θ ∈ Rn×d : ∥Θ∥F ≤ ¯Θ} for some ¯Θ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Then we have  E[l(Dnew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' ˆΘ)] ≤ min  Θ∈K E[l(Dnew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Θ)] + 28 + 8 ¯Θ2 + 7m¯σ2  √  T + 1  ,  where the expectation is taken with respect to both the new sample Dnew and the training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  The proposition follows the same analysis as Proposition 1 and the right-hand-side involves an addi-  tional term which will become zero under the separability condition of Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' In the previous  section, we motivate the optimization formulation (3) of MOM in its own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The following corollary  reveals an interesting connection between its objective function and the suboptimality loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Specifically,  the margin violation loss optimized in MOM can be viewed as an upper bound of the suboptimality loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The following inequality holds for any Θ ∈ Rn×d,  lsub(Θ) ≤ E[l(Dnew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Θ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  8  Therefore, under Assumptions 1 and 2, and the same setup as Proposition 2,  lsub(ˆΘ) ≤ min  Θ∈K E[l(Dnew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Θ)] + 28 + 8 ¯Θ2 + 7m¯σ2  √  T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  The additional term on the right-hand-side in both Proposition 2 and Proposition 3 can be viewed as  a measure of separability in terms of the LP’s optimality condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Specifically, if one can predict the  objective c very well with the covariates z, then this term will become close to zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' otherwise, this term  will become large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' While the standard measurement of the predictive power of z concerns the minimum  error in predicting c, it does not account for the structure of the downstream LP optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='3  Online maximum optimality margin  The MOM formulation also enables an online algorithm which reduces the quadratic program of (3) to  a subgradient descent step per sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Specifically, we can view the margin violation of the t-th sample  as a piecewise-linear function of Θ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=',  l(Dt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Θ) = ∥st∥1 =  ���1|N ∗  t | − ˆc⊤  t,N ∗  t − ˆc⊤  t,B∗  t A−1  t,B∗  t At,N ∗  t  ���  1  where Dt represents the t-th sample in the dataset D(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The function l(Dt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Θ) is a convex function with  respect to Θ and the piecewise linearity enables an efficient calculation of the subgradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Algorithm  2 arises from an online subgradient descent algorithm with respect to the cumulative loss  T  �  t=1  l(Dt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Algorithm 2 Online MOM Algorithm  1: Input: Dataset D(T ) = {(x∗  t , At, bt, zt, B∗  t , N ∗  t )}T  t=1, step size η, K  2: Initialize Θ1 = 0 ∈ Rn×d  3: for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', T do  4:  Predict the objective by ˆct = Θtzt  5:  Solve the following LP and denote its optimal solution by xt  min ˆc⊤  t x,  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Atx = bt, x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  6:  Update  Θt+1 = ProjK (Θt − η · ∂Θl(Dt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Θt))  7: end for  8: Output: {xt}T  t=1, {ΘT }T +1  t=1  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Under Assumptions 1 and 2, with the choice of η =  2 ¯Θ  (√n+¯σ·mn)  √  T and K = {Θ ∈ Rn×d :  ∥Θ∥F ≤ ¯Θ} for some ¯Θ > 0, the outputs of Algorithm 2 satisfy  1  T E  � T  �  t=1  ˆc⊤  t (x∗  t − xt)  �  ≤ E  �  min  Θ∈K  T  �  t=1  l(Dt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Θ)  �  + 3 ¯Θ√n + 3¯σ ¯Θ · mn  √  T  (8)  where Dnew = (cnew, Anew, bnew, znew) denotes a new sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  9  Moreover, under Assumption 3, let ˆcnew = ˆΘznew where ˆΘ is uniformly randomly sampled from  {Θt}T +1  t=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Let x∗  new and ˆxnew denote the optimal solutions of LP(cnew, Anew, bnew) and LP(ˆcnew, Anew, bnew).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Then we have  E [P (x∗  new = ˆxnew)] ≥ 1 − 3 ¯Θ√n + 3¯σ ¯Θ · mn  √  T + 1  (9)  where the expectation is taken with respect to both training data samples and the new sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  The proposition states that the results in Proposition 1 and Proposition 2 can also be achieved by a  subgradient descent algorithm from an online perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The bound (8) also recovers the regret bounds  in the online inverse optimization literature (Bärmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Chen and Kılınç-Karzan, 2020) under  a contextual setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' However, the algorithms therein require the convex set K not containing the origin  (which will significantly restrict the predictive power of zt → ct).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Otherwise, we find numerically that  these existing algorithms will drive the parameter Θt → 0 and ˆct → 0, and consequently provide no  meaningful prediction of the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' This reinforces the point of scale invariance raised earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  As mentioned earlier, the margin-based formulation of MOM resolves the issue of scale invariance, and  as a result, Algorithm 2 can provide an additional bound (9) and also perform stably well in numerical  experiments, which we refer to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='4  Tighter bound under separability – optimality-driven perceptron  We conclude this section with Algorithm 3 that achieves a faster rate under the separability condition  (Assumption 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Specifically, the algorithm is an online algorithm that utilizes the knowledge of sepa-  rability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The idea is to treat each inequality constraint in (3) as an individual binary classification task  of distinguishing non-basic variables from basic variables, and view the margin of optimality condition  as the margin for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Then when the margin (the reduced cost in Algorithm 3) drops below a  threshold of 1/2, we perform an update of the parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Under Assumptions 1, 2 and 3, the number of mistakes made by Algorithm 3 is inde-  pendent of T ,  # {t ∈ [T ] : x∗  t ̸= xt} ≤ ¯Θ2 + ¯σ2 ¯Θ2m2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  In addition, suppose (cnew, Anew, bnew, znew) is a new sample from P and denote ˆcnew = ˆΘznew where  ˆΘ is uniformly randomly sampled from {Θt}T +1  t=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Let x∗  new and ˆxnew denote the optimal solutions of  LP(cnew, Anew, bnew) and LP(ˆcnew, Anew, bnew),  E [P(x∗  new = ˆxnew)] ≥ 1 −  ¯Θ2 + ¯σ2 ¯Θ2m2n  T + 1  ,  where the expectation is taken with respect to both the training data and the new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Proposition 5 provides an upper bound on the number of mistakes made by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  The  algorithm improves the dependency on T compared with the previous bounds in Proposition 1 and  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  However, we note this improvement sacrifices the dependency on either the problem  size m and n or the constants of ¯Θ and ¯σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' In this light, the result is more of theoretical interests  that completes our discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  In the numerical experiments in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='4, we observe that the  performance of Algorithm 3 is indeed worse than Algorithm 1 and Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  4  Numerical Experiments and Discussions  We conduct numerical experiments for two underlying LP problems – the shortest path problem and the  fractional knapsack problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We present one experiment here and defer the remaining ones to Appendix  10  Algorithm 3 Optimality-driven Perceptron Algorithm  1: Input: Dataset D(T ) = {(x∗  t , At, bt, zt, B∗  t , N ∗  t )}T  t=1  2: Let Θ1 = 0 ∈ Rn×d  3: %% We use (M)i to denote the i-th row of the matrix M  4: for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', T do  5:  Predict the objective by ˆct = Θtzt  6:  Solve the following LP and denote its optimal solution by xt  min ˆc⊤  t x,  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Atx = bt, x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  7:  Let Θtmp = Θt  8:  for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', n do  9:  if i /∈ N ∗  t then  10:  Continue  11:  end if  12:  Predict the objective by ˆc(i)  t  = Θtmpzt  13:  Compute the reduced cost  r(i)⊤  t  = ˆc(i)⊤  t  − ˆc(i)⊤  t,B∗  t A−1  t,B∗  t At  14:  if r(i)  t,i ≤ 1  2 then  15:  Update  (Θtmp)i ← (Θtmp)i + z⊤  t  (Θtmp)B∗  t ← (Θtmp)B∗  t − A−1  B∗  t Atzt  16:  end if  17:  end for  18:  Let Θt+1 = Θtmp  19: end for  20: Output: {xt}T  t=1, {Θt}T +1  t=1  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The experiments compare the proposed MOM algorithms against several benchmarks and illustrate  different aspects, such as loss performance, computational time, scale consistency/invariance, sample  complexity, kernelized version of MOM, and online setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Specifically, we consider a shortest path (SP) problem here following the setup of Elmachtoub and Grigas  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The SP problem is defined on a 5 × 5 grid network with n = 40 directed edges associated with  a cost vector c ∈ Rn, and the covariates z ∈ Rd with d = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' For the training data, the covariates  are generated from the Gaussian distribution, and the objective vector is generated from a polynomial  mapping from the covariates with additional white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' For this numerical experiment, we consider  the performance measure of relative loss defined by  Relative Loss := c⊤  new(xnew − x∗  new)  c⊤x∗new  where cnew is the objective vector of a new test sample, xnew is the predicted optimal solution, and x∗  new  is the true optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Indeed, this relative loss normalizes the estimate loss lest with the optimal  objective value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We refer more details on the experiment setup to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Figure 1 presents the experiment results for two MOM algorithms and a few benchmarks, and each  panel represents a different degree of the polynomial that governs the true mapping from the covariates  to the objective vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' a higher degree indicates a stronger non-linearity between the covariates and the  objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We make the following observations:  11  RF  OLS  Ridge  SPO+  MOM  MOM-OGD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='15  Relative Loss  Degree 1  RF  OLS  Ridge  SPO+  MOM  MOM-OGD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='5  Degree 2  RF  OLS  Ridge  SPO+  MOM  MOM-OGD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='0  Degree 4  RF  OLS  Ridge  SPO+  MOM  MOM-OGD  0  10  20  30  40  Degree 6  Random Forest  Ordinary Least Squares  Ridge Regression  SPO+  Maximum Optimality Margin  MOM-Online Gradient Descent  Figure 1: Relative loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The plot (means and confidence intervals) is generated based on 30 random  trials each with 1000 training samples and 1000 test samples (20% of training samples are used for tuning  hyper-parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The degree indicates the degree of the true polynomial function that generates the  objective c from the covariates z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' RF denotes random forests, OLS denotes ordinary least squares, Ridge  denotes the ridge linear regression, SPO+ implements the algorithm of Elmachtoub and Grigas (2022),  MOM implements Algorithm 1, and MOM-OGD implements Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' For all these methods, they  first predict the objective vector for a test sample, and solve an LP with the predicted objective for the  optimal solution as the predicted solution for this test sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  First, from an information viewpoint, all four benchmark algorithms utilize the observations of ct’s on  the training data, while the two MOM algorithms only observe x∗  t ’s on the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' In this sense,  our algorithms give a better predictive performance with less amount of information, and therefore our  algorithms are the only ones applicable to the inverse LP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Second, other than the two MOM algorithms, the SPO+ performs the best among the four benchmark  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' However, the SPO+ takes significantly longer training time than all the other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Although SPO+ is a stochastic gradient descent-based algorithm, the calculation of the stochastic gra-  dient at each step requires solving k LPs with k being the number of training samples in the mini-batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Comparatively, our MOM Algorithm 1 only solves a quadratic program for once, and its online version  of Algorithm 2 allows a direct calculation of the online sub-gradient which makes it even faster than  Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Third, the three ML-based methods (RF, OLS, Ridge) perform generally worse than the other three  methods (SPO+, MOM, MOM-OGD) because they do not take into account the optimization structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  To see this, for the shortest path problem, there may be some large entries for the cost vector c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The ML  models treat all the dimensions of c equally and spend too much effort in fitting those large entries (as  they cause large L2 errors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' However, from the optimization perspective, an accurate estimation of those  large entries is not useful in that the optimal path will avoid those edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' That highlights the intuition  behind the SPO+ and our MOM algorithms – one needs to incorporate the optimization structure into  the learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Concluding remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' In this paper, we develop a new approach of maximum optimality margin  for the problems of predict-then-optimize and inverse LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The MOM framework leads to new charac-  terizations of the difficulty of the problem through the separability condition of Assumption 3 and the  violation loss l(Dnew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Θ) which appears in Propositions 2, 3, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' With the MOM perspective, we  derive bounds on correctly predicting the optimal solution,  P(x∗  new = ˆxnew)  12  in Propositions 1, 4 and 5, which is rarely seen in the existing literature even under strong conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Without the separability condition of Assumption 3, we draw a connection of our MOM objective value  with the suboptimality loss lsub in the inverse LP literature and derive performance bounds in terms  of lsub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Numerically, we observe the MOM algorithms also perform well under the estimate loss lest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  To theoretically quantify the performance of MOM under lest, we conjecture that one needs to impose  stronger structures on the dual LPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Besides, another interesting future direction to pursue is to generalize  the algorithms and analyses of MOM to non-linear optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  References  Ahuja, Ravindra K, James B Orlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
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+page_content=' Operations Research 49(5) 771–783.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Bärmann, Andreas, Alexander Martin, Sebastian Pokutta, Oskar Schneider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' An online-learning  approach to inverse optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' arXiv preprint arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='12997 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Bertsimas, Dimitris, Vishal Gupta, Ioannis Ch Paschalidis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Data-driven estimation in equilibrium  using inverse optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Mathematical Programming 153 595–633.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Bertsimas, Dimitris, Nathan Kallus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' From predictive to prescriptive analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Management Science  66(3) 1025–1044.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Besbes, Omar, Yuri Fonseca, Ilan Lobel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Contextual inverse optimization: Offline and online  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Available at SSRN 3863366 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Bousquet, Olivier, André Elisseeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Stability and generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The Journal of Machine Learning  Research 2 499–526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Boyd, Stephen, Lieven Vandenberghe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Convex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Cambridge university press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Chen, Violet Xinying, Fatma Kılınç-Karzan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Online convex optimization perspective for learning  from dynamically revealed preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' arXiv preprint arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='10460 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  El Balghiti, Othman, Adam N Elmachtoub, Paul Grigas, Ambuj Tewari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Generalization bounds  in the predict-then-optimize framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Advances in neural information processing systems 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Elmachtoub, Adam, Jason Cheuk Nam Liang, Ryan McNellis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Decision trees for decision-making  under the predict-then-optimize framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' PMLR,  2858–2867.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Elmachtoub, Adam N, Paul Grigas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Smart “predict, then optimize”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Management Science 68(1)  9–26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Feldman, Vitaly, Jan Vondrak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' High probability generalization bounds for uniformly stable algo-  rithms with nearly optimal rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Conference on Learning Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' PMLR, 1270–1279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Hazan, Elad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Introduction to online convex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Foundations and Trends in Optimization  2(3-4) 157–325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Ho-Nguyen, Nam, Fatma Kılınç-Karzan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Risk guarantees for end-to-end prediction and optimiza-  tion processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Management Science .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Hofmann, Thomas, Bernhard Schölkopf, Alexander J Smola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Kernel methods in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  The annals of statistics 36(3) 1171–1220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  13  Hu, Yichun, Nathan Kallus, Xiaojie Mao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Fast rates for contextual linear optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Manage-  ment Science .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Joachims, Thorsten, Thomas Hofmann, Yisong Yue, Chun-Nam Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Predicting structured objects  with support vector machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Communications of the ACM 52(11) 97–104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Luenberger, David G, Yinyu Ye, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
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+page_content=' Linear and nonlinear programming, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Megiddo, Nimrod, Ramaswamy Chandrasekaran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  On the ε-perturbation method for avoiding  degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Operations Research Letters 8(6) 305–308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Mohajerin Esfahani, Peyman, Soroosh Shafieezadeh-Abadeh, Grani A Hanasusanto, Daniel Kuhn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Data-driven inverse optimization with imperfect information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Mathematical Programming 167(1) 191–  234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Shalev-Shwartz, Shai, Ohad Shamir, Nathan Srebro, Karthik Sridharan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Learnability, stability and  uniform convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The Journal of Machine Learning Research 11 2635–2670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Tan, Yingcong, Daria Terekhov, Andrew Delong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Learning linear programs from optimal decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Advances in Neural Information Processing Systems 33 19738–19749.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Zhang, Jianzhong, Zhenhong Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Calculating some inverse linear programming problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Journal  of Computational and Applied Mathematics 72(2) 261–273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  14  A  Numerical Experiments  In this section, we provide the details of the experiment setup and more numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='1  Basic Setup  For the numerical experiments, we compare the performance of our proposed algorithms against several  benchmark methods, and we test the performance under both online and offline settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Specifically,  we consider the following benchmark methods where all of them can only be used to solve the predict-  then-optimize problem (but not the contextual linear programming) as they all require the observations  of ct on the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Linear regression models where we consider ordinary least squares (OLS) and ridge regression (RR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Accordingly, the parameter estimation follows the following loss functions:  lOLS = 1  2∥Θz − c∥2  2,  lRidge = 1  2∥Θz − c∥2  2 + λ  2 ∥Θ∥2  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Random Forest (RF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We apply RF algorithm with squared error criterion  lRF = ∥ˆc − c∥2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Predict-then-optimize method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We take SPO+ algorithm (Elmachtoub and Grigas, 2022) as an  example, where the loss function is as follows  lSPO+ = (2Θz − c)⊤x∗(c) −  min  Ax=b,x≥0{(2Θz − c)⊤x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Following the previous implementations, we optimize the SPO+ loss under stochastic gradient  descent (SGD) approach and Frobenius regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Underlying LP problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  We study two canonical LP problems for the numerical experiments, namely the shortest path (SP)  problem and the fractional knapsack (FK) problem, and we present their results in the following two  sections A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='2 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The experiment setup follows that of Elmachtoub and Grigas (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Ho-Nguyen and Kılınç-Karzan (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Besbes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Performance Measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  We compare different algorithms by the performance measure of relative loss for the offline setting  and cumulative Regret in the online setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Under the offline setting, the relative loss normalizes the estimate loss lest by the optimal value,  Rel-LossSP := c⊤(x − x∗)  c⊤x∗  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Specifically, for the shortest path (SP) problem, the optimal value is always non-zero (unless for trivial  c) so the loss is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  For the fractional knapsack (FK) problem, due to the random noise, the optimal value may be zero  with positive probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' So we consider another normalization which leads to  Rel-LossFK := c⊤(x∗ − x)  ∥c∥2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  15  Under the online setting, we define the cumulative regret by the cumulative sum of the relative loss  of each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Overview of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  In Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='2, we demonstrate the algorithm performance under the effect of degree (see Figure 1)  and also investigate numerically the issue of scale inconsistency/invariance (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' In Appendix  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='3, we examine the sample complexity under the offline setting (see Figure 3) and compare the perfor-  mance of several online algorithms (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' In addition, we consider the kernelized version of the  MOM formulation and present its performance in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='2  Shortest Path Problem  For the shortest path probelm, we follow the experiment setup of (Elmachtoub and Grigas, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Con-  sider a 5 × 5 grid network with n = 40 directed edges associated with a cost vector c ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Each edge  is either from south to north or west to east, where the goal is to minimize the cost to traverse from  the southwest corner to the northeast corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' For each node of the network, there is a flow balance  constraint, encoded by Ax = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The problem can then be formulated as the standard-form LP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  For the methods of ordinary least square, ridge regression, and our maximum optimality margin,  we solve the corresponding optimization problems using cvxpy package with GUROBI solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' For the  random forest algorithm, we implement the sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='RandomForestRegressor package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' For  the SPO+ algorithm and the online version of our MOM approach, we implement gradient descent  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' For each simulation trial, the training sets and test sets both consist of N = 1000 sample  sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' All regularization parameters are chosen from 10−6 to 102, and all step sizes are chosen from  10−3 to 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  All projection radius (the diameter of K) are chosen from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='8∥B∥F to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='0∥B∥F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  The  hyper-parameters are chosen via a validation subset of N/4 samples from the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Data generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Now we describe the data generation mechanism for Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' For each trial of our experiments,  we generate a gound-truth coefficient matrix Θ∗ ∈ Rn×d independently, where each component of Θ∗ is  drawn from Bernoulli(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Here n = 40 and d = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' For the feature vector zt: the first d − 1 components are generated independently from a Unif[0, 1]  distribution, and the last component of each zt is set to be 1 (to model the intercept term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The true cost vectors are generated as follows:  ct,j =  � 1  √  d  (Θ∗zt)j + 3  �deg  + 1  where deg is a fixed integer to be chosen to represent the non-linearity of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Scale inconsistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Figure 2 presents an experiment on scale attack that aims to illustrate the robustness of the underlying  methods in terms of scale invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Specifically, in generating the training data, we follow the same  mechanism as Figure 1 except that we multiply some objective vectors ct’s with a factor αt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Specifically,  we have  ct,j =  �� 1  √  d  (Θ∗zt)j + 3  �deg  + 1  �  αt,  16  where the scaling factor  αt =  \uf8f1  \uf8f2  \uf8f3  1 + ¯α,  if zt,1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='5,  1,  else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  RF  OLS  Ridge  SPO+  MOM  MOM-OGD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='075  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='125  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='150  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='175  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='200  Relative Loss  Attack Power 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='1  RF  OLS  Ridge  SPO+  MOM  MOM-OGD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='20  Attack Power 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='0  RF  OLS  Ridge  SPO+  MOM  MOM-OGD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='25  Attack Power 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='0  Random Forest  Ordinary Least Squares  Ridge Regression  SPO+  Maximum Optimality Margin  MOM-Online Gradient Descent  Figure 2: Relative loss under different scale attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  In Figure 2, we note that the performances of three ML models are more or less affected by this  scale attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' However, the scale attack will not change the optimal solutions on the training data, so  the method of SPO+ and our MOM algorithms which all rely on the optimal solutions, give a stable  performance under this scale attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Interestingly, the scale noise αt here is dependent on the covariates  zt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' When these two are independent, we establish that the linear regression model also has the property  of scale invariance, which we refer to Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='3  Fractional Knapsack Problem  For this experiment, we follow the problem setup of (Ho-Nguyen and Kılınç-Karzan, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Specifically,  each decision maker is presented with a bunch of items, and their aim is to maximize the utility (or  equivalently, to minimize the cost) under a knapsack constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Different from the discrete case, frac-  tional knapsack problem allows the decision maker to select a fraction of some certain item, making it a  linear program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Mathematically, the decision maker solves the following LP  min  x,s1,s2 c⊤x,  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' p⊤x + s1 = B, x + s2 = 1, x, s1, s2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Here p ∈ Rn can be interpreted as the price vector, and B ∈ R is the budget of the decision maker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The  corresponding utility vector should be interpreted as the opposite of the objective vector c since the LP  considers a minimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The slack variables s1 and s2 are introduced simply to convert the  problem into a standard form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Data generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  For each simulation trial, we generate a true coefficient matrix Θ∗ ∈ Rn×d, where each component  is a Bernoulli random variable with expectation 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Here n = 10 and d = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We then generate the  price vector, where each component is selected to be a random integer between 1 and 1000 in the offline  setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' To ease the pain of large constants, we re-scale the price vector in the online setting of the next  17  subsection, where each component is uniformly distributed between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We generate two auxiliary  variables low and high, where low = maxj pj and high = 1⊤p − u · low, and u ∼ Unif[0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Then the  budget is chosen as Unif[low, high].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' For the feature vector zt: the first d − 1 components are generated independently from a Unif[0, 1]  distribution, and the last component of each zt is set to be 1 (to model the intercept term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The true cost vectors are generated as follows:  ct,j = (Θ∗zt)deg  j  ǫt,j + ¯ηηt,j,  where deg is a fixed integer that represents the intensity of the non-linearity dependency, ǫt,j  represents the intrinsic diverse noise, and ηt,j is the additive noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The constants ¯ǫ and ¯η are  non-negative parameters chosen to control the power of each noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Specifically,  ǫt,j ∼ Unif[1 − ¯ǫ, 1 + ¯ǫ],  2ηt,j + 1 ∼ Exponential(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  100  200  500  1000  2000  Number of Training Samples  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='2  Relative Loss  RF  OLS  Ridge  SPO+  MOM  MOM-OGD  Figure 3: Sample complexity of different algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Figure 3 illustrates the sample efficiency of different algorithms, and the plot is based on an average  over 30 random trails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The parameters are chosen by deg = 1, ¯ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='1, ¯η = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='0, and ¯α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' For  training sets, we generate different numbers of training samples N = 100, 200, 500, 1000, 5000, and test  the performance against 1000 test samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' As the underlying true model is a linear model, the two  linear ML models unsurprisingly achieve the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The performance of the random forest  algorithm is similar to that of offline MOM algorithm (Algorithm 1), and both are slightly worse than  that of linear regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' As for SPO+ method, their averaged performance is undermined by the  noise to a fairly large extent, compared with its performance under noiseless environment (see Figure 1  and Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The performance difference between SPO+ and MOM can be partially explained by the  different ways of dealing with noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The SPO+ method directly utilizes the optimal solution of  the noisy data to the gradient, while the optimal solution could probably vary drastically due to some  18  noise to the cost vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' In comparison, the margin-based MOM performs more steadily: for those noisy  cost vectors, their optimal solutions could be far from the noiseless cost vectors, but the change with  respect to optimal basis will usually be only a few components which only affects a few corresponding  lines in estimated Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Compared to these ML models which utilize the observations of ct, although the  MOM algorithm (Algorithm 1) consumes less information in the training phase, it does not exhibit a  higher requirement of the number of training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The online version of MOM (Algorithm 2) does  see a decreasing error with more training samples, and this is because the nature of the gradient-based  algorithm does not take the full advantage of observed samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Kernelization and non-linear MOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  In Figure 1, the performances of all the methods deteriorate significantly as there is a stronger non-  linearity between the objective c and the covariates z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Here we explore the kernelized version of MOM  and find that it effectively improves the performance of MOM when the underlying dependency is non-  linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Specifically, we consider three kernelized versions of MOM by different ways to kernelize the  covariates: Linear MOM (Linear), Polynomial Kernelized MOM (PolyKer), and Radial Basis Function  Kernelized MOM (RbfKer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The degree of the polynomial kernelized MOM is chosen from {1, 2, 3, 4}  and the scale parameters γ’s of both kernelized MOM methods are chosen from {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='5, 1, 2, 3, 4, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  All regularization parameters are chosen from 10−6 to 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The hyper-parameters are determined via a  validation set of 25% training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Linear  PolyKer  RbfKer  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='3  Relative Loss  Degree 1  Linear  PolyKer  RbfKer  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='25  Degree 2  Linear  PolyKer  RbfKer  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='20  Degree 4  Linear  PolyKer  RbfKer  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='25  Degree 6  Linear  Polynomial Kernelized  Radial Basis Function Kernelized  Figure 4: Relative loss for kernelized MOMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Figure 4 presents the performance of the three kernelized MOM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The performance of linear  MOM (Algorithm 1) is worsened as the degree of the model grows (just as Figure 1), since the linear  mapping induces a stronger model misspecificaion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' But kernelized MOM methods give good performance  under high degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' In comparison of Figure 1, we note that the kernelized versions of MOM can be  quite effective in handling nonlinear dependency, and the loss does not increase much with the increase  of the degree of non-linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='4  Online Setting  Next we analyze the online setting for the MOM algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We compare the performance of the OGD  version of MOM (Algorithm 2), the perceptron version of MOM (Algorithm 3), and also a follow-the-  regularized-leader (FTRL) version of the MOM Algorithm 1 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Basically, Algorithm 4 solves the  19  MOM optimization formulation 3 at each time t and use the estimated parameter to predict the optimal  solution of the next time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Algorithm 4 Follow-the-Regularized-Leader MOM  1: Input: Dataset D(T ) = {(x∗  t , At, bt, zt, B∗  t , N ∗  t )}T  t=1, the set K  2: Initialize Θ1 = 0 ∈ Rn×d  3: for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', T do  4:  Predict the objective by ˆct = Θtzt  5:  Solve the following LP and denote its optimal solution by xt  min ˆc⊤  t x,  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Atx = bt, x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  6:  Solve the optimization problem (3) with {(x∗  s, As, bs, zs, B∗  s, N ∗  s )}t  s=1, λ = 1/  √  t and K  7:  Let Θt+1 be the optimal solution  8: end for  9: Output: {xt}T  t=1  Algorithm 4 gives another interpretation of the MOM formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Under an online setting, if we  solve the MOM optimization problem (3) at each time, this is equivalent to a follow-the-regularized  online algorithm to solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The theoretical analysis of Algorithm 4 can be extended from  Proposition 1 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  0  200  400  600  800  1000  Time Step  0  20  40  60  80  100  120  Averaged Regret  Alg3: Perceptron  Alg1/4: MOM-FTRL  Alg2: MOM-OGD  Figure 5: Cumulative regret curve (averaged over 30 trials) for online algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Figure 5 presents the cumulative regret of several algorithms for a fractional knapsack problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' In  this numerical experiment, we let the objective vector c = Θ∗z almost surely for some fixed Θ∗ so as  to achieve the separability condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We emphasize that it only ensures the existence of a parameter  Θ∗ to meet the optimality condition (2) but not with a margin of 1 as Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' From the plot,  we observe that Algorithm 3 has the worst performance though it features for the best theoretical  dependency on T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We remark that this regret curve does not contradict the bounded number of mistakes  in Proposition 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' we find that when we increase the horizon to T = 100, 000, the regret curve of Algorithm  3 becomes flattened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' For Algorithm 2 and Algorithm 4, both achieve significantly better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Comparatively, 4 performs better than Algorithm 2, with the price of more computation cost (to solve  a quadratic program at each time period).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Importantly, we also implement the online algorithm of (Bärmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Chen and Kılınç-Karzan,  2020) under the same setting, and it incurs a regret linearly increasing with T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' so we do not include its  regret curve as it will clap all the regret curves of Figure 5 to x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' A closer investigation shows that  20  the online algorithm of (Bärmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Chen and Kılınç-Karzan, 2020) will render the estimate  Θt → 0, as noted by the scale inconsistency issue in earlier sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' In contrast, our margin-based  formulation plays an important rule to ensure that a small optimality condition violation is caused by  discovering the correct mapping from z to c but not by the scale of the predicted ˆc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  B  Proofs  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='1  Proof of Lemma 1  Here we show a stronger version of Lemma 1 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Consider an LP of the standard form (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' For any feasible basis B ⊂ [n] satisfying |B| = m  and its complement N = [n]\\B, denote x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', xn)⊤ and r = (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', rn)⊤ as the solution and reduced  cost vector corresponding the basis B, respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=',  xB = A−1  B b, xN = 0, r = c − A⊤(A−1  B )⊤cB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Denote x∗ = (x∗  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', x∗  n)⊤ as one optimal solution of LP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Then, we have that rB = 0, and  c⊤x − c⊤x∗ ≤ max  i∈[n] x∗  i ·  �  i∈N  (−ri)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  (10)  Furthermore, (10) implies that x is an optimal solution if r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' First, it is easy to verify that rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Then, we only need to show that the inequality (10) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  For any feasible solution x′ = (x′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', x′  n) ≥ 0 satisfying Ax′ = b, we have  x′  B = A−1  B b − A−1  B AN x′  N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  It implies  c⊤x′ = c⊤  BA−1  B b − c⊤  BA−1  B AN x′  N + c⊤  N x′  N = c⊤  BA−1  B b + r⊤  N x′  N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  (11)  Next, from (11), we have  c⊤x − c⊤x′ = −r⊤  N x′  N  ≤  �  i∈N  x′  i(−ri)+  ≤ max  i∈[n] x′  i ·  �  i∈N  (−ri)+,  where the first line comes from equality (11) directly, and the last two lines come from the non-negativity  of x′ and (ri)+ for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Note that the above inequality holds for any feasible solution x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' by plugging in  x′ = x∗, we obtain (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='2  Proof of Proposition 2  We first show Proposition 2 and then utilize the result for the proof of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The key is to utilize the algorithm  stability analysis where we cite the result from (Shalev-Shwartz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', 2010) as the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  We refer to (Bousquet and Elisseeff, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Shalev-Shwartz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Feldman and Vondrak, 2019) for  more related analysis such as the high probability bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  21  Theorem 1 (Theorem 3 in Shalev-Shwartz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' (2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Let f : H× Z → R be such that H is bounded  by B and f(h, z) is convex and L-Lipschitz with respect to h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Let , z0, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', zT be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' samples and let  ˆhλ = arg min  h∈H  � T  �  t=1  f(h, zi) + λ  2 ∥h∥2  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Then, we have  E  �  f(ˆhλ, z0)  �  ≤ inf  h∈H E [f(h, z0)] + λ  2 B2 + 4(L + λB)2  λT  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We refer to Theorem 2 and Theorem 3 in Shalev-Shwartz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Now we show Proposition 2 with Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Recall that  Dnew = (c, A, b, z, B∗, N ∗)  denotes a new sample from P, and the loss function  l(Dnew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Θ) := ∥s∥1 where s := (1|N ∗| − ˆc⊤  N ∗ − ˆc⊤  B∗A−1  B∗ AN ∗)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Here, (·)+ denotes the entry-wise positive part function, and B∗ and N ∗ are defined as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Here, with a slight abuse of notation, we drop the sample tuple Dnew and let l(Θ) = l (Dnew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Then,  under Assumption 2, l(Θ) is (2 + √m¯σ)-Lipschitz with respect to Θ in the Frobenius norm for a fixed  Dnew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' To see this, for any two parameters Θ = (Θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', Θn)⊤, ˆΘ = (ˆΘ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' ˆΘn)⊤ ∈ Rn×d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='|l(Θ) − l(Θ′)| =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='�����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='i∈N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='(1 − Θ⊤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='i z + A⊤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='i (A−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='B∗ )⊤ΘB∗z)+ − (1 − ˆΘ⊤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='i z + A⊤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='i (A−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='B∗ )⊤ ˆΘB∗z)+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='≤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='i∈N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='���(1 − Θ⊤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='i z + A⊤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='i (A−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='B∗ )⊤ΘB∗z)+ − (1 − ˆΘ⊤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='i z + A⊤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='i (A−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='B∗ )⊤ ˆΘB∗z)+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='≤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='i∈N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='���−Θ⊤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='i z + A⊤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='i (A−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='B∗ )⊤ΘB∗z + ˆΘ⊤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='i z − A⊤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='i (A−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='B∗ )⊤ ˆΘB∗z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='≤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='i∈N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='����(Θi − ˆΘi)⊤z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='��� +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='���A⊤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='i (A−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='B∗ )⊤(ΘB∗ − ˆΘB∗)z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='≤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='i∈N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='����Θi − ˆΘi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='2 ∥z∥2 + σmax  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='(A−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='B∗ )⊤(ΘB∗ − ˆΘB∗)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='∥Ai∥2 ∥z∥2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='≤ 2∥Θ − ˆΘ∥F + √mσmax  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='(A−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='B∗ )⊤�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='σmax  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='ΘB∗ − ˆΘB∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='≤ (2 + √m¯σ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='���Θ − ˆΘ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Here the first line comes from the definition of l(·), the second and fourth lines are obtained by the  convexity of the absolute value function, the third line comes from a direct computation of the positive  part function, the fifth line is obtained by Cauchy’s inequality and the definition of σmax, the sixth line  comes from the inequality that  σmax(XY ) ≤ σmax(X)σmax(Y )  for any two matrices X, Y , and the last line comes from Assumption 2 and the inequality σmax(X) ≤  ∥X∥F for any matrix X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  By Theorem 1 and Assumption 2, we have  E[l(ˆΘ)] ≤ min  Θ∈K E[l(Θ)] + λ  2  ¯Θ2 + 4(2 + √m¯σ + λ¯Θ)2  λT  ,  (12)  22  where  ˆΘ = arg min  Θ∈K  � T  �  t=1  l(Dnew, Θ) + λ  2 ∥Θ∥F  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Finally, plugging λ =  1  √  T into (12), we have  E[l(ˆΘ)] ≤ min  Θ∈K E[l(Θ)] + 28 + 8 ¯Θ2 + 7m¯σ2  √  T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='3  Proof of Proposition 1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Under Assumption 3, there exists a Θ∗ ∈ K such that the following inequality holds almost surely  Θ∗  N ∗z − A⊤  N ∗(A−1  B∗ )⊤Θ∗  B∗z ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Thus we have  min  Θ∈K E[l(Θ)] = 0,  where l(Θ) is defined following the previous proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Then, from Proposition 2,  E[l(ˆΘ)] ≤ 28 + 8 ¯Θ2 + 7m¯σ2  √  T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  (13)  For a new sample (cnew, Anew, bnew, znew) with ˆcnew = ˆΘznew, we can utilize Lemma 2 and bound the  suboptimality loss as follows  E  �  ˆc⊤  newx∗  new − ˆc⊤  newˆxnew  �  ≤ E  \uf8ee  \uf8f0max  i∈[n](ˆxnew)i ·  �  i∈N ∗  new  (−ri)+  \uf8f9  \uf8fb  ≤ E  \uf8ee  \uf8f0 �  i∈N ∗  new  (−ri)+  \uf8f9  \uf8fb  ≤ E  \uf8ee  \uf8f0 �  i∈N ∗  new  (1 − ri)+  \uf8f9  \uf8fb  ≤ 28 + 8 ¯Θ2 + 7m¯σ2  √  T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  (14)  Here the first line comes from Lemma 2, the second line comes from Assumption 2, the third line comes  from the monotonicity of the positive part function, and the last line comes from (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Now, we note that if ˆΘ renders any non-basic variables in N ∗ as basic variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=',  ˆc⊤  new,N ∗  new − ˆc⊤  new,B∗newA−1  new,B∗newAnew,N ∗  new ≥ 0  does not hold, we have l(Dnew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' ˆΘ) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Thus, by applying Markov’s inequality to , we have that with  probability no less than 1− 28+8 ¯Θ2+7m¯σ2  √  T  , ˆΘ can identify both the true optimal basis and the true optimal  solution correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='4  Proof of Proposition 4  The proof in this part is basically an application of the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  23  Lemma 3 (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='1 in Hazan (2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Let {ft(x)}T  t=1 be a sequence of convex functions defined  on {x : ∥x∥2 ≤ K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Suppose ∥∇ft(x)∥2 ≤ G for all x such that ∥x∥2 ≤ K and all t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Let  xt+1 = xt −  2K  G  √  t∇ft(xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Then, the following inequality holds  T  �  t=1  ft(xt) −  min  x:∥x∥2≤K  T  �  t=1  ft(x) ≤ 3KG  √  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We refer to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='1 in Hazan (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Next, we show Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Recall the definition of lt(Dt, Θ) as follows  l(Dt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Θ) =  ���(1|N ∗  t | − ˆc⊤  t,N ∗  t − ˆc⊤  t,B∗  t A−1  t,B∗  t At,N ∗  t )+  ���  1 ,  where ˆc = Θzt, and (·)+ denotes the entry-wise positive part function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' For the sake of simplicity, we use  lt(Θ) to denote l(Dt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' By calculating the derivative of lt(Θ), we have  ∇ΘNtlt(Θ) = −gtz⊤  t , ∇ΘBt lt(Θ) = (A−1  t,Bt)⊤At,Ntgtz⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  (15)  Here, gt ∈ Rn−m is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' For the i-th element in Nt for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', m, correspondingly, we  define the i-th element of gt as  gt,i =  \uf8f1  \uf8f2  \uf8f3  1,  if  ��  eNt − ΘNtzt + A⊤  t,Nt(A−1  t,Bt)⊤ΘBtzt  �  +  �  i  > 0  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' we have  ∥∇lt(Θ)∥F ≤ ∥gtz⊤  t ∥F + ∥(A−1  t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='Bt)⊤At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='Ntgtz⊤∥F  ≤ ∥gt∥2∥zt∥2 + ∥A−1  t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='Bt∥F∥At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='Nt∥F ∥gt∥2∥zt∥2  ≤ √n + ∥A−1  t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='Bt∥F ∥At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='Nt∥F · √n  ≤ √n + ¯σmn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  where the first line comes from the equalities in (15) and the triangle inequality inequality ∥A + B∥F ≤  ∥A∥F + ∥B∥F for any two matrices A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' B with the same size,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' the second line comes from the inequality  ∥AB∥F ≤ ∥A∥F ∥B∥F for any two matrices A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' B and the fact that ∥g∥F = ∥g∥2 for any vector g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' the  third line comes from the definition of gt and Assumption 2 that ∥zt∥2 ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' and the last line comes from  Assumption 2 that all entries of At are in [−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' 1] and the inequality ∥A∥F ≤ √mσmax(A) for any matrix  A ∈ Rm×m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Next, by Lemma 3, with η =  2 ¯Θ  (√n+¯σmn)  √  T ,  T  �  t=1  lt(Θt) −  min  Θ:∥Θ∥F ≤ ˆΘ  T  �  t=1  lt(Θ) ≤  �  3 ¯Θ√n + 3¯σ ¯Θmn  � √  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  (16)  24  Then, we show inequality (8) by  1  T E[z⊤  t Θ⊤  t (x∗  t − xt)] ≤ E  � T  �  t=1  lt(Θt)  �  ≤ E  �  min  Θ:∥Θ∥F ≤ ˆΘ  T  �  t=1  lt(Θ)  �  + 3 ¯Θ√n + 3¯σ ¯Θ · mn  √  T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Here, we can obtain the first line by Lemma 2 and a similar proof as in the proof of Proposition 1, and  obtain the second line by inequality (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Furthermore, under Assumption 3, we have with probability 1,  min  Θ:∥Θ∥F ≤ ˆΘ  T  �  t=1  lt(Θ) = 0,  which implies  E  � T  �  t=1  lt(Θt)  �  ≤ 3 ¯Θ√n + 3¯σ ¯Θ · mn  √  T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  (17)  Recall that ˆΘ denotes the matrix sampled uniformly from {Θt}T  t=1, x∗  new denotes the optimal solution  of LP(cnew, Anew, bnew), ˆxnew denotes the optimal solution of LP(ˆcnew, Anew, bnew), and ˆcnew = ˆΘznew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Similar to the last paragraph of the proof of Proposition 1, we have if ˆΘ misclassifies any non-basic  variable in Nnew, we have l(Dnew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' ˆΘ) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Thus,  E[P(x∗  new ̸= ˆxnew)] ≤ E[P(l(Dnew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' ˆΘ) ≥ 1)]  ≤ E[l(Dnew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' ˆΘ)]  =  1  T + 1E  �T +1  �  t=1  l(Dnew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Θt)  �  =  1  T + 1E  �T +1  �  t=1  l(Dt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Θt)  �  ≤ 3 ¯Θ√n + 3¯σ ¯Θ · mn  √  T + 1  ,  by which we have shown that, with probability no less than 1 − 3 ¯Θ√n+3¯σ ¯Θ·mn  √T +1  , Algorithm 2 can identify  both the true optimal basis and the true optimal solution correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Here, the first line comes from the  fact that l(Dnew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' ˆΘ) ≥ 1 if x∗  new ̸= ˆxnew, the second line comes from Markov’s inequality, the third  line comes from the definition of ˆΘ, the forth line comes from the fact that Dt and Dnew are two i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  samples that are also independent of Θt, and the last line comes from inequality (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='5  Proof of Proposition 5  The proof integrates the standard analysis of the perceptron algorithm with the structure of the linear  program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' In this part, we denote Θt,i as the value of matrix Θtmp at the beginning on the t-th iteration of  the outer loop and the i-th iteration of the inner loop for t ∈ [T ] and i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Also, we view Θt,n+1 and  25  Θt+1,1 as the same to avoid undefined boundary cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Recall the definition of the reduced cost vector  rt(Θ) = Θzt − A⊤  t (A−1  t,Bt)⊤ΘBtzt, for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', T,  which is a linear function of Θ entry-wisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Thus, we have that for each t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', T and i ∈ [n],  there exists a matrix Wt,i ∈ Rn∗d such that r(i)  t,i (Θ) = Trace(W ⊤  t,iΘ) and ∥Wt,i∥F ≤ 1 + ¯σ√mn, where  Trace(W) =  n�  i=1  wii for any square matrix W = (wij)n  i,j=1 ∈ Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Then, we define  ht,i(Θ) = sign(Trace(W ⊤  t,iΘ) − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='5),  where sign(·) denotes the sign function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Moreover, if there is an i ∈ Nt at some time t such that  ht,i(Θt,i) = −1, we have Θt,i+1 = Θt,i+Wt,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The updating rule in Algorithm 3 is obtained as mentioned  above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Specifically, as in Algorithm 3, for any i ∈ Nt and t ∈ [T ], Wt,i is defined as follows  (Wt,i)i = z⊤  t , (Wt,i)B∗  t = A−1  B∗  t Atzt,  and all other entries are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Then, we have  ∥Wt,i∥2  F = ∥zt∥2  F + ∥A−1  B∗  t Atzt∥2  F  ≤ ∥zt∥2  F + ∥AB∗  t ∥2  F ∥At∥2  F ∥zt∥2  F ,  (18)  ≤ 1 + ¯σ2m2n  where the first line comes directly from the definition of Wt,i, the second line comes from the inequality  that ∥AB∥F ≤ ∥A∥F ∥B∥F for any two matrices A, B, and the last line comes from Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  We say that Algorithm 3 misclassifies one non-basic variable i ∈ Nt if ht,i(Θt,i) ≤ 0, and it misclas-  sifies one basic variable i ∈ Bt if ht,i(ΘT ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Since the values of entries of the reduced cost vector  corresponding to the true basis Bt are 0 for all t and i, Algorithm 3 makes a mistake only if it misclassi-  fies one non-basic variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Denote the number of identification mistakes at the t-th iteration as Kt for  t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Let K =  T�  t=1  Kt be the number of all mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' From the updating rule, inequality (18) and  the triangle inequality of the Frobenius norm, we have ∥Θt,i+1∥2  F ≤ Kt(1 + ¯σ2m2n), and then,  ∥ΘT +1∥2  F ≤ K(1 + ¯σ2m2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  (19)  Moreover, under Assumption 3, there exists a matrix Θ∗ such that  Trace(W ⊤  t,iΘ∗) ≥ 1,  if i ∈ Nt,  Trace(W ⊤  t,iΘ∗) ≤ 0,  if i ∈ Bt,  for all t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Then, we have once a mistake is made for some i ∈ Nt at time t  Trace((Θt,i+1 − Θt,i)⊤Θ∗) = Trace(W ⊤  t,iΘ∗) ≥ 1,  which implies  trace(Θ⊤  T,n+1Θ∗) ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  (20)  26  Then, combining inequalities (19) and (20), we have  K ≤ trace(Θ⊤  T,n+1Θ∗)  ≤ ¯Θ∥ΘT,n+1∥F  ≤ ¯Θ  �  K(1 + ¯σ2m2n),  where the first line comes from (20), the second line comes from Assumption 3 that ∥Θ∗∥F ≤ ¯Θ and the  Cauchy-Schwartz inequality, and the last line comes from (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Dividing both sides by  √  K and taking  square,  K ≤ ¯Θ2 + ¯σ2 ¯Θ2m2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Moreover, Lemma 1 tells that one can recover the optimal solution if the optimal basis is identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  This statement implies that K is an upper bound of times that we cannot identify the true optimal  solutions by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Thus, we have  |{t ∈ [T ] : x∗  t ̸= xt}| ≤ ¯Θ2 + ¯σ2 ¯Θ2m2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  For the generalization bound, we apply the symmetry of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Following a similar argument as  in the last paragraph of the proof of Proposition 4, we obtain  E [P(x∗  new ̸= ˆxnew)] ≤  ¯Θ2 + ¯σ2 ¯Θ2m2n  T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  C  Additional Discussions  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='1  Why Structured Prediction Not Work  We argued earlier that the inequality constraints in (3) is critical in that alternative formulations such  as (4) and (5) will fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Here we explore another alternative – structured support vector machine (SVM)  (Joachims et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=', 2009) and show that it leads to an computationally intractable formulation of the  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  The goal of a structured SVM classifier under our context to estimate the optimal basis (or equiv-  alently, the non-basic variables) by observing {(z, A)} and a predictor trained on {(zt, At)}’s and their  corresponding labels {yt}’s, where we define (yt)i = +1 for i ∈ Nt and (yt)i = −1 for i ∈ Bt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' The  Maximum Optimality Margin approach is a principle to maximize the estimated reduced cost vectors for  non-basic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' To express this principle more explicitly, the margin term one wants to maximize in  structured SVM can be written as  max  Θ∈K  �  j∈N  ˆrj,  where ˆrj’s are to be specified later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  For the simplicity of notations, we define some auxiliary vectors and matrices named as 1, JBt, JNt,  and Φt,yt, where  1 ∈ Rn−m, 1j = 1, ∀j ∈ [n − m],  JBtv = vBt, ∀v ∈ Rn,  JNtv = vNt, ∀v ∈ Rn,  27  Φt,yt := JNt − A⊤  t,NtA−⊤  t,BtJBt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Specifically, the estimated reduced cost with respect to yt can be written as  ˆrt,yt = Φt,ytΘzt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Hence the margin term (where the subscript t is sometimes omitted for simplicity) is  �  j∈N  ˆrj = 1⊤ˆr = ⟨Θ, Φ⊤  y 1z⊤⟩,  which implies the feature map  φ((zt, At), yt) = Φ⊤  t,yt1z⊤  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Note that the corresponding label space  Y = {y ∈ {−1, +1}n, #{i, yi = +1} = m}  is exponentially large in n in general cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We also define some measurement of difference ∆(·, ·) ∈  Y × Y → R, where ∆(y, y′) ≥ 0 and ∆(y, y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' For example, the ∆(y, y′) function can be  1{y ̸= y′}  and we retrieve the multiclass Hinge loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Such ∆(y, y′) can also be defined to be the Hamming distance  and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Equipped with those notations, the structured SVM problem can now be formulated as:  min  Θ∈K,s∈RN,y∈Y  λ  2 ∥Θ∥2 + 1  N  N  �  i=1  si,  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' si ≥ ∆(yi, y) − ⟨Θ, φ((zi, Ai), yi)⟩ + ⟨Θ, φ((zi, Ai), y)⟩,  ∀i ∈ [N], ∀y ∈ Y,  si ≥ 0,  ∀i ∈ [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  We thereby note that solving the above problem requires solving another sub-problem where  gi(Θ) := s∗  i = max  y∈Y {∆(yi, y) − ⟨Θ, φ((zi, Ai), yi)⟩ + ⟨Θ, φ((zi, Ai), y)⟩} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  But the above sub-problem is computationally intractable, since the precise evaluation of gi(Θ)  requires solving a discrete optimization problem with exponentially large feasible set Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Such an obstacle  makes the training hard to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Besides, even if we get the training result ˜Θ, the following inference  problem  ˜f(z, A) = arg max  y∈Y  ⟨˜Θ, φ((z, A), y)⟩  is still highly intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='2  Scale Consistency of Least Squares Linear Regression  In the previous sections, we raise the point of scale consistency several times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Specifically, the key point  made is that the objective vectors of c and αc for α > 0 produce the same optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Therefore,  the algorithm should account for this scale invariance as there might be some scale contamination of the  training data in application contexts such as revealed preference/stated preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Here we present a  self-contained result on the scale consistency of the linear regression model that might be of independent  interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Consider the linear regression model where one wants to estimate the true coefficient matrix using  28  cost vectors ci ∈ Rn and feature vectors zi ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Instead of observing true ci’s, we only observe  their perturbed/contaminated versions with scale noises (1 + αi)ci’s, where αi’s here are some random  variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We claim that if αi’s are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' generated and independent of zi and ci, then the ordinary least  squares method is still consistent if E[α] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Proposition 6 (Scale Consistency of Ordinary Least Squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Assume c = Θ∗z + ǫ, where E[ǫ|z] = 0,  Θ∗ ∈ Rn×d is the true underlying coefficient matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Assume α is independent of z and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Assume we  observe N i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' samples (zi, (1 + αi)ci)’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Assume zi, ci, and αi have finite second order moments,  which implies that they follow the strong law of large numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Further assume that ci and αi have finite  fourth order moment, which implies the strong law of large numbers for c2  i and α2  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' Assume E[ziz⊤  i ] = Σ,  and Σ is non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We estimate the underlying coefficient matrix by ordinary least squares method:  ˆΘN = arg min  Θ  �  fN(Θ) :=  N  �  i=1  1  N ∥(1 + αi)ci − Θzi∥2  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Then  ˆΘN  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  −−→ (1 + E[α])Θ∗,  as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' we only prove the one-dimensional case c ∈ R, since multi-dimensional cases can be  proved similarly by breaking Θ = (Θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' , Θn)⊤ into n independent Θi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' For one-dimensional case, the  estimator is now minimizing  fN(Θ) = 1  N  N  �  i=1  ∥z⊤  i Θ − (1 + αi)ci∥2  2  = Θ⊤  �  1  N  N  �  i=1  ziz⊤  i  �  Θ −  �  1  N  N  �  i=1  2(1 + αi)ciz⊤  i  �  Θ + 1  N  N  �  i=1  (1 + αi)2c2  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Note that we assume that zi’s, ci’s, c2  i ’s, αi’s, α2  i ’s all have finite second order moments, which implies  1  N  N  �  i=1  ziz⊤  i  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  −−→ Σ,  as N → ∞,  1  N  N  �  i=1  2(1 + αi)ciz⊤  i  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  −−→ 2E[(1 + α)cz⊤],  as N → ∞,  1  N  N  �  i=1  (1 + αi)2c2  i  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  −−→ E[(1 + α)2c2] = (1 + 2E[α] + E[α2])E[c2] < ∞,  as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Combining the boundedness of the last component with the fact that Σ is non-singular and positive  semidefinite and the fact that 2E[(1 + α)cz⊤] is bounded (which will be shown later), we have  fN  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  −−→ f,  as N → ∞,  where f is a positive definite quadratic function of Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Therefore, the unique minimizer of f must be its first order stationary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We now compute the  29  partial derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' We have  E[(1 + α)cz⊤] = E[1 + α]E[cz⊤]  = (1 + E[α])E[E[cz⊤|z]]  = (1 + E[α])E[E[Θ∗⊤zz⊤ + ǫz⊤|z]]  = (1 + E[α])E[Θ∗⊤zz⊤],  = (1 + E[α])Θ∗⊤Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  where the first equality is from the fact that αi’s are independent of zi’s and ci’s, the second equality  follows the tower law of conditional expectation, the third equality comes from the linear assumption,  and the fourth equality comes from E[ǫ|z] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content=' It follows immediately that  ∂f  ∂Θ = 2ΣΘ − 2(1 + E[α])ΣΘ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Since Σ is non-singular, we have proved that the unique minimizer of f is exactly (1 + E[α])Θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Then from the fact that fN  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  −−→ f and f is positive definite quadratic function, we have  ˆΘN = arg min fN  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  −−→ arg min f = (1 + E[α])Θ∗,  as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  Specifically, if E[α] = 0, we retrieve the true Θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
+page_content='  30' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFIT4oBgHgl3EQfbCu_/content/2301.11260v1.pdf'}
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+STABILITY OF FINITE RECEDING HORIZON CONTROL: A
+COMPLEMENTARY APPROACH ∗
+Wen-Hua Chen
+Department of Aeronautical and Automotive Engineering
+Loughborough University
+Loughborough LE11 3TU, U.K.
+w.chen@lboro.ac.uk
+Yunda Yan
+School of Engineering and Sustainable Development
+De Montfort University
+Leicester, LE1 9BH, U.K.
+yunda.yan@dmu.ac.uk
+ABSTRACT
+This paper presents a complementary approach to establish stability of finite receding horizon control
+with a terminal cost. First a new augmented stage cost is defined by rotating the terminal cost. Then
+a one-step optimisation problem is defined based on this augmented stage cost. It is shown that a
+slightly modified Model Predictive Control (MPC) algorithm is stable if the value function of the
+augmented one-step cost (OSVF) is a control Lyapunov function. The proposed stability condition is
+completely complementary to the existing terminal cost based MPC stability conditions in the sense
+that they are mutually excluded with each other. By using this approach, we are able to establish
+stability for MPC algorithms with zero terminal cost or even negative terminal cost as special cases.
+Combining this new approach with the existing MPC stability theory, we are able to significantly
+relax the stability requirement on MPC and extend the design space where stability are guaranteed.
+The proposed approach will help to further reduce the gap between stability theory and practical
+applications of MPC and other optimisation based control methods.
+Keywords Finite horizon · optimisation, stability · model predictive control · constraints
+1
+Introduction
+It is well known that stability of Receding Horizon Control (RHC), also known as Model Predictive Control (MPC),
+with a finite horizon can not be guaranteed. The most widely used approach to guarantee the stability of a receding
+horizon controller is to add a terminal cost, Vf(xk+N|k), in its cost function where xk+N|k is the terminal state in the
+prediction horizon [1–4]. Consequently, a well-established sufficient condition is that the system x+ = f(x, u) is stable
+under an MPC algorithm if there exists any control u in the admissible set U such that
+−Vf(x) + l(x, u) + Vf(f(x, u)) < 0
+(1)
+for any x ̸= 0 where l(x, u) is the stage cost. That is, a terminal cost Vf(xk+N|k) that covers the time-go-cost (i.e.
+from the terminal state to the final equilibrium) shall be added into the cost function for on-line optimisation. A natural
+question arising is, what happens if condition (1) is violated, or more precisely, if there does not exist an admissible
+control u ∈ U such that condition (1) holds. Formally, this can be represented as, for all u ∈ U and x ̸= 0,
+ℓ(x, u, x+) := −Vf(x) + l(x, u) + Vf(f(x, u)) > 0
+(2)
+This question is more interesting since it includes the case that Vf(x) = 0, i.e. no terminal cost. In the case of zero
+terminal cost, condition (2) is satisfied as long as the stage cost, l(x, u), is positive definite. Unfortunately, numerous
+counterexamples exist where condition (2) holds but the resulted MPC is unstable, e.g. [5,6]. It is noted that significant
+effort and progress have been made in relaxing the requirement on the terminal cost in satisfying (1) [7–10]. For
+example, stability can still be achieved if the receding horizon is sufficiently long even in the case when the terminal
+∗Corresponding author: Wen-Hua Chen.
+arXiv:2301.12024v1  [math.OC]  27 Jan 2023
+
+Stability of Finite Receding Horizon Control: A Complementary Approach
+weight is zero [8]. So far, there is little work devoted to directly exploring the space defined by condition (2) which is
+opposite to the design space defined by (1).
+The motivation of this paper is to answer the question under what condition a finite receding horizon controller satisfying
+(2) is stable. Condition (2) is equivalent to m(x) > 0, where
+m(x) := min
+u∈U ℓ(x, u, x+)
+(3)
+is referred as the One-Step Value Function (OSVF). That plays a key role in establishing stability of MPC in this paper.
+The main contribution of this paper is to present a slightly modified MPC algorithm which is stable as long as the
+corresponding OSVF is a control Lyapunov function (CLF).
+Since condition (2) is precisely opposite to condition (1) used in establishing stability in the current MPC literature for
+more than two decades [1,9,11], our results provide a complementary approach to the existing terminal cost based MPC
+framework. That is, all cases covered by our new conditions do not satisfy the existing stability conditions developed
+based on (1). Furthermore, our conditions allow stability to be established if there is no terminal cost or even a negative
+terminal cost (see numerical examples). It clearly shows that, despite the huge success in developing stability theory for
+MPC in the last three decades [1,10,12], there is still a significant, unexplored, design space where stability is possible
+to achieve by optimising a finite horizon cost function. By combining the new analysis approach in this paper with
+the existing ones, particularly the existing terminal cost based MPC stability conditions, we are in a much stronger
+position in analysing stability of MPC and other optimisation based control methods and designing stability guaranteed
+optimisation based algorithms with a much less restricted freedom and space. This is the main motivation behind this
+paper.
+There are two key ideas behind the proposed approach. The first is to define a new stage cost, ℓ(x, u, x+), as in (2) by
+rotating the terminal cost. It shall be pointed out that the stage cost in (2) is somehow similar to the rotation stage cost
+technique widely used in economic MPC (EMPC) [10]. To establish stability of EMPC, the stage cost is first rotated by
+a storage function, λ(x), (normally being non-negative [10]). Subsequently, it is required that the modified terminal
+cost, ˜
+Vf(x) = Vf(x) + λ(x), is a CLF. This is the same technique as in the current tracking MPC (1); for example,
+see [10,13,14]. It is interesting to notice that the widely used dissipativity inequality is equivalent to condition (2) when
+the storage function λ(x) is specifically chosen as −Vf(x). However, we cannot derive any stability results under this
+specific choice as discussed in Remark 1.
+The second idea is that it is observed that the change of the initial state related cost in a cost function involved in online
+optimisation involved in MPC does not change the optimal control sequence and the MPC algorithm. By combining
+these two ideas, we are able to modify the MPC cost function but without changing its behaviour. Fundamentally,
+when the optimal value function of the modified cost is used as a Lyapunov function candidate to establish its stability,
+the combination of these two ideas enables a wide range of possible Lyapunov function candidates to be used to
+establish stability for the same MPC algorithm. This breaks down the rigid link between the cost function used in
+optimisation in an MPC algorithm and the optimal value function used as a Lyapunov function candidate in the current
+approaches [1,11]. The fact that changing initial state cost does not alter the optimisation involved in MPC has been
+exploited in [9,15]. By further exploiting this property in this paper, we are able to explore new design space for MPC
+stability that was not possible previously.
+Technically, to establish stability by the virtue of the OSVF being a CLF, we have to slightly modify the MPC algorithm.
+That is, rather than using a fixed terminal constraint as in the current terminal cost based framework, we construct
+a contractive terminal set by making use of the OSVF being a CLF. The idea of using a contractive terminal set can
+be found in [16]; however, the whole optimal control sequence (not only the first element) is applied into the system
+in [16]. This does not conform with the MPC setup in this paper.
+This paper is organised as follows. In Section 2, a new stage cost is proposed by rotating the terminal cost in the
+conventional cost function without affecting the optimal trajectories. Based on this new augmented stage cost and some
+mild assumptions, a new MPC algorithm with stability guarantee is proposed in Section 3. In Section 4, a procedure for
+finding a terminal cost satisfying the new stability conditions is developed. Results are provided by two illustrative
+examples, giving further insight into the proposed approach, in Section 5. Finally, Section 6 concludes the paper.
+Notation. I and R are integers and real numbers, respectively, where subscripts may be added to give specific ranges.
+For any vector x ∈ Rn, |x| denotes the 2-norm and |x|P is defined by |x|P := xT Px, where P ∈ Rn×n is a symmetric
+matrix, but not necessary to be positive-definite here. The superscript ∗ is used to indicate the optimal solution, the
+optimal control sequence or state under the optimal control sequence.
+2
+
+Stability of Finite Receding Horizon Control: A Complementary Approach
+2
+Augmented stage cost and equivalent MPC problems
+Consider a nonlinear discrete-time system described by
+xk+1 = f(xk, uk), k ∈ I≥0
+(4)
+or, more succinctly x+ = f(x, u), where x or xk ∈ X is the current state, u or uk ∈ U is the current control, and
+x+ or xk+1 is the successor state. It is assumed in this paper that
+(A1) f(x, u) : X × U → X is continuous and f(0, 0) = 0. X ⊆ Rn and U ⊆ Rm are closed and contain the origin in
+the interior.
+We start from a classic finite horizon optimisation problem, with the cost function
+J(xk, uk) :=
+N−1
+�
+i=0
+l(xk+i|k, uk+i|k) + Vf(xk+N|k),
+(5)
+where xk|k = xk and xk+i+1|k = f(xk+i|k, uk+i|k), i ∈ I0:N−1; the control and state sequences are1 uk :=
+�
+uk|k, · · · , uk+N−1|k
+�
+and xk :=
+�
+xk|k, · · · , xk+N|k
+�
+, subject to the constraints U and X, N ≥ 1 is the length of the
+control horizon, l(x, u) and Vf(x) are the stage cost and the terminal cost respectively. This satisfies the following
+assumption.
+(A2) l(x, u) : X × U → R and Vf(x) : X → R are continuous. Furthermore, l(0, 0) = 0 and Vf(0) = 0.
+It shall be pointed out that there is no requirement that (l, x) or Vf(x) shall be positive definite in this paper.
+Following the conventional MPC framework, the first control action of the optimal control sequence u∗
+k is applied into
+the system, uk = u∗
+k|k, which implies that
+xk+1 = f(xk, uk) = f
+�
+xk|k, u∗
+k|k
+�
+= x∗
+k+1|k.
+(6)
+For the widely used cost function in (5), it is observed that the cost that is only associated with the initial state, xk|k, is
+never changed by any control sequence. That is, this term is completely independent from the optimisation process.
+Motivated by this intuitive but important observation, we rewrite the cost function as
+J(xk, uk)
+=
+N−1
+�
+i=0
+l(xk+i|k, uk+i|k) + Vf(xk+N|k)
+= Vf(xk|k) −Vf(xk|k) + l(xk|k, uk|k) + Vf(xk+1|k)
+�
+��
+�
+ℓ(xk|k,uk|k,xk+1|k)
+...
+−Vf(xk+N−1|k) + l(xk+N−1|k, uk+N−1|k) + Vf(xk+N|k)
+�
+��
+�
+ℓ(xk+N−1|k,uk+N−1|k,xk+N|k)
+= Vf(xk|k) +
+N−1
+�
+i=0
+ℓ(xk+i|k, uk+i|k, xk+i+1|k),
+(7)
+where ℓ(x, u, x+) is the new augmented stage cost and has been defined in (2) by rotating the terminal cost. It shall be
+highlighted that the information of the system dynamics is embedded in this new stage cost. By ignoring the initial one,
+Vf(xk|k), since it does not affect the solution of the optimisation, we can define the new cost function for MPC online
+optimisation as
+J (xk, uk) :=
+N−1
+�
+i=0
+ℓ(xk+i|k, uk+i|k, xk+i+1|k).
+(8)
+1For simplicity, when used in algebraic expressions, xk denotes the column vector
+�
+xT
+k|k, · · · , xT
+k+N|k
+�T ; similarly in algebraic
+expressions of uk.
+3
+
+Stability of Finite Receding Horizon Control: A Complementary Approach
+Considering the optimisation problems based on different cost functions, J(xk, uk) and J (xk, uk), with the same
+initial state xk and the same constraints, the optimal trajectories should be the same, regardless of the different value
+functions. That is
+min
+uk J (xk, uk) = −Vf(xk) + min
+uk J(xk, uk)
+arg min
+uk J (xk, uk) = arg min
+uk J(xk, uk).
+(9)
+Considering (9), the design and analysis on the conventional MPC are equivalent to that with a new cost function
+J (xk, uk), which is given in a form without any terminal cost.
+Remark 1 The stage cost rotation technique has been widely used for EMPC in establishing its stability [13,14,17,18].
+More specifically, a dissipativity inequality is defined with a non-negative storage function λ(x) (see Definition 3.1
+in [10]), that is,
+−λ(x) + λ(f(x, u)) − l(x, u) ≤ −l(xs, us)
+(10)
+is satisfied for all x and u, where xs and us are in steady state operating condition and l(xs, us) can be treated as
+zero when the equilibrium point is at the origin. Then stability of EMPC can be established by mainly resorting to
+standard stabilising MPC theory. That is, to establish stability, the modified terminal cost ˜Vf(x) = Vf(x) + λ(x) is
+required to be a Control Lyapunov Function (CLF), satisfying condition (1). The authors are aware that significant
+effort and progress has been made in further developing stability analysis of EMPC with various extensions (see, for
+example, [10,17–20]).
+Condition (2) can be interpreted as a dissipativity inequality with the corresponding storage function as λ(x) = −Vf(x).
+It is interesting to notice that the resulted storage function λ(x) is quite often negative definite since the terminal cost is,
+in general, positive definite, which makes it different from a standard dissipation definition. Most importantly, with
+the choice of λ(x) = −Vf(x), the modified terminal cost becomes ˜Vf(x) = 0, which does not satisfy the requirement
+that the terminal cost must be a CLF, i.e. condition (1). We are not able to assert stability of EMPC or any other MPC
+schemes by following this approach.
+Furthermore, based on the dissipativity condition, similar functions have recently been defined in EMPC using a general
+storage function which is referred to as a control dissipativity function in [21] and control storage function in [19].
+Remark 2 The approach proposed in this paper is applicable to both tracking MPC and EMPC directly. It follows
+from Assumption (A3) that the stage cost l(x, u) could be positive or negative as long as condition (2) is satisfied. In
+that sense, we also present an alternative stability condition for EMPC where the rotated terminal cost ˜
+Vf(x) does not
+satisfy the requirements in the current stability conditions, e.g. to be a CLF.
+3
+MPC algorithm and stability
+3.1
+New terminal weight based MPC algorithm
+Here we define an auxiliary optimisation problem to minimise the augmented stage cost ℓ(x, u, x+) in (2) or (7)
+m(x) := min
+u∈U ℓ(x, u, x+)
+= min
+u∈U − Vf(x) + l(x, u) + Vf(x+)
+s.t. x ∈ Ω ⊆ X, u ∈ U, f(x, u) ∈ Ω,
+(11)
+where m(x) : Ω ⊆ X �→ R is the One-Step Value Function (OSVF) of the augmented stage cost and Ω is a control
+invariant set.
+It shall be highlighted that m(x) > 0 iff condition (2) holds. In other words, m(x) > 0 only if there does not exist any
+control u ∈ U such that condition (1) holds. We will investigate the design space where m(x) > 0 is positive definite.
+This is entirely complementary to the design space investigated by the existing MPC stability theories and has not been
+explored previously.
+We define the sublevel set of m(x) > 0 as
+Ω(α) := {x ∈ Rn : m(x) ≤ α} ⊆ Ω ⊆ X,
+(12)
+where α is a positive constant. As briefly mentioned in the introduction, the proposed MPC in this paper is based on the
+condition that OSVF is a CLF. That is, the following assumption is imposed in this paper.
+4
+
+Stability of Finite Receding Horizon Control: A Complementary Approach
+(A3) The OSVF m(x) is a CLF for system (4) with respect to a set Ω(α0), i.e., for any x ∈ Ω(α0), there exist a control
+u ∈ U and K∞ functions β1(·), β2(·), β3(·) such that
+β1(|x|) ≤ m(x) ≤ β2(|x|)
+m(f(x, u)) − m(x) ≤ −β3(|x|),
+(13)
+(14)
+where α0 > 0.
+Based on Assumption (A3), we are now able to construct a contractive terminal set, which is the only difference from
+the conventional MPC with terminal cost. The new algorithm is described below.
+Algorithm:
+Offline:
+Step 1
+For the given system (4) subject to the constraints X and U, define the stage cost l(x, u)
+and the terminal cost Vf(x), and calculate terminal constraint α0. Initialise the time as
+k = 0.
+Online:
+Step 2
+At time k, measure the current state xk. Solve the following optimisation problem
+V (xk, αk) := min
+uk J (xk, uk)
+= min
+uk
+N−1
+�
+i=0
+ℓ
+�
+xk+i|k, uk+i|k, xk+i+1|k
+�
+s.t. xk|k = xk
+xk+i+1|k = f(xk+i|k, uk+i|k)
+xk+i|k ∈ X, uk+i|k ∈ U, i ∈ I0:N−1
+xk+N|k ∈ Ω(αk)
+(15)
+and obtain the optimal solution u∗
+k and x∗
+k. Apply the optimal control uk = u∗
+k|k to system
+(4).
+Step 3
+Denote the minimum between m
+�
+x∗
+k+N|k
+�
+and m
+�
+x∗
+k+1|k
+�
+as
+m
+�
+x∗
+k+s|k
+�
+:= min
+�
+m
+�
+x∗
+k+1|k
+�
+, m
+�
+x∗
+k+N|k
+��
+, s ∈ {1, N}.
+(16)
+Update the terminal set by the following rule
+αk+1 =
+�
+�
+�
+�
+�
+m
+�
+x∗
+k+s|k
+�
+− δ, if m
+�
+x∗
+k+s|k
+�
+≥ δ
+0, if m
+�
+x∗
+k+s|k
+�
+< δ,
+(17)
+where δ > 0 is a sufficient small constant.
+Step 4
+k ← k + 1 and go to Step 2.
+Remark 3 Compared with the conventional MPC setup, Step 3 is the only difference. To establish stability, there are
+also three key terminal elements in this algorithm: terminal cost, terminal control and terminal constraints. Instead
+of using a fixed terminal constraint, a contractive terminal constraint is constructed in Step 3 where one-step ahead
+optimisation is involved in (16) and (17). It will be shown that under the condition that, for a given terminal cost, if the
+corresponding OSVF is a CLF, then it is always feasible to construct a contractive terminal constraint. That is, Step 3
+and the new MPC algorithm is always feasible as long as it is feasible at the beginning. We will discuss how to choose
+a terminal cost such that the corresponding OSVF is a CLF in Section 4.
+Remark 4 One of the drawbacks in the current terminal weight based MPC framework is how to strike a good balance
+between performance and stability, particularly for nonlinear or uncertain system. For a given stage cost, it is quite
+difficult to estimate the optimal cost-to-go for these systems. Since, in order to achieve stability, it requires the terminal
+cost to cover the cost to go, quite often a conservative terminal cost function has to be employed to ensure the unknown
+optimal value function is covered. This may lead to poor performance. The total cost to be optimised in MPC consists
+of the summed stage cost and the terminal cost. If the chosen terminal cost is much larger than the summed stage cost,
+5
+
+Stability of Finite Receding Horizon Control: A Complementary Approach
+the optimisation process actually puts more effort on the terminal cost since the former carries a higher relative weight.
+This is despite the fact that the stage cost may represent the performance requirements better. Our approach avoids
+this problem by exploring the design space where the terminal cost cannot cover the cost-to-go, even if it is zero. Our
+hypothesis is that this may give a better MPC performance for nonlinear or uncertain systems, as shown in the second
+example in Section 5.
+The main result is given by the following theorem while the detailed proof is given by the subsequent sections.
+Theorem 1 Suppose that
+• Assumptions (A1)-(A3) are satisfied,
+• the optimisation problem (15) is feasible at time k = 0,
+• the parameter α0 is chosen such that the terminal set Ω(α0) ⊆ Ω
+Then, the closed-loop system with the proposed MPC is recursively feasible and, eventually, asymptotically stable.
+3.2
+Recursive feasibility
+Recursive feasibility of the proposed MPC is mainly based on the condition of OSVF being a CLF, seeing Assumption
+(A3).
+Proof of recursive feasibility in Theorem 1. Supposing that the optimisation (15) is feasible at time k ∈ I≥0, the optimal
+control and state sequences exist, which can be denoted as
+u∗
+k =
+�
+u∗
+k|k, · · · , u∗
+k+N−1|k
+�
+x∗
+k =
+�
+x∗
+k|k, · · · , x∗
+k+N|k
+�
+,
+where the terminal state satisfies that
+x∗
+k+N|k ∈ Ω(αk) ⇔ m
+�
+x∗
+k+N|k
+�
+≤ αk.
+(18)
+In what follows, we will construct the feasible control and state sequences at time k + 1 based on the current optimal
+sequences. The case when the terminal set shrinks to the origin is ignored as it is degenerated into the conventional
+MPC with an equality terminal constraint. It is straightforward to show its recursive feasibility. To make the discussion
+clear, we use the superscripts, a and b, to represent two cases due to the different results of (16).
+In the case of m
+�
+x∗
+k+1|k
+�
+≥ m
+�
+x∗
+k+N|k
+�
+, based on (17), if we choose δ ∈ (0, β3(|x∗
+k+N|k|)], then
+αk+1 = m
+�
+x∗
+k+N|k
+�
+− δ
+≥ m
+�
+x∗
+k+N|k
+�
+− β3
+����x∗
+k+N|k
+���
+�
+.
+(19)
+By Assumption (A3) and since x∗
+k+N|k
+∈
+Ω(αk), there exists a control ua
+k+N|k
+∈
+U, xa
+k+N+1|k
+=
+f
+�
+x∗
+k+N|k, ua
+k+N|k
+�
+such that
+m
+�
+xa
+k+N+1|k
+�
+≤ m
+�
+x∗
+k+N|k
+�
+− β3
+����x∗
+k+N|k
+���
+�
+≤ αk+1
+⇒xa
+k+N+1|k ∈ Ω(αk+1).
+(20)
+The feasible sequences at time k + 1 can be constructed as
+ua
+k+1 =
+�
+u∗
+k+1|k, · · · , u∗
+k+N−1|k, ua
+k+N|k
+�
+xa
+k+1 =
+�
+x∗
+k+1|k, · · · , x∗
+k+N|k, xa
+k+N+1|k
+�
+.
+In the case of m
+�
+x∗
+k+1|k
+�
+< m
+�
+x∗
+k+N|k
+�
+, and noting (18), we have that
+m
+�
+x∗
+k+1|k
+�
+< m
+�
+x∗
+k+N|k
+�
+≤ αk
+⇒x∗
+k+1|k ∈ Ω(αk).
+(21)
+6
+
+Stability of Finite Receding Horizon Control: A Complementary Approach
+Based on (17), we have that
+αk+1 ≤ m
+�
+x∗
+k+1|k
+�
+< m
+�
+x∗
+k+N|k
+�
+≤ αk.
+(22)
+By Assumption (A3), there exists a control ub
+k+1|k ∈ U, xb
+k+2|k = f
+�
+x∗
+k+1|k, ub
+k+1|k
+�
+such that
+m
+�
+xb
+k+2|k
+�
+≤ m
+�
+x∗
+k+1|k
+�
+− β3
+����x∗
+k+1|k
+���
+�
+≤ m
+�
+x∗
+k+1|k
+�
+− δ
+= αk+1, if δ ≤ β3
+����x∗
+k+1|k
+���
+�
+⇒xb
+k+2|k ∈ Ω(αk+1) ⊆ Ω(αk).
+(23)
+Using Assumption (A3) again and noting (23), there exists a ub
+k+2|k ∈ U, xb
+k+3|k = f
+�
+xb
+k+2|k, ub
+k+2|k
+�
+such that
+m
+�
+xb
+k+3|k
+�
+≤ m
+�
+xb
+k+2|k
+�
+− β3
+����xb
+k+2|k
+���
+�
+≤ m
+�
+xb
+k+2|k
+�
+≤ αk+1
+⇒xb
+k+3|k ∈ Ω(αk+1).
+(24)
+Combing (23) and (24) together, we have that the set Ω(αk+1) is also control invariant and hence there exists a control
+sequence ub
+k+i|k ∈ U, i ∈ I0:N such that
+xb
+k+1+i|k = f
+�
+xb
+k+i|k, ub
+k+i|k
+�
+∈ Ω(αk+1).
+(25)
+The feasible sequences of this case can be conducted as
+ub
+k+1 =
+�
+ub
+k+1|k, · · · , ub
+k+N−1|k, ub
+k+N|k
+�
+xb
+k+1 =
+�
+x∗
+k+1|k, · · · , xb
+k+N|k, xb
+k+N+1|k
+�
+.
+In conclusion, based on the above discussion on the two cases, the proposed algorithm is recursively feasible, which
+completes the proof.
+■
+Recursive feasibility guarantees that once the proposed algorithm is feasible at time k = 0, the optimisation algorithm
+associated with MPC and the closed-loop system is always feasible, which implies that all the signals in the control
+system (e.g., αk) are well defined, even as time goes to infinity.
+3.3
+Asymptotic stability
+To establish stability in Theorem 1, several lemmas are necessary. Lemma 1 shows the upper bounds of the new value
+function V (x, α) in (15) while Lemma 2 establishes the condition for its monotonicity.
+Lemma 1 Suppose that Assumption (A3) holds. Then, for any feasible state x of the optimisation problem (15) with
+Ω(α) ⊆ Ω, there exists a K∞ function β4(·) such that V (x, α) ≤ β4(|x|).
+Proof. We first consider a simple case, where x ∈ Ω(α). Then, we replace the optimisation problem (15) by a
+suboptimal control problem, i.e. splitting it into N one-step optimisation problems (11). Recursively solving these N
+one-step optimisation problems as in (11) and using the optimal states and inputs as the feasible ones of the optimisation
+problem (15), it will give that
+V (x, α) ≤ Nβ2(|x|), ∀x ∈ Ω(α).
+(26)
+Therefore, V (x, α) is continuous at zero as 0 ∈ Ω(α). Following the similar analysis given in [12, Chap. 2.4.2], we
+know that the set of feasible states is closed and V (x, α) is locally bounded by it. By applying [12, Proposition B.25],
+the upper bound conclusion can be extended to any feasible state, which completes the proof.
+■
+Lemma 2 Suppose that there exists a control uk+N|k ∈ U such that xk+N+1|k = f
+�
+x∗
+k+N|k, uk+N|k
+�
+∈ Ω(αk+1).
+Then
+V (xk+1, αk+1) − V (xk, αk) ≤ ℓ
+�
+x∗
+k+N|k, uk+N|k, xk+N+1|k
+�
+− ℓ
+�
+x∗
+k|k, u∗
+k|k, x∗
+k+1|k
+�
+.
+7
+
+Stability of Finite Receding Horizon Control: A Complementary Approach
+Proof. We denote the optimal control and state sequences at time k as
+u∗
+k =
+�
+u∗
+k|k, · · · , u∗
+k+N−1|k
+�
+x∗
+k =
+�
+x∗
+k|k, · · · , x∗
+k+N|k
+�
+.
+The value function V (xk, αk) can be rewritten as
+V (xk, αk) =
+N−1
+�
+i=0
+ℓ
+�
+x∗
+k+i|k, u∗
+k+i|k, x∗
+k+i+1|k
+�
+.
+(27)
+If there exists a control uk+N|k ∈ U such that xk+N+1|k = f
+�
+x∗
+k+N|k, uk+N|k
+�
+∈ Ω(αk+1), we could construct the
+feasible sequences at time k + 1 as
+uk+1 =
+�
+u∗
+k+1|k, · · · , u∗
+k+N−1|k, uk+N|k
+�
+xk+1 =
+�
+x∗
+k+1|k, · · · , x∗
+k+N|k, xk+N+1|k
+�
+.
+Thus, we have that
+V (xk+1, αk+1) − V (xk, αk)
+≤ J (xk+1, uk+1) − V (xk, αk)
+= J
+�
+x∗
+k+1|k, uk+1
+�
+− V (xk, αk)
+= ℓ
+�
+x∗
+k+N|k, uk+N|k, xk+N+1|k
+�
+− ℓ
+�
+x∗
+k|k, u∗
+k|k, x∗
+k+1|k
+�
+,
+(28)
+which completes the proof.
+■
+We are now ready to prove Theorem 5 for asymptotic stability of the proposed MPC algorithm. It is established through
+a two stage process. In the first process, it is shown that the terminal set continuously shrinks to the origin under the
+constructed update rule. In the second stage, asymptotic stability is established using V (x, 0) as a Lyapunov function.
+Proof of asymptotic stability in Theorem 1. Recursive feasibility established in Section 3.2 shows that
+αk+1 ≤ m
+�
+x∗
+k+s|k
+�
+− δ ≤ m
+�
+x∗
+k+N|k
+�
+− δ ≤ αk − δ,
+(29)
+which implies that the sequence {αk} is continuously decreasing until reaching the origin since δk > 0 until x∗
+k+N|k = 0.
+Consequently, once the terminal constraint set approaches the origin, one has
+x∗
+k+N|k = 0.
+(30)
+It follows from Assumption (A3) that
+ℓ
+�
+x∗
+k+N|k, uk+N|k, xk+N+1|k
+�
+= 0.
+(31)
+Following Lemma 2 with αk+1 = αk = 0, we have
+V (xk+1, 0) − V (xk, 0)
+≤ −ℓ
+�
+x∗
+k|k, u∗
+k|k, x∗
+k+1|k
+�
+≤ −m(xk|k)
+≤ −β1(|xk|k|).
+(32)
+Combining Lemma 1 and noting V (x, α) ≥ m(x) ≥ β1(|x|), the closed-loop system is asymptotically stable under the
+proposed MPC algorithm which completes the proof.
+■
+Remark 5 Similar to the existing MPC theories, asymptotic stability is established also by employing an optimal value
+function V (x, 0) as a Lyapunov function candidate in our approaches. However there are three key differences in these
+two approaches. First, the new value function V (x, α) is different from the the original optimal value function obtained
+by optimising the original cost function (5) used in the existing MPC stability theories. This is not only because the
+8
+
+Stability of Finite Receding Horizon Control: A Complementary Approach
+modification of the cost function by removing the cost associated with initial state, but also the contractive constraints
+in the online optimisation, caused by shirking αk. Secondly, stability of the MPC algorithm is established in the existing
+MPC theories by exploiting the monotonicity of the optimal value function during the whole MPC control process.
+However we are not able to establish the monotonicity of the value function, V (x, α), at the beginning of the contracive
+MPC algorithm proposed in this paper. Actually V (x, α) may increase if the terminal set shrinks quite quickly based on
+the proposed update rule. We rely on the contractive terminal constraints enabled by the property of OSVF being a
+CLF to force the system state to approach the origin. Once the terminal set shrinks to the origin, we are able to show
+the monotonicity of the value function V (x, 0). Finally, we establish stability by imposing a certain property on the
+modified stage cost (i.e. OSVF to be a CLF) while the existing stability theories ensure stability by imposing a condition
+on the terminal cost (i.e. the terminal cost to be a CLF).
+4
+Terminal cost calculation
+All the assumptions used in establishing stability of MPC are quite similar to that in the existing terminal cost based
+MPC framework except Assumption (A3). This section is devoted to developing a procedure for finding a terminal cost
+such that the corresponding OSVF m(x) is a CLF. As discussed previously, the condition m(x) being positive definite
+is opposite to the existing well-established MPC stability conditions. Furthermore, it is possible for Vf(x) to be zero
+which includes the case of no terminal weight. Therefore, the results [6] for stability analysis of MPC without terminal
+cost can be considered as a special case in this paper. More importantly, we are able to explore the freedom of adding
+an appropriate terminal cost to guarantee the stability of a MPC algorithm of concern.
+In what follows, we propose a design procedure to find a terminal cost to satisfy Assumption (A3) for linear systems,
+which can be extended to nonlinear systems by the similar approach of the current MPC framework. Linearisation
+of the nonlinear system at the operation condition [12] is one such approach, or using another approach like linear
+differential inclusion as in [22]. It also provides more insight about the definition and the properties of OSVF m(x).
+Here, we consider a linear system as
+x+ = Ax + Bu,
+(33)
+with the stage cost l(x, u) = |x|2
+Q + |u|2
+R. We aim to find a terminal cost Vf(x) = |x|2
+P satisfying Assumption (A3).
+The modified stage cost in (2) can be specified as
+ℓ
+�
+x, u, x+�
+= −|x|2
+P + |x|2
+Q + |u|2
+R + |Ax + Bu|2
+P
+= |x|2
+AT P A+Q−P + |u|2
+R+BT P B
++ 2xT AT PBu
+=
+�
+xT
+uT �
+M
+�
+x
+u
+�
+,
+(34)
+where
+M :=
+�
+AT PA + Q − P
+AT PB
+BT PA
+R + BT PB
+�
+.
+(35)
+If R + BT PB > 0, solving the one-step optimisation problem (11) gives
+arg min
+u ℓ(x) = −(R + BT PB)−1BT PAx
+m(x) = min
+u ℓ(x) = |x|2
+MP ,
+(36)
+where
+MP := AT PA + Q − P − AT PB(R + BT PB)−1BT PA.
+(37)
+Since that m(x) serves as a CLF in Assumption (A3), MP should at least be positive definite. By the Schur Complement
+Lemma, R + BT PB > 0 and MP > 0 is equivalent to M > 0.
+This result is summarised in the following lemma.
+Lemma 3 Consider a linear system (33) with a quadratic cost function. Then its OSVF m(x) is positive definite if
+there exists a terminal cost Vf(x) = |x|2
+P with a matrix P such that M defined in (35) is positive definite.
+By now, we only need to consider condition (14). Since in this case, m(x) is in a quadratic form, condition (14) can be
+relaxed since there exists a control uk such that m(xk+1) < m(xk) holds for any xk.
+9
+
+Stability of Finite Receding Horizon Control: A Complementary Approach
+Lemma 4 m(xk+1) < m(xk) holds, if and only if there exist uk and uk+1 such that ℓ(xk+1, uk+1, xk+2) < m(xk)
+holds.
+Proof. ⇐: If there exist uk and uk+1 such that ℓ(xk+1, uk+1, xk+2) < m(xk), we then have that m(xk+1) ≤
+ℓ(xk+1, uk+1, xk+2) < m(xk).
+⇒: If m(xk+1) < m(xk), the existence of uk is obvious. As for uk+1, we can choose
+uk+1 = arg min
+uk+1 ℓ(xk+1, uk+1, xk+2).
+Hence, there exist uk and uk+1 such that
+ℓ(xk+1, uk+1, xk+2) = m(xk+1) < m(xk).
+This completes the proof.
+■
+Lemma 5 If R > 0, then
+�
+Q − SR−1ST
+A
+AT
+B
+�
+> 0 ⇔
+�
+�
+Q
+S
+A
+ST
+R
+0
+AT
+0
+B
+�
+� > 0.
+(38)
+Proof. By using congruent transformation, we have that
+�
+�
+I
+−SR−1
+0
+0
+I
+0
+0
+0
+I
+�
+�
+�
+�
+Q
+S
+A
+ST
+R
+0
+AT
+0
+B
+�
+�
+�
+�
+I
+0
+0
+−R−1ST
+I
+0
+0
+0
+I
+�
+�
+=
+�
+�
+Q − SR−1ST
+0
+A
+0
+R
+0
+AT
+0
+B
+�
+� .
+This completes the proof.
+■
+Theorem 2 If there exists a terminal cost P and two control gains K1 and K2 such that the following bilinear matrix
+inequality (BMI) is satisfied
+�
+���
+M
+�
+(A + BK1)T
+KT
+2
+0
+0
+�
+M
+M
+�
+(A + BK1)
+0
+K2
+0
+�
+M
+�
+��� > 0
+(39)
+then m(x) is a CLF for system (33).
+Proof. Note that if condition (39) is satisfied, it follows that M > 0, which further guarantees R + BT PB > 0 and
+MP > 0 by using the Schur Complement Lemma. R + BT PB > 0 implies that the one-step optimisation problem
+(11) is well-defined.
+Letting uk = K1xk and uk+1 = ˜K2xk+1, a quick calculation gives
+ℓ(xk+1, uk+1, xk+2)
+=
+�
+xT
+k (A + BK1)T
+xT
+k KT
+2
+�
+M
+�
+(A + BK1)xk
+K2xk
+�
+= xT
+k
+�
+(A + BK1)T
+KT
+2
+�
+M
+�
+A + BK1
+K2
+�
+xk
+(40)
+where K2 = ˜K2(A + BK1). Using the Schur Complement Lemma and Lemma 4, noting R + BT PB > 0, the
+condition ℓ(xk+1, uk+1, xk+2) < m(xk) is satisfied if the following matrix inequality holds
+10
+
+Stability of Finite Receding Horizon Control: A Complementary Approach
+�
+(A + BK1)T
+KT
+2
+�
+M
+�
+A + BK1
+K2
+�
+< MP
+⇔
+�
+(A + BK1)T
+KT
+2
+�
+MM −1M
+�
+A + BK1
+K2
+�
+< MP
+⇔
+�
+�
+MP
+�
+(A + BK1)T
+KT
+2
+�
+M
+M
+�
+A + BK1
+K2
+�
+M
+�
+� > 0
+⇔ (39).
+This completes the proof.
+■
+Remark 6 The proposed BMI (39) can be solved by using the PENLAB package [23], which is an open source software
+package implemented in MATLAB for matrix inequalities.
+5
+Illustrative examples
+5.1
+A first order example
+This section is to illustrate the significance of the proposed new stability analysis approach in this paper and highlight
+the main differences with the existing stability theory of MPC by using a simple first order system
+x+ = ax + bu, b ̸= 0
+(41)
+with l(x, u) = qx2 + ru2 and Vf(x) = px2, where the lowercase letters represent scalars.
+More specifically, this example is used to highlight the following two points.
+1. The proposed MPC algorithm may be stable with a zero terminal cost or a negative terminal cost with the new
+condition.
+2. The new stability condition is complementary to the existing terminal cost based MPC stability conditions.
+It follows from the definition of the augmented stage cost that ℓ(x, u, x+) = −px2 + qx2 + ru2 + p(ax + bu)2 for
+this example. The corresponding OSVF m(x) is ready to be calculated. For this first order linear system, according to
+Lemma 3, the positive definite condition in Assumption (A3) reduce to
+r + b2p > 0
+a2p + q − p − a2p2b2
+r + b2p > 0.
+(42)
+Furthermore, Lemma 4 shows that the CLF condition (14) is met for any positive definite OSVF that satisfies condition
+(42).
+Inspecting condition (42) gives
+q > (1 − a2)p + a2p2b2
+r + b2p = 1
+b2
+�
+z1 + a2r2
+z1
+−
+�
+1 + a2�
+r
+�
+,
+(43)
+where z1 := r + b2p > 0. Eq.(43) gives a lower bound of q by
+q > q := 2|ar| − (1 + a2)r
+b2
+=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+−(|a| − 1)2r
+b2
+, if r > 0
+0,
+if r = 0
+−(|a| + 1)2r
+b2
+, if r < 0.
+(44)
+It shall be highlighted that Eq.(44) reveals q is possible to be negative or zero if r ̸= 0,.
+11
+
+Stability of Finite Receding Horizon Control: A Complementary Approach
+Next, for the purpose of comparison, we present the conventional MPC stability conditions for this simple MPC problem.
+First, all the weightings are required to be at least positive semi-definite, i.e., q > 0, r ≥ 0, and p ≥ 0. Secondly, the
+current condition states that the conventional MPC for this unconstrained system is stable if the famous Fake Algebraic
+Riccati Equation (FARE) [24] holds
+b2p2 + (1 − b2q − a2)p − q ≥ 0
+⇔ q ≤ 1
+b2
+�
+z2 + a2r2
+z2
+−
+�
+1 + a2�
+r
+�
+(45)
+where z2 = r + b2p ≥ r.
+If we focus on the right-side functions of (43) and (45), when p is large enough, both functions go to a linear one
+q = p − ra2
+b2 , which is parallel with but below the line q = p if r ̸= 0.
+In what follows, we will make an explicit comparison of the stability conditions and regions between the proposed new
+approach and the existing terminal weight based MPC framework, in terms of the control weight r > 0, r < 0, and
+r = 0.
+5.1.1
+Case with control weight r > 0
+We only consider the case that r = 1, as the same results hold for a different r by normalizing state and terminal
+weighting with q/r and p/r. Regarding stability condition (44), there are three cases to discuss, |a| < 1, |a| = 1, and
+|a| > 1, which correspond to open-loop stable, marginally stable, and unstable systems, respectively. The stability
+regions defined by the new condition (44) are given by Fig. 1. These regions actually define the possible parameter
+combinations of the system and cost functions including the stage and terminal cost so we refer them as the feasible
+design spaces. It is interesting to see when |a| < 1, the proposed MPC could be stable even when both p and q are
+negative, which covers the situation widely studied in EMPC.
+It shall be noted that the blue curve corresponds to the solution of the Riccati equations under different stage costs, i.e.
+q. Therefore, to ensure stability, the existing MPC stability condition requires that the terminal cost p is larger than or
+equal to the solution of the Riccati equation. This is shown by the region on the right side of the blue curve; that is, the
+system under MPC is stable when the corresponding terminal cost p covers the cost-to-go. In contrast, the stability
+region given by our new stability condition lies on the left side of the blue curve so covers the design space where the
+terminal weight p is less than the Riccati solution under the corresponding stage and control weights q and r.
+It is observed that there is no overlap between the stability regions derived by the existing terminal weight based MPC
+framework and the new condition proposed in this paper. Furthermore, by combining the stability regions yielded by
+these two methods, we are able to significantly extend the MPC design space (i.e. in terms of possible terminal costs)
+where stability could be guaranteed. Therefore, our approach is complementary to the existing MPC framework.
+Finally, is is also observed that both the lines p = 0 and p = q are covered by the proposed MPC stability condition as
+shown in Fig 1. p = 0 and p = q are both the special cases of MPC where there is no terminal cost, but with slightly
+different Lyapunov functions. To well illustrate the difference, we take an example when N = 2.
+• If p = 0, the optimisation in (15) reduces to
+min
+uk|k, uk+1|k
+�
+qx2
+k|k + ru2
+k|k
+�
++
+�
+qx2
+k+1|k + ru2
+k+1|k
+�
+s.t. xk|k = xk
+xk+i+1|k = axk+i|k + buk+i|k, i = 0, 1
+xk+2|k ∈ Ω(αk),
+(46)
+where MP = a2p + q − p − a2p2b2(r + b2p)−1 = q.
+12
+
+Stability of Finite Receding Horizon Control: A Complementary Approach
+• If p = q, the optimisation in (15) becomes
+min
+uk|k, uk+1|k
+�
+qx2
+k|k + ru2
+k|k
+�
++
+�
+qx2
+k+1|k + ru2
+k+1|k
+�
++ px2
+k+2|k
+=
+min
+uk|k, uk+1|k, uk+2|k
+2
+�
+i=0
+�
+qx2
+k+i|k + ru2
+k+i|k
+�
+s.t. xk|k = xk
+xk+i+1|k = axk+i|k + buk+i|k, i = 0, 1
+xk+2|k ∈ Ω(αk),
+(47)
+where MP = a2p+q−p−a2p2b2(r+b2p)−1 = a2q−a2q2b2(r+b2q)−1 = a2qr(r+b2q)−1. The equivalent
+cost function holds as the decision variable uk+2|k would never affect the control and state sequences.
+Therefore, they correspond to the different choices of the augmented stage cost and the associated OSVF. This
+also implies different terminal constraints are used in the algorithm and different Lyapunov functions employed in
+establishing stability. This special case also provides more insight into the proposed new approach.
+5.1.2
+Case with control weight r = 0
+The case of r = 0 is also interesting as both methods occupy the perfect half of the whole design space, as shown by
+Fig. 2 a). This is because when r = 0, condition (43) is equivalent to q > p > 0 while condition (45) is 0 < q ≤ p.
+5.1.3
+Case with control weight r < 0
+In this case, in a similar fashion as r > 0, we fix r = −1 and the result is given by Fig. 2 b). It is worth noting that the
+design space of the conventional MPC framework is empty since the control weight is normally required to be positive
+definite.
+In summary, for the cases of the control weight r < 0 or r = 0, we are able to draw the same conclusion from Fig 2
+about the proposed new stability condition, and its comparison and connection with the existing terminal weight based
+MPC framework, as in the case of r > 0. More simply, for each pair of state and control weights q, r, the existing MPC
+stability theory ascertains that the corresponding MPC algorithm with the terminal cost p in the yellow region is stable
+while our stability condition states that the corresponding MPC algorithm with the terminal cost p in the light green
+area is also stable. Thus, we are able to state that an MPC algorithm is stable with any terminal cost from the combined
+areas of these two stability approaches. This substantially relaxes the stability requirement on MPC algorithms and
+increase the freedom in designing and tuning the key parameters involved in MPC. This numerical example and detailed
+comparison give more insight into the proposed new augmented stage cost method and its links with the existing MPC
+stability theory.
+5.2
+Simple nonlinear system study
+This section is to apply the new proposed MPC algorithm with a contractive terminal constraint on a nonlinear system
+and compare it with the existing terminal weight based MPC approach. It aims to highlight their differences in terminal
+regions and total running costs. Here, these two MPC laws are applied to a benchmark system [25], i.e, a cart with a
+mass moving on a plane, as shown in Fig. 3. The nonlinearity in this example comes from the elasticity in the spring,
+ks = k0e−x1, where x1 is the displacement of the carriage from the equilibrium position associated with the external
+force u = 0. The system is described by
+˙x1 = x2
+˙x2 = − k0
+M e−x1x1 − hd
+M x2 + u
+M ,
+(48)
+where x2 is its velocity and hd is the damping coefficient. The system is discretised into a discrete-time system using
+the Euler approximation with sampling time of 0.4sec, given by
+x1,k+1 = x1,k + 0.4x2,k
+x2,k+1 = −0.132e−x1,k + 0.56x2,k + 0.4uk,
+(49)
+13
+
+Stability of Finite Receding Horizon Control: A Complementary Approach
+-1
+0
+1
+2
+3
+4
+5
+p
+-1
+0
+1
+2
+3
+4
+5
+q
+a=0.5, b=1, r=1
+The new design space
+The conventional design space
+p=0
+p=q
+q=0
+a)
+-1
+0
+1
+2
+3
+4
+5
+p
+-1
+0
+1
+2
+3
+4
+5
+q
+a=1, b=1, r=1
+The new design space
+The conventional design space
+p=0
+p=q
+q=0
+b)
+-1
+0
+1
+2
+3
+4
+5
+p
+-1
+0
+1
+2
+3
+4
+5
+q
+a=1.5, b=1, r=1
+The new design space
+The conventional design space
+p=0
+p=q
+q=0
+c)
+Figure 1: Design spaces of the proposed and conventional MPC for different open-loop systems when r > 0. a) Stable,
+i.e. |a| < 1. b) Marginally stable, i.e. |a| = 1. c) Unstable, i.e. |a| > 1.
+where the specific system parameters are all given in [25]. The state and input constrains are X = [−2, 2] × [−3, 3] and
+U = [−4, 4] and the initial state is x0 = [−2, 1]T . The linearization at the origin gives the controllable pair
+A =
+�
+1
+0.4
+−0.132
+0.56
+�
+, B =
+�
+0
+0.4
+�
+.
+The prediction horizon is chosen to be short enough, N = 3, due to the fast calculation time by the motion control
+system. The stage and terminal costs are both given in a quadratic form, i.e., l(x, u) = |x|2
+Q + |u|2
+R and Vf(x) = |x|2
+P ,
+where
+Q =
+�
+2
+0
+0
+4
+�
+, R = 1.
+In what follows, we will calculate the terminal costs and regions and assess their stability and performance using these
+two methods respectively.
+14
+
+Stability of Finite Receding Horizon Control: A Complementary Approach
+-1
+0
+1
+2
+3
+4
+5
+p
+-1
+0
+1
+2
+3
+4
+5
+q
+a=1, b=1, r=0
+The new design space
+The conventional design space
+p=0
+p=q
+q=0
+a)
+-1
+0
+1
+2
+3
+4
+5
+p
+-1
+0
+1
+2
+3
+4
+5
+q
+a=0.5, b=1, r=-1
+The new design space
+p=0
+p=q
+q=0
+b)
+Figure 2: Design spaces of the proposed and conventional MPC. a) r = 0. b) r < 0.
+𝑘!
+ℎ"
+𝑀
+𝑢
+𝑥#
+Figure 3: Cart and spring example
+• We first consider the conventional MPC design. The terminal cost is obtained by the Riccati equation for the
+linearization
+P =
+�
+10.9153
+4.5604
+4.5604
+7.5023
+�
+and the terminal region is given by
+Ω =
+�
+x ∈ R2 : xT Px ≤ α
+�
+where α = 6.3076 is calculated by the optimization method introduced in [22].
+• As for the proposed MPC design, following the procedure in Section 4, the terminal cost is obtained by solving
+the BMI (39)
+P =
+�
+3.5249
+−0.3522
+−0.3522
+1.5731
+�
+and the corresponding terminal region is given by
+Ω =
+�
+x ∈ R2 : xT MP x ≤ α
+�
+where MP is computed by (37) and α = 5.4823 is also calculated by the same method in [22].
+15
+
+Stability of Finite Receding Horizon Control: A Complementary Approach
+The terminal regions are given by Fig. 4, where the region of the proposed MPC covers that of the conventional one.
+The states and inputs are given by Fig. 5. It shall be highlighted that different from stability regions plotted in the
+parameter space as in Figs. 1 and 2, the terminal regions are given in state space in Fig. 5. It is worth noting that the
+terminal cost P calculated by our approach does not satisfy the condition (1), i.e. covering the cost to go as required in
+the existing terminal weight based MPC framework.
+Although the original stage costs used in these two MPC algorithms are the same, the final cost functions are different
+since their terminal costs are different. The total running cost is used for a fair comparison of the performance of these
+two methods, that is,
+Jrun =
+Ts
+�
+k=0
+l (xk, uk) =
+Ts
+�
+k=0
+�
+|xk|2
+Q + |uk|2
+R
+�
+,
+where Ts is sufficiently large such that the system reaches its steady state, which is chosen as 50 sec in this study. The
+total running cost of the proposed MPC algorithm is 47.2148, which is less than that of the conventional one at 49.1587.
+Therefore, for this example, comparing with the existing terminal weight based MPC scheme, the proposed MPC
+algorithm has a larger terminal region and exhibits slightly better performance.
+-3
+-2
+-1
+0
+1
+2
+3
+-3
+-2
+-1
+0
+1
+2
+3
+Terminal region
+The conventional MPC
+The proposed MPC
+Figure 4: Terminal regions of the proposed and conventional MPC.
+6
+Conclusion
+Stability is an important property of any control method since it helps to ensure the safety of a system. Comparing with
+other optimisation based methods, one distinctive feature of finite horizon optimisation based control is that it is possible
+to establish its stability under certain conditions. However, despite its huge success, the current stability conditions are
+still conservative, which restricts the design space of MPC algorithms. This paper explores the design space where the
+current terminal weight based MPC framework is not applicable. We propose a completely complementary stability
+condition in this unexplored design space. We show that, by constructing a contractive terminal constraint, rather than a
+fixed terminal constraint as in the current MPC framework, we are able to guarantee stability in a significant region that
+has not been covered previously. To achieve this, we construct an augmented stage cost using a terminal cost. That is,
+the augmented stage cost consists of a nominal stage cost and a rotated terminal cost. It is shown that feasibility and
+the stability of the proposed MPC algorithm can be established under the condition that the one-step optimal value
+function of the augmented stage cost is a CLF. Due to the nature of the proposed stability condition, it complements to
+the existing stability theory and is not intended to replace the conventional stability conditions. The stability region
+of MPC can be significantly enlarged by combining the new condition with the existing ones; e.g. guarantee stability
+under a much wider range of combination of state, control and terminal weights. The work in this paper only concerns
+a basic MPC regulation problem without disturbance or uncertainty.
+Our future work includes extending the proposed approach to other areas including robust MPC, stochastic MPC
+or economic MPC where stability has been established with the current terminal weight based MPC approach and
+comparing their differences.
+16
+
+Stability of Finite Receding Horizon Control: A Complementary Approach
+0
+10
+20
+30
+40
+50
+-2
+-1.5
+-1
+-0.5
+0
+0.5
+The conventional MPC
+The proposed MPC
+a)
+0
+10
+20
+30
+40
+50
+-0.5
+0
+0.5
+1
+1.5
+2
+The conventional MPC
+The proposed MPC
+b)
+0
+10
+20
+30
+40
+50
+-2.5
+-2
+-1.5
+-1
+-0.5
+0
+0.5
+The conventional MPC
+The proposed MPC
+c)
+Figure 5: States and inputs of the proposed and conventional MPC. a) x1,k. b) x2,k. c) uk.
+Acknowledgments
+This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) Established
+Career Fellowship “Goal-Oriented Control Systems: Disturbance, Uncertainty and Constarints” under the grant number
+EP/T005734/1.
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+18
+
diff --git a/N9FLT4oBgHgl3EQfOi8m/content/tmp_files/load_file.txt b/N9FLT4oBgHgl3EQfOi8m/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf,len=527
+page_content='STABILITY OF FINITE RECEDING HORIZON CONTROL: A  COMPLEMENTARY APPROACH ∗  Wen-Hua Chen  Department of Aeronautical and Automotive Engineering  Loughborough University  Loughborough LE11 3TU, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='chen@lboro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='uk  Yunda Yan  School of Engineering and Sustainable Development  De Montfort University  Leicester, LE1 9BH, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  yunda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='yan@dmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='uk  ABSTRACT  This paper presents a complementary approach to establish stability of finite receding horizon control  with a terminal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' First a new augmented stage cost is defined by rotating the terminal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Then  a one-step optimisation problem is defined based on this augmented stage cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' It is shown that a  slightly modified Model Predictive Control (MPC) algorithm is stable if the value function of the  augmented one-step cost (OSVF) is a control Lyapunov function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The proposed stability condition is  completely complementary to the existing terminal cost based MPC stability conditions in the sense  that they are mutually excluded with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' By using this approach, we are able to establish  stability for MPC algorithms with zero terminal cost or even negative terminal cost as special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Combining this new approach with the existing MPC stability theory, we are able to significantly  relax the stability requirement on MPC and extend the design space where stability are guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  The proposed approach will help to further reduce the gap between stability theory and practical  applications of MPC and other optimisation based control methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Keywords Finite horizon · optimisation, stability · model predictive control · constraints  1  Introduction  It is well known that stability of Receding Horizon Control (RHC), also known as Model Predictive Control (MPC),  with a finite horizon can not be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The most widely used approach to guarantee the stability of a receding  horizon controller is to add a terminal cost, Vf(xk+N|k), in its cost function where xk+N|k is the terminal state in the  prediction horizon [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Consequently, a well-established sufficient condition is that the system x+ = f(x, u) is stable  under an MPC algorithm if there exists any control u in the admissible set U such that  −Vf(x) + l(x, u) + Vf(f(x, u)) < 0  (1)  for any x ̸= 0 where l(x, u) is the stage cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' That is, a terminal cost Vf(xk+N|k) that covers the time-go-cost (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  from the terminal state to the final equilibrium) shall be added into the cost function for on-line optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' A natural  question arising is, what happens if condition (1) is violated, or more precisely, if there does not exist an admissible  control u ∈ U such that condition (1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Formally, this can be represented as, for all u ∈ U and x ̸= 0,  ℓ(x, u, x+) := −Vf(x) + l(x, u) + Vf(f(x, u)) > 0  (2)  This question is more interesting since it includes the case that Vf(x) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' no terminal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' In the case of zero  terminal cost, condition (2) is satisfied as long as the stage cost, l(x, u), is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Unfortunately, numerous  counterexamples exist where condition (2) holds but the resulted MPC is unstable, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' [5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' It is noted that significant  effort and progress have been made in relaxing the requirement on the terminal cost in satisfying (1) [7–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' For  example, stability can still be achieved if the receding horizon is sufficiently long even in the case when the terminal  ∗Corresponding author: Wen-Hua Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='12024v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='OC]  27 Jan 2023  Stability of Finite Receding Horizon Control: A Complementary Approach  weight is zero [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' So far, there is little work devoted to directly exploring the space defined by condition (2) which is  opposite to the design space defined by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  The motivation of this paper is to answer the question under what condition a finite receding horizon controller satisfying  (2) is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Condition (2) is equivalent to m(x) > 0, where  m(x) := min  u∈U ℓ(x, u, x+)  (3)  is referred as the One-Step Value Function (OSVF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' That plays a key role in establishing stability of MPC in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  The main contribution of this paper is to present a slightly modified MPC algorithm which is stable as long as the  corresponding OSVF is a control Lyapunov function (CLF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Since condition (2) is precisely opposite to condition (1) used in establishing stability in the current MPC literature for  more than two decades [1,9,11], our results provide a complementary approach to the existing terminal cost based MPC  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' That is, all cases covered by our new conditions do not satisfy the existing stability conditions developed  based on (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Furthermore, our conditions allow stability to be established if there is no terminal cost or even a negative  terminal cost (see numerical examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' It clearly shows that, despite the huge success in developing stability theory for  MPC in the last three decades [1,10,12], there is still a significant, unexplored, design space where stability is possible  to achieve by optimising a finite horizon cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' By combining the new analysis approach in this paper with  the existing ones, particularly the existing terminal cost based MPC stability conditions, we are in a much stronger  position in analysing stability of MPC and other optimisation based control methods and designing stability guaranteed  optimisation based algorithms with a much less restricted freedom and space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' This is the main motivation behind this  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  There are two key ideas behind the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The first is to define a new stage cost, ℓ(x, u, x+), as in (2) by  rotating the terminal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' It shall be pointed out that the stage cost in (2) is somehow similar to the rotation stage cost  technique widely used in economic MPC (EMPC) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' To establish stability of EMPC, the stage cost is first rotated by  a storage function, λ(x), (normally being non-negative [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Subsequently, it is required that the modified terminal  cost, ˜  Vf(x) = Vf(x) + λ(x), is a CLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' This is the same technique as in the current tracking MPC (1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' for example,  see [10,13,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' It is interesting to notice that the widely used dissipativity inequality is equivalent to condition (2) when  the storage function λ(x) is specifically chosen as −Vf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' However, we cannot derive any stability results under this  specific choice as discussed in Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  The second idea is that it is observed that the change of the initial state related cost in a cost function involved in online  optimisation involved in MPC does not change the optimal control sequence and the MPC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' By combining  these two ideas, we are able to modify the MPC cost function but without changing its behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Fundamentally,  when the optimal value function of the modified cost is used as a Lyapunov function candidate to establish its stability,  the combination of these two ideas enables a wide range of possible Lyapunov function candidates to be used to  establish stability for the same MPC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' This breaks down the rigid link between the cost function used in  optimisation in an MPC algorithm and the optimal value function used as a Lyapunov function candidate in the current  approaches [1,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The fact that changing initial state cost does not alter the optimisation involved in MPC has been  exploited in [9,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' By further exploiting this property in this paper, we are able to explore new design space for MPC  stability that was not possible previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Technically, to establish stability by the virtue of the OSVF being a CLF, we have to slightly modify the MPC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  That is, rather than using a fixed terminal constraint as in the current terminal cost based framework, we construct  a contractive terminal set by making use of the OSVF being a CLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The idea of using a contractive terminal set can  be found in [16];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' however, the whole optimal control sequence (not only the first element) is applied into the system  in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' This does not conform with the MPC setup in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  This paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' In Section 2, a new stage cost is proposed by rotating the terminal cost in the  conventional cost function without affecting the optimal trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Based on this new augmented stage cost and some  mild assumptions, a new MPC algorithm with stability guarantee is proposed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' In Section 4, a procedure for  finding a terminal cost satisfying the new stability conditions is developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Results are provided by two illustrative  examples, giving further insight into the proposed approach, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Finally, Section 6 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' I and R are integers and real numbers, respectively, where subscripts may be added to give specific ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  For any vector x ∈ Rn, |x| denotes the 2-norm and |x|P is defined by |x|P := xT Px, where P ∈ Rn×n is a symmetric  matrix, but not necessary to be positive-definite here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The superscript ∗ is used to indicate the optimal solution, the  optimal control sequence or state under the optimal control sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  2  Stability of Finite Receding Horizon Control: A Complementary Approach  2  Augmented stage cost and equivalent MPC problems  Consider a nonlinear discrete-time system described by  xk+1 = f(xk, uk), k ∈ I≥0  (4)  or, more succinctly x+ = f(x, u), where x or xk ∈ X is the current state, u or uk ∈ U is the current control, and  x+ or xk+1 is the successor state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' It is assumed in this paper that  (A1) f(x, u) : X × U → X is continuous and f(0, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' X ⊆ Rn and U ⊆ Rm are closed and contain the origin in  the interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  We start from a classic finite horizon optimisation problem, with the cost function  J(xk, uk) :=  N−1  �  i=0  l(xk+i|k, uk+i|k) + Vf(xk+N|k),  (5)  where xk|k = xk and xk+i+1|k = f(xk+i|k, uk+i|k), i ∈ I0:N−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' the control and state sequences are1 uk :=  �  uk|k, · · · , uk+N−1|k  �  and xk :=  �  xk|k, · · · , xk+N|k  �  , subject to the constraints U and X, N ≥ 1 is the length of the  control horizon, l(x, u) and Vf(x) are the stage cost and the terminal cost respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' This satisfies the following  assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  (A2) l(x, u) : X × U → R and Vf(x) : X → R are continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Furthermore, l(0, 0) = 0 and Vf(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  It shall be pointed out that there is no requirement that (l, x) or Vf(x) shall be positive definite in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Following the conventional MPC framework, the first control action of the optimal control sequence u∗  k is applied into  the system, uk = u∗  k|k, which implies that  xk+1 = f(xk, uk) = f  �  xk|k, u∗  k|k  �  = x∗  k+1|k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  (6)  For the widely used cost function in (5), it is observed that the cost that is only associated with the initial state, xk|k, is  never changed by any control sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' That is, this term is completely independent from the optimisation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Motivated by this intuitive but important observation, we rewrite the cost function as  J(xk, uk)  =  N−1  �  i=0  l(xk+i|k, uk+i|k) + Vf(xk+N|k)  = Vf(xk|k) −Vf(xk|k) + l(xk|k, uk|k) + Vf(xk+1|k)  �  ��  �  ℓ(xk|k,uk|k,xk+1|k)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  −Vf(xk+N−1|k) + l(xk+N−1|k, uk+N−1|k) + Vf(xk+N|k)  �  ��  �  ℓ(xk+N−1|k,uk+N−1|k,xk+N|k)  = Vf(xk|k) +  N−1  �  i=0  ℓ(xk+i|k, uk+i|k, xk+i+1|k),  (7)  where ℓ(x, u, x+) is the new augmented stage cost and has been defined in (2) by rotating the terminal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' It shall be  highlighted that the information of the system dynamics is embedded in this new stage cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' By ignoring the initial one,  Vf(xk|k), since it does not affect the solution of the optimisation, we can define the new cost function for MPC online  optimisation as  J (xk, uk) :=  N−1  �  i=0  ℓ(xk+i|k, uk+i|k, xk+i+1|k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  (8)  1For simplicity, when used in algebraic expressions, xk denotes the column vector  �  xT  k|k, · · · , xT  k+N|k  �T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' similarly in algebraic  expressions of uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  3  Stability of Finite Receding Horizon Control: A Complementary Approach  Considering the optimisation problems based on different cost functions, J(xk, uk) and J (xk, uk), with the same  initial state xk and the same constraints, the optimal trajectories should be the same, regardless of the different value  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' That is  min  uk J (xk, uk) = −Vf(xk) + min  uk J(xk, uk)  arg min  uk J (xk, uk) = arg min  uk J(xk, uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  (9)  Considering (9), the design and analysis on the conventional MPC are equivalent to that with a new cost function  J (xk, uk), which is given in a form without any terminal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Remark 1 The stage cost rotation technique has been widely used for EMPC in establishing its stability [13,14,17,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  More specifically, a dissipativity inequality is defined with a non-negative storage function λ(x) (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='1  in [10]), that is,  −λ(x) + λ(f(x, u)) − l(x, u) ≤ −l(xs, us)  (10)  is satisfied for all x and u, where xs and us are in steady state operating condition and l(xs, us) can be treated as  zero when the equilibrium point is at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Then stability of EMPC can be established by mainly resorting to  standard stabilising MPC theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' That is, to establish stability, the modified terminal cost ˜Vf(x) = Vf(x) + λ(x) is  required to be a Control Lyapunov Function (CLF), satisfying condition (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The authors are aware that significant  effort and progress has been made in further developing stability analysis of EMPC with various extensions (see, for  example, [10,17–20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Condition (2) can be interpreted as a dissipativity inequality with the corresponding storage function as λ(x) = −Vf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  It is interesting to notice that the resulted storage function λ(x) is quite often negative definite since the terminal cost is,  in general, positive definite, which makes it different from a standard dissipation definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Most importantly, with  the choice of λ(x) = −Vf(x), the modified terminal cost becomes ˜Vf(x) = 0, which does not satisfy the requirement  that the terminal cost must be a CLF, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' condition (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' We are not able to assert stability of EMPC or any other MPC  schemes by following this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Furthermore, based on the dissipativity condition, similar functions have recently been defined in EMPC using a general  storage function which is referred to as a control dissipativity function in [21] and control storage function in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Remark 2 The approach proposed in this paper is applicable to both tracking MPC and EMPC directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' It follows  from Assumption (A3) that the stage cost l(x, u) could be positive or negative as long as condition (2) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' In  that sense, we also present an alternative stability condition for EMPC where the rotated terminal cost ˜  Vf(x) does not  satisfy the requirements in the current stability conditions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' to be a CLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  3  MPC algorithm and stability  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='1  New terminal weight based MPC algorithm  Here we define an auxiliary optimisation problem to minimise the augmented stage cost ℓ(x, u, x+) in (2) or (7)  m(x) := min  u∈U ℓ(x, u, x+)  = min  u∈U − Vf(x) + l(x, u) + Vf(x+)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' x ∈ Ω ⊆ X, u ∈ U, f(x, u) ∈ Ω,  (11)  where m(x) : Ω ⊆ X �→ R is the One-Step Value Function (OSVF) of the augmented stage cost and Ω is a control  invariant set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  It shall be highlighted that m(x) > 0 iff condition (2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' In other words, m(x) > 0 only if there does not exist any  control u ∈ U such that condition (1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' We will investigate the design space where m(x) > 0 is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  This is entirely complementary to the design space investigated by the existing MPC stability theories and has not been  explored previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  We define the sublevel set of m(x) > 0 as  Ω(α) := {x ∈ Rn : m(x) ≤ α} ⊆ Ω ⊆ X,  (12)  where α is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' As briefly mentioned in the introduction, the proposed MPC in this paper is based on the  condition that OSVF is a CLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' That is, the following assumption is imposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  4  Stability of Finite Receding Horizon Control: A Complementary Approach  (A3) The OSVF m(x) is a CLF for system (4) with respect to a set Ω(α0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=', for any x ∈ Ω(α0), there exist a control  u ∈ U and K∞ functions β1(·), β2(·), β3(·) such that  β1(|x|) ≤ m(x) ≤ β2(|x|)  m(f(x, u)) − m(x) ≤ −β3(|x|),  (13)  (14)  where α0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Based on Assumption (A3), we are now able to construct a contractive terminal set, which is the only difference from  the conventional MPC with terminal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The new algorithm is described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Algorithm:  Offline:  Step 1  For the given system (4) subject to the constraints X and U, define the stage cost l(x, u)  and the terminal cost Vf(x), and calculate terminal constraint α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Initialise the time as  k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Online:  Step 2  At time k, measure the current state xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Solve the following optimisation problem  V (xk, αk) := min  uk J (xk, uk)  = min  uk  N−1  �  i=0  ℓ  �  xk+i|k, uk+i|k, xk+i+1|k  �  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' xk|k = xk  xk+i+1|k = f(xk+i|k, uk+i|k)  xk+i|k ∈ X, uk+i|k ∈ U, i ∈ I0:N−1  xk+N|k ∈ Ω(αk)  (15)  and obtain the optimal solution u∗  k and x∗  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Apply the optimal control uk = u∗  k|k to system  (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Step 3  Denote the minimum between m  �  x∗  k+N|k  �  and m  �  x∗  k+1|k  �  as  m  �  x∗  k+s|k  �  := min  �  m  �  x∗  k+1|k  �  , m  �  x∗  k+N|k  ��  , s ∈ {1, N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  (16)  Update the terminal set by the following rule  αk+1 =  �  �  �  �  �  m  �  x∗  k+s|k  �  − δ, if m  �  x∗  k+s|k  �  ≥ δ  0, if m  �  x∗  k+s|k  �  < δ,  (17)  where δ > 0 is a sufficient small constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Step 4  k ← k + 1 and go to Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Remark 3 Compared with the conventional MPC setup, Step 3 is the only difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' To establish stability, there are  also three key terminal elements in this algorithm: terminal cost, terminal control and terminal constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Instead  of using a fixed terminal constraint, a contractive terminal constraint is constructed in Step 3 where one-step ahead  optimisation is involved in (16) and (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' It will be shown that under the condition that, for a given terminal cost, if the  corresponding OSVF is a CLF, then it is always feasible to construct a contractive terminal constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' That is, Step 3  and the new MPC algorithm is always feasible as long as it is feasible at the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' We will discuss how to choose  a terminal cost such that the corresponding OSVF is a CLF in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Remark 4 One of the drawbacks in the current terminal weight based MPC framework is how to strike a good balance  between performance and stability, particularly for nonlinear or uncertain system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' For a given stage cost, it is quite  difficult to estimate the optimal cost-to-go for these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Since, in order to achieve stability, it requires the terminal  cost to cover the cost to go, quite often a conservative terminal cost function has to be employed to ensure the unknown  optimal value function is covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' This may lead to poor performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The total cost to be optimised in MPC consists  of the summed stage cost and the terminal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' If the chosen terminal cost is much larger than the summed stage cost,  5  Stability of Finite Receding Horizon Control: A Complementary Approach  the optimisation process actually puts more effort on the terminal cost since the former carries a higher relative weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  This is despite the fact that the stage cost may represent the performance requirements better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Our approach avoids  this problem by exploring the design space where the terminal cost cannot cover the cost-to-go, even if it is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Our  hypothesis is that this may give a better MPC performance for nonlinear or uncertain systems, as shown in the second  example in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  The main result is given by the following theorem while the detailed proof is given by the subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Theorem 1 Suppose that  Assumptions (A1)-(A3) are satisfied,  the optimisation problem (15) is feasible at time k = 0,  the parameter α0 is chosen such that the terminal set Ω(α0) ⊆ Ω  Then, the closed-loop system with the proposed MPC is recursively feasible and, eventually, asymptotically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='2  Recursive feasibility  Recursive feasibility of the proposed MPC is mainly based on the condition of OSVF being a CLF, seeing Assumption  (A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Proof of recursive feasibility in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Supposing that the optimisation (15) is feasible at time k ∈ I≥0, the optimal  control and state sequences exist, which can be denoted as  u∗  k =  �  u∗  k|k, · · · , u∗  k+N−1|k  �  x∗  k =  �  x∗  k|k, · · · , x∗  k+N|k  �  ,  where the terminal state satisfies that  x∗  k+N|k ∈ Ω(αk) ⇔ m  �  x∗  k+N|k  �  ≤ αk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  (18)  In what follows, we will construct the feasible control and state sequences at time k + 1 based on the current optimal  sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The case when the terminal set shrinks to the origin is ignored as it is degenerated into the conventional  MPC with an equality terminal constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' It is straightforward to show its recursive feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' To make the discussion  clear, we use the superscripts, a and b, to represent two cases due to the different results of (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  In the case of m  �  x∗  k+1|k  �  ≥ m  �  x∗  k+N|k  �  , based on (17), if we choose δ ∈ (0, β3(|x∗  k+N|k|)], then  αk+1 = m  �  x∗  k+N|k  �  − δ  ≥ m  �  x∗  k+N|k  �  − β3  ����x∗  k+N|k  ���  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  (19)  By Assumption (A3) and since x∗  k+N|k  ∈  Ω(αk), there exists a control ua  k+N|k  ∈  U, xa  k+N+1|k  =  f  �  x∗  k+N|k, ua  k+N|k  �  such that  m  �  xa  k+N+1|k  �  ≤ m  �  x∗  k+N|k  �  − β3  ����x∗  k+N|k  ���  �  ≤ αk+1  ⇒xa  k+N+1|k ∈ Ω(αk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  (20)  The feasible sequences at time k + 1 can be constructed as  ua  k+1 =  �  u∗  k+1|k, · · · , u∗  k+N−1|k, ua  k+N|k  �  xa  k+1 =  �  x∗  k+1|k, · · · , x∗  k+N|k, xa  k+N+1|k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  In the case of m  �  x∗  k+1|k  �  < m  �  x∗  k+N|k  �  , and noting (18), we have that  m  �  x∗  k+1|k  �  < m  �  x∗  k+N|k  �  ≤ αk  ⇒x∗  k+1|k ∈ Ω(αk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  (21)  6  Stability of Finite Receding Horizon Control: A Complementary Approach  Based on (17), we have that  αk+1 ≤ m  �  x∗  k+1|k  �  < m  �  x∗  k+N|k  �  ≤ αk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  (22)  By Assumption (A3), there exists a control ub  k+1|k ∈ U, xb  k+2|k = f  �  x∗  k+1|k, ub  k+1|k  �  such that  m  �  xb  k+2|k  �  ≤ m  �  x∗  k+1|k  �  − β3  ����x∗  k+1|k  ���  �  ≤ m  �  x∗  k+1|k  �  − δ  = αk+1, if δ ≤ β3  ����x∗  k+1|k  ���  �  ⇒xb  k+2|k ∈ Ω(αk+1) ⊆ Ω(αk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  (23)  Using Assumption (A3) again and noting (23), there exists a ub  k+2|k ∈ U, xb  k+3|k = f  �  xb  k+2|k, ub  k+2|k  �  such that  m  �  xb  k+3|k  �  ≤ m  �  xb  k+2|k  �  − β3  ����xb  k+2|k  ���  �  ≤ m  �  xb  k+2|k  �  ≤ αk+1  ⇒xb  k+3|k ∈ Ω(αk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  (24)  Combing (23) and (24) together, we have that the set Ω(αk+1) is also control invariant and hence there exists a control  sequence ub  k+i|k ∈ U, i ∈ I0:N such that  xb  k+1+i|k = f  �  xb  k+i|k, ub  k+i|k  �  ∈ Ω(αk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  (25)  The feasible sequences of this case can be conducted as  ub  k+1 =  �  ub  k+1|k, · · · , ub  k+N−1|k, ub  k+N|k  �  xb  k+1 =  �  x∗  k+1|k, · · · , xb  k+N|k, xb  k+N+1|k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  In conclusion, based on the above discussion on the two cases, the proposed algorithm is recursively feasible, which  completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  ■  Recursive feasibility guarantees that once the proposed algorithm is feasible at time k = 0, the optimisation algorithm  associated with MPC and the closed-loop system is always feasible, which implies that all the signals in the control  system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=', αk) are well defined, even as time goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='3  Asymptotic stability  To establish stability in Theorem 1, several lemmas are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Lemma 1 shows the upper bounds of the new value  function V (x, α) in (15) while Lemma 2 establishes the condition for its monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Lemma 1 Suppose that Assumption (A3) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Then, for any feasible state x of the optimisation problem (15) with  Ω(α) ⊆ Ω, there exists a K∞ function β4(·) such that V (x, α) ≤ β4(|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' We first consider a simple case, where x ∈ Ω(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Then, we replace the optimisation problem (15) by a  suboptimal control problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' splitting it into N one-step optimisation problems (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Recursively solving these N  one-step optimisation problems as in (11) and using the optimal states and inputs as the feasible ones of the optimisation  problem (15), it will give that  V (x, α) ≤ Nβ2(|x|), ∀x ∈ Ω(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  (26)  Therefore, V (x, α) is continuous at zero as 0 ∈ Ω(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Following the similar analysis given in [12, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='2], we  know that the set of feasible states is closed and V (x, α) is locally bounded by it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' By applying [12, Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='25],  the upper bound conclusion can be extended to any feasible state, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  ■  Lemma 2 Suppose that there exists a control uk+N|k ∈ U such that xk+N+1|k = f  �  x∗  k+N|k, uk+N|k  �  ∈ Ω(αk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Then  V (xk+1, αk+1) − V (xk, αk) ≤ ℓ  �  x∗  k+N|k, uk+N|k, xk+N+1|k  �  − ℓ  �  x∗  k|k, u∗  k|k, x∗  k+1|k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  7  Stability of Finite Receding Horizon Control: A Complementary Approach  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' We denote the optimal control and state sequences at time k as  u∗  k =  �  u∗  k|k, · · · , u∗  k+N−1|k  �  x∗  k =  �  x∗  k|k, · · · , x∗  k+N|k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  The value function V (xk, αk) can be rewritten as  V (xk, αk) =  N−1  �  i=0  ℓ  �  x∗  k+i|k, u∗  k+i|k, x∗  k+i+1|k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  (27)  If there exists a control uk+N|k ∈ U such that xk+N+1|k = f  �  x∗  k+N|k, uk+N|k  �  ∈ Ω(αk+1), we could construct the  feasible sequences at time k + 1 as  uk+1 =  �  u∗  k+1|k, · · · , u∗  k+N−1|k, uk+N|k  �  xk+1 =  �  x∗  k+1|k, · · · , x∗  k+N|k, xk+N+1|k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Thus, we have that  V (xk+1, αk+1) − V (xk, αk)  ≤ J (xk+1, uk+1) − V (xk, αk)  = J  �  x∗  k+1|k, uk+1  �  − V (xk, αk)  = ℓ  �  x∗  k+N|k, uk+N|k, xk+N+1|k  �  − ℓ  �  x∗  k|k, u∗  k|k, x∗  k+1|k  �  ,  (28)  which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  ■  We are now ready to prove Theorem 5 for asymptotic stability of the proposed MPC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' It is established through  a two stage process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' In the first process, it is shown that the terminal set continuously shrinks to the origin under the  constructed update rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' In the second stage, asymptotic stability is established using V (x, 0) as a Lyapunov function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Proof of asymptotic stability in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Recursive feasibility established in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='2 shows that  αk+1 ≤ m  �  x∗  k+s|k  �  − δ ≤ m  �  x∗  k+N|k  �  − δ ≤ αk − δ,  (29)  which implies that the sequence {αk} is continuously decreasing until reaching the origin since δk > 0 until x∗  k+N|k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Consequently, once the terminal constraint set approaches the origin, one has  x∗  k+N|k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  (30)  It follows from Assumption (A3) that  ℓ  �  x∗  k+N|k, uk+N|k, xk+N+1|k  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  (31)  Following Lemma 2 with αk+1 = αk = 0, we have  V (xk+1, 0) − V (xk, 0)  ≤ −ℓ  �  x∗  k|k, u∗  k|k, x∗  k+1|k  �  ≤ −m(xk|k)  ≤ −β1(|xk|k|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  (32)  Combining Lemma 1 and noting V (x, α) ≥ m(x) ≥ β1(|x|), the closed-loop system is asymptotically stable under the  proposed MPC algorithm which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  ■  Remark 5 Similar to the existing MPC theories, asymptotic stability is established also by employing an optimal value  function V (x, 0) as a Lyapunov function candidate in our approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' However there are three key differences in these  two approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' First, the new value function V (x, α) is different from the the original optimal value function obtained  by optimising the original cost function (5) used in the existing MPC stability theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' This is not only because the  8  Stability of Finite Receding Horizon Control: A Complementary Approach  modification of the cost function by removing the cost associated with initial state, but also the contractive constraints  in the online optimisation, caused by shirking αk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Secondly, stability of the MPC algorithm is established in the existing  MPC theories by exploiting the monotonicity of the optimal value function during the whole MPC control process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  However we are not able to establish the monotonicity of the value function, V (x, α), at the beginning of the contracive  MPC algorithm proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Actually V (x, α) may increase if the terminal set shrinks quite quickly based on  the proposed update rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' We rely on the contractive terminal constraints enabled by the property of OSVF being a  CLF to force the system state to approach the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Once the terminal set shrinks to the origin, we are able to show  the monotonicity of the value function V (x, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Finally, we establish stability by imposing a certain property on the  modified stage cost (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' OSVF to be a CLF) while the existing stability theories ensure stability by imposing a condition  on the terminal cost (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' the terminal cost to be a CLF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  4  Terminal cost calculation  All the assumptions used in establishing stability of MPC are quite similar to that in the existing terminal cost based  MPC framework except Assumption (A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' This section is devoted to developing a procedure for finding a terminal cost  such that the corresponding OSVF m(x) is a CLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' As discussed previously, the condition m(x) being positive definite  is opposite to the existing well-established MPC stability conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Furthermore, it is possible for Vf(x) to be zero  which includes the case of no terminal weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Therefore, the results [6] for stability analysis of MPC without terminal  cost can be considered as a special case in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' More importantly, we are able to explore the freedom of adding  an appropriate terminal cost to guarantee the stability of a MPC algorithm of concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  In what follows, we propose a design procedure to find a terminal cost to satisfy Assumption (A3) for linear systems,  which can be extended to nonlinear systems by the similar approach of the current MPC framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Linearisation  of the nonlinear system at the operation condition [12] is one such approach, or using another approach like linear  differential inclusion as in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' It also provides more insight about the definition and the properties of OSVF m(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Here, we consider a linear system as  x+ = Ax + Bu,  (33)  with the stage cost l(x, u) = |x|2  Q + |u|2  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' We aim to find a terminal cost Vf(x) = |x|2  P satisfying Assumption (A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  The modified stage cost in (2) can be specified as  ℓ  �  x, u, x+�  = −|x|2  P + |x|2  Q + |u|2  R + |Ax + Bu|2  P  = |x|2  AT P A+Q−P + |u|2  R+BT P B  + 2xT AT PBu  =  �  xT  uT �  M  �  x  u  �  ,  (34)  where  M :=  �  AT PA + Q − P  AT PB  BT PA  R + BT PB  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  (35)  If R + BT PB > 0, solving the one-step optimisation problem (11) gives  arg min  u ℓ(x) = −(R + BT PB)−1BT PAx  m(x) = min  u ℓ(x) = |x|2  MP ,  (36)  where  MP := AT PA + Q − P − AT PB(R + BT PB)−1BT PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  (37)  Since that m(x) serves as a CLF in Assumption (A3), MP should at least be positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' By the Schur Complement  Lemma, R + BT PB > 0 and MP > 0 is equivalent to M > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  This result is summarised in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Lemma 3 Consider a linear system (33) with a quadratic cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Then its OSVF m(x) is positive definite if  there exists a terminal cost Vf(x) = |x|2  P with a matrix P such that M defined in (35) is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  By now, we only need to consider condition (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Since in this case, m(x) is in a quadratic form, condition (14) can be  relaxed since there exists a control uk such that m(xk+1) < m(xk) holds for any xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  9  Stability of Finite Receding Horizon Control: A Complementary Approach  Lemma 4 m(xk+1) < m(xk) holds, if and only if there exist uk and uk+1 such that ℓ(xk+1, uk+1, xk+2) < m(xk)  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' ⇐: If there exist uk and uk+1 such that ℓ(xk+1, uk+1, xk+2) < m(xk), we then have that m(xk+1) ≤  ℓ(xk+1, uk+1, xk+2) < m(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  ⇒: If m(xk+1) < m(xk), the existence of uk is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' As for uk+1, we can choose  uk+1 = arg min  uk+1 ℓ(xk+1, uk+1, xk+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Hence, there exist uk and uk+1 such that  ℓ(xk+1, uk+1, xk+2) = m(xk+1) < m(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  ■  Lemma 5 If R > 0, then  �  Q − SR−1ST  A  AT  B  �  > 0 ⇔  �  �  Q  S  A  ST  R  0  AT  0  B  �  � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  (38)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' By using congruent transformation, we have that  �  �  I  −SR−1  0  0  I  0  0  0  I  �  �  �  �  Q  S  A  ST  R  0  AT  0  B  �  �  �  �  I  0  0  −R−1ST  I  0  0  0  I  �  �  =  �  �  Q − SR−1ST  0  A  0  R  0  AT  0  B  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  ■  Theorem 2 If there exists a terminal cost P and two control gains K1 and K2 such that the following bilinear matrix  inequality (BMI) is satisfied  �  ���  M  �  (A + BK1)T  KT  2  0  0  �  M  M  �  (A + BK1)  0  K2  0  �  M  �  ��� > 0  (39)  then m(x) is a CLF for system (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Note that if condition (39) is satisfied, it follows that M > 0, which further guarantees R + BT PB > 0 and  MP > 0 by using the Schur Complement Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' R + BT PB > 0 implies that the one-step optimisation problem  (11) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Letting uk = K1xk and uk+1 = ˜K2xk+1, a quick calculation gives  ℓ(xk+1, uk+1, xk+2)  =  �  xT  k (A + BK1)T  xT  k KT  2  �  M  �  (A + BK1)xk  K2xk  �  = xT  k  �  (A + BK1)T  KT  2  �  M  �  A + BK1  K2  �  xk  (40)  where K2 = ˜K2(A + BK1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Using the Schur Complement Lemma and Lemma 4, noting R + BT PB > 0, the  condition ℓ(xk+1, uk+1, xk+2) < m(xk) is satisfied if the following matrix inequality holds  10  Stability of Finite Receding Horizon Control: A Complementary Approach  �  (A + BK1)T  KT  2  �  M  �  A + BK1  K2  �  < MP  ⇔  �  (A + BK1)T  KT  2  �  MM −1M  �  A + BK1  K2  �  < MP  ⇔  �  �  MP  �  (A + BK1)T  KT  2  �  M  M  �  A + BK1  K2  �  M  �  � > 0  ⇔ (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  ■  Remark 6 The proposed BMI (39) can be solved by using the PENLAB package [23], which is an open source software  package implemented in MATLAB for matrix inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  5  Illustrative examples  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='1  A first order example  This section is to illustrate the significance of the proposed new stability analysis approach in this paper and highlight  the main differences with the existing stability theory of MPC by using a simple first order system  x+ = ax + bu, b ̸= 0  (41)  with l(x, u) = qx2 + ru2 and Vf(x) = px2, where the lowercase letters represent scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  More specifically, this example is used to highlight the following two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The proposed MPC algorithm may be stable with a zero terminal cost or a negative terminal cost with the new  condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The new stability condition is complementary to the existing terminal cost based MPC stability conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  It follows from the definition of the augmented stage cost that ℓ(x, u, x+) = −px2 + qx2 + ru2 + p(ax + bu)2 for  this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The corresponding OSVF m(x) is ready to be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' For this first order linear system, according to  Lemma 3, the positive definite condition in Assumption (A3) reduce to  r + b2p > 0  a2p + q − p − a2p2b2  r + b2p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  (42)  Furthermore, Lemma 4 shows that the CLF condition (14) is met for any positive definite OSVF that satisfies condition  (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Inspecting condition (42) gives  q > (1 − a2)p + a2p2b2  r + b2p = 1  b2  �  z1 + a2r2  z1  −  �  1 + a2�  r  �  ,  (43)  where z1 := r + b2p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' (43) gives a lower bound of q by  q > q := 2|ar| − (1 + a2)r  b2  =  �  �  �  �  �  �  �  �  �  �  �  −(|a| − 1)2r  b2  , if r > 0  0,  if r = 0  −(|a| + 1)2r  b2  , if r < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  (44)  It shall be highlighted that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' (44) reveals q is possible to be negative or zero if r ̸= 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  11  Stability of Finite Receding Horizon Control: A Complementary Approach  Next, for the purpose of comparison, we present the conventional MPC stability conditions for this simple MPC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  First, all the weightings are required to be at least positive semi-definite, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=', q > 0, r ≥ 0, and p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Secondly, the  current condition states that the conventional MPC for this unconstrained system is stable if the famous Fake Algebraic  Riccati Equation (FARE) [24] holds  b2p2 + (1 − b2q − a2)p − q ≥ 0  ⇔ q ≤ 1  b2  �  z2 + a2r2  z2  −  �  1 + a2�  r  �  (45)  where z2 = r + b2p ≥ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  If we focus on the right-side functions of (43) and (45), when p is large enough, both functions go to a linear one  q = p − ra2  b2 , which is parallel with but below the line q = p if r ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  In what follows, we will make an explicit comparison of the stability conditions and regions between the proposed new  approach and the existing terminal weight based MPC framework, in terms of the control weight r > 0, r < 0, and  r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='1  Case with control weight r > 0  We only consider the case that r = 1, as the same results hold for a different r by normalizing state and terminal  weighting with q/r and p/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Regarding stability condition (44), there are three cases to discuss, |a| < 1, |a| = 1, and  |a| > 1, which correspond to open-loop stable, marginally stable, and unstable systems, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The stability  regions defined by the new condition (44) are given by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' These regions actually define the possible parameter  combinations of the system and cost functions including the stage and terminal cost so we refer them as the feasible  design spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' It is interesting to see when |a| < 1, the proposed MPC could be stable even when both p and q are  negative, which covers the situation widely studied in EMPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  It shall be noted that the blue curve corresponds to the solution of the Riccati equations under different stage costs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Therefore, to ensure stability, the existing MPC stability condition requires that the terminal cost p is larger than or  equal to the solution of the Riccati equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' This is shown by the region on the right side of the blue curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' that is, the  system under MPC is stable when the corresponding terminal cost p covers the cost-to-go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' In contrast, the stability  region given by our new stability condition lies on the left side of the blue curve so covers the design space where the  terminal weight p is less than the Riccati solution under the corresponding stage and control weights q and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  It is observed that there is no overlap between the stability regions derived by the existing terminal weight based MPC  framework and the new condition proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Furthermore, by combining the stability regions yielded by  these two methods, we are able to significantly extend the MPC design space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' in terms of possible terminal costs)  where stability could be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Therefore, our approach is complementary to the existing MPC framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Finally, is is also observed that both the lines p = 0 and p = q are covered by the proposed MPC stability condition as  shown in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' p = 0 and p = q are both the special cases of MPC where there is no terminal cost, but with slightly  different Lyapunov functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' To well illustrate the difference, we take an example when N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  If p = 0, the optimisation in (15) reduces to  min  uk|k, uk+1|k  �  qx2  k|k + ru2  k|k  �  +  �  qx2  k+1|k + ru2  k+1|k  �  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' xk|k = xk  xk+i+1|k = axk+i|k + buk+i|k, i = 0, 1  xk+2|k ∈ Ω(αk),  (46)  where MP = a2p + q − p − a2p2b2(r + b2p)−1 = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  12  Stability of Finite Receding Horizon Control: A Complementary Approach  If p = q, the optimisation in (15) becomes  min  uk|k, uk+1|k  �  qx2  k|k + ru2  k|k  �  +  �  qx2  k+1|k + ru2  k+1|k  �  + px2  k+2|k  =  min  uk|k, uk+1|k, uk+2|k  2  �  i=0  �  qx2  k+i|k + ru2  k+i|k  �  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' xk|k = xk  xk+i+1|k = axk+i|k + buk+i|k, i = 0, 1  xk+2|k ∈ Ω(αk),  (47)  where MP = a2p+q−p−a2p2b2(r+b2p)−1 = a2q−a2q2b2(r+b2q)−1 = a2qr(r+b2q)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The equivalent  cost function holds as the decision variable uk+2|k would never affect the control and state sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Therefore, they correspond to the different choices of the augmented stage cost and the associated OSVF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' This  also implies different terminal constraints are used in the algorithm and different Lyapunov functions employed in  establishing stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' This special case also provides more insight into the proposed new approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='2  Case with control weight r = 0  The case of r = 0 is also interesting as both methods occupy the perfect half of the whole design space, as shown by  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' 2 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' This is because when r = 0, condition (43) is equivalent to q > p > 0 while condition (45) is 0 < q ≤ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='3  Case with control weight r < 0  In this case, in a similar fashion as r > 0, we fix r = −1 and the result is given by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' 2 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' It is worth noting that the  design space of the conventional MPC framework is empty since the control weight is normally required to be positive  definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  In summary, for the cases of the control weight r < 0 or r = 0, we are able to draw the same conclusion from Fig 2  about the proposed new stability condition, and its comparison and connection with the existing terminal weight based  MPC framework, as in the case of r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' More simply, for each pair of state and control weights q, r, the existing MPC  stability theory ascertains that the corresponding MPC algorithm with the terminal cost p in the yellow region is stable  while our stability condition states that the corresponding MPC algorithm with the terminal cost p in the light green  area is also stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Thus, we are able to state that an MPC algorithm is stable with any terminal cost from the combined  areas of these two stability approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' This substantially relaxes the stability requirement on MPC algorithms and  increase the freedom in designing and tuning the key parameters involved in MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' This numerical example and detailed  comparison give more insight into the proposed new augmented stage cost method and its links with the existing MPC  stability theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='2  Simple nonlinear system study  This section is to apply the new proposed MPC algorithm with a contractive terminal constraint on a nonlinear system  and compare it with the existing terminal weight based MPC approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' It aims to highlight their differences in terminal  regions and total running costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Here, these two MPC laws are applied to a benchmark system [25], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='e, a cart with a  mass moving on a plane, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The nonlinearity in this example comes from the elasticity in the spring,  ks = k0e−x1, where x1 is the displacement of the carriage from the equilibrium position associated with the external  force u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The system is described by  ˙x1 = x2  ˙x2 = − k0  M e−x1x1 − hd  M x2 + u  M ,  (48)  where x2 is its velocity and hd is the damping coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The system is discretised into a discrete-time system using  the Euler approximation with sampling time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='4sec, given by  x1,k+1 = x1,k + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='4x2,k  x2,k+1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='132e−x1,k + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='56x2,k + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='4uk,  (49)  13  Stability of Finite Receding Horizon Control: A Complementary Approach  1  0  1  2  3  4  5  p  1  0  1  2  3  4  5  q  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='5, b=1, r=1  The new design space  The conventional design space  p=0  p=q  q=0  a)  1  0  1  2  3  4  5  p  1  0  1  2  3  4  5  q  a=1, b=1, r=1  The new design space  The conventional design space  p=0  p=q  q=0  b)  1  0  1  2  3  4  5  p  1  0  1  2  3  4  5  q  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='5, b=1, r=1  The new design space  The conventional design space  p=0  p=q  q=0  c)  Figure 1: Design spaces of the proposed and conventional MPC for different open-loop systems when r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' a) Stable,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' |a| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' b) Marginally stable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' |a| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' c) Unstable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' |a| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  where the specific system parameters are all given in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The state and input constrains are X = [−2, 2] × [−3, 3] and  U = [−4, 4] and the initial state is x0 = [−2, 1]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The linearization at the origin gives the controllable pair  A =  �  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='4  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='132  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='56  �  , B =  �  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  The prediction horizon is chosen to be short enough, N = 3, due to the fast calculation time by the motion control  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The stage and terminal costs are both given in a quadratic form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=', l(x, u) = |x|2  Q + |u|2  R and Vf(x) = |x|2  P ,  where  Q =  �  2  0  0  4  �  , R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  In what follows, we will calculate the terminal costs and regions and assess their stability and performance using these  two methods respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  14  Stability of Finite Receding Horizon Control: A Complementary Approach  1  0  1  2  3  4  5  p  1  0  1  2  3  4  5  q  a=1, b=1, r=0  The new design space  The conventional design space  p=0  p=q  q=0  a)  1  0  1  2  3  4  5  p  1  0  1  2  3  4  5  q  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='5, b=1, r=-1  The new design space  p=0  p=q  q=0  b)  Figure 2: Design spaces of the proposed and conventional MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' a) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' b) r < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  ℎ"  𝑀  𝑢  𝑥#  Figure 3: Cart and spring example  We first consider the conventional MPC design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The terminal cost is obtained by the Riccati equation for the  linearization  P =  �  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='9153  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='5604  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='5604  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='5023  �  and the terminal region is given by  Ω =  �  x ∈ R2 : xT Px ≤ α  �  where α = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='3076 is calculated by the optimization method introduced in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  As for the proposed MPC design, following the procedure in Section 4, the terminal cost is obtained by solving  the BMI (39)  P =  �  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='5249  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='3522  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='3522  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='5731  �  and the corresponding terminal region is given by  Ω =  �  x ∈ R2 : xT MP x ≤ α  �  where MP is computed by (37) and α = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='4823 is also calculated by the same method in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  15  Stability of Finite Receding Horizon Control: A Complementary Approach  The terminal regions are given by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' 4, where the region of the proposed MPC covers that of the conventional one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  The states and inputs are given by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' It shall be highlighted that different from stability regions plotted in the  parameter space as in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' 1 and 2, the terminal regions are given in state space in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' It is worth noting that the  terminal cost P calculated by our approach does not satisfy the condition (1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' covering the cost to go as required in  the existing terminal weight based MPC framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Although the original stage costs used in these two MPC algorithms are the same, the final cost functions are different  since their terminal costs are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The total running cost is used for a fair comparison of the performance of these  two methods, that is,  Jrun =  Ts  �  k=0  l (xk, uk) =  Ts  �  k=0  �  |xk|2  Q + |uk|2  R  �  ,  where Ts is sufficiently large such that the system reaches its steady state, which is chosen as 50 sec in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The  total running cost of the proposed MPC algorithm is 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='2148, which is less than that of the conventional one at 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='1587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Therefore, for this example, comparing with the existing terminal weight based MPC scheme, the proposed MPC  algorithm has a larger terminal region and exhibits slightly better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  3  2  1  0  1  2  3  3  2  1  0  1  2  3  Terminal region  The conventional MPC  The proposed MPC  Figure 4: Terminal regions of the proposed and conventional MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  6  Conclusion  Stability is an important property of any control method since it helps to ensure the safety of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Comparing with  other optimisation based methods, one distinctive feature of finite horizon optimisation based control is that it is possible  to establish its stability under certain conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' However, despite its huge success, the current stability conditions are  still conservative, which restricts the design space of MPC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' This paper explores the design space where the  current terminal weight based MPC framework is not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' We propose a completely complementary stability  condition in this unexplored design space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' We show that, by constructing a contractive terminal constraint, rather than a  fixed terminal constraint as in the current MPC framework, we are able to guarantee stability in a significant region that  has not been covered previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' To achieve this, we construct an augmented stage cost using a terminal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' That is,  the augmented stage cost consists of a nominal stage cost and a rotated terminal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' It is shown that feasibility and  the stability of the proposed MPC algorithm can be established under the condition that the one-step optimal value  function of the augmented stage cost is a CLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' Due to the nature of the proposed stability condition, it complements to  the existing stability theory and is not intended to replace the conventional stability conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The stability region  of MPC can be significantly enlarged by combining the new condition with the existing ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' guarantee stability  under a much wider range of combination of state, control and terminal weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' The work in this paper only concerns  a basic MPC regulation problem without disturbance or uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  Our future work includes extending the proposed approach to other areas including robust MPC, stochastic MPC  or economic MPC where stability has been established with the current terminal weight based MPC approach and  comparing their differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='  16  Stability of Finite Receding Horizon Control: A Complementary Approach  0  10  20  30  40  50  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='5  The conventional MPC  The proposed MPC  a)  0  10  20  30  40  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='5  2  The conventional MPC  The proposed MPC  b)  0  10  20  30  40  50  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='5  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content='5  The conventional MPC  The proposed MPC  c)  Figure 5: States and inputs of the proposed and conventional MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
+page_content=' a) x1,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FLT4oBgHgl3EQfOi8m/content/2301.12024v1.pdf'}
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index 0000000000000000000000000000000000000000..2529a9c233ade9ef5e8960c13dade7e53fb7d2c9
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+ 
+1 
+Self-Consistent Hopping Theory of Activated Relaxation and Diffusion of 
+Dilute Penetrants in Dense Crosslinked Polymer Networks 
+   
+Baicheng Mei a,d, Tsai-Wei Lin c,d, Charles E. Sing a,c,d  and Kenneth S. Schweizer *a,b,c,d 
+ 
+a Department of Materials Science, University of Illinois, Urbana, IL 61801 
+b. Department of Chemistry, University of Illinois, Urbana, IL 61801 
+c. Department of Chemical & Biomolecular Engineering, University of Illinois, Urbana, IL 61801 
+d. Materials Research Laboratory, University of Illinois, Urbana, IL 61801 
+ 
+* kschweiz@illinois.edu 
+ 
+ 
+ 
+ 
+ 
+
+ 
+2 
+Abstract 
+ 
+We generalize and apply a microscopic force-level statistical mechanical theory of the activated 
+dynamics of dilute spherical penetrants in glass-forming liquids to study the influence of 
+permanent crosslinking in polymer networks on the relaxation time and diffusivity over a wide 
+range of temperature and crosslink density. The theory treats network crosslinkers as locally 
+pinned or vibrating sites. Calculations are performed for model parameters relevant to recent 
+experimental studies of a nm-sized organic dye diffusing in crosslinked poly(n-butyl methacrylate) 
+networks. The theory predicts the penetrant alpha time increases exponentially with the crosslink 
+fraction (fn) dependent glass transition temperature, Tg, which grows roughly linearly with the 
+square root of crosslink density. Moreover, we find that Tg is also proportional to an entropic 
+geometric confinement parameter defined as the ratio of the penetrant diameter to the mean mesh 
+size. The decoupling of the ratio of the penetrant and Kuhn segment alpha relaxation times displays 
+a complex non-monotonic dependence on crosslink density and temperature via the variable 
+Tg(fn)/T, but a remarkably good collapse is predicted. The microscopic mechanism for activated 
+penetrant relaxation is elucidated based on the theoretically predicted crosslink fraction dependent 
+coupled local cage and nonlocal collective elastic barriers. A model for the penetrant diffusion 
+constant that combines glassy physics and entropic mesh confinement is proposed which results 
+in a significantly stronger suppression of mass transport with degree of effective supercooling than 
+predicted for the penetrant alpha relaxation time. This represents a unique polymer network-based 
+type of “decoupling” of diffusion and relaxation. In contrast to the diffusion of larger nanoparticles 
+in high temperature rubbery polymer networks, our analysis suggests that for the penetrants studied 
+the mesh confinement effects are of secondary importance in the highly supercooled regimes of 
+interest relative to the consequences on mass transport of crosslink-induced slowing down of 
+activated segmental relaxation. Support for the theoretical predictions from our complementary 
+simulation study is briefly discussed.  
+ 
+ 
+
+ 
+3 
+I. Introduction 
+Understanding and controlling molecular (“penetrant”) transport in dense and cold polymer 
+melts, crosslinked networks, and glasses is a problem of high basic polymer physics interest[1-14] 
+that also underpins advanced separation membrane applications[10-27]. However, even in the dilute 
+penetrant limit of interest in this work, existing theoretical models are highly phenomenological, 
+often built on the difficult to uniquely define and unambiguously predict concept of “free 
+volume”.[13, 27-33]  
+Recently, two of us have pursued the construction of a statistical mechanical theory (the 
+Self Consistent Cooperative Hopping (SCCH) theory) for penetrant activated dynamics in polymer 
+melts [34]. The foundation is the Elastically Collective Nonlinear Langevin Equation (ECNLE) 
+theory[35, 36] for the thermally activated structural (alpha) relaxation process on the Kuhn segment 
+scale in one-component melts[37]. This approach is microscopic in the sense that it causally relates 
+pair interactions, thermodynamic state, and packing structure as embedded in a dynamic free 
+energy which quantifies kinetic constraints. The alpha relaxation is a coupled local-nonlocal 
+activated process characterized by a local cage barrier (FB,K) associated with large-amplitude 
+activated hopping coupled with collective long-range (nonlocal) small displacements of all 
+segments outside the cage characterized by an elastic barrier (Fel,K). Recent studies support the 
+predictions of this approach for a large number of real-world polymeric liquids, including olefins, 
+dienes, siloxanes, and vinyl polymer chemistries.[38-40] Chemical crosslinking introduces new 
+kinetic constraints and additional dynamical slowing down. ECNLE theory has been very recently 
+extended to polymer networks and shown to provide an excellent understanding of experiments 
+on poly(n-butylacrylate) (PnBA) networks and coarse-grained simulations over a wide range of 
+crosslink densities and temperatures.[41]  
+
+ 
+4 
+SCCH theory for dilute hard sphere penetrants (diameter, d) in a hard-sphere matrix [42, 43] 
+or semi-flexible polymer melts [34] under structurally equilibrated conditions has been extensively 
+applied, and many of its predictions agree well with simulations [7, 42] and experiments [44]. The 
+penetrant activation barrier and mean hopping time, and the extent of coupling of penetrant 
+transport with the early, medium, and late stages of the matrix structural relaxation process, are 
+self-consistently predicted based on two coupled dynamic free energies.[34, 42, 43, 45]  
+The goal of the present work is to formulate and apply SCCH theory to permanent 
+crosslinked polymer networks. We are especially motivated by recent measurements of Evans and 
+coworkers on the diffusion of a dilute aromatic molecule [N,N’-Bis(2,5-di-tert-butylphenyl)-
+3,4,9,10-perylenedicarboximide (BTBP)] in crosslinked PnBA networks over a wide range of 
+crosslink densities at fixed temperature.[9] The experiments found that the penetrant alpha time 
+and diffusion constant both vary exponentially with the crosslink fraction (𝑓n) dependent network 
+glass transition temperature 𝑇g(𝑓n) , but their quantitative dependences on 𝑇g(𝑓n)  and T are 
+different, a form of “decoupling” in the language of glass physics. The precise physical 
+mechanisms at play remain unclear. For example, is the apparent decoupling a consequence of 
+entropic mesh confinement constraints on penetrant motion in crosslinked networks?  
+Alternatively, is the glass physics mechanism associated with the dynamic heterogeneity (DH)[46-
+48] driven breakdown of the Stokes-Einstein breakdown relation [49-54] in supercooled liquids 
+relevant? Or is there another explanation?   
+The mesh confinement effect has been studied theoretically[55], experimentally[9], and with 
+simulation[56, 57] for the problem of the motion of a dilute spherical nanoparticle (larger than typical 
+molecules) in polymer networks. A key variable is the confinement parameter,  𝐶 = 𝑑/𝑎x, where 
+𝑎x is the mean mesh size that decreases with crosslink density. For example, the polymer physics 
+
+ 
+5 
+model of Cai-Panykov-Rubinstein (CPR) [55] predicts that when mesh confinement is strong 
+enough (𝐶>1) the nanoparticle diffusion constant is: 
+𝐷p ∝
+𝑑2
+𝜏R exp⁡(−𝐶2)/𝐶 ≡
+𝑑2
+𝜏R 𝑋                                                         (1) 
+where 𝜏R~𝑁x
+2𝜏𝛼 is the Rouse time of a network strand composed of 𝑁x segments, the segmental 
+alpha time, 𝜏𝛼, depends on temperature but is assumed to not vary with crosslink density, and the 
+quantity 𝑋(𝐶) is defined which quantifies the explicit entropic effect of a physical mesh. Such a 
+model is most germane to the high temperature rubbery regime. Recently, a simulation study of 
+such a semi-dilute solution model in the high temperature rubbery regime has been performed [57], 
+and finds log⁡(1/𝐶𝐷p) ∝ 𝐶2 for both C > 1 and C < 1.  By revisiting their data, we find the 
+prefactor between log⁡(1/𝐶𝐷p) and 𝐶2 is significantly larger than log(e) in the CPR entropic mesh 
+model, suggesting the kinetic factor  
+𝑑2
+𝜏R  in Eq.(1) must also contribute significantly to the crosslink 
+density dependence of the penetrant diffusion constant and may also vary exponentially with  −𝐶2 
+for unknown reasons. Qualitatively, this is what we expect within the SCCH theory of penetrant 
+diffusivity given our recent ECNLE theory of segmental relaxation in polymer networks (and the 
+complementary experiment and simulations) found the alpha time of polymer Kuhn segments 
+increases strongly with crosslink fraction[41] which must have a significant impact on the penetrant 
+alpha relaxation process. 
+Given the above, in this article we propose and test the idea that the penetrant diffusion 
+constant in Eq.(1) should scale inversely with the activated “penetrant alpha time” yielding: 
+𝐷p ∝
+𝑑2
+𝜏𝛼,p exp⁡(−𝐶2)/𝐶 ≡
+𝑑2
+𝜏𝛼,p 𝑋 ∝ exp⁡(−𝐵𝐶2)/𝐶                                 (2) 
+where 𝜏𝛼,p is both temperature and crosslink density dependent, and B is a chemistry specific 
+numerical factor. We note that since in most real world situations the molecular penetrants do not 
+
+ 
+6 
+obey the strong mesh confinement inequality C>1, other workers [56] have considered, and 
+explored with simulation, an alternative model of the effect of the mesh confinement 
+corresponding to a penetrant entropic barrier scaling linearly with the confinement parameter, 
+𝐷p ∝
+𝑑2
+𝜏R⁡ exp(−𝑏′𝐶). If one adopts this description of mesh confinement effects, then our proposal 
+in Eq.(2) becomes: 
+𝐷p ∝
+𝑑2
+𝜏𝛼,p⁡ exp(−𝑏′𝐶) ∝ exp(−𝐸𝐶)                                                  (3) 
+Here, and in our companion simulation paper [58], we adopt SCCH theory to quantitatively 
+analyze the effect of crosslinking effect on the penetrant alpha relaxation time and diffusion 
+constant in the specific context of the well-studied PnBA networks. In general, the penetrant alpha 
+relaxation time and diffusion constant are high-dimensional functions, depending on variables such 
+as temperature, penetrant chemistry (size and shape), penetrant-polymer interaction, polymer 
+persistence or Kuhn length, and degree of crosslinking (or mesh size). To render the analysis 
+tractable, we consider a single spherical penetrant of diameter that mimics BTBP [9, 59] (having the 
+same volume, 735.88 Å3, corresponding to d=1.12nm) and a purely repulsive penetrant-polymer 
+interaction (reasonable for the nonpolar dye molecule) and a polymer model calibrated for PnBA 
+networks, and focus on how temperature, crosslink density, and mesh size impact 𝜏𝛼,p and 𝐷p. This 
+choice of penetrant size and the range of crosslink fractions studied allows us to probe confinement 
+parameters from well below unity to modestly above unity, as germane to the typical molecular 
+penetrant problem. When C>>1, penetrant motion in the melt will be slaved to the polymer 
+segmental relaxation time, and in permanent networks will be effectively quenched.    
+Section II briefly describes ECNLE theory and its extension to crosslinked polymer 
+networks, and presents some new calculations relevant to the penetrant work in this article. SCCH 
+theory is briefly reviewed in section III, and the calculation of foundational theoretical quantities 
+
+ 
+7 
+required to predict penetrant alpha relaxation time and diffusion constant are presented. The 
+penetrant relaxation time and its relationship to the network glass transition temperature and mesh 
+confinement parameter are systematically studied in section IV. The question of the degree of 
+decoupling between the penetrant and polymer segmental relaxation times is analyzed in section 
+V.  The role of crosslinking on the penetrant diffusion constant is studied in section VI, and the 
+effects of crosslink-induced mesh confinement versus prolongation of network segmental 
+relaxation on penetrant transports are disentangled. The article concludes in section VII with a 
+summary and future outlook. Throughout the paper we comment on the comparison of our 
+theoretical results with our companion simulation study at the level of qualitative and semi-
+quantitative trends.  
+Finally, we emphasize that nearly all the theoretical technical details, equations, 
+derivations, numerical implementations, and discussion of limitations of the dynamic theories 
+(ECNLE[35-37, 41] and SCCH[34, 42, 43, 45]) at the Kuhn scale and the Polymer Reference Interaction 
+Site Model (PRISM) integral equation theory[34, 37, 41] for the required structural pair correlation 
+functions have been thoroughly documented in the literature and are not repeated here. The present 
+paper is the first to include crosslinks in SCCH theory, but its analog for polymer melts has been 
+discussed in detail [34]. Thus, our review of ECNLE theory in section II and SCCH theory in 
+section III focuses on their physical content and key parameters, with mathematical development 
+presented only for the new aspects of SCCH theory required to treat crosslinked networks. 
+II. ECNLE Theory of Structural Relaxation in Crosslinked Polymer Networks 
+A. Theoretical Background for Homopolymer Melts and Crosslinked Networks 
+The foundational starting point for predicting the activated dynamics of one-component 
+spherical particle fluids is the single particle scalar displacement (𝑟) - dependent dynamic free 
+
+ 
+8 
+energy, 𝐹dyn(𝑟). The negative gradient of this quantity determines an effective force in a stochastic 
+nonlinear Langevin equation that controls the spatially local aspect of activated barrier hopping 
+trajectories and a local cage barrier (right panel of Fig.1).[60] At sufficiently high (low) packing 
+fraction (temperature), a particle jump distance is predicted to be sufficiently large (
+Kr
+
+)(see Fig.1)  
+that in order for it to occur all the particles outside the cage must elastically displace in a dilational 
+(cage expansion) collective manner by a small amount. This cage expansion sets the amplitude of 
+the collective elastic displacement field.[35, 36] and costs elastic energy. The resultant physical 
+picture is that the alpha relaxation is a coupled local-nonlocal in space process (see middle panel 
+of Fig.1), characterized by local cage and longer-range collective elastic barriers.  
+Recently, we extended ECNLE theory to homopolymer melts including the consequences 
+of local chain connectivity and backbone stiffness on packing structure and dynamics [37, 41, 61]. 
+Motivated by the theoretical arguments of Zhou and two of us [37], the Kuhn segment is adopted 
+as the relevant dynamical length scale for the alpha process, and thus the scalar displacement of 
+the rigid Kuhn segment (consisting of several bonded elementary sites) center-of-mass from its 
+initial position, rK(t), is the central dynamical variable. The Kuhn segment is characterized by an 
+effective dynamically correlated number of beads, 𝑁K (see left panel of Fig.1) in a semiflexible 
+Koyama model[62, 63]. For typical polymers, 𝑁K ~2-3, and in the present work we set 𝑁K = 2𝑙p 
+with 𝑙p the persistence length. The sensibility of this model construction has been quantitatively 
+tested in prior combined simulation-theory work [37].  
+
+ 
+9 
+σ
+Bare bead
+Kuhn bead
+Polymer network
+pinned
+unpinned
+dyn,K
+K
+(
+)
+F
+r
+
+B,K
+F
+
+Kuhn Bead
+B,K
+r
+loc,K
+r
+Jump distance
+Kr
+
+Kr
+
+
+increasing fn
+Schematic of ECNLE theory 
+ 
+Fig.1. Left panel: Schematic of coarse-graining a polymer melt to the Kuhn scale and introduction 
+of crosslinks based on pinning interaction sites in a regular manner along the chain. Middle panel: 
+Schematic of the physical ideas of local-nonlocal relaxation mechanism within ECNLE theory 
+shown for simplicity for a spherical particle liquid: activated particle hopping involves a local cage 
+barrier and a longer-range cooperative elastic displacement of all particles outside the cage which 
+induces an additional nonlocal elastic barrier. Right panel: Schematic of key features of the 
+dynamic free energy as a function of dimensionless displacement of a Kuhn segment and its 
+evolution with increasing crosslink fraction, fn. The localization length rloc,K decreases with fn, 
+while the local cage barrier, barrier location rB,K, and jump distance ΔrK= rB,K-rloc,K all increase. 
+ 
+The local cage barrier 𝐹B,K is obtained directly from the dynamic free energy which is 
+constructed solely from knowledge of the inter- and intra- molecular equilibrium pair correlation 
+functions. The elastic barrier has been derived to be:  
+3
+2
+el,K
+cage,K
+eff,K
+K
+0,K
+2
+F
+r
+r
+K
+
+
+
+
+
+, where K0,K is 
+the harmonic curvature or spring constant at the localization length and the angularly averaged 
+effective cage dilation amplitude is 
+2
+eff,K
+K
+cage,K
+3
+32
+r
+r
+r
+
+ 
+; both these quantities are computed 
+from the dynamic free energy. The Kuhn scale “cage radius” is 
+1/3
+cage,K
+K
+cage,m
+r
+N
+r
+=
+ with cage,m
+r
+ being 
+the location of the first minimum of polymer matrix radial distribution function 𝑔mm(𝑟) at the 
+elementary interaction site level and 𝜌K = 𝜌/𝑁K the number density of Kuhn segments. 
+Chemical crosslinking is modeled as follows [41]. First, it is assumed to not change the site-
+site intramolecular and intermolecular structural pair correlations. This simplification is consistent 
+with a recent simulation study [64] that found the dimensionless compressibility of crosslink 
+networks remains essentially unchanged compared to the melt. Second, a “neutral confinement” 
+model of partial random pinning of particles (as widely studied in the glassy dynamics area) is 
+
+ 
+10 
+adopted which in the polymer network context mimics chemical crosslinking as a fraction (fn) of 
+dynamically immobile segments (zero localization length) distributed in a regular manner along a 
+chain, per the schematic in Fig.1. The ratio 𝑛crosslink/(𝑛crosslink + 𝑛monomer) defines the crosslink 
+fraction fn, where 𝑛crosslink and 𝑛monomer are the site (bare bead) level number of crosslinkers and 
+normal mobile beads, respectively. Determination of the corresponding dynamic partial structure 
+factors which enter as collective Debye-Waller factors [41, 61, 65] in the construction of the dynamic 
+free energy is identical to prior work [41]. Crosslinkers modeled as pinned sites modify all aspects 
+of ECNLE theory via the dynamic free energy. 
+The mean barrier hopping time 𝜏hop,K is then computed based on the Kramers mean first 
+passage time theory as [41, 42, 66, 67] 
+ 
+loc,K
+K
+el,K
+K
+dyn,K
+K
+dyn,K
+K
+loc,K
+loc,K
+(
+)
+(
+)
+s,K
+hop,K
+K
+K
+2
+K
+2
+r
+r
+F
+r
+F
+r
+F
+r
+r
+r
+e
+dr e
+dr e
+
+
+
+
+
+
++
+
+−
+
+=
+
+
+  
+(4) 
+where 
+2
+s,K
+K
+s,K
+D
+
+
+=
+ is a characteristic very short-time scale unaffected by crosslinking for a 
+Kuhn unit (
+1/3
+K
+K
+N
+
+
+=
+) and the formula for 𝐷s,K  is given elsewhere [35, 41, 42]. Using the PRISM 
+theory of equilibrium structural pair correlation functions as input to the dynamic free energy and 
+hence Eq.(4), the mean Kuhn segment alpha relaxation time is computed numerically as 
+,K
+s,K
+hop,K
+
+
+
+
+=
++
+. To confront our theoretical results with experimental or simulation results, we 
+adopt the elementary timescale of a Newtonian liquid, τ0=16ϕσ(βM/π)1/2 as the time unit, which is 
+typically ~ 1 ps, per prior studies [35, 41].  
+B. Mapping to Thermal Polymer Melts and Networks 
+To determine the required equilibrium site-site pair correlation functions, PRISM integral 
+equation theory with the Percus-Yevick (PY) closure is employed.[41, 61] The penetrant and 
+polymer interactions sites are modeled as hard spheres. The intramolecular structure factor of an 
+
+ 
+11 
+ideal semiflexible chain 𝜔mm(𝑞), penetrant-to-matrix size ratio 𝑑/𝜎, and polymer site reduced 
+density or packing fraction enter as inputs to PRISM theory, and 𝜔mm(𝑞) is determined using the 
+semiflexible tangent discrete Koyama model[41, 62, 67]. This single chain statistical model is 
+characterized by the bond length 𝑙b, bead diameter 𝑙d, and persistence length 𝑙p; we adopt its 
+“tangent” version corresponding to 𝑙b = 𝑙d = 𝜎. In our recent work on pure polymer network 
+relaxation, a generic flexible chain persistence length of 𝑙p =
+4
+3 𝜎 was adopted [41]. In the present 
+work, for the PnBA network the Kuhn length and monomer effective diameter are evaluated from 
+experimental information to be 𝑙K = 1.72nm and 𝑙b = 0.72nm, respectively, corresponding to a 
+local aspect ratio 𝑙K/𝑙b = 2.4, or in terms of the persistence length  𝑙p/𝜎 = 1.2. Since in the tangent 
+Koyama model one has 
+𝑙K
+𝜎 =
+2𝑙p
+𝜎 − 1, the adopted mapping yields 𝜎 = 1.23nm, which is slightly 
+larger than 𝜎 = 1.03nm if 𝑙p =
+4
+3 𝜎 is chosen. The core theoretical predictions for the 𝑙p/𝜎 = 1.2 
+and 4/3 models are very similar (see Appendix). In the main text we adopt the 𝑙p/𝜎 = 1.2 model. 
+To address experimental systems under isobaric (1 atm) conditions where temperature is 
+the control variable, we adopt the well developed and tested mapping approach that relates in a 
+chemistry specific manner the polymer packing fraction to temperature.[39-41] Specifically, the 
+dimensionless compressibility, 𝑆0(𝑇), from PRISM theory for the semiflexible Koyama chain 
+model[41, 62, 67]  is equated to that obtained from experimental equation of state (EOS) data[44, 68] 
+for PnBA melts.[39-41] The result of this mapping is shown in the inset of Fig.2. A nearly linear 
+dependence is valid from the very high temperature regime (> 400K) through the modestly 
+supercooled regime, down to the glass transition temperature. The main frame of Fig.2 shows S0 
+can be accurately represented by the analytic formula 𝑆0
+−1 = 𝑁s(𝐵/𝑇 − 𝐴′)2 derived from the van 
+der Waals model[69] EOS. As discussed previously[38, 69, 70], reasonable variation of Ns does not 
+
+ 
+12 
+affect our analysis since the crucial aspect is the T-dependence of S0 that is determined by the EOS 
+parameters 𝐴′  and B, and Ns enters only as a T-independent prefactor. The final step of our model 
+construction is to set Ns = 7.75 to reproduce the experimental glass transition temperature of 𝑇g = 
+-45 °C of pure PnBA melts [9]. 
+0.5
+0.6
+0.7
+4
+6
+8
+10
+S-1/2
+0
+Tg/T
+ expt
+ S-1/2
+0  = 3(1141.53K/T-0.5271)
+200
+300
+400
+0.60
+0.64
+0.68
+0.72
+T/K
+feff
+ 
+Fig. 2. Main: Experimental dimensionless compressibility data of PnBA melt versus scaled inverse 
+temperature in a representation suggested by the van der Waals model [69]. One sees the data agrees 
+very well with expectations that  𝑆0
+−1/2 is a linear function of Tg/T in the range Tg/T=0.46-0.74. 
+The red line corresponds to the van der Waals model analytic representation [69] 𝑆0
+−1 =
+9(1141.53K/𝑇 − 0.5271)2, which is employed in the main text to map the hard polymer chain 
+model to its thermal PnBA analog at 1 atm. Inset: Relationship between effective packing fraction 
+from PRISM theory and temperature for PnBA deduced from the mapping procedure [39-41]. 
+C. Dynamic Relaxation of Kuhn Segments and Glass Transition Temperature.  
+0.0
+0.1
+0.2
+0.3
+1.0
+1.1
+1.2
+Tg(fn)/Tg(0)
+f 1/2
+n
+ Th: ,K(Tg)/0
+     1014
+     109
+     105
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+f 1/2
+cross
+ Expt
+  Thermo
+  (Tg)=10-4s
+ Sim
+  Thermo
+  (Tg)=105 *
+0.0
+0.5
+1.0
+240
+280
+320
+Tg(fn)/K
+a-1
+x =C/d [nm-1]
+ Expt:
+Thermo;
+,K(Tg)=108 ps
+ Sim: 
+Thermo;
+,K(Tg) / *=105
+  TH ,K(Tg) /0
+    1014
+    108
+    105
+ 
+Fig.3. (a) Comparisons between scaled glass transition temperatures from experiment, simulation, 
+and theory (𝑙p/𝜎 =1.2 model) extracted and defined in two ways as indicated (dynamic and 
+
+ 
+13 
+thermodynamic), Tg(fn)/Tg(0), plotted as a function of the square root of crosslink fraction. The 
+experimental and simulated crosslink fractions are taken to be exactly the same and the theory 
+analog obeys fn = 0.19fcross [41]. (b) Glass transition temperature 𝑇g(𝑓n) in Kelvin as a function of 
+inverse mesh size.  Experimental and simulation mesh sizes are directly measured where the bond 
+length 𝑙b = 0.72nm was used in simulation as the length unit [41] while in the theory analysis the 
+mesh size is calculated using 𝑎x
+−1 = 𝐴(𝑓n/0.19)1/2/𝜎 with A=1.94 (see main text).  
+We first consider how the degree of crosslinking affects the glass transition temperature 
+𝑇g(𝑓n) defined in the theory as when Kuhn segment alpha time reaches a fixed value relevant to a 
+specific experiment or simulation; here we employ 𝜏𝛼,K(𝑇g)/𝜏0 = 1014, 108, and 105 to illustrate 
+our predictions based on a vitrification criterion in the laboratory deeply supercooled, intermediate 
+supercooled, and weakly supercooled (per simulations) regimes, respectively. This issue was 
+studied in our previous publication for the 𝑙p/𝜎 = 4/3 Koyama model where we found 𝑇g(𝑓n) 
+grows non-linearly with 𝑓n, but no quantitative relationship was identified.[41] Here, as shown in 
+Fig.3a, we report a new discovery of a square-root relationship, 𝑇g(𝑓n) ∝ 𝑓n
+1/2, with only small 
+deviations evident in the very low crosslink density regime. Adoption of different 𝑇g criteria do 
+not change the robustness of this relation. Fig.3a also shows that the previously obtained 
+experimental and simulation 𝑇g(𝑓cross) data also obey well the relation 𝑇g(𝑓cross) ∝ 𝑓cross
+1/2 . We 
+note that in our prior joint experiment-simulation-theory study for PnBA based on the 𝑙p/𝜎 = 4/3 
+model, the experimental crosslink fraction was given by 𝑓cross = 𝑓n/0.19.[41] This prefactor of 
+0.19 remains unchanged for the present 𝑙p/𝜎 = 1.2 model, and the numerical scaling factor likely 
+reflects the difference between our model construction based on the neutral pinning idea of coarse-
+grained polymer model and real crosslinking of chemically complex polymers.  
+In polymer networks, an alternative variable to the crosslink fraction 𝑓n  is the mean 
+geometric mesh size, 𝑎x. In our theory, this quantity does not play any direct role. However, based 
+on its classic definition in experiment and simulation, 𝑎x is proportional to the square root of the 
+
+ 
+14 
+average number of bare beads between two neighboring crosslinks 𝑁x, i.e., 𝑎x ∝ 𝜎𝑁x
+1/2, where 
+𝑓cross = 1/𝑁x ≡ 𝑛crosslink/(𝑛crosslink + 𝑛monomer). As such, one has 𝑎x ∝ 𝜎/𝑓cross
+1/2 , and we quantify 
+this connection as  𝑎x = 𝜎/𝐴𝑓cross
+1/2 = 𝜎/𝐴(𝑓n/0.19)1/2 . Since 𝐴/𝜎 = 1/(𝑎x𝑓cross
+1/2 ), one can 
+compute 𝐴 using purely experimental results for 𝑎x and 𝑓cross. However, to be consistent with a 
+previous simulation of penetrant diffusion in semi-dilute solutions based in a coarse grained 
+polymer bead model (as we also employ) [57] where 𝐴 = 1.94 was adopted to achieve the best 
+agreement of simulation data with the formula log⁡(𝐷p)~(𝑑/𝑎x)2, we a priori fix 𝐴 = 1.94. To 
+check the sensitivity of our theoretical results concerning this choice, we demonstrate below that 
+using 𝐴 = 4.0 does not modify any of our qualitative findings. 
+  
+Based on 𝑎x = 𝜎/𝐴(𝑓n/0.19)1/2 ∝ 1/𝑓n
+1/2 and our above discovery that 𝑇g(𝑓n) ∝ 𝑓n
+1/2, 
+we deduce that 𝑇g(𝑓n) and 𝑎x
+−1(𝑓n) are proportional. In experiment and simulation, the average 
+mesh size 𝑎x can be directly measured [9] instead of using 𝑎x = 𝜎/𝐴𝑓cross
+1/2 . Here based on previous 
+experimental results of PnBA for mesh size 𝑎x [9] and glass transition temperature 𝑇g(𝑓cross) [9, 
+41], and our published 𝑇g(𝑓cross) [41] and 𝑎x estimates from simulations [58], we test the linearity 
+between 𝑇g(𝑓n) and 𝑎x
+−1(𝑓n) in Fig.3b. We find excellent agreement over the entire range of data.  
+We now consider the predictions of ECNLE theory for the polymer network alpha 
+relaxation time for a choice of aspect ratio (1.2) modestly different than in prior work [41]. Figure 
+4a shows the foundational results for the nonlocal collective elastic (solid) and local cage (open) 
+barriers as a function of Tg(fn)/T over a wide range of crosslink densities (covering all ranges that 
+was achieved in experiments [9]) at various fixed temperatures. The theoretical results are plotted 
+based on two very different dynamic Tg criteria appropriate for the deeply and weakly supercooled 
+regimes in the main frame and inset of Fig.4a, respectively. Different choices of vitrification 
+
+ 
+15 
+criterion do not, at zeroth order, change the qualitative finding of a good collapse and shape of the 
+master curves. Of course, quantitatively the barriers cannot be identical, and must be larger at the 
+same Tg(fn)/T based on defining glass formation with a longer alpha time criterion. For the classic 
+laboratory criterion that the network vitrifies when the alpha time is 100s, the elastic barrier results 
+as a function of Tg(fn)/T collapse very well over the entire Tg/T range for all temperatures that 
+corresponds to ~16 decades growth of the alpha time. The local cage barriers at different 
+temperatures do not collapse as well, although the deviations are not large. Moreover, we find the 
+local cage barrier grows linearly with Tg(fn)/T for each fixed temperature, while the elastic barrier 
+grows in a non-linear manner, a feature that has been discussed in great depth previously [41]. 
+0.6
+0.7
+0.8
+0.9
+1.0
+0
+4
+8
+12
+16
+20
+24
+FB(el),K
+Tg(fn)/T
+,K(Tg) /0  = 1014
+  FB     Fel      T/K
+|
+ 273.15
+|
+ 296.15
+|
+ 323.15
+|
+ 348.15
+|
+ 400.15
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+0
+4
+8
+12
+16
+FB(el),K
+Tg(fn)/T
+,K(Tg) /0  = 105
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+101
+104
+107
+1010
+1013
+1016
+,K /0
+Tg/T
+SIM
+EXPT
+TH
+|
+|
+|
+|
+|
+|
+|
+|
+|
+increase
+   fcross
+TH
+EXPT
+0.6
+0.8
+1.0
+100
+104
+108
+1012
+ 
+Fig.4. (a) Kuhn segment local cage and collective elastic barriers and (b) mean alpha relaxation 
+time based on different Tg definition criteria as a function of Tg/T over a wide range of crosslink 
+densities for the 𝑙p/𝜎 = 1.2 model. In (b) the experiment and simulation results are also shown. 
+The Tg criteria in the main frame and inset of (a) are 𝜏𝛼,K(𝑇g)/𝜏0 = 1014 and 105 (using an 
+elementary time scale 𝜏0~1ps per typical viscous liquids), respectively, while for the main frame 
+of (b) they are 𝜏𝛼,K(𝑇g) = 108ps, 𝜏̃𝛼,K(𝑇g) = 105 corresponding to ~320ns, and 𝜏𝛼,K(𝑇g)/𝜏0 =
+1014, for experiment, simulation, and theory, respectively, and for the inset of (b) they are the 
+same for both experiment and theory, i.e., 𝜏𝛼,K(𝑇g) = 108ps or 108𝜏0. Note the variable Tg/T in 
+(a) is crosslink fraction dependent at fixed temperature (T/K = 273.15, 296.15, 323.15, 348.15, 
+400.15, respectively), i.e., Tg(fn), while that in (b) is temperature at fixed crosslink fraction (fn = 0, 
+0.02, 0.04, and 0.1, respectively). The vertical line in the inset of (a) or (b) represents an estimate 
+of a higher kinetic glass transition temperature, which is far higher than the normal glass transition 
+temperature deduced thermodynamically or dynamically in experiment of ~100 seconds. 
+
+ 
+16 
+Given the predicted collapse behavior for the elastic barriers and near collapse of the local 
+cage barriers for different temperatures and crosslink densities, a nontrivial collapse of the total 
+barrier and hence the Kuhn scale alpha times for all temperatures and crosslink densities is 
+expected. Figure 4b verifies this is the case for the 𝑙p/𝜎 = ⁡1.2 model. This behavior provides a 
+theoretical basis for the collapsed Angell plots observed in experiment and simulation, as shown 
+in the main frame of Fig.4b. Finally, we emphasize that the theoretical results in Fig.4 provide 
+foundational input to the SCCH theory since the penetrant dynamic free energy is coupled with 
+the polymer network dynamic free energy [34, 42, 43]. 
+III. SCCH Theory in Crosslinked Polymer Networks 
+A. Theoretical Background for Penetrant Dynamics  
+We now briefly recall the key physical elements of the SCCH theory[34, 42, 43, 45] for the 
+mean penetrant hopping time which describes an activated process (see the corresponding 
+penetrant dynamic free energy in the left panel of Fig.5) which is self-consistently coupled with 
+an activated dynamic “facilitation displacement” of polymer segments determined by the Kuhn 
+segment dynamic free energy (see the middle panel of Fig.5). Crucially, the penetrant dynamic 
+free energy depends on both its displacement and that of the Kuhn segment (
+pr  and 
+Kr , 
+respectively). The mathematical formulation of SCCH theory has been discussed in great detail 
+previously for a dilute penetrant in a hard sphere fluid[42, 43, 45] and in a polymer melt[34]. The 
+derivative of the penetrant dynamic free energy in polymer melts is given as [34]: 
+ 
+2 2
+2 2
+p
+K
+K
+mm
+6
+dyn,p
+p
+K
+p
+( ) 6
+( )
+2
+2
+mp
+mm
+3
+p
+p
+( ,
+)
+3
+( )
+( )
+3
+(2 )
+q r
+q r
+q
+S
+q
+F
+r r
+r
+d
+q
+C
+q S
+q e
+e
+r
+r
+
+
+
+
+−
+−
+
+= −
++
+
+
+q
+ 
+(5) 
+
+ 
+17 
+where 𝜌 is the number density of matrix interaction sites or beads (see Fig.1), Cmp(q) is the Fourier 
+transform of the penetrant-polymer site-site direct correlation function, Smm(q) is the Fourier space 
+static collective structure factor of the polymer network, and ωK(q) is the intramolecular structure 
+factor of the elementary Kuhn segment that in Eq.(5) (it can be set to unity without significantly 
+changing the numerical predictions [34]). The subscript “m” represents the elementary bead or site 
+level (see the left panel of Fig.1) description of the polymer and is employed to distinguish it from 
+the Kuhn segment scale (denoted by the subscript “K”). Based on the pinned bead model of 
+polymer crosslinks[41], Eq.(5) for polymer melt is modified as: 
+ 
+
+
+2 2
+p 6
+dyn,p
+p
+K
+p
+2
+2
+mp
+3
+p
+p
+uu
+Ku
+nn
+Ku
+un
+Ku
+nu
+Ku
+( ,
+)
+3
+( )
+3
+(2 )
+( ,
+)
+( ,
+)
+( ,
+)
+( ,
+)
+q r
+F
+r r
+r
+d
+q C
+q e
+r
+r
+S
+q r
+S
+q r
+S
+q r
+S
+q r
+
+
+−
+
+= −
++
+
+
++
++
++
+
+q
+ 
+(6) 
+The presence of pinned segments enters via the form of the collective partial Debye-Waller factors 
+of the polymer matrix, Sαβ(q,rKu), which differs from a melt of fully mobile segments as previously 
+derived and discussed []. We additionally note that the subscript Ku refers to the mobile Kuhn 
+segments, and the subscripts u and n indicate unpinned and pinned sites, respectively.  
+To render SCCH theory predictive and tractable, the degree of Kuhn segment dynamic 
+displacement that facilitates the penetrant hopping event is described by introducing a single 
+trajectory coupling variable, γ, defined as:  
+Ku
+loc,Ku
+p
+loc,p
+( )
+(
+)
+r
+r
+r
+r
+
+
+=
++
+−
+.[34] The “pinned” Kuhn 
+segments (crosslinkers) are also allowed to vibrate locally with an amplitude set by the predicted 
+localization length of unpinned Kuhn segments, i.e., loc,Kn
+loc,Ku
+r
+r
+=
+. The parameter γ is determined 
+based on enforcing a temporal self-consistency condition[34, 42, 43]: 
+hop,p
+dis,K
+Ku,c
+( )
+(
+( ))
+r
+
+
+
+
+=
+
+, 
+where 
+Ku,c
+p
+( )
+( )
+r
+r
+
+
+
+
+= 
+ and both 
+hop,p( )
+
+  and 
+dis,K
+K,c
+(
+( ))
+r
+
+
+
+ are computed using Kramers 
+theory [66, 67] as sketched in section II. Solving the self-consistency condition numerically yields γ 
+
+ 
+18 
+which depends on all control parameters of the problem. Using it, one can then compute the 
+penetrant alpha time as 
+,p
+s,p
+hop,p
+
+
+
+
+=
++
+, where 
+2
+s,p
+s,p
+d
+D
+
+=
+ is the characteristic elementary 
+short-time scale for the penetrant given elsewhere [34, 42]. As in ECNLE theory of one-component 
+liquids, at high enough matrix packing fractions the penetrant jump length can become (depending 
+on penetrant size) sufficiently large that a collective elastic distortion of matrix particles outside 
+the cage region is required to allow the penetrant to hop.[34, 42, 43] This physics enters via a common 
+elastic barrier (
+el,p
+el,Ku,c
+F
+F
+=
+) in the computation of the timescales 
+hop,p( )
+
+  and 
+dis,K
+Ku,c
+(
+( ))
+r
+
+
+
+. 
+The elastic barrier is determined following the same ideas and methods employed for Kuhn 
+segment in the pure polymer system as discussed above and elsewhere, with the result previously 
+derived 
+as: 
+3
+2
+el,p
+n
+cage,p
+eff,p
+K
+0,K
+2 (1
+)
+F
+f r
+r
+K
+
+
+
+
+−
+
+ where 
+2
+eff,p
+Ku,c
+cage,p
+( )
+3
+( ) 32
+r
+r
+r
+
+
+
+= 
+, 
+1/3
+cage,p
+min,mp
+K
+(
+) (
+)
+r
+r
+N
+d
+d
+
+
+=
++
++
+, and 
+min,mp
+r
+ is the cage radius corresponding to the first 
+minimum of cross radial distribution function
+mp( )
+g
+r .[34, 41] Everywhere below the notation Fel 
+(FB) is employed to indicate the penetrant elastic (local cage) barrier. 
+ 
+Fig.5. Schematic of the dynamic free energies for the coupled penetrant (left) and Kuhn segment 
+(middle) displacements in SCCH theory. The trajectory level matrix facilitation of penetrant 
+hopping is formulated based on the simplifying concept of a dimensionless dynamic coupling 
+parameter, 𝛾 , defined as the ratio of the penetrant jump distance Δ𝑟p(𝛾) to the facilitating 
+displacement of the matrix Kuhn segments, Δ𝑟K,c(𝛾). Relevant length and energy scales are 
+indicated. Right panel: Schematic of the physical ideas of SCCH theory for a single smaller 
+spherical penetrant in a bulk liquid of spheres. The image shows the matrix as disconnected spheres 
+solely for visual clarity and is literally relevant only for the hard sphere mixture model [42, 43]. 
+
+BF
+(rk
+IBK
+penetrant
+Kuhn matrix
+'?
+rioc.K
+BF
+B,K
+BF.
+BF
+B,P
+B,K,c
+Ar,(y) /o
+Ark. (r) /o
+Jump
+Cooperative displacement to
+distance
+facilitate penetrant hopping 
+19 
+This completes our brief review of the construction of SCCH theory that includes 
+crosslinking. From knowledge of the coupling parameter, penetrant dynamic free energy, and its 
+local cage and elastic barriers, the SCCH theory then predicts the mean penetrant alpha time, 𝜏𝛼,p, 
+as a function of size ratio, polymer packing fraction or temperature, crosslink fraction, etc.  
+As discussed in the Introduction, we will apply the theory motivated by a recent 
+experimental study of the relaxation and diffusivity of BTBP in PnBA networks [9].  Modeling 
+BTBP as a sphere, its effective hard sphere diameter is 1.12nm. For the adopted 𝑙p/𝜎 = 1.2 
+Koyama model 𝜎=1.23nm, while for the 𝑙p/𝜎 = 4/3 model one has 𝜎=1.03nm, thereby yielding 
+a penetrant-to-matrix size ratio of order unity, 𝑑/𝜎=0.91 and 1.09, respectively. 
+B. Crosslink Fraction Dependence of Jump Displacements and Associated Properties 
+In SCCH theory[34], the penetrant jump distance and the facilitation displacement of Kuhn 
+segments are very important for the activated penetrant hopping event since they (i) directly enter 
+calculation of the penetrant elastic barrier, and (ii) the difference between displacements of a Kuhn 
+segment at the penetrant and polymer Kuhn segment alpha times provides a mechanistic picture 
+of the degree of trajectory level coupling between the penetrant and Kuhn segment alpha processes. 
+Thus, we first consider these quantities. 
+Both increasing crosslink fraction and decreasing temperature induces a significant 
+increase of the penetrant and Kuhn segment alpha times. This results in an increased jump distance 
+of both the penetrants and Kuhn segments, as shown in Figs.6a and 6b, respectively. The coupling 
+between penetrant and Kuhn segment activated displacements at the penetrant alpha time 
+corresponds to a facilitation displacement Δ𝑟Ku,c of a Kuhn segment, which grows with crosslink 
+density or cooling (longer alpha time) as shown in Fig.6b. We note the crosslinking dependence 
+of Δ𝑟Ku,c is weaker than its temperature dependence, which differs from that of Δ𝑟p and Δ𝑟Ku where 
+
+ 
+20 
+both the temperature and crosslinking dependences are significant. We also find that the crosslink 
+density dependence of Δ𝑟p is nearly unchanged with increasing the temperature, i.e., calculations 
+of Δ𝑟p(𝑓n) − Δ𝑟p(0) collapse (vertical shift of the data in Fig.6a) for different fixed temperatures 
+(not shown). 
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.30
+0.35
+0.40
+0.45
+rp /
+fn
+           T/K
+ 273.15
+ 296.15
+ 323.15
+ 348.15
+ 400.15
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.3
+0.4
+0.5
+rX /
+fn
+solid: rX = rKu
+dash: rX = rKu,c
+           T/K
+ 273.15
+ 296.15
+ 323.15
+ 348.15
+ 400.15
+ 
+Fig.6. Jump distance of (a) the penetrant, Δ𝑟p and (b) mobile Kuhn segment (solid curves), Δ𝑟Ku, 
+in units of 𝜎 at their corresponding alpha time scales as a function of crosslink fraction for a wide 
+range of temperatures. The facilitation displacement of unpinned Kuhn segments, Δ𝑟Ku,c, at the 
+penetrant alpha time scale is shown in (b) as dashed curves. The similarity of the absolute 
+magnitude of the penetrant and unpinned Kuhn segment jump distances reflects the strong 
+coupling of their motions as a consequence of the very similar penetrant and Kuhn segment sizes.  
+Fig.6b also shows that the absolute value of Δ𝑟Ku,c (corresponding to the penetrant alpha 
+time scale) is much lower than that of Δ𝑟Ku (corresponding to the Kuhn segment alpha time scale), 
+and the difference increases significantly with degree of effective supercooling (lowering 
+temperature or increasing crosslink fraction). This suggests that the dynamical coupling between 
+penetrant and polymer matrix weakens at lower temperatures and higher crosslink densities. For 
+the highest temperature T = 400K and lower crosslink density regime shown in Fig.6b, the 
+crosslink dependence of Δ𝑟Ku,c is very similar to that of Δ𝑟Ku, corresponding to a relatively stronger 
+coupling between the alpha relaxations of the penetrant and polymer matrix.  
+One can also see from Fig.6b that the absolute value of the Kuhn segment facilitation 
+displacement is not much smaller than the location of its barrier (Kuhn segment jump distance). 
+
+ 
+21 
+This implies there is a strong coupling between penetrant and Kuhn segment alpha relaxations and 
+suggests the dynamic coupling parameters, 𝛾, defined as the ratio of penetrant jump distance Δrp(γ) 
+to the facilitating Kuhn segment displacement ΔrKu,c(γ), should not be much larger than unity. 
+Figure 7a shows this to be the case. The smaller 𝛾 is, the larger is the dynamic coupling of activated 
+penetrant and polymer matrix displacements. Figure 7a also shows that the coupling decreases 
+with cooling and crosslink fraction as physically expected, but the dependences are weak. The 
+reason is that the penetrant jump distance has a significant temperature dependence while that of 
+Kuhn segment facilitation displacement is more limited (per Figs.6a and 6b), while both their 
+crosslinking dependences are rather weak. Since the temporal self-consistency condition is 
+hop,p
+dis,K
+K,c
+( )
+(
+( ))
+r
+
+
+
+
+=
+
+ with 
+el,p
+el,K,c
+F
+F
+=
+, the coupling parameter 𝛾  encodes dynamical 
+displacement physics associated only with local caging. Therefore, variation of 𝛾 with crosslink 
+fraction is physically intuitive, even in the large crosslink fraction regime it is nearly constant. 
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+1.20
+1.25
+1.30
+1.35
+
+fn
+T/K = 273.15
+296.15
+323.15
+348.15
+400.15
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+1
+2
+3
+X(fn)/X(0)
+fn
+solid: X = K0,Ku
+dash: X = r 4
+Ku,c
+           T/K
+ 273.15
+ 296.15
+ 323.15
+ 348.15
+ 400.15
+ 
+Fig.7. (a) Self-consistently determined dynamic coupling parameter, 𝛾 ≡ Δ𝑟p/Δ𝑟Ku,c, as a function 
+of crosslink fraction at various temperatures. (b) Relative importance of the key factors Δ𝑟Ku,⁡c
+4
+ 
+(dash) and 𝐾0,⁡Ku (solid) reduced by their values at zero crosslink fraction that enter the penetrant 
+elastic barrier as a function of crosslink fraction at various temperatures. 
+Finally, in Fig.7b we compare the relative importance of the two key factors Δ𝑟Ku,⁡c
+4
+ and 
+𝐾0,⁡Ku  that enter the calculation of penetrant elastic barrier. The harmonic curvature 𝐾0,⁡Ku 
+(proportional to the inverse dynamic localization length squared) weakly increases with cooling 
+
+ 
+22 
+and crosslink fraction, with 𝐾0,⁡Ku(𝑓n)/𝐾0,⁡Ku(0) varying only from ~1.0-1.4 for all ranges of 
+temperature and crosslink fractions studied. For pure hard sphere fluids, increases of the localized 
+state spring constant with packing fraction is the most important origin for the corresponding 
+increase of the elastic barrier[35].  However, for networks, Fig.7b shows that as crosslink fraction 
+increases, the jump distance contribution Δ𝑟Ku,⁡c
+4
+ dominates the change of elastic barrier. The 
+reason is that introducing crosslinking increases the penetrant hopping time, and hence penetrants 
+can displace significantly more corresponding to a larger jump distance. 
+IV. Penetrant Alpha Relaxation Time  
+We now consider theoretical predictions for quantities that are directly measurable in 
+experiment and simulation, specifically the penetrant alpha relaxation time in this section. In the 
+context of empirical experimental and simulation data[9, 56, 57], one can consider two different ways 
+of displaying its dependence on a control parameter depending on which physical perspective one 
+thinks is the primary origin of effect of network crosslinking on penetrant motion: crosslink 
+dependent changes of segmental dynamics (i.e., activated glassy physics) versus entropic mesh 
+confinement. In this section we only pursue an objective empirical analysis based on plotting the 
+theoretical data for the penetrant alpha time in two different ways. Of course, we know the physics 
+in SCCH theory is all about activated hopping dynamics. Our goal is to test whether a “degeneracy 
+of interpretation” of the theoretical results might exist due to the discovered connection between 
+Tg and mesh size discussed in section II. This exercise does not answer the question whether the 
+physics of penetrant diffusion is dominated by segmental scale glassy effect or entropic mesh 
+confinement, the answer to which must depend to some extent on Tg/T, crosslink density, and 
+penetrant-to-matrix size ratio. This question is explored in section VI, and in our companion 
+simulation article.  
+
+ 
+23 
+As a preview of the combined insights gleaned from the results in the remainder of our 
+present theoretical work and the complementary simulations, we find for the penetrant sizes 
+studied that although the crosslink fraction dependent (but temperature independent) entropic mesh 
+confinement does affect penetrant diffusivity, it is an increasingly secondary (and ultimately very 
+minor) effect as the temperature is lowered in the supercooled regime of interest  compared to the 
+coupling of penetrant hopping to the temperature and crosslink fraction dependent activated barrier 
+hopping associated with glassy dynamics. 
+A. Crosslinking Modified Segmental Dynamics Perspective 
+Figure 8 shows results for the inverse penetrant mean alpha time as a function of the 
+temperature scaled glass transition temperature, Tg(𝑓n)/T, over a wide range of crosslink fractions 
+at various fixed values of temperature. The results are presented based on a typical laboratory 
+timescale criterion for vitrification (main panel) and a much smaller simulation-like timescale 
+criterion (inset). The predicted behaviors are very similar for the two criteria. For each fixed 
+temperature, the penetrant alpha relaxation time 𝜏𝛼,p/𝜏0 (𝜏0 ~1 ps per a typical viscous liquid) is 
+predicted to increase exponentially with 𝑇g(𝑓n), in the spirit of an “Arrhenius-like” law. Moreover, 
+we observe that the corresponding apparent activation energy [defined as the slope of log(𝜏𝛼,p/𝜏0) 
+versus 𝑇g(𝑓n)/𝑇,  see Table 1] and the absolute value of 𝜏0/𝜏𝛼,p decreases and increases with 
+temperature, respectively. Overall, results for different temperatures are rather well collapsed. 
+There is a weak trend of lower penetrant hopping rate upon cooling, which is more prominent in 
+the main frame that is relevant to laboratory experiments. As discussed in detail below, the reason 
+for this difference, and for the roughly Arrhenius-like form of the penetrant alpha time, is the 
+relative unimportance of the elastic barrier compared to its local cage analog. The latter is 
+
+ 
+24 
+quantitatively more true in the simulation based choice for Tg results which do not probe the deeply 
+supercooled regime. 
+0.6
+0.7
+0.8
+0.9
+1.0
+10-10
+10-9
+10-8
+10-7
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+0 /,p
+Tg(fn)/T
+,K(Tg) /0  = 1014
+TH
+           T/K
+ 273.15
+ 296.15
+ 323.15
+ 348.15
+ 400.15
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+10-9
+10-7
+10-5
+10-3
+10-1
+0 /,p
+Tg(fn)/T
+,K(Tg) /0  = 105
+ 
+Fig.8. Penetrant inverse mean alpha time based on SCCH theory as a function of Tg(fn)/T at various 
+fixed temperatures for vitrification criteria 𝜏𝛼,K(𝑇g)/𝜏0 = 1014  (main) and 105  (inset). The 
+vertical line in the inset is an estimate of a higher kinetic glass transition temperature, which 
+corresponds to a simulation criterion of vitrification that delivers a glass transition temperature far 
+higher than its laboratory based analog.  
+ 
+Table 1. Slope of the logarithm of the penetrant alpha time versus 𝑇g/𝑇 or C plot for various fixed 
+temperatures and 𝑇g criteria where A=1.94 is adopted in theoretical evaluation of C. 
+ 
+T/K 
+𝑻𝐠 criterion 
+273 
+296 
+323 
+348 
+400 
+𝐥𝐨𝐠⁡(𝝉𝜶,𝐩/𝝉𝟎) vs 𝑻𝐠/𝑻 
+105 
+18.4 
+15.3 
+13.5 
+11.7 
+7.3 
+𝐥𝐨𝐠⁡(𝝉𝜶,𝐩/𝝉𝟎) vs 𝑻𝐠/𝑻 
+1014 
+19.4 
+17.1 
+14.8 
+12.8 
+7.5 
+𝐥𝐨𝐠⁡(𝟏/𝑫𝐩) vs 𝑻𝐠/𝑻 
+105 
+26.2 
+23.7 
+22.2 
+20.6 
+17.2 
+𝐥𝐨𝐠⁡(𝟏/𝑫𝐩) vs 𝑻𝐠/𝑻 
+1014 
+28.0 
+25.6 
+24.2 
+22.3 
+19.2 
+𝐥𝐨𝐠⁡(𝝉𝜶,𝐩/𝝉𝟎) vs 𝑪 
+ 
+3.3 
+2.8 
+2.2 
+1.8 
+0.77 
+𝐥𝐨𝐠⁡(𝟏/𝑫𝐩) vs 𝑪 
+ 
+4.5 
+3.8 
+3.2 
+2.8 
+2.1 
+ 
+SCCH theory can provide the physical mechanism to understand how crosslinking 
+modifies the penetrant alpha relaxation process by carefully examining the penetrant local cage 
+and collective elastic barriers. Figure 9 presents theoretical results for these quantities in an Angell-
+
+ 
+25 
+like representation, where the dynamic 𝑇g(𝑓n) criterion appropriate for the deeply supercooled 
+regime of highest experimental relevance is adopted. For all fixed temperatures studied, the elastic 
+barriers are far smaller than the local cage barriers. There are several reasons for this: (i) for 
+polymer systems, the relative importance of the elastic barrier is modestly weaker because of the 
+influence of chain connectivity [37, 38], and (ii) the penetrant size is relatively small which is known 
+to induce significant decoupling between the penetrant and matrix dynamics and a lower elastic 
+barrier. Figure 9 also shows that the elastic barrier increases more slowly with 𝑇g(𝑓n)/𝑇 relative 
+to that of local cage barrier. Hence, the total penetrant barrier and its growth with 𝑇g(𝑓n)/𝑇 is 
+largely dominated by the local cage barrier, and for 𝑇g(𝑓n)/𝑇 ≤ 1 it is more so for the simulation 
+timescale vitrification criterion (inset) than its laboratory analog. 
+0.6
+0.7
+0.8
+0.9
+1.0
+0
+4
+8
+12
+16
+20
+24
+FB(el)
+Tg(fn)/T
+,K(Tg) /0  = 1014
+  FB     Fel      T/K
+|
+ 273.15
+|
+ 296.15
+|
+ 323.15
+|
+ 348.15
+|
+ 400.15
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+0
+4
+8
+12
+16
+FB(el)
+Tg(fn)/T
+,K(Tg) /0  = 105
+ 
+Fig.9. Penetrant local cage (open) and collective elastic (solid) barriers as a function of Tg(fn)/T at 
+various temperatures with the 𝑇g criteria in main frame and inset corresponding to  𝜏𝛼,K(𝑇g)/𝜏0 =
+1014 and 105, respectively. The vertical line in the inset represents an estimate of a higher kinetic 
+glass transition temperature corresponding to a typical simulation criterion.   
+ 
+More insight is obtained by comparing Figs.9 and 4a, which shows that the penetrant local 
+cage barrier is comparable to its Kuhn segment analog, as expected from the predicted strong 
+coupling between activated hopping of the penetrant and Kuhn segment displacements as 
+manifested in the close to unity value of the coupling parameter 𝛾 in Fig.7a. On the other hand, 
+
+ 
+26 
+the penetrant elastic barrier is far smaller than its Kuhn segment analog because its facilitation 
+displacement is much smaller than the jump distance associated with the pure network alpha 
+relaxation. The latter result can be seen from Fig.6b and the fact that the jump distance enters the 
+penetrant elastic barrier as the 4th fourth power (see section III). 
+The local cage barrier varies linearly with 𝑇g(𝑓n)/𝑇 which is the same behavior predicted 
+for the Kuhn segment (Fig.4a). The fact that this linearity is sufficiently well obeyed over the entire 
+range of 𝑇g/𝑇 is why an exponential relationship between alpha time and 𝑇g(𝑓n)/𝑇 is found in 
+Fig.8. Note also from Figs.4a and 9 that at fixed 𝑇g(𝑓n)/𝑇 the local cage barriers for both the 
+penetrant and Kuhn segments are slightly larger at lower temperature. This implies that normal 
+glass-like physics effects play an important role in determining the temperature dependence of the 
+penetrant alpha time, which is intensified in the high crosslink density regime. This behavior also 
+explains why the apparent activation energy (Table 1) increases as temperature decreases. 
+B. Confinement Mesh Perspective  
+We now explore the extent to which the SCCH theory results for the penetrant alpha time 
+can be empirically interpreted in a purely confinement mesh scenario,  𝐷p ∝ exp⁡(−𝐵𝐶2)/𝐶 or 
+𝐷p ∝ exp(−𝐸𝐶), per Eqs (2) and (3). Specifically, we ask if the crosslink fraction dependent 
+penetrant alpha time data can be “correlated” or “described” as:  1/𝜏𝛼,p ∝ exp⁡[−(𝐵 − 1)𝐶2] 
+and/or 1/𝜏𝛼,p ∝ exp[−(𝐸 − 𝑏′)𝐶]. Obviously, the entropic mesh perspective cannot account for 
+explicit temperature dependences of the theoretical penetrant alpha time. 
+
+ 
+27 
+0.0
+0.4
+0.8
+1.2
+10-17
+10-13
+10-9
+10-5
+10-1
+0 /,p
+C
+C = 4(d/)(fn/0.19)1/2
+C = 1.94(d/)(fn/0.19)1/2
+           T/K
+ 273.15
+ 296.15
+ 323.15
+ 348.15
+ 400.15
+0
+1
+2
+10-8
+10-6
+10-4
+10-2
+0 /,p
+C
+ 
+Fig.10. Inverse penetrant mean alpha time 𝜏0/𝜏𝛼,p as a function of the confinement parameter at 
+various fixed temperatures. The main frame uses  𝐶 = 1.94(𝑑/𝜎)(𝑓n/0.19)1/2, while the inset 
+adopts 𝐶 = 4(𝑑/𝜎)(𝑓n/0.19)1/2. 
+ 
+Figure 10 plots in a log-linear manner the theoretical penetrant relaxation rate as a function 
+of confinement parameter 𝐶 = 𝑑/𝑎x over a wide range of crosslink densities and temperatures. A 
+linear relationship is observed, albeit with a slope that decreases strongly with temperature. The 
+latter trend is due to activated glassy dynamics effects, and is inconsistent with a literal [55-57] 
+entropic confinement mesh picture (see Table 1). Thus, as a cautionary comment, if simulation or 
+experimental data is taken at only one temperature or over a narrow range, one could incorrectly 
+conclude that the confinement mesh idea is “verified”. Indeed, the slope changes documented in 
+Table 1 prove the importance of T-dependent physics in crosslinked polymer networks, and 
+necessarily implies the rate of increase of the alpha time with 𝑇g/𝑇 is significantly larger than that 
+with C. Physically the reason is simply the very different rate of change of 𝑇g and 𝐶 with crosslink 
+density 𝑓n. As discussed in section II, 𝑇g(𝑓n) is proportional to inverse mesh size 𝑎x
+−1 and hence 
+𝐶(𝑓n) for a fixed penetrant size and different vitrification criteria, which is the non-causal reason 
+the curves in Fig.10 are linearized. Our conclusions remain robust with varying the constant A in 
+defining the confinement parameter, as shown in the inset of Fig.10 where 𝐴 = 4.0 is adopted.  
+
+ 
+28 
+We also tested whether a linear relationship between log(𝜏α,p/𝜏0) and 𝐶2 can empirically 
+“work” (not shown), i.e., 1/𝜏α,p~ exp[−(𝐵 − 1)𝐶2]  , and find it is also a reasonable description 
+in the limited C regime examined in Fig.10. However, for large enough 𝐶 , SCCH theory predicts 
+that the exponential relationship 1/𝜏α,p~ exp[−(𝐸 − 𝑏′)𝐶] will break down since the elastic 
+barrier of penetrant becomes more and more important and it does not grow linearly with 
+confinement parameter but rather quadratically (see Fig.9).  Simulation tests of the relations 
+1/𝜏𝛼,p ∝ exp⁡[−(𝐵 − 1)𝐶2]  and 1/𝜏𝛼,p ∝ exp[−(𝐸 − 𝑏′)𝐶]  are reported in our companion 
+paper, and good agreement between theory and simulation is found.[58] 
+V. Decoupling Ratio of Penetrant and Network Alpha Times 
+As previously studied using SCCH theory for hard sphere mixtures [42, 43] and dilute 
+spherical penetrants in polymer melts [34], the ratio of the penetrant to Kuhn segment alpha times 
+can be predicted which quantifies the degree of “dynamic decoupling”, a quantity that is 
+measurable in simulations and experiments. This quantity is especially sensitive to collective 
+elasticity given the hopping of penetrants and Kuhn segments couples to this longer range effect 
+generally to very extents. In prior work 34, 42], the decoupling time ratio, τα,p/τα,K, was plotted as a 
+function of packing fraction, and found to initially increase very slightly with matrix packing 
+fraction in the lower packing fraction regime (corresponding to high temperature where barriers 
+are low and the timescale of the short time dynamical process, τs,p and τs,K, matter), and then sharply 
+decrease with further increase of packing fraction indicating a strong decoupling between 
+penetrant and matrix activated dynamics when the penetrant and matrix barriers become larger and 
+dominate the alpha relaxation times. The crucial physical effect is the very different rate at which 
+the penetrant and matrix elastic barriers grow with increasing packing fraction or decreasing 
+
+ 
+29 
+temperature, a difference significantly larger than that for their local cage barrier analogs (see 
+results in Fig.4a and Fig.9).  
+0.7
+0.8
+0.9
+1.0
+1.1
+10-4
+10-3
+10-2
+10-1
+100
+Tg(fn)/T
+Tg(fn)/T = 1
+,p /,K
+ SIM          TH     T/K
+ 300 
+ 273.15
+ 310 
+ 296.15
+ 320 
+ 323.15
+                 
+ 348.15
+                 
+ 400.15
+,K(Tg) /0  = 105
+TH
+0.6
+0.7
+0.8
+0.9
+1.0
+10-4
+10-3
+10-2
+10-1
+100
+,p /,K
+Tg(fn)/T
+,K(Tg) /0  = 1014
+ 
+Fig.11. Decoupling time ratio as a function of 𝑇g(𝑓n)/𝑇 for various values of crosslink density 𝑓n 
+using 𝜏𝛼,K(𝑇g)/𝜏0 = 105 (main) and 1014 (inset) 𝑇g criteria and for four temperatures in theory 
+and for three temperatures for the 𝑑/𝜎 = 1.0 simulations [58]. The simulation data of mean alpha 
+relaxation times are based on the 1/e decay definition criterion, and the time ratio is vertically 
+shifted up by a factor of 480 in the main panel for comparison of their functional form with the 
+theory results. The vertical line in the main frame represents an estimate of a higher kinetic glass 
+transition temperature based on the simulation vitrification criterion. 
+ 
+The inset of Figure 11 shows theoretical predictions for the decoupling time ratio plotted 
+versus 𝑇g(𝑓n)/𝑇 over a wide range of 𝑇g(𝑓n)/𝑇 from the rubbery regime, through the deeply 
+supercooled regime, to the laboratory kinetic glass transition. For the four lower temperatures 
+studied (273, 296, 323, 348 K) the time ratio decreases monotonically with the effective degree of 
+supercooling parameter, 𝑇g(𝑓n)/𝑇 . In contrast, at high temperature (400 K) where barriers are 
+relatively small and only the local cage barrier is non-negligible, a nearly constant (weakly 
+decreasing) behavior is predicted. Overall, for all five temperatures (which span a change of the 
+polymer network alpha time of ~14 decades), a nonmonotonic behavior is found, qualitatively 
+consistent with the previous predictions for other systems [34, 42]. A new finding is that for all 
+temperatures and crosslink densities, the decoupling time ratio tends to collapse in 𝑇g(𝑓n)/𝑇 space 
+with only slight deviations visible in the high 𝑇g(𝑓n)/𝑇 regime. We hope future experiments can 
+
+ 
+30 
+test our results in the deeply supercooled regime where a significant decrease of the decoupling 
+ratio is predicted. 
+We have verified that all of the above findings remain unchanged if a different 𝑇g criterion 
+is adopted. The main frame of Fig.11 shows an example based on a vitrification criterion relevant 
+to simulations. For the three temperatures simulated (300, 310, and 320 K based on the calibrated 
+PnBA coarse-grained network model employed) the decoupling time ratio slightly decreases 
+monotonically, and the data is in good agreement with our prediction in this weakly supercooled 
+regime.  We hope future simulations can better test the predicted weakly non-monotonic behavior 
+with T and probe lower temperatures. 
+VI. Penetrant Diffusion Constant  
+A. Introductory Comments and High-Level Picture 
+We now consider the predictions of Eqs.(2) and (3) which combine in a simple 
+multiplicative manner the activated physics (at the level of mean alpha relaxation times) of SCCH 
+theory with the entropic mesh confinement effect. We showed in section IV that the glass physics 
+based theoretical local hopping rate can be empirically described using the mathematical relations 
+of the entropic mesh models formulated solely in terms of the mesh confinement variable C ( 
+1/𝜏𝛼,p ∝ exp⁡[−(𝐵 − 1)𝐶2] and 1/𝜏𝛼,p ∝ exp[−(𝐸 − 𝑏′)𝐶]). Thus, logically, introducing the 
+latter mechanism into Eqs. (2) or (3) [i.e., 𝑋(𝐶) = exp⁡(−𝐶2)/𝐶 or 𝑋(𝐶) = exp(−𝑏′𝐶)] cannot 
+change the basic form of our results for how crosslink fraction modifies the diffusion constant if 
+expressed in terms of C.  
+Now, adopting Eqs.(2) or (3) for the penetrant diffusion constant, our theoretical finding 
+that the penetrant alpha relaxation rate can be empirically described in a mesh confinement 
+parameter framework implies it will be very difficult or impossible to dissect how important the 
+
+ 
+31 
+factors 1/𝜏𝛼,p and 𝑋(𝐶) are at fixed temperature from experimental or simulation data. Of course, 
+insight can be obtained from by studying the temperature dependence of the penetrant relaxation 
+time and diffusion constant. Here, in the next subsection we consider a quantitative comparison of 
+the theoretical predictions with and without the mesh confinement effect. In addition,  as discussed 
+in more detail in our companion simulation paper[58], reasonable, albeit not in general quantitative, 
+agreement between SCCH theory and simulation for both the crosslink fraction and temperature 
+dependences of the penetrant diffusion constant based on Eq.(2) is found, which systematically 
+improves as the polymer network becomes increasingly supercooled. The latter trend is expected 
+given mesh confinement is of entropic origin in contrast to the crosslinking-induced prolongation 
+of segmental relaxation which becomes increasingly more important in colder polymer networks.  
+Some of the key deductions reported in our companion simulation paper [58] can also be 
+drawn from the purely theoretical effective slope data in Table 1. One sees that under the same 
+conditions, the effective slope determined from the diffusion constant is larger than that 
+determined from the inverse alpha time, consistent with the existence of an additional mesh 
+confinement contribution. Of course, in general, one a priori expects that the relative importance 
+of glassy physics and mesh confinement depends on both temperature and penetrant size, with the 
+latter (former) expected to dominate at sufficiently high (low) temperatures. Importantly, “high” 
+and “low” depends on penetrant size since the degree of coupling of penetrant activated hopping 
+with the polymer network dynamics depends on the penetrant size relative to the size of the 
+polymer monomer or Kuhn segment. An additional generic physical effect in pure glass-forming 
+liquids is “dynamic heterogeneity” (DH) which induces a distribution of activation barriers and 
+hopping times in the lower temperature supercooled regime, and an associated decoupling of 
+diffusivity and relaxation corresponding to a weaker temperature dependence of the self-diffusion 
+
+ 
+32 
+constant than the alpha time. The consequences of such a DH effect is not well established for 
+penetrant dynamics, but it expected to be present at some level, and would go in the opposite 
+direction of the entropic mesh confinement induced slowing down of diffusion relative to 
+relaxation. The version of ECNLE and SCCH theories employed in this article focus solely on the 
+mean barriers and relaxation times, and do not consider this type of DH. Thus, its absence can lead 
+to deviations of some theoretical predictions with simulation at lower temperatures and especially 
+for smaller penetrants, as confirmed in our companion paper.[58] On the other hand, at higher 
+temperatures and/or for large penetrants where DH effects are either intrinsically small or are 
+sufficiently averaged over, respectively, one expects better agreement between theory and 
+simulation, as we have verified.[58]   
+We note that the ECNLE theory for the alpha relaxation in 1-component glass-forming 
+atomic, molecular and polymer liquids has been extended to include DH via a distribution of 
+collective elastic barriers and relaxation times[71, 72]. It captures well the Stokes-Einstein 
+breakdown or diffusion-relaxation decoupling and temperature-dependent stretched exponential 
+relaxation in molecular liquids, and also the nonuniversal failure of time-temperature-
+superposition in polymer melts corresponding to the decoupling of the temperature dependences 
+of the segmental alpha and chain scale end-to-end relaxation times. The ideas developed in these 
+prior works can be extended to the penetrant dynamics problem in the framework of SCCH theory.   
+Finally, we emphasize that given the structure of Eq.(2), penetrant diffusivity will be 
+slowed down more with crosslinking than that of penetrant alpha relaxation. This represents a 
+qualitatively distinct form of “decoupling” of penetrant mass transport and relaxation in polymer 
+networks compared to one-component glass forming liquids, as discussed above. 
+B. Quantitative Theoretical Predictions  
+
+ 
+33 
+Theoretical results are presented in Fig.12 for the penetrant diffusion constant based on  
+𝐷p ≡
+𝑑2
+𝜏𝛼,p exp⁡(−𝐶2)/𝐶 plotted as a function of Tg(fn)/T over a wide range of crosslink fractions 
+for 𝑇g criteria of 𝜏𝛼,K(𝑇g)/𝜏0 = 1014 (main) and 105 (inset). As found for the penetrant inverse 
+alpha time in Fig.8, and as expected based on our insights that connect Tg(fn) and 𝐶 discussed in 
+section II, we predict (i) the diffusion constant also varies exponentially with 𝑇g(𝑓n) with a linearity 
+even better than that shown in Fig.8, and (ii) the slope of the logarithm of diffusion constant versus 
+𝑇g(𝑓n)/𝑇 plot decreases with heating. However, the absolute value of diffusion constants at fixed 
+Tg(fn)/T weakly decreases with temperature, opposite to our findings for inverse alpha time (see 
+Fig.8). This difference is a rather subtle 2nd order effect, but it is an objective signature of the 
+importance of mesh confinement for the diffusion constant that is amenable to experimental and 
+simulation tests. Hence, we now discuss its origin in more detail. 
+0.6
+0.7
+0.8
+0.9
+1.0
+10-12
+10-9
+10-6
+10-3
+100
+Dp=X0 /,p
+Tg(fn)/T
+           T/K
+ 273.15
+ 296.15
+ 323.15
+ 348.15
+ 400.15
+,K(Tg) /0  = 1014
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+10-9
+10-7
+10-5
+10-3
+10-1
+Dp
+Tg(fn)/T
+,K(Tg)/0  = 105
+ 
+Fig.12. Penetrant diffusion constant as a function of Tg(fn)/T at various temperatures for 
+vitrification criteria of 𝜏𝛼,K(𝑇g)/𝜏0 = 1014 (main) and 𝜏𝛼,K(𝑇g)/𝜏0 = 105 (inset). The vertical 
+line in the inset represents an estimate of a higher kinetic glass transition temperature that 
+corresponds to a simulation-like vitrification criterion. 
+Per Eq.(2), the difference between the diffusion constant and inverse alpha time is 
+proportional to exp⁡(−𝐶2)/𝐶. At fixed Tg(fn)/T, a lower T corresponds to a lower Tg(fn), fewer 
+crosslinks, a larger mesh, and hence a smaller confinement parameter C. This trend results in an 
+
+ 
+34 
+increase of the diffusion constant with cooling in the representation of Fig.12. In contrast, the 
+inverse alpha time decreases with cooling, resulting in the opposite trend with T in Fig.8.  
+The corresponding slope (proportional to an effective activation barrier) for the logarithm 
+of the diffusion constant versus 𝑇g(𝑓cross)/𝑇 based on SCCH theory is shown in Table 1. Relative 
+to its analog for the inverse penetrant alpha time, its absolute value increases significantly to a 
+degree that depends on temperature, 𝑇g criterion, and the chemistry-specific parameter A that 
+enters the quantification of the confinement parameter C. In our companion simulation paper [58], 
+we show that by using the commonly adopted 1/e decay criterion of the time correlation functions 
+employed to define the penetrant alpha relaxation time, the degree of DH is small, and it is thus 
+appropriate to quantitatively compare such simulation results to our theory.  We find from 
+simulation that the effective slope change with temperature for 
+𝑑
+𝜎 = 1⁡is significant for the inverse 
+alpha time (from 8.0 to 2.6), but moderate/minimal for the diffusion constant (from 9.5 to 6.1). In 
+the present theory, the slope change for both the inverse alpha time and diffusion constant with 
+temperature is moderate (Table 1). We suspect this difference arises from the approximate nature 
+of Eq.(2) with 𝑋(𝐶) = exp⁡(−𝐶2)/𝐶 which reflects our simple ansatz for how to combine glassy 
+activated physics and mesh confinement to predict diffusivity. Indeed, our complementary 
+simulations[58] do find modest deviations from the simple formula 𝐷p~exp⁡(−𝐶2)/𝐶𝜏α,⁡p, though 
+they decrease with cooling as the glass physics effects become relatively more important. Both 
+theory and simulation[58] find the slope change with temperature for the alpha time is significantly 
+larger than that for the diffusion constant, emphasizing the additional crosslinking effect on 
+penetrant diffusion beyond its coupling with the activated segmental relaxation process. 
+As mentioned in section VIA, if  𝐷p~exp⁡(−𝐶2)/𝐶𝜏α,⁡p is qualitatively correct then the 
+diffusion constant will decrease more rapidly with Tg(fn)/T than the penetrant alpha time increases, 
+
+ 
+35 
+opposite to “normal” Stokes-Einstein relation failure[49-54] in one-component liquids associated 
+with  DH[46-48]. However, DH in pure liquids generally results in a distribution of barriers or 
+relaxation times[71, 72], which is likely is present to some degree for the problem of penetrant 
+diffusion in glass-forming polymer networks. The pristine DH physics in pure supercooled liquids 
+is typically manifested in a decoupling relation of the form 𝐷p𝜏𝛼,p ∝ 𝜏𝛼,p
+𝛿 , where the exponent 𝛿 is 
+modestly greater than zero. Hence, there may be two competing effects of different physical origins 
+that can induce decoupling of penetrant diffusivity and relaxation, and which go in opposite 
+directions with increased supercooling. As shown in our companion simulation paper [58], if the 
+penetrant alpha time extracted from the intermediate scattering function (ISF) adopts the 1/e decay 
+criterion to define a relaxation time, then the DH effects are relatively weak as indicated by the 
+product 𝐷p𝜏𝛼,p decreasing with Tg/T due to the mesh confinement effect per Eq(2). However, if 
+the penetrant alpha time is extracted using a longer timescale criterion (the ISF decays to 0.1), then 
+larger DH effects are present resulting in our finding that the product 𝐷p𝜏𝛼,p increases with Tg/T 
+corresponding to the entropic mesh confinement effect becoming of second order importance. 
+0.0
+0.4
+0.8
+1.2
+10-10
+10-7
+10-4
+10-1
+Dp=X0 /,p
+C
+           T/K
+ 273.15
+ 296.15
+ 323.15
+ 348.15
+ 400.15
+0.0
+0.5
+1.0
+1.5
+10-9
+10-8
+10-7
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+CDp=CX0 /,p
+C2
+           T/K
+ 273.15
+ 296.15
+ 323.15
+ 348.15
+ 400.15
+ 
+Fig.13. (a) Penetrant diffusion constant 𝐷p as a function of confinement parameter C at various 
+temperatures with 𝐶 = 1.94(𝑑/𝜎)(𝑓𝑛/0.19)1/2. (b) Same display as (a) but for the square power 
+law relationship between 𝐶𝐷p and C.  
+
+ 
+36 
+Finally, we consider our diffusion constant predictions as a function of the mesh 
+confinement variable. Combining the exponential relation between log(𝜏α,p)  and 𝐶  or 𝐶2 
+predicted by SCCH theory, with an additional mesh confinement contribution per X = exp⁡(−𝑏′𝐶) 
+or exp⁡(−𝐶2)/𝐶, one deduces from Eqs.(3) or (2) a net exponential relationship of the form  
+1/𝐷p~ exp(−𝐸𝐶) or 1/𝐶𝐷p~ exp(−𝐵𝐶2), respectively. Figures 13a and 13b plots the theoretical 
+penetrant diffusion constants as a function of 𝐶 and 𝐶2, respectively, over a wide range of 
+temperatures and crosslink densities. One finds the diffusion constant data behaves nearly the same 
+as that of inverse alpha time observed in Fig.10 in the following sense: (i) the diffusion constant 
+varies exponentially with 𝐶 or 𝐶2, and the corresponding linearity is even better (slightly worse 
+for the 𝐶2 relation since the local cage barrier dominates the dynamics of penetrant while its elastic 
+barrier is negligible) than that of inverse penetrant alpha time shown in Fig.10; (ii) the effective 
+slope  (Table 1) of the logarithm of inverse diffusion constant versus 𝐶 or 𝐶2 decreases with 
+temperature; and (iii) the diffusion constants increase with temperature. SCCH theory predicts that 
+the elastic barrier will increase significantly as the degree of effective supercooling grows 
+sufficiently large (increasing penetrant size or crosslink density and/or decreasing temperature), 
+and then both the local cage and elastic barriers are crucial for penetrant hopping. In this situation, 
+the net consequence of these two barriers results in a predicted square law relation between the 
+total barrier and 𝐶, and hence one expects in practice a log(𝐶𝐷p) ~ 𝐶2 law will eventually emerge 
+in the high degree of supercooling regime. 
+We have verified that all our conclusions above deduced from numerical computations for 
+the diffusion constant remain qualitatively robust regardless of the Tg criterion and choice of the 
+prefactor A (1.94 is adopted in Figs.12 and 13) in defining confinement parameter C. 
+
+ 
+37 
+VII. Summary and Conclusions 
+ 
+We have carried out a detailed theoretical study of how permanent crosslinking in polymer 
+networks impacts the relaxation and diffusivity of dilute spherical penetrants over a wide range of 
+temperatures, crosslink densities, and Tg criteria. The key new methodological aspect is a general 
+extension of the SCCH theory of penetrant relaxation to address the dynamic consequences of 
+chemical crosslinking based on a segment pinning model where the local structure remains 
+unchanged but the microscopic dynamic constraints encountered by a penetrant from its Kuhn 
+segment neighbors is significantly intensified as crosslink density increases. This results in a rich 
+behavior of penetrant relaxation and diffusivity as a function of temperature, crosslink density, and 
+penetrant size. 
+The penetrant local cage barrier is predicted to dominate its alpha process while the elastic 
+barrier remains relatively small. The local cage barrier varies linearly with the network glass 
+transition temperature Tg, leading to an exponential increase (apparent “Arrhenius”) of the 
+isothermal penetrant alpha relaxation time with Tg(fn)/T. Numerical calculations based on the 
+ECNLE theory of polymer networks reveals a proportionality between Tg and the square root of 
+the crosslink fraction. We note that both Tg and fn are properties of the pure polymer matrix, and 
+hence the activated dynamics of a penetrant of fixed size is dominated by its coupling with the 
+polymer network alpha process. 
+Glassy segmental scale dynamics and mesh confinement glass physics are in general both 
+important for penetrant diffusivity, and their effects on the crosslink fraction dependence of the 
+activation barrier are predicted to obey a remarkable degeneracy (linear in C or linear in Tg(fn)/T) 
+due to the relation 𝐶 = 𝑑/𝑎x = 𝐴(𝑑/𝜎)𝑓n
+1/2. However, at fixed crosslink density, the penetrant 
+diffusivity is a strong function of temperature, a hallmark of the importance of glassy segmental 
+
+ 
+38 
+dynamics absents in the entropy-based mesh confinement effect. Overall, the latter plays an 
+increasing minor role compared to the consequence of crosslinking induced slowing down of 
+polymer segmental relaxation as the degree of polymer network supercooling is increased and/or 
+(we expect) when penetrants become sufficiently small compared to the mesh size. However, mesh 
+confinement does slow down mass transport to varying degrees, but seemingly not the more 
+spatially local penetrant alpha relaxation process. This results in diffusion being more strongly 
+suppressed with increasing crosslink density than penetrant alpha relaxation, a novel form of 
+“decoupling” of an opposite nature than the generic Stokes-Einstein breakdown[49-54] phenomenon 
+in glass-forming liquids.   
+The mechanistic coupling between activated dynamics of the penetrant and network Kuhn 
+segments at the correlated displacement level has been elucidated, and further quantified by the 
+decoupling time ratio defined as the ratio of the penetrant to Kuhn segment alpha relaxation times. 
+This decoupling ratio varies non-monotonically as the degree of effective supercooling degree 
+grows (increasing Tg(fn)/T via lowering temperature or increasing crosslink density), and sharply 
+decreases in the deeply supercooled regime. The latter behavior arises because of the relative 
+unimportance of the elastic barrier for penetrant hopping versus its high importance for Kuhn 
+segment relaxation in fragile polymer networks. A remarkably good (but not perfect) collapse of 
+the decoupling ratio as a function of Tg(fn)/T for all crosslink densities and temperatures studied is 
+predicted, independent of the vitrification timescale adopted to define the glass transition 
+temperature and the precise value of the polymer network strand persistence length. 
+Overall, we believe that the generic theoretical insights and specific predictions obtained 
+will be useful for designing new polymer networks to optimize the absolute and relative (e.g., for 
+selectivity applications) penetrant diffusion rates relevant to polymer-based membrane 
+
+ 
+39 
+separations. Extensions of the theory to other penetrant sizes, inclusion of penetrant-polymer 
+attractions, under pressure, and in the glassy state (thermosets) and vitrimers[73-76] is possible and 
+will be studied in future work.   
+Acknowledgement 
+This research was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, 
+Division of Materials Sciences and Engineering (Award No. DE-SC0020858), through the 
+Materials Research Laboratory at the University of Illinois at Urbana-Champaign. Helpful 
+discussions with Christopher Evans are gratefully acknowledged. 
+ 
+Appendix: Effect of Persistence Length Choice  
+ 
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+10-9
+10-7
+10-5
+10-3
+10-1
+0 /,p
+Tg(fn)/T
+TH,4/3
+           T/K
+ 273.15
+ 296.15
+ 323.15
+ 348.15
+ 400.15
+,K(Tg) /0  = 1014
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+10-8
+10-6
+10-4
+10-2
+0 /,p
+Tg(fn)/T
+,K(Tg) /0  = 105
+0.6
+0.7
+0.8
+0.9
+1.0
+10-15
+10-12
+10-9
+10-6
+10-3
+100
+Dp=X0 /,p
+Tg(fn)/T
+TH, 4/3
+           T/K
+ 273.15
+ 296.15
+ 323.15
+ 348.15
+ 400.15
+,K(Tg) /0  = 1014
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+10-10
+10-7
+10-4
+10-1
+Dp
+Tg(fn)/T
+,K(Tg)/0  = 105
+ 
+Fig.S1. Penetrant (a) inverse mean alpha time and (b) diffusion constant as a function of Tg(fn)/T 
+at various temperatures using a 𝜏𝛼,K(𝑇g)/τ0 = 1014 criterion. Same displays as that of the main 
+text Fig.8 and Fig.12, respectively, but using the 𝑙p/𝜎 = 4/3 local aspect ratio model. Both insets 
+are the corresponding results for a simulation-like 𝜏𝛼,K(𝑇g)/τ0 = 105 vitrification criterion. 
+ 
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf,len=1574
+page_content='1  Self-Consistent Hopping Theory of Activated Relaxation and Diffusion of  Dilute Penetrants in Dense Crosslinked Polymer Networks  Baicheng Mei a,d, Tsai-Wei Lin c,d, Charles E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Sing a,c,d  and Kenneth S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Schweizer *a,b,c,d  a Department of Materials Science, University of Illinois, Urbana, IL 61801  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Department of Chemistry, University of Illinois, Urbana, IL 61801  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Department of Chemical & Biomolecular Engineering, University of Illinois, Urbana, IL 61801  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Materials Research Laboratory, University of Illinois, Urbana, IL 61801  kschweiz@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='edu  2  Abstract  We generalize and apply a microscopic force-level statistical mechanical theory of the activated  dynamics of dilute spherical penetrants in glass-forming liquids to study the influence of  permanent crosslinking in polymer networks on the relaxation time and diffusivity over a wide  range of temperature and crosslink density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The theory treats network crosslinkers as locally  pinned or vibrating sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Calculations are performed for model parameters relevant to recent  experimental studies of a nm-sized organic dye diffusing in crosslinked poly(n-butyl methacrylate)  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The theory predicts the penetrant alpha time increases exponentially with the crosslink  fraction (fn) dependent glass transition temperature, Tg, which grows roughly linearly with the  square root of crosslink density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Moreover, we find that Tg is also proportional to an entropic  geometric confinement parameter defined as the ratio of the penetrant diameter to the mean mesh  size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The decoupling of the ratio of the penetrant and Kuhn segment alpha relaxation times displays  a complex non-monotonic dependence on crosslink density and temperature via the variable  Tg(fn)/T, but a remarkably good collapse is predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The microscopic mechanism for activated  penetrant relaxation is elucidated based on the theoretically predicted crosslink fraction dependent  coupled local cage and nonlocal collective elastic barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' A model for the penetrant diffusion  constant that combines glassy physics and entropic mesh confinement is proposed which results  in a significantly stronger suppression of mass transport with degree of effective supercooling than  predicted for the penetrant alpha relaxation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' This represents a unique polymer network-based  type of “decoupling” of diffusion and relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' In contrast to the diffusion of larger nanoparticles  in high temperature rubbery polymer networks, our analysis suggests that for the penetrants studied  the mesh confinement effects are of secondary importance in the highly supercooled regimes of  interest relative to the consequences on mass transport of crosslink-induced slowing down of  activated segmental relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Support for the theoretical predictions from our complementary  simulation study is briefly discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  3  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Introduction  Understanding and controlling molecular (“penetrant”) transport in dense and cold polymer  melts, crosslinked networks, and glasses is a problem of high basic polymer physics interest[1-14]  that also underpins advanced separation membrane applications[10-27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' However, even in the dilute  penetrant limit of interest in this work, existing theoretical models are highly phenomenological,  often built on the difficult to uniquely define and unambiguously predict concept of “free  volume”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' [13, 27-33]  Recently, two of us have pursued the construction of a statistical mechanical theory (the  Self Consistent Cooperative Hopping (SCCH) theory) for penetrant activated dynamics in polymer  melts [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The foundation is the Elastically Collective Nonlinear Langevin Equation (ECNLE)  theory[35, 36] for the thermally activated structural (alpha) relaxation process on the Kuhn segment  scale in one-component melts[37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' This approach is microscopic in the sense that it causally relates  pair interactions, thermodynamic state, and packing structure as embedded in a dynamic free  energy which quantifies kinetic constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The alpha relaxation is a coupled local-nonlocal  activated process characterized by a local cage barrier (FB,K) associated with large-amplitude  activated hopping coupled with collective long-range (nonlocal) small displacements of all  segments outside the cage characterized by an elastic barrier (Fel,K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Recent studies support the  predictions of this approach for a large number of real-world polymeric liquids, including olefins,  dienes, siloxanes, and vinyl polymer chemistries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' [38-40] Chemical crosslinking introduces new  kinetic constraints and additional dynamical slowing down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' ECNLE theory has been very recently  extended to polymer networks and shown to provide an excellent understanding of experiments  on poly(n-butylacrylate) (PnBA) networks and coarse-grained simulations over a wide range of  crosslink densities and temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' [41]  4  SCCH theory for dilute hard sphere penetrants (diameter, d) in a hard-sphere matrix [42, 43]  or semi-flexible polymer melts [34] under structurally equilibrated conditions has been extensively  applied, and many of its predictions agree well with simulations [7, 42] and experiments [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The  penetrant activation barrier and mean hopping time, and the extent of coupling of penetrant  transport with the early, medium, and late stages of the matrix structural relaxation process, are  self-consistently predicted based on two coupled dynamic free energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' [34, 42, 43, 45]  The goal of the present work is to formulate and apply SCCH theory to permanent  crosslinked polymer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' We are especially motivated by recent measurements of Evans and  coworkers on the diffusion of a dilute aromatic molecule [N,N’-Bis(2,5-di-tert-butylphenyl)-  3,4,9,10-perylenedicarboximide (BTBP)] in crosslinked PnBA networks over a wide range of  crosslink densities at fixed temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' [9] The experiments found that the penetrant alpha time  and diffusion constant both vary exponentially with the crosslink fraction (𝑓n) dependent network  glass transition temperature 𝑇g(𝑓n) , but their quantitative dependences on 𝑇g(𝑓n)  and T are  different, a form of “decoupling” in the language of glass physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The precise physical  mechanisms at play remain unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' For example, is the apparent decoupling a consequence of  entropic mesh confinement constraints on penetrant motion in crosslinked networks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Alternatively, is the glass physics mechanism associated with the dynamic heterogeneity (DH)[46-  48] driven breakdown of the Stokes-Einstein breakdown relation [49-54] in supercooled liquids  relevant?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Or is there another explanation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  The mesh confinement effect has been studied theoretically[55], experimentally[9], and with  simulation[56, 57] for the problem of the motion of a dilute spherical nanoparticle (larger than typical  molecules) in polymer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' A key variable is the confinement parameter,  𝐶 = 𝑑/𝑎x, where  𝑎x is the mean mesh size that decreases with crosslink density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' the polymer physics  5  model of Cai-Panykov-Rubinstein (CPR) [55] predicts that when mesh confinement is strong  enough (𝐶>1) the nanoparticle diffusion constant is:  𝐷p ∝  𝑑2  𝜏R exp\u2061(−𝐶2)/𝐶 ≡  𝑑2  𝜏R 𝑋                                                         (1)  where 𝜏R~𝑁x  2𝜏𝛼 is the Rouse time of a network strand composed of 𝑁x segments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' the segmental  alpha time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' 𝜏𝛼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' depends on temperature but is assumed to not vary with crosslink density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' and the  quantity 𝑋(𝐶) is defined which quantifies the explicit entropic effect of a physical mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Such a  model is most germane to the high temperature rubbery regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Recently, a simulation study of  such a semi-dilute solution model in the high temperature rubbery regime has been performed [57],  and finds log\u2061(1/𝐶𝐷p) ∝ 𝐶2 for both C > 1 and C < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' By revisiting their data, we find the  prefactor between log\u2061(1/𝐶𝐷p) and 𝐶2 is significantly larger than log(e) in the CPR entropic mesh  model, suggesting the kinetic factor  𝑑2  𝜏R  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' (1) must also contribute significantly to the crosslink  density dependence of the penetrant diffusion constant and may also vary exponentially with  −𝐶2  for unknown reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Qualitatively, this is what we expect within the SCCH theory of penetrant  diffusivity given our recent ECNLE theory of segmental relaxation in polymer networks (and the  complementary experiment and simulations) found the alpha time of polymer Kuhn segments  increases strongly with crosslink fraction[41] which must have a significant impact on the penetrant  alpha relaxation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Given the above, in this article we propose and test the idea that the penetrant diffusion  constant in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' (1) should scale inversely with the activated “penetrant alpha time” yielding:  𝐷p ∝  𝑑2  𝜏𝛼,p exp\u2061(−𝐶2)/𝐶 ≡  𝑑2  𝜏𝛼,p 𝑋 ∝ exp\u2061(−𝐵𝐶2)/𝐶                                 (2)  where 𝜏𝛼,p is both temperature and crosslink density dependent, and B is a chemistry specific  numerical factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' We note that since in most real world situations the molecular penetrants do not  6  obey the strong mesh confinement inequality C>1, other workers [56] have considered, and  explored with simulation, an alternative model of the effect of the mesh confinement  corresponding to a penetrant entropic barrier scaling linearly with the confinement parameter,  𝐷p ∝  𝑑2  𝜏R\u2061 exp(−𝑏′𝐶).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' If one adopts this description of mesh confinement effects, then our proposal  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' (2) becomes:  𝐷p ∝  𝑑2  𝜏𝛼,p\u2061 exp(−𝑏′𝐶) ∝ exp(−𝐸𝐶)                                                  (3)  Here, and in our companion simulation paper [58], we adopt SCCH theory to quantitatively  analyze the effect of crosslinking effect on the penetrant alpha relaxation time and diffusion  constant in the specific context of the well-studied PnBA networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' In general, the penetrant alpha  relaxation time and diffusion constant are high-dimensional functions, depending on variables such  as temperature, penetrant chemistry (size and shape), penetrant-polymer interaction, polymer  persistence or Kuhn length, and degree of crosslinking (or mesh size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' To render the analysis  tractable, we consider a single spherical penetrant of diameter that mimics BTBP [9, 59] (having the  same volume, 735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='88 Å3, corresponding to d=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='12nm) and a purely repulsive penetrant-polymer  interaction (reasonable for the nonpolar dye molecule) and a polymer model calibrated for PnBA  networks, and focus on how temperature, crosslink density, and mesh size impact 𝜏𝛼,p and 𝐷p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' This  choice of penetrant size and the range of crosslink fractions studied allows us to probe confinement  parameters from well below unity to modestly above unity, as germane to the typical molecular  penetrant problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' When C>>1, penetrant motion in the melt will be slaved to the polymer  segmental relaxation time, and in permanent networks will be effectively quenched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Section II briefly describes ECNLE theory and its extension to crosslinked polymer  networks, and presents some new calculations relevant to the penetrant work in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' SCCH  theory is briefly reviewed in section III, and the calculation of foundational theoretical quantities  7  required to predict penetrant alpha relaxation time and diffusion constant are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The  penetrant relaxation time and its relationship to the network glass transition temperature and mesh  confinement parameter are systematically studied in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The question of the degree of  decoupling between the penetrant and polymer segmental relaxation times is analyzed in section  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  The role of crosslinking on the penetrant diffusion constant is studied in section VI, and the  effects of crosslink-induced mesh confinement versus prolongation of network segmental  relaxation on penetrant transports are disentangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The article concludes in section VII with a  summary and future outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Throughout the paper we comment on the comparison of our  theoretical results with our companion simulation study at the level of qualitative and semi-  quantitative trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Finally, we emphasize that nearly all the theoretical technical details, equations,  derivations, numerical implementations, and discussion of limitations of the dynamic theories  (ECNLE[35-37, 41] and SCCH[34, 42, 43, 45]) at the Kuhn scale and the Polymer Reference Interaction  Site Model (PRISM) integral equation theory[34, 37, 41] for the required structural pair correlation  functions have been thoroughly documented in the literature and are not repeated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The present  paper is the first to include crosslinks in SCCH theory, but its analog for polymer melts has been  discussed in detail [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Thus, our review of ECNLE theory in section II and SCCH theory in  section III focuses on their physical content and key parameters, with mathematical development  presented only for the new aspects of SCCH theory required to treat crosslinked networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' ECNLE Theory of Structural Relaxation in Crosslinked Polymer Networks  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Theoretical Background for Homopolymer Melts and Crosslinked Networks  The foundational starting point for predicting the activated dynamics of one-component  spherical particle fluids is the single particle scalar displacement (𝑟) - dependent dynamic free  8  energy, 𝐹dyn(𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The negative gradient of this quantity determines an effective force in a stochastic  nonlinear Langevin equation that controls the spatially local aspect of activated barrier hopping  trajectories and a local cage barrier (right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' [60] At sufficiently high (low) packing  fraction (temperature), a particle jump distance is predicted to be sufficiently large (  Kr  \uf044  )(see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='1)  that in order for it to occur all the particles outside the cage must elastically displace in a dilational  (cage expansion) collective manner by a small amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' This cage expansion sets the amplitude of  the collective elastic displacement field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' [35, 36] and costs elastic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The resultant physical  picture is that the alpha relaxation is a coupled local-nonlocal in space process (see middle panel  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='1), characterized by local cage and longer-range collective elastic barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Recently, we extended ECNLE theory to homopolymer melts including the consequences  of local chain connectivity and backbone stiffness on packing structure and dynamics [37, 41, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Motivated by the theoretical arguments of Zhou and two of us [37], the Kuhn segment is adopted  as the relevant dynamical length scale for the alpha process, and thus the scalar displacement of  the rigid Kuhn segment (consisting of several bonded elementary sites) center-of-mass from its  initial position, rK(t), is the central dynamical variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The Kuhn segment is characterized by an  effective dynamically correlated number of beads, 𝑁K (see left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='1) in a semiflexible  Koyama model[62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' For typical polymers, 𝑁K ~2-3, and in the present work we set 𝑁K = 2𝑙p  with 𝑙p the persistence length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The sensibility of this model construction has been quantitatively  tested in prior combined simulation-theory work [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  9  σ  Bare bead  Kuhn bead  Polymer network  pinned  unpinned  dyn,K  K  (  )  F  r  \uf062  B,K  F  \uf062  Kuhn Bead  B,K  r  loc,K  r  Jump distance  Kr  \uf073  Kr  \uf073  \uf044  increasing fn  Schematic of ECNLE theory  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Left panel: Schematic of coarse-graining a polymer melt to the Kuhn scale and introduction  of crosslinks based on pinning interaction sites in a regular manner along the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Middle panel:  Schematic of the physical ideas of local-nonlocal relaxation mechanism within ECNLE theory  shown for simplicity for a spherical particle liquid: activated particle hopping involves a local cage  barrier and a longer-range cooperative elastic displacement of all particles outside the cage which  induces an additional nonlocal elastic barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Right panel: Schematic of key features of the  dynamic free energy as a function of dimensionless displacement of a Kuhn segment and its  evolution with increasing crosslink fraction, fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The localization length rloc,K decreases with fn,  while the local cage barrier, barrier location rB,K, and jump distance ΔrK= rB,K-rloc,K all increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  The local cage barrier 𝐹B,K is obtained directly from the dynamic free energy which is  constructed solely from knowledge of the inter- and intra- molecular equilibrium pair correlation  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The elastic barrier has been derived to be:  3  2  el,K  cage,K  eff,K  K  0,K  2  F  r  r  K  \uf062  \uf070  \uf072  \uf0bb  \uf044  , where K0,K is  the harmonic curvature or spring constant at the localization length and the angularly averaged  effective cage dilation amplitude is  2  eff,K  K  cage,K  3  32  r  r  r  \uf044  \uf0bb \uf044  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' both these quantities are computed  from the dynamic free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The Kuhn scale “cage radius” is  1/3  cage,K  K  cage,m  r  N  r  =  with cage,m  r  being  the location of the first minimum of polymer matrix radial distribution function 𝑔mm(𝑟) at the  elementary interaction site level and 𝜌K = 𝜌/𝑁K the number density of Kuhn segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Chemical crosslinking is modeled as follows [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' First, it is assumed to not change the site-  site intramolecular and intermolecular structural pair correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' This simplification is consistent  with a recent simulation study [64] that found the dimensionless compressibility of crosslink  networks remains essentially unchanged compared to the melt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Second, a “neutral confinement”  model of partial random pinning of particles (as widely studied in the glassy dynamics area) is  10  adopted which in the polymer network context mimics chemical crosslinking as a fraction (fn) of  dynamically immobile segments (zero localization length) distributed in a regular manner along a  chain, per the schematic in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The ratio 𝑛crosslink/(𝑛crosslink + 𝑛monomer) defines the crosslink  fraction fn, where 𝑛crosslink and 𝑛monomer are the site (bare bead) level number of crosslinkers and  normal mobile beads, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Determination of the corresponding dynamic partial structure  factors which enter as collective Debye-Waller factors [41, 61, 65] in the construction of the dynamic  free energy is identical to prior work [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Crosslinkers modeled as pinned sites modify all aspects  of ECNLE theory via the dynamic free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  The mean barrier hopping time 𝜏hop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='K is then computed based on the Kramers mean first  passage time theory as [41,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' 42,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' 66,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' 67]  loc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='K  K  el,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='K  K  dyn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='K  K  dyn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='K  K  loc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='K  loc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='K  (  )  (  )  s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='K  hop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='K  K  K  2  K  2  r  r  F  r  F  r  F  r  r  r  e  dr e  dr e  \uf062  \uf062  \uf062  \uf074  \uf074  \uf073  +\uf044  \uf0a2  −  \uf0a2  =  \uf0f2  \uf0f2  (4)  where  2  s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='K  K  s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='K  D  \uf074  \uf073  =  is a characteristic very short-time scale unaffected by crosslinking for a  Kuhn unit (  1/3  K  K  N  \uf073  \uf073  =  ) and the formula for 𝐷s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='K  is given elsewhere [35,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' 41,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Using the PRISM  theory of equilibrium structural pair correlation functions as input to the dynamic free energy and  hence Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' (4), the mean Kuhn segment alpha relaxation time is computed numerically as  ,K  s,K  hop,K  \uf061  \uf074  \uf074  \uf074  =  +  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' To confront our theoretical results with experimental or simulation results, we  adopt the elementary timescale of a Newtonian liquid, τ0=16ϕσ(βM/π)1/2 as the time unit, which is  typically ~ 1 ps, per prior studies [35, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Mapping to Thermal Polymer Melts and Networks  To determine the required equilibrium site-site pair correlation functions, PRISM integral  equation theory with the Percus-Yevick (PY) closure is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' [41, 61] The penetrant and  polymer interactions sites are modeled as hard spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The intramolecular structure factor of an  11  ideal semiflexible chain 𝜔mm(𝑞), penetrant-to-matrix size ratio 𝑑/𝜎, and polymer site reduced  density or packing fraction enter as inputs to PRISM theory, and 𝜔mm(𝑞) is determined using the  semiflexible tangent discrete Koyama model[41, 62, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' This single chain statistical model is  characterized by the bond length 𝑙b, bead diameter 𝑙d, and persistence length 𝑙p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' we adopt its  “tangent” version corresponding to 𝑙b = 𝑙d = 𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' In our recent work on pure polymer network  relaxation, a generic flexible chain persistence length of 𝑙p =  4  3 𝜎 was adopted [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' In the present  work, for the PnBA network the Kuhn length and monomer effective diameter are evaluated from  experimental information to be 𝑙K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='72nm and 𝑙b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='72nm, respectively, corresponding to a  local aspect ratio 𝑙K/𝑙b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='4, or in terms of the persistence length  𝑙p/𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Since in the tangent  Koyama model one has  𝑙K  𝜎 =  2𝑙p  𝜎 − 1, the adopted mapping yields 𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='23nm, which is slightly  larger than 𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='03nm if 𝑙p =  4  3 𝜎 is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The core theoretical predictions for the 𝑙p/𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2  and 4/3 models are very similar (see Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' In the main text we adopt the 𝑙p/𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  To address experimental systems under isobaric (1 atm) conditions where temperature is  the control variable, we adopt the well developed and tested mapping approach that relates in a  chemistry specific manner the polymer packing fraction to temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' [39-41] Specifically, the  dimensionless compressibility, 𝑆0(𝑇), from PRISM theory for the semiflexible Koyama chain  model[41, 62, 67]  is equated to that obtained from experimental equation of state (EOS) data[44, 68]  for PnBA melts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' [39-41] The result of this mapping is shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' A nearly linear  dependence is valid from the very high temperature regime (> 400K) through the modestly  supercooled regime, down to the glass transition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The main frame of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2 shows S0  can be accurately represented by the analytic formula 𝑆0  −1 = 𝑁s(𝐵/𝑇 − 𝐴′)2 derived from the van  der Waals model[69] EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' As discussed previously[38, 69, 70], reasonable variation of Ns does not  12  affect our analysis since the crucial aspect is the T-dependence of S0 that is determined by the EOS  parameters 𝐴′  and B, and Ns enters only as a T-independent prefactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The final step of our model  construction is to set Ns = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='75 to reproduce the experimental glass transition temperature of 𝑇g =  45 °C of pure PnBA melts [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='7  4  6  8  10  S-1/2  0  Tg/T  expt  S-1/2  0  = 3(1141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='53K/T-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='5271)  200  300  400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='68  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='72  T/K  feff  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Main: Experimental dimensionless compressibility data of PnBA melt versus scaled inverse  temperature in a representation suggested by the van der Waals model [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' One sees the data agrees  very well with expectations that  𝑆0  −1/2 is a linear function of Tg/T in the range Tg/T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='46-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  The red line corresponds to the van der Waals model analytic representation [69] 𝑆0  −1 =  9(1141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='53K/𝑇 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='5271)2, which is employed in the main text to map the hard polymer chain  model to its thermal PnBA analog at 1 atm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Inset: Relationship between effective packing fraction  from PRISM theory and temperature for PnBA deduced from the mapping procedure [39-41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Dynamic Relaxation of Kuhn Segments and Glass Transition Temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2  Tg(fn)/Tg(0)  f 1/2  n  Th: \uf074\uf061,K(Tg)/\uf0740  1014  109  105  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='7  f 1/2  cross  Expt  Thermo  \uf074\uf061(Tg)=10-4s  Sim  Thermo  \uf074\uf061(Tg)=105\uf074 *  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  240  280  320  Tg(fn)/K  a-1  x =C/d [nm-1]  Expt:  Thermo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  \uf074\uf061,K(Tg)=108 ps  Sim:  Thermo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  \uf074\uf061,K(Tg) /\uf074 *=105  TH \uf074\uf061,K(Tg) /\uf0740  1014  108  105  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' (a) Comparisons between scaled glass transition temperatures from experiment, simulation,  and theory (𝑙p/𝜎 =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2 model) extracted and defined in two ways as indicated (dynamic and  13  thermodynamic), Tg(fn)/Tg(0), plotted as a function of the square root of crosslink fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The  experimental and simulated crosslink fractions are taken to be exactly the same and the theory  analog obeys fn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='19fcross [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' (b) Glass transition temperature 𝑇g(𝑓n) in Kelvin as a function of  inverse mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Experimental and simulation mesh sizes are directly measured where the bond  length 𝑙b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='72nm was used in simulation as the length unit [41] while in the theory analysis the  mesh size is calculated using 𝑎x  −1 = 𝐴(𝑓n/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='19)1/2/𝜎 with A=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='94 (see main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  We first consider how the degree of crosslinking affects the glass transition temperature  𝑇g(𝑓n) defined in the theory as when Kuhn segment alpha time reaches a fixed value relevant to a  specific experiment or simulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' here we employ 𝜏𝛼,K(𝑇g)/𝜏0 = 1014, 108, and 105 to illustrate  our predictions based on a vitrification criterion in the laboratory deeply supercooled, intermediate  supercooled, and weakly supercooled (per simulations) regimes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' This issue was  studied in our previous publication for the 𝑙p/𝜎 = 4/3 Koyama model where we found 𝑇g(𝑓n)  grows non-linearly with 𝑓n, but no quantitative relationship was identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' [41] Here, as shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='3a, we report a new discovery of a square-root relationship, 𝑇g(𝑓n) ∝ 𝑓n  1/2, with only small  deviations evident in the very low crosslink density regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Adoption of different 𝑇g criteria do  not change the robustness of this relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='3a also shows that the previously obtained  experimental and simulation 𝑇g(𝑓cross) data also obey well the relation 𝑇g(𝑓cross) ∝ 𝑓cross  1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' We  note that in our prior joint experiment-simulation-theory study for PnBA based on the 𝑙p/𝜎 = 4/3  model, the experimental crosslink fraction was given by 𝑓cross = 𝑓n/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' [41] This prefactor of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='19 remains unchanged for the present 𝑙p/𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2 model, and the numerical scaling factor likely  reflects the difference between our model construction based on the neutral pinning idea of coarse-  grained polymer model and real crosslinking of chemically complex polymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  In polymer networks, an alternative variable to the crosslink fraction 𝑓n  is the mean  geometric mesh size, 𝑎x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' In our theory, this quantity does not play any direct role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' However, based  on its classic definition in experiment and simulation, 𝑎x is proportional to the square root of the  14  average number of bare beads between two neighboring crosslinks 𝑁x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=', 𝑎x ∝ 𝜎𝑁x  1/2, where  𝑓cross = 1/𝑁x ≡ 𝑛crosslink/(𝑛crosslink + 𝑛monomer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' As such, one has 𝑎x ∝ 𝜎/𝑓cross  1/2 , and we quantify  this connection as  𝑎x = 𝜎/𝐴𝑓cross  1/2 = 𝜎/𝐴(𝑓n/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='19)1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Since 𝐴/𝜎 = 1/(𝑎x𝑓cross  1/2 ), one can  compute 𝐴 using purely experimental results for 𝑎x and 𝑓cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' However, to be consistent with a  previous simulation of penetrant diffusion in semi-dilute solutions based in a coarse grained  polymer bead model (as we also employ) [57] where 𝐴 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='94 was adopted to achieve the best  agreement of simulation data with the formula log\u2061(𝐷p)~(𝑑/𝑎x)2, we a priori fix 𝐴 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' To  check the sensitivity of our theoretical results concerning this choice, we demonstrate below that  using 𝐴 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0 does not modify any of our qualitative findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Based on 𝑎x = 𝜎/𝐴(𝑓n/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='19)1/2 ∝ 1/𝑓n  1/2 and our above discovery that 𝑇g(𝑓n) ∝ 𝑓n  1/2,  we deduce that 𝑇g(𝑓n) and 𝑎x  −1(𝑓n) are proportional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' In experiment and simulation, the average  mesh size 𝑎x can be directly measured [9] instead of using 𝑎x = 𝜎/𝐴𝑓cross  1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Here based on previous  experimental results of PnBA for mesh size 𝑎x [9] and glass transition temperature 𝑇g(𝑓cross) [9,  41], and our published 𝑇g(𝑓cross) [41] and 𝑎x estimates from simulations [58], we test the linearity  between 𝑇g(𝑓n) and 𝑎x  −1(𝑓n) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' We find excellent agreement over the entire range of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  We now consider the predictions of ECNLE theory for the polymer network alpha  relaxation time for a choice of aspect ratio (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2) modestly different than in prior work [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Figure  4a shows the foundational results for the nonlocal collective elastic (solid) and local cage (open)  barriers as a function of Tg(fn)/T over a wide range of crosslink densities (covering all ranges that  was achieved in experiments [9]) at various fixed temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The theoretical results are plotted  based on two very different dynamic Tg criteria appropriate for the deeply and weakly supercooled  regimes in the main frame and inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='4a, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Different choices of vitrification  15  criterion do not, at zeroth order, change the qualitative finding of a good collapse and shape of the  master curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Of course, quantitatively the barriers cannot be identical, and must be larger at the  same Tg(fn)/T based on defining glass formation with a longer alpha time criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' For the classic  laboratory criterion that the network vitrifies when the alpha time is 100s, the elastic barrier results  as a function of Tg(fn)/T collapse very well over the entire Tg/T range for all temperatures that  corresponds to ~16 decades growth of the alpha time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The local cage barriers at different  temperatures do not collapse as well, although the deviations are not large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Moreover, we find the  local cage barrier grows linearly with Tg(fn)/T for each fixed temperature, while the elastic barrier  grows in a non-linear manner, a feature that has been discussed in great depth previously [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  0  4  8  12  16  20  24  \uf062FB(el),K  Tg(fn)/T  \uf074\uf061,K(Tg) /\uf0740  = 1014  FB     Fel      T/K  |  273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  |  296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  |  323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  |  348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  |  400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2  0  4  8  12  16  \uf062FB(el),K  Tg(fn)/T  \uf074\uf061,K(Tg) /\uf0740  = 105  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  101  104  107  1010  1013  1016  \uf074\uf061,K /\uf0740  Tg/T  SIM  EXPT  TH  |  |  |  |  |  |  |  |  |  increase  fcross  TH  EXPT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  100  104  108  1012  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' (a) Kuhn segment local cage and collective elastic barriers and (b) mean alpha relaxation  time based on different Tg definition criteria as a function of Tg/T over a wide range of crosslink  densities for the 𝑙p/𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' In (b) the experiment and simulation results are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  The Tg criteria in the main frame and inset of (a) are 𝜏𝛼,K(𝑇g)/𝜏0 = 1014 and 105 (using an  elementary time scale 𝜏0~1ps per typical viscous liquids), respectively, while for the main frame  of (b) they are 𝜏𝛼,K(𝑇g) = 108ps, 𝜏̃𝛼,K(𝑇g) = 105 corresponding to ~320ns, and 𝜏𝛼,K(𝑇g)/𝜏0 =  1014, for experiment, simulation, and theory, respectively, and for the inset of (b) they are the  same for both experiment and theory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=', 𝜏𝛼,K(𝑇g) = 108ps or 108𝜏0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Note the variable Tg/T in  (a) is crosslink fraction dependent at fixed temperature (T/K = 273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15, 296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15, 323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15, 348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15,  400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15, respectively), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=', Tg(fn), while that in (b) is temperature at fixed crosslink fraction (fn = 0,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='04, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='1, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The vertical line in the inset of (a) or (b) represents an estimate  of a higher kinetic glass transition temperature, which is far higher than the normal glass transition  temperature deduced thermodynamically or dynamically in experiment of ~100 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  16  Given the predicted collapse behavior for the elastic barriers and near collapse of the local  cage barriers for different temperatures and crosslink densities, a nontrivial collapse of the total  barrier and hence the Kuhn scale alpha times for all temperatures and crosslink densities is  expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Figure 4b verifies this is the case for the 𝑙p/𝜎 = \u20611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' This behavior provides a  theoretical basis for the collapsed Angell plots observed in experiment and simulation, as shown  in the main frame of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Finally, we emphasize that the theoretical results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='4 provide  foundational input to the SCCH theory since the penetrant dynamic free energy is coupled with  the polymer network dynamic free energy [34, 42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' SCCH Theory in Crosslinked Polymer Networks  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Theoretical Background for Penetrant Dynamics  We now briefly recall the key physical elements of the SCCH theory[34, 42, 43, 45] for the  mean penetrant hopping time which describes an activated process (see the corresponding  penetrant dynamic free energy in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='5) which is self-consistently coupled with  an activated dynamic “facilitation displacement” of polymer segments determined by the Kuhn  segment dynamic free energy (see the middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Crucially, the penetrant dynamic  free energy depends on both its displacement and that of the Kuhn segment (  pr  and  Kr ,  respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The mathematical formulation of SCCH theory has been discussed in great detail  previously for a dilute penetrant in a hard sphere fluid[42, 43, 45] and in a polymer melt[34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The  derivative of the penetrant dynamic free energy in polymer melts is given as [34]:  2 2  2 2  p  K  K  mm  6  dyn,p  p  K  p  ( ) 6  ( )  2  2  mp  mm  3  p  p  ( ,  )  3  ( )  ( )  3  (2 )  q r  q r  q  S  q  F  r r  r  d  q  C  q S  q e  e  r  r  \uf077  \uf062  \uf072  \uf070  −  −  \uf0b6  = −  +  \uf0b6  \uf0f2  q  (5)  17  where 𝜌 is the number density of matrix interaction sites or beads (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='1), Cmp(q) is the Fourier  transform of the penetrant-polymer site-site direct correlation function, Smm(q) is the Fourier space  static collective structure factor of the polymer network, and ωK(q) is the intramolecular structure  factor of the elementary Kuhn segment that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' (5) (it can be set to unity without significantly  changing the numerical predictions [34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The subscript “m” represents the elementary bead or site  level (see the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='1) description of the polymer and is employed to distinguish it from  the Kuhn segment scale (denoted by the subscript “K”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Based on the pinned bead model of  polymer crosslinks[41], Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' (5) for polymer melt is modified as:  \uf05b  \uf05d  2 2  p 6  dyn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='p  p  K  p  2  2  mp  3  p  p  uu  Ku  nn  Ku  un  Ku  nu  Ku  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  )  3  ( )  3  (2 )  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  )  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  )  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  )  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  )  q r  F  r r  r  d  q C  q e  r  r  S  q r  S  q r  S  q r  S  q r  \uf062  \uf070  −  \uf0b6  = −  +  \uf0b6  \uf0b4  +  +  +  \uf0f2  q  (6)  The presence of pinned segments enters via the form of the collective partial Debye-Waller factors  of the polymer matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Sαβ(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='rKu),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' which differs from a melt of fully mobile segments as previously  derived and discussed [].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' We additionally note that the subscript Ku refers to the mobile Kuhn  segments, and the subscripts u and n indicate unpinned and pinned sites, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  To render SCCH theory predictive and tractable, the degree of Kuhn segment dynamic  displacement that facilitates the penetrant hopping event is described by introducing a single  trajectory coupling variable, γ, defined as:  Ku  loc,Ku  p  loc,p  ( )  (  )  r  r  r  r  \uf067  \uf067  =  +  −  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' [34] The “pinned” Kuhn  segments (crosslinkers) are also allowed to vibrate locally with an amplitude set by the predicted  localization length of unpinned Kuhn segments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=', loc,Kn  loc,Ku  r  r  =  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The parameter γ is determined  based on enforcing a temporal self-consistency condition[34, 42, 43]:  hop,p  dis,K  Ku,c  ( )  (  ( ))  r  \uf074  \uf067  \uf074  \uf067  =  \uf044  ,  where  Ku,c  p  ( )  ( )  r  r  \uf067  \uf067  \uf067  \uf044  = \uf044  and both  hop,p( )  \uf074  \uf067  and  dis,K  K,c  (  ( ))  r  \uf074  \uf067  \uf044  are computed using Kramers  theory [66, 67] as sketched in section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Solving the self-consistency condition numerically yields γ  18  which depends on all control parameters of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Using it, one can then compute the  penetrant alpha time as  ,p  s,p  hop,p  \uf061  \uf074  \uf074  \uf074  =  +  , where  2  s,p  s,p  d  D  \uf074  =  is the characteristic elementary  short-time scale for the penetrant given elsewhere [34, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' As in ECNLE theory of one-component  liquids, at high enough matrix packing fractions the penetrant jump length can become (depending  on penetrant size) sufficiently large that a collective elastic distortion of matrix particles outside  the cage region is required to allow the penetrant to hop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' [34, 42, 43] This physics enters via a common  elastic barrier (  el,p  el,Ku,c  F  F  =  ) in the computation of the timescales  hop,p( )  \uf074  \uf067  and  dis,K  Ku,c  (  ( ))  r  \uf074  \uf067  \uf044  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  The elastic barrier is determined following the same ideas and methods employed for Kuhn  segment in the pure polymer system as discussed above and elsewhere,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' with the result previously  derived  as:  3  2  el,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='p  n  cage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='p  eff,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='p  K  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='K  2 (1  )  F  f r  r  K  \uf062  \uf070  \uf072  \uf0bb  −  \uf044  where  2  eff,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='p  Ku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='c  cage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='p  ( )  3  ( ) 32  r  r  r  \uf067  \uf067  \uf044  = \uf044  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  1/3  cage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='p  min,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='mp  K  (  ) (  )  r  r  N  d  d  \uf073  \uf073  =  +  +  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' and  min,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='mp  r  is the cage radius corresponding to the first  minimum of cross radial distribution function  mp( )  g  r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' [34, 41] Everywhere below the notation Fel  (FB) is employed to indicate the penetrant elastic (local cage) barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Schematic of the dynamic free energies for the coupled penetrant (left) and Kuhn segment  (middle) displacements in SCCH theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The trajectory level matrix facilitation of penetrant  hopping is formulated based on the simplifying concept of a dimensionless dynamic coupling  parameter, 𝛾 , defined as the ratio of the penetrant jump distance Δ𝑟p(𝛾) to the facilitating  displacement of the matrix Kuhn segments, Δ𝑟K,c(𝛾).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Relevant length and energy scales are  indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Right panel: Schematic of the physical ideas of SCCH theory for a single smaller  spherical penetrant in a bulk liquid of spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The image shows the matrix as disconnected spheres  solely for visual clarity and is literally relevant only for the hard sphere mixture model [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content="  BF  (rk  IBK  penetrant  Kuhn matrix  '?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  rioc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='K  BF  B,K  BF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  BF  B,P  B,K,c  Ar,(y) /o  Ark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' (r) /o  Jump  Cooperative displacement to  distance  facilitate penetrant hopping  19  This completes our brief review of the construction of SCCH theory that includes  crosslinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' From knowledge of the coupling parameter, penetrant dynamic free energy, and its  local cage and elastic barriers, the SCCH theory then predicts the mean penetrant alpha time, 𝜏𝛼,p,  as a function of size ratio, polymer packing fraction or temperature, crosslink fraction, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  As discussed in the Introduction, we will apply the theory motivated by a recent  experimental study of the relaxation and diffusivity of BTBP in PnBA networks [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Modeling  BTBP as a sphere, its effective hard sphere diameter is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='12nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' For the adopted 𝑙p/𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2  Koyama model 𝜎=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='23nm, while for the 𝑙p/𝜎 = 4/3 model one has 𝜎=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='03nm, thereby yielding  a penetrant-to-matrix size ratio of order unity, 𝑑/𝜎=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='91 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='09, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Crosslink Fraction Dependence of Jump Displacements and Associated Properties  In SCCH theory[34],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' the penetrant jump distance and the facilitation displacement of Kuhn  segments are very important for the activated penetrant hopping event since they (i) directly enter  calculation of the penetrant elastic barrier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' and (ii) the difference between displacements of a Kuhn  segment at the penetrant and polymer Kuhn segment alpha times provides a mechanistic picture  of the degree of trajectory level coupling between the penetrant and Kuhn segment alpha processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Thus, we first consider these quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Both increasing crosslink fraction and decreasing temperature induces a significant  increase of the penetrant and Kuhn segment alpha times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' This results in an increased jump distance  of both the penetrants and Kuhn segments, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='6a and 6b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The coupling  between penetrant and Kuhn segment activated displacements at the penetrant alpha time  corresponds to a facilitation displacement Δ𝑟Ku,c of a Kuhn segment, which grows with crosslink  density or cooling (longer alpha time) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' We note the crosslinking dependence  of Δ𝑟Ku,c is weaker than its temperature dependence, which differs from that of Δ𝑟p and Δ𝑟Ku where  20  both the temperature and crosslinking dependences are significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' We also find that the crosslink  density dependence of Δ𝑟p is nearly unchanged with increasing the temperature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=', calculations  of Δ𝑟p(𝑓n) − Δ𝑟p(0) collapse (vertical shift of the data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='6a) for different fixed temperatures  (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content='45  \uf044rp /\uf073  fn  T/K  273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='5  \uf044rX /\uf073  fn  solid: \uf044rX = \uf044rKu  dash: \uf044rX = \uf044rKu,c  T/K  273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content='15  400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Jump distance of (a) the penetrant, Δ𝑟p and (b) mobile Kuhn segment (solid curves), Δ𝑟Ku,  in units of 𝜎 at their corresponding alpha time scales as a function of crosslink fraction for a wide  range of temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The facilitation displacement of unpinned Kuhn segments, Δ𝑟Ku,c, at the  penetrant alpha time scale is shown in (b) as dashed curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The similarity of the absolute  magnitude of the penetrant and unpinned Kuhn segment jump distances reflects the strong  coupling of their motions as a consequence of the very similar penetrant and Kuhn segment sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='6b also shows that the absolute value of Δ𝑟Ku,c (corresponding to the penetrant alpha  time scale) is much lower than that of Δ𝑟Ku (corresponding to the Kuhn segment alpha time scale),  and the difference increases significantly with degree of effective supercooling (lowering  temperature or increasing crosslink fraction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' This suggests that the dynamical coupling between  penetrant and polymer matrix weakens at lower temperatures and higher crosslink densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' For  the highest temperature T = 400K and lower crosslink density regime shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='6b, the  crosslink dependence of Δ𝑟Ku,c is very similar to that of Δ𝑟Ku, corresponding to a relatively stronger  coupling between the alpha relaxations of the penetrant and polymer matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  One can also see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='6b that the absolute value of the Kuhn segment facilitation  displacement is not much smaller than the location of its barrier (Kuhn segment jump distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  21  This implies there is a strong coupling between penetrant and Kuhn segment alpha relaxations and  suggests the dynamic coupling parameters, 𝛾, defined as the ratio of penetrant jump distance Δrp(γ)  to the facilitating Kuhn segment displacement ΔrKu,c(γ), should not be much larger than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Figure 7a shows this to be the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The smaller 𝛾 is, the larger is the dynamic coupling of activated  penetrant and polymer matrix displacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Figure 7a also shows that the coupling decreases  with cooling and crosslink fraction as physically expected, but the dependences are weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The  reason is that the penetrant jump distance has a significant temperature dependence while that of  Kuhn segment facilitation displacement is more limited (per Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='6a and 6b), while both their  crosslinking dependences are rather weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Since the temporal self-consistency condition is  hop,p  dis,K  K,c  ( )  (  ( ))  r  \uf074  \uf067  \uf074  \uf067  =  \uf044  with  el,p  el,K,c  F  F  =  , the coupling parameter 𝛾  encodes dynamical  displacement physics associated only with local caging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Therefore, variation of 𝛾 with crosslink  fraction is physically intuitive, even in the large crosslink fraction regime it is nearly constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content='20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content='35  \uf067  fn  T/K = 273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content='10  1  2  3  X(fn)/X(0)  fn  solid: X = K0,Ku  dash: X = \uf044r 4  Ku,c  T/K  273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content='15  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' (a) Self-consistently determined dynamic coupling parameter, 𝛾 ≡ Δ𝑟p/Δ𝑟Ku,c, as a function  of crosslink fraction at various temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' (b) Relative importance of the key factors Δ𝑟Ku,\u2061c  4  (dash) and 𝐾0,\u2061Ku (solid) reduced by their values at zero crosslink fraction that enter the penetrant  elastic barrier as a function of crosslink fraction at various temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Finally, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='7b we compare the relative importance of the two key factors Δ𝑟Ku,\u2061c  4  and  𝐾0,\u2061Ku  that enter the calculation of penetrant elastic barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The harmonic curvature 𝐾0,\u2061Ku  (proportional to the inverse dynamic localization length squared) weakly increases with cooling  22  and crosslink fraction, with 𝐾0,\u2061Ku(𝑓n)/𝐾0,\u2061Ku(0) varying only from ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='4 for all ranges of  temperature and crosslink fractions studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' For pure hard sphere fluids, increases of the localized  state spring constant with packing fraction is the most important origin for the corresponding  increase of the elastic barrier[35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' However, for networks, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='7b shows that as crosslink fraction  increases, the jump distance contribution Δ𝑟Ku,\u2061c  4  dominates the change of elastic barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The  reason is that introducing crosslinking increases the penetrant hopping time, and hence penetrants  can displace significantly more corresponding to a larger jump distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Penetrant Alpha Relaxation Time  We now consider theoretical predictions for quantities that are directly measurable in  experiment and simulation, specifically the penetrant alpha relaxation time in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' In the  context of empirical experimental and simulation data[9, 56, 57], one can consider two different ways  of displaying its dependence on a control parameter depending on which physical perspective one  thinks is the primary origin of effect of network crosslinking on penetrant motion: crosslink  dependent changes of segmental dynamics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=', activated glassy physics) versus entropic mesh  confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' In this section we only pursue an objective empirical analysis based on plotting the  theoretical data for the penetrant alpha time in two different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Of course, we know the physics  in SCCH theory is all about activated hopping dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Our goal is to test whether a “degeneracy  of interpretation” of the theoretical results might exist due to the discovered connection between  Tg and mesh size discussed in section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' This exercise does not answer the question whether the  physics of penetrant diffusion is dominated by segmental scale glassy effect or entropic mesh  confinement, the answer to which must depend to some extent on Tg/T, crosslink density, and  penetrant-to-matrix size ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' This question is explored in section VI, and in our companion  simulation article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  23  As a preview of the combined insights gleaned from the results in the remainder of our  present theoretical work and the complementary simulations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' we find for the penetrant sizes  studied that although the crosslink fraction dependent (but temperature independent) entropic mesh  confinement does affect penetrant diffusivity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' it is an increasingly secondary (and ultimately very  minor) effect as the temperature is lowered in the supercooled regime of interest  compared to the  coupling of penetrant hopping to the temperature and crosslink fraction dependent activated barrier  hopping associated with glassy dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Crosslinking Modified Segmental Dynamics Perspective  Figure 8 shows results for the inverse penetrant mean alpha time as a function of the  temperature scaled glass transition temperature, Tg(𝑓n)/T, over a wide range of crosslink fractions  at various fixed values of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The results are presented based on a typical laboratory  timescale criterion for vitrification (main panel) and a much smaller simulation-like timescale  criterion (inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The predicted behaviors are very similar for the two criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' For each fixed  temperature, the penetrant alpha relaxation time 𝜏𝛼,p/𝜏0 (𝜏0 ~1 ps per a typical viscous liquid) is  predicted to increase exponentially with 𝑇g(𝑓n), in the spirit of an “Arrhenius-like” law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Moreover,  we observe that the corresponding apparent activation energy [defined as the slope of log(𝜏𝛼,p/𝜏0)  versus 𝑇g(𝑓n)/𝑇,  see Table 1] and the absolute value of 𝜏0/𝜏𝛼,p decreases and increases with  temperature, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Overall, results for different temperatures are rather well collapsed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  There is a weak trend of lower penetrant hopping rate upon cooling, which is more prominent in  the main frame that is relevant to laboratory experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' As discussed in detail below, the reason  for this difference, and for the roughly Arrhenius-like form of the penetrant alpha time, is the  relative unimportance of the elastic barrier compared to its local cage analog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The latter is  24  quantitatively more true in the simulation based choice for Tg results which do not probe the deeply  supercooled regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  10-10  10-9  10-8  10-7  10-6  10-5  10-4  10-3  10-2  10-1  \uf0740 /\uf074\uf061,p  Tg(fn)/T  \uf074\uf061,K(Tg) /\uf0740  = 1014  TH  T/K  273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2  10-9  10-7  10-5  10-3  10-1  \uf0740 /\uf074\uf061,p  Tg(fn)/T  \uf074\uf061,K(Tg) /\uf0740  = 105  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Penetrant inverse mean alpha time based on SCCH theory as a function of Tg(fn)/T at various  fixed temperatures for vitrification criteria 𝜏𝛼,K(𝑇g)/𝜏0 = 1014  (main) and 105  (inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The  vertical line in the inset is an estimate of a higher kinetic glass transition temperature, which  corresponds to a simulation criterion of vitrification that delivers a glass transition temperature far  higher than its laboratory based analog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Slope of the logarithm of the penetrant alpha time versus 𝑇g/𝑇 or C plot for various fixed  temperatures and 𝑇g criteria where A=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='94 is adopted in theoretical evaluation of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  T/K  𝑻𝐠 criterion  273  296  323  348  400  𝐥𝐨𝐠\u2061(𝝉𝜶,𝐩/𝝉𝟎) vs 𝑻𝐠/𝑻  105  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='4  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='3  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='5  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='7  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='3  𝐥𝐨𝐠\u2061(𝝉𝜶,𝐩/𝝉𝟎) vs 𝑻𝐠/𝑻  1014  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='4  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='1  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='5  𝐥𝐨𝐠\u2061(𝟏/𝑫𝐩) vs 𝑻𝐠/𝑻  105  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='7  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='6  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2  𝐥𝐨𝐠\u2061(𝟏/𝑫𝐩) vs 𝑻𝐠/𝑻  1014  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='6  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='3  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2  𝐥𝐨𝐠\u2061(𝝉𝜶,𝐩/𝝉𝟎) vs 𝑪  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='77  𝐥𝐨𝐠\u2061(𝟏/𝑫𝐩) vs 𝑪  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='1  SCCH theory can provide the physical mechanism to understand how crosslinking  modifies the penetrant alpha relaxation process by carefully examining the penetrant local cage  and collective elastic barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Figure 9 presents theoretical results for these quantities in an Angell-  25  like representation, where the dynamic 𝑇g(𝑓n) criterion appropriate for the deeply supercooled  regime of highest experimental relevance is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' For all fixed temperatures studied, the elastic  barriers are far smaller than the local cage barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' There are several reasons for this: (i) for  polymer systems, the relative importance of the elastic barrier is modestly weaker because of the  influence of chain connectivity [37, 38], and (ii) the penetrant size is relatively small which is known  to induce significant decoupling between the penetrant and matrix dynamics and a lower elastic  barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Figure 9 also shows that the elastic barrier increases more slowly with 𝑇g(𝑓n)/𝑇 relative  to that of local cage barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Hence, the total penetrant barrier and its growth with 𝑇g(𝑓n)/𝑇 is  largely dominated by the local cage barrier, and for 𝑇g(𝑓n)/𝑇 ≤ 1 it is more so for the simulation  timescale vitrification criterion (inset) than its laboratory analog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  0  4  8  12  16  20  24  \uf062FB(el)  Tg(fn)/T  \uf074\uf061,K(Tg) /\uf0740  = 1014  FB     Fel      T/K  |  273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  |  296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  |  323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  |  348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  |  400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2  0  4  8  12  16  \uf062FB(el)  Tg(fn)/T  \uf074\uf061,K(Tg) /\uf0740  = 105  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Penetrant local cage (open) and collective elastic (solid) barriers as a function of Tg(fn)/T at  various temperatures with the 𝑇g criteria in main frame and inset corresponding to  𝜏𝛼,K(𝑇g)/𝜏0 =  1014 and 105, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The vertical line in the inset represents an estimate of a higher kinetic  glass transition temperature corresponding to a typical simulation criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  More insight is obtained by comparing Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='9 and 4a, which shows that the penetrant local  cage barrier is comparable to its Kuhn segment analog, as expected from the predicted strong  coupling between activated hopping of the penetrant and Kuhn segment displacements as  manifested in the close to unity value of the coupling parameter 𝛾 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' On the other hand,  26  the penetrant elastic barrier is far smaller than its Kuhn segment analog because its facilitation  displacement is much smaller than the jump distance associated with the pure network alpha  relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The latter result can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='6b and the fact that the jump distance enters the  penetrant elastic barrier as the 4th fourth power (see section III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  The local cage barrier varies linearly with 𝑇g(𝑓n)/𝑇 which is the same behavior predicted  for the Kuhn segment (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The fact that this linearity is sufficiently well obeyed over the entire  range of 𝑇g/𝑇 is why an exponential relationship between alpha time and 𝑇g(𝑓n)/𝑇 is found in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Note also from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='4a and 9 that at fixed 𝑇g(𝑓n)/𝑇 the local cage barriers for both the  penetrant and Kuhn segments are slightly larger at lower temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' This implies that normal  glass-like physics effects play an important role in determining the temperature dependence of the  penetrant alpha time, which is intensified in the high crosslink density regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' This behavior also  explains why the apparent activation energy (Table 1) increases as temperature decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Confinement Mesh Perspective  We now explore the extent to which the SCCH theory results for the penetrant alpha time  can be empirically interpreted in a purely confinement mesh scenario,  𝐷p ∝ exp\u2061(−𝐵𝐶2)/𝐶 or  𝐷p ∝ exp(−𝐸𝐶), per Eqs (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Specifically, we ask if the crosslink fraction dependent  penetrant alpha time data can be “correlated” or “described” as:  1/𝜏𝛼,p ∝ exp\u2061[−(𝐵 − 1)𝐶2]  and/or 1/𝜏𝛼,p ∝ exp[−(𝐸 − 𝑏′)𝐶].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Obviously, the entropic mesh perspective cannot account for  explicit temperature dependences of the theoretical penetrant alpha time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2  10-17  10-13  10-9  10-5  10-1  \uf0740 /\uf074\uf061,p  C  C = 4(d/\uf073)(fn/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='19)1/2  C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='94(d/\uf073)(fn/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='19)1/2  T/K  273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  0  1  2  10-8  10-6  10-4  10-2  \uf0740 /\uf074\uf061,p  C  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Inverse penetrant mean alpha time 𝜏0/𝜏𝛼,p as a function of the confinement parameter at  various fixed temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The main frame uses  𝐶 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='94(𝑑/𝜎)(𝑓n/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='19)1/2, while the inset  adopts 𝐶 = 4(𝑑/𝜎)(𝑓n/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='19)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Figure 10 plots in a log-linear manner the theoretical penetrant relaxation rate as a function  of confinement parameter 𝐶 = 𝑑/𝑎x over a wide range of crosslink densities and temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' A  linear relationship is observed, albeit with a slope that decreases strongly with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The  latter trend is due to activated glassy dynamics effects, and is inconsistent with a literal [55-57]  entropic confinement mesh picture (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Thus, as a cautionary comment, if simulation or  experimental data is taken at only one temperature or over a narrow range, one could incorrectly  conclude that the confinement mesh idea is “verified”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Indeed, the slope changes documented in  Table 1 prove the importance of T-dependent physics in crosslinked polymer networks, and  necessarily implies the rate of increase of the alpha time with 𝑇g/𝑇 is significantly larger than that  with C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Physically the reason is simply the very different rate of change of 𝑇g and 𝐶 with crosslink  density 𝑓n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' As discussed in section II, 𝑇g(𝑓n) is proportional to inverse mesh size 𝑎x  −1 and hence  𝐶(𝑓n) for a fixed penetrant size and different vitrification criteria, which is the non-causal reason  the curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='10 are linearized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Our conclusions remain robust with varying the constant A in  defining the confinement parameter, as shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='10 where 𝐴 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0 is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  28  We also tested whether a linear relationship between log(𝜏α,p/𝜏0) and 𝐶2 can empirically  “work” (not shown), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=', 1/𝜏α,p~ exp[−(𝐵 − 1)𝐶2]  , and find it is also a reasonable description  in the limited C regime examined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' However, for large enough 𝐶 , SCCH theory predicts  that the exponential relationship 1/𝜏α,p~ exp[−(𝐸 − 𝑏′)𝐶] will break down since the elastic  barrier of penetrant becomes more and more important and it does not grow linearly with  confinement parameter but rather quadratically (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Simulation tests of the relations  1/𝜏𝛼,p ∝ exp\u2061[−(𝐵 − 1)𝐶2]  and 1/𝜏𝛼,p ∝ exp[−(𝐸 − 𝑏′)𝐶]  are reported in our companion  paper, and good agreement between theory and simulation is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' [58]  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Decoupling Ratio of Penetrant and Network Alpha Times  As previously studied using SCCH theory for hard sphere mixtures [42, 43] and dilute  spherical penetrants in polymer melts [34], the ratio of the penetrant to Kuhn segment alpha times  can be predicted which quantifies the degree of “dynamic decoupling”, a quantity that is  measurable in simulations and experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' This quantity is especially sensitive to collective  elasticity given the hopping of penetrants and Kuhn segments couples to this longer range effect  generally to very extents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' In prior work 34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' 42],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' the decoupling time ratio,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' τα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='p/τα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' was plotted as a  function of packing fraction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' and found to initially increase very slightly with matrix packing  fraction in the lower packing fraction regime (corresponding to high temperature where barriers  are low and the timescale of the short time dynamical process,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' τs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='p and τs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' matter),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' and then sharply  decrease with further increase of packing fraction indicating a strong decoupling between  penetrant and matrix activated dynamics when the penetrant and matrix barriers become larger and  dominate the alpha relaxation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The crucial physical effect is the very different rate at which  the penetrant and matrix elastic barriers grow with increasing packing fraction or decreasing  29  temperature, a difference significantly larger than that for their local cage barrier analogs (see  results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='4a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='1  10-4  10-3  10-2  10-1  100  Tg(fn)/T  Tg(fn)/T = 1  \uf074\uf061,p /\uf074\uf061,K  SIM          TH     T/K  300  273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  310  296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  320  323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  \uf074\uf061,K(Tg) /\uf0740  = 105  TH  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  10-4  10-3  10-2  10-1  100  \uf074\uf061,p /\uf074\uf061,K  Tg(fn)/T  \uf074\uf061,K(Tg) /\uf0740  = 1014  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Decoupling time ratio as a function of 𝑇g(𝑓n)/𝑇 for various values of crosslink density 𝑓n  using 𝜏𝛼,K(𝑇g)/𝜏0 = 105 (main) and 1014 (inset) 𝑇g criteria and for four temperatures in theory  and for three temperatures for the 𝑑/𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0 simulations [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The simulation data of mean alpha  relaxation times are based on the 1/e decay definition criterion, and the time ratio is vertically  shifted up by a factor of 480 in the main panel for comparison of their functional form with the  theory results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The vertical line in the main frame represents an estimate of a higher kinetic glass  transition temperature based on the simulation vitrification criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  The inset of Figure 11 shows theoretical predictions for the decoupling time ratio plotted  versus 𝑇g(𝑓n)/𝑇 over a wide range of 𝑇g(𝑓n)/𝑇 from the rubbery regime, through the deeply  supercooled regime, to the laboratory kinetic glass transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' For the four lower temperatures  studied (273, 296, 323, 348 K) the time ratio decreases monotonically with the effective degree of  supercooling parameter, 𝑇g(𝑓n)/𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' In contrast, at high temperature (400 K) where barriers are  relatively small and only the local cage barrier is non-negligible, a nearly constant (weakly  decreasing) behavior is predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Overall, for all five temperatures (which span a change of the  polymer network alpha time of ~14 decades), a nonmonotonic behavior is found, qualitatively  consistent with the previous predictions for other systems [34, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' A new finding is that for all  temperatures and crosslink densities, the decoupling time ratio tends to collapse in 𝑇g(𝑓n)/𝑇 space  with only slight deviations visible in the high 𝑇g(𝑓n)/𝑇 regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' We hope future experiments can  30  test our results in the deeply supercooled regime where a significant decrease of the decoupling  ratio is predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  We have verified that all of the above findings remain unchanged if a different 𝑇g criterion  is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The main frame of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='11 shows an example based on a vitrification criterion relevant  to simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' For the three temperatures simulated (300, 310, and 320 K based on the calibrated  PnBA coarse-grained network model employed) the decoupling time ratio slightly decreases  monotonically, and the data is in good agreement with our prediction in this weakly supercooled  regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' We hope future simulations can better test the predicted weakly non-monotonic behavior  with T and probe lower temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Penetrant Diffusion Constant  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Introductory Comments and High-Level Picture  We now consider the predictions of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' (2) and (3) which combine in a simple  multiplicative manner the activated physics (at the level of mean alpha relaxation times) of SCCH  theory with the entropic mesh confinement effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' We showed in section IV that the glass physics  based theoretical local hopping rate can be empirically described using the mathematical relations  of the entropic mesh models formulated solely in terms of the mesh confinement variable C (  1/𝜏𝛼,p ∝ exp\u2061[−(𝐵 − 1)𝐶2] and 1/𝜏𝛼,p ∝ exp[−(𝐸 − 𝑏′)𝐶]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Thus, logically, introducing the  latter mechanism into Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' (2) or (3) [i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=', 𝑋(𝐶) = exp\u2061(−𝐶2)/𝐶 or 𝑋(𝐶) = exp(−𝑏′𝐶)] cannot  change the basic form of our results for how crosslink fraction modifies the diffusion constant if  expressed in terms of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Now, adopting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' (2) or (3) for the penetrant diffusion constant, our theoretical finding  that the penetrant alpha relaxation rate can be empirically described in a mesh confinement  parameter framework implies it will be very difficult or impossible to dissect how important the  31  factors 1/𝜏𝛼,p and 𝑋(𝐶) are at fixed temperature from experimental or simulation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Of course,  insight can be obtained from by studying the temperature dependence of the penetrant relaxation  time and diffusion constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Here, in the next subsection we consider a quantitative comparison of  the theoretical predictions with and without the mesh confinement effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' In addition,  as discussed  in more detail in our companion simulation paper[58], reasonable, albeit not in general quantitative,  agreement between SCCH theory and simulation for both the crosslink fraction and temperature  dependences of the penetrant diffusion constant based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' (2) is found, which systematically  improves as the polymer network becomes increasingly supercooled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The latter trend is expected  given mesh confinement is of entropic origin in contrast to the crosslinking-induced prolongation  of segmental relaxation which becomes increasingly more important in colder polymer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Some of the key deductions reported in our companion simulation paper [58] can also be  drawn from the purely theoretical effective slope data in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' One sees that under the same  conditions, the effective slope determined from the diffusion constant is larger than that  determined from the inverse alpha time, consistent with the existence of an additional mesh  confinement contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Of course, in general, one a priori expects that the relative importance  of glassy physics and mesh confinement depends on both temperature and penetrant size, with the  latter (former) expected to dominate at sufficiently high (low) temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Importantly, “high”  and “low” depends on penetrant size since the degree of coupling of penetrant activated hopping  with the polymer network dynamics depends on the penetrant size relative to the size of the  polymer monomer or Kuhn segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' An additional generic physical effect in pure glass-forming  liquids is “dynamic heterogeneity” (DH) which induces a distribution of activation barriers and  hopping times in the lower temperature supercooled regime, and an associated decoupling of  diffusivity and relaxation corresponding to a weaker temperature dependence of the self-diffusion  32  constant than the alpha time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The consequences of such a DH effect is not well established for  penetrant dynamics, but it expected to be present at some level, and would go in the opposite  direction of the entropic mesh confinement induced slowing down of diffusion relative to  relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The version of ECNLE and SCCH theories employed in this article focus solely on the  mean barriers and relaxation times, and do not consider this type of DH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Thus, its absence can lead  to deviations of some theoretical predictions with simulation at lower temperatures and especially  for smaller penetrants, as confirmed in our companion paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' [58] On the other hand, at higher  temperatures and/or for large penetrants where DH effects are either intrinsically small or are  sufficiently averaged over, respectively, one expects better agreement between theory and  simulation, as we have verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' [58]  We note that the ECNLE theory for the alpha relaxation in 1-component glass-forming  atomic, molecular and polymer liquids has been extended to include DH via a distribution of  collective elastic barriers and relaxation times[71, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' It captures well the Stokes-Einstein  breakdown or diffusion-relaxation decoupling and temperature-dependent stretched exponential  relaxation in molecular liquids, and also the nonuniversal failure of time-temperature-  superposition in polymer melts corresponding to the decoupling of the temperature dependences  of the segmental alpha and chain scale end-to-end relaxation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The ideas developed in these  prior works can be extended to the penetrant dynamics problem in the framework of SCCH theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Finally, we emphasize that given the structure of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' (2), penetrant diffusivity will be  slowed down more with crosslinking than that of penetrant alpha relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' This represents a  qualitatively distinct form of “decoupling” of penetrant mass transport and relaxation in polymer  networks compared to one-component glass forming liquids, as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Quantitative Theoretical Predictions  33  Theoretical results are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='12 for the penetrant diffusion constant based on  𝐷p ≡  𝑑2  𝜏𝛼,p exp\u2061(−𝐶2)/𝐶 plotted as a function of Tg(fn)/T over a wide range of crosslink fractions  for 𝑇g criteria of 𝜏𝛼,K(𝑇g)/𝜏0 = 1014 (main) and 105 (inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' As found for the penetrant inverse  alpha time in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8, and as expected based on our insights that connect Tg(fn) and 𝐶 discussed in  section II, we predict (i) the diffusion constant also varies exponentially with 𝑇g(𝑓n) with a linearity  even better than that shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8, and (ii) the slope of the logarithm of diffusion constant versus  𝑇g(𝑓n)/𝑇 plot decreases with heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' However, the absolute value of diffusion constants at fixed  Tg(fn)/T weakly decreases with temperature, opposite to our findings for inverse alpha time (see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' This difference is a rather subtle 2nd order effect, but it is an objective signature of the  importance of mesh confinement for the diffusion constant that is amenable to experimental and  simulation tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Hence, we now discuss its origin in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  10-12  10-9  10-6  10-3  100  Dp=X\uf0740 /\uf074\uf061,p  Tg(fn)/T  T/K  273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  \uf074\uf061,K(Tg) /\uf0740  = 1014  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2  10-9  10-7  10-5  10-3  10-1  Dp  Tg(fn)/T  \uf074\uf061,K(Tg)/\uf0740  = 105  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Penetrant diffusion constant as a function of Tg(fn)/T at various temperatures for  vitrification criteria of 𝜏𝛼,K(𝑇g)/𝜏0 = 1014 (main) and 𝜏𝛼,K(𝑇g)/𝜏0 = 105 (inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The vertical  line in the inset represents an estimate of a higher kinetic glass transition temperature that  corresponds to a simulation-like vitrification criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Per Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' (2), the difference between the diffusion constant and inverse alpha time is  proportional to exp\u2061(−𝐶2)/𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' At fixed Tg(fn)/T, a lower T corresponds to a lower Tg(fn), fewer  crosslinks, a larger mesh, and hence a smaller confinement parameter C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' This trend results in an  34  increase of the diffusion constant with cooling in the representation of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' In contrast, the  inverse alpha time decreases with cooling, resulting in the opposite trend with T in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  The corresponding slope (proportional to an effective activation barrier) for the logarithm  of the diffusion constant versus 𝑇g(𝑓cross)/𝑇 based on SCCH theory is shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Relative  to its analog for the inverse penetrant alpha time, its absolute value increases significantly to a  degree that depends on temperature, 𝑇g criterion, and the chemistry-specific parameter A that  enters the quantification of the confinement parameter C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' In our companion simulation paper [58],  we show that by using the commonly adopted 1/e decay criterion of the time correlation functions  employed to define the penetrant alpha relaxation time, the degree of DH is small, and it is thus  appropriate to quantitatively compare such simulation results to our theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' We find from  simulation that the effective slope change with temperature for  𝑑  𝜎 = 1\u2061is significant for the inverse  alpha time (from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='6), but moderate/minimal for the diffusion constant (from 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='5 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' In  the present theory, the slope change for both the inverse alpha time and diffusion constant with  temperature is moderate (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' We suspect this difference arises from the approximate nature  of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' (2) with 𝑋(𝐶) = exp\u2061(−𝐶2)/𝐶 which reflects our simple ansatz for how to combine glassy  activated physics and mesh confinement to predict diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Indeed, our complementary  simulations[58] do find modest deviations from the simple formula 𝐷p~exp\u2061(−𝐶2)/𝐶𝜏α,\u2061p, though  they decrease with cooling as the glass physics effects become relatively more important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Both  theory and simulation[58] find the slope change with temperature for the alpha time is significantly  larger than that for the diffusion constant, emphasizing the additional crosslinking effect on  penetrant diffusion beyond its coupling with the activated segmental relaxation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  As mentioned in section VIA, if  𝐷p~exp\u2061(−𝐶2)/𝐶𝜏α,\u2061p is qualitatively correct then the  diffusion constant will decrease more rapidly with Tg(fn)/T than the penetrant alpha time increases,  35  opposite to “normal” Stokes-Einstein relation failure[49-54] in one-component liquids associated  with  DH[46-48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' However, DH in pure liquids generally results in a distribution of barriers or  relaxation times[71, 72], which is likely is present to some degree for the problem of penetrant  diffusion in glass-forming polymer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The pristine DH physics in pure supercooled liquids  is typically manifested in a decoupling relation of the form 𝐷p𝜏𝛼,p ∝ 𝜏𝛼,p  𝛿 , where the exponent 𝛿 is  modestly greater than zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Hence, there may be two competing effects of different physical origins  that can induce decoupling of penetrant diffusivity and relaxation, and which go in opposite  directions with increased supercooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' As shown in our companion simulation paper [58], if the  penetrant alpha time extracted from the intermediate scattering function (ISF) adopts the 1/e decay  criterion to define a relaxation time, then the DH effects are relatively weak as indicated by the  product 𝐷p𝜏𝛼,p decreasing with Tg/T due to the mesh confinement effect per Eq(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' However, if  the penetrant alpha time is extracted using a longer timescale criterion (the ISF decays to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='1), then  larger DH effects are present resulting in our finding that the product 𝐷p𝜏𝛼,p increases with Tg/T  corresponding to the entropic mesh confinement effect becoming of second order importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2  10-10  10-7  10-4  10-1  Dp=X\uf0740 /\uf074\uf061,p  C  T/K  273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='5  10-9  10-8  10-7  10-6  10-5  10-4  10-3  10-2  10-1  CDp=CX\uf0740 /\uf074\uf061,p  C2  T/K  273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' (a) Penetrant diffusion constant 𝐷p as a function of confinement parameter C at various  temperatures with 𝐶 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='94(𝑑/𝜎)(𝑓𝑛/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='19)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' (b) Same display as (a) but for the square power  law relationship between 𝐶𝐷p and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  36  Finally, we consider our diffusion constant predictions as a function of the mesh  confinement variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Combining the exponential relation between log(𝜏α,p)  and 𝐶  or 𝐶2  predicted by SCCH theory, with an additional mesh confinement contribution per X = exp\u2061(−𝑏′𝐶)  or exp\u2061(−𝐶2)/𝐶, one deduces from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' (3) or (2) a net exponential relationship of the form  1/𝐷p~ exp(−𝐸𝐶) or 1/𝐶𝐷p~ exp(−𝐵𝐶2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Figures 13a and 13b plots the theoretical  penetrant diffusion constants as a function of 𝐶 and 𝐶2, respectively, over a wide range of  temperatures and crosslink densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' One finds the diffusion constant data behaves nearly the same  as that of inverse alpha time observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='10 in the following sense: (i) the diffusion constant  varies exponentially with 𝐶 or 𝐶2, and the corresponding linearity is even better (slightly worse  for the 𝐶2 relation since the local cage barrier dominates the dynamics of penetrant while its elastic  barrier is negligible) than that of inverse penetrant alpha time shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' (ii) the effective  slope  (Table 1) of the logarithm of inverse diffusion constant versus 𝐶 or 𝐶2 decreases with  temperature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' and (iii) the diffusion constants increase with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' SCCH theory predicts that  the elastic barrier will increase significantly as the degree of effective supercooling grows  sufficiently large (increasing penetrant size or crosslink density and/or decreasing temperature),  and then both the local cage and elastic barriers are crucial for penetrant hopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' In this situation,  the net consequence of these two barriers results in a predicted square law relation between the  total barrier and 𝐶, and hence one expects in practice a log(𝐶𝐷p) ~ 𝐶2 law will eventually emerge  in the high degree of supercooling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  We have verified that all our conclusions above deduced from numerical computations for  the diffusion constant remain qualitatively robust regardless of the Tg criterion and choice of the  prefactor A (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='94 is adopted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='12 and 13) in defining confinement parameter C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  37  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Summary and Conclusions  We have carried out a detailed theoretical study of how permanent crosslinking in polymer  networks impacts the relaxation and diffusivity of dilute spherical penetrants over a wide range of  temperatures, crosslink densities, and Tg criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The key new methodological aspect is a general  extension of the SCCH theory of penetrant relaxation to address the dynamic consequences of  chemical crosslinking based on a segment pinning model where the local structure remains  unchanged but the microscopic dynamic constraints encountered by a penetrant from its Kuhn  segment neighbors is significantly intensified as crosslink density increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' This results in a rich  behavior of penetrant relaxation and diffusivity as a function of temperature, crosslink density, and  penetrant size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  The penetrant local cage barrier is predicted to dominate its alpha process while the elastic  barrier remains relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The local cage barrier varies linearly with the network glass  transition temperature Tg, leading to an exponential increase (apparent “Arrhenius”) of the  isothermal penetrant alpha relaxation time with Tg(fn)/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Numerical calculations based on the  ECNLE theory of polymer networks reveals a proportionality between Tg and the square root of  the crosslink fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' We note that both Tg and fn are properties of the pure polymer matrix, and  hence the activated dynamics of a penetrant of fixed size is dominated by its coupling with the  polymer network alpha process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Glassy segmental scale dynamics and mesh confinement glass physics are in general both  important for penetrant diffusivity, and their effects on the crosslink fraction dependence of the  activation barrier are predicted to obey a remarkable degeneracy (linear in C or linear in Tg(fn)/T)  due to the relation 𝐶 = 𝑑/𝑎x = 𝐴(𝑑/𝜎)𝑓n  1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' However, at fixed crosslink density, the penetrant  diffusivity is a strong function of temperature, a hallmark of the importance of glassy segmental  38  dynamics absents in the entropy-based mesh confinement effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Overall, the latter plays an  increasing minor role compared to the consequence of crosslinking induced slowing down of  polymer segmental relaxation as the degree of polymer network supercooling is increased and/or  (we expect) when penetrants become sufficiently small compared to the mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' However, mesh  confinement does slow down mass transport to varying degrees, but seemingly not the more  spatially local penetrant alpha relaxation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' This results in diffusion being more strongly  suppressed with increasing crosslink density than penetrant alpha relaxation, a novel form of  “decoupling” of an opposite nature than the generic Stokes-Einstein breakdown[49-54] phenomenon  in glass-forming liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  The mechanistic coupling between activated dynamics of the penetrant and network Kuhn  segments at the correlated displacement level has been elucidated, and further quantified by the  decoupling time ratio defined as the ratio of the penetrant to Kuhn segment alpha relaxation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  This decoupling ratio varies non-monotonically as the degree of effective supercooling degree  grows (increasing Tg(fn)/T via lowering temperature or increasing crosslink density), and sharply  decreases in the deeply supercooled regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' The latter behavior arises because of the relative  unimportance of the elastic barrier for penetrant hopping versus its high importance for Kuhn  segment relaxation in fragile polymer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' A remarkably good (but not perfect) collapse of  the decoupling ratio as a function of Tg(fn)/T for all crosslink densities and temperatures studied is  predicted, independent of the vitrification timescale adopted to define the glass transition  temperature and the precise value of the polymer network strand persistence length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Overall, we believe that the generic theoretical insights and specific predictions obtained  will be useful for designing new polymer networks to optimize the absolute and relative (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=', for  selectivity applications) penetrant diffusion rates relevant to polymer-based membrane  39  separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Extensions of the theory to other penetrant sizes, inclusion of penetrant-polymer  attractions, under pressure, and in the glassy state (thermosets) and vitrimers[73-76] is possible and  will be studied in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Acknowledgement  This research was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Department of Energy, Office of Basic Energy Sciences,  Division of Materials Sciences and Engineering (Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' DE-SC0020858), through the  Materials Research Laboratory at the University of Illinois at Urbana-Champaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Helpful  discussions with Christopher Evans are gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Appendix: Effect of Persistence Length Choice  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  10-9  10-7  10-5  10-3  10-1  \uf0740 /\uf074\uf061,p  Tg(fn)/T  TH,4/3  T/K  273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  \uf074\uf061,K(Tg) /\uf0740  = 1014  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2  10-8  10-6  10-4  10-2  \uf0740 /\uf074\uf061,p  Tg(fn)/T  \uf074\uf061,K(Tg) /\uf0740  = 105  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  10-15  10-12  10-9  10-6  10-3  100  Dp=X\uf0740 /\uf074\uf061,p  Tg(fn)/T  TH, 4/3  T/K  273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='15  \uf074\uf061,K(Tg) /\uf0740  = 1014  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='2  10-10  10-7  10-4  10-1  Dp  Tg(fn)/T  \uf074\uf061,K(Tg)/\uf0740  = 105  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Penetrant (a) inverse mean alpha time and (b) diffusion constant as a function of Tg(fn)/T  at various temperatures using a 𝜏𝛼,K(𝑇g)/τ0 = 1014 criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Same displays as that of the main  text Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='8 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='12, respectively, but using the 𝑙p/𝜎 = 4/3 local aspect ratio model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Both insets  are the corresponding results for a simulation-like 𝜏𝛼,K(𝑇g)/τ0 = 105 vitrification criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' Zaccarelli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' Tartaglia, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' Narayanan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Kanduč, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Kim, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Roa, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Dzubiella, ACS nano 15, 614-624 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Xue, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Shi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Tian, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Zheng, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' 20, 3895-3904 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Sheridan and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Evans, Macromolecules 54, 11198-11208 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Smith, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Guo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Robeson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' McGrath, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Paul, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  Freeman, Polymer 54, 4729-4761 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Li and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Mooney, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Vrentas and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Vrentas, Diffusion and mass transfer (CRC Press, Boca Raton, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Galizia, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Chi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Smith, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Merkel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Baker, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Freeman,  Macromolecules 50, 7809-7843 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' Jhalaria, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Huang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Abbas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Midya, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Benedetti, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Parisi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Egger,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Dickmann, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content='  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' Drioli, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Golemme, Ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content='  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Freeman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content='  Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' Lively, Nature 532, 435-437 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content='  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content='  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' Mahajan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' Zhang and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Kumar, ACS Macro Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Schweizer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Schweizer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Zhou, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Mei, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' Schweizer, Macromolecules 54, 10086-10099 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Schweizer, Macromolecules 49, 9655-9664 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' Lin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' Mei, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' Saltzman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' Xie and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content=' Xie and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+page_content='  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Perego and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Khabaz, Macromolecules 53, 8406-8416 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Perego, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Lazarenko, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Cloitre, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Khabaz, Macromolecules 55, 7605-7613 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content='  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Porath, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Soman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Jing, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' Evans, ACS Macro Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
+page_content=' 11, 475-483 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE4T4oBgHgl3EQf9Q4M/content/2301.05353v1.pdf'}
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+The Iteration Number of the Weisfeiler-Leman Algorithm
+Martin Grohe
+RWTH Aachen University
+grohe@informatik.rwth-aachen.de
+Moritz Lichter
+TU Darmstadt
+lichter@mathematik.tu-darmstadt.de
+Daniel Neuen
+Simon Fraser University
+dneuen@sfu.ca
+Abstract
+We prove new upper and lower bounds on the number of iterations the k-dimensional
+Weisfeiler-Leman algorithm (k-WL) requires until stabilization. For k ≥ 3, we show that
+k-WL stabilizes after at most O(knk−1 log n) iterations (where n denotes the number of
+vertices of the input structures), obtaining the first improvement over the trivial upper
+bound of nk − 1 and extending a previous upper bound of O(n log n) for k = 2 [Lichter et
+al., LICS 2019].
+We complement our upper bounds by constructing k-ary relational structures on which
+k-WL requires at least nΩ(k) iterations to stabilize. This improves over a previous lower
+bound of nΩ(k/ log k) [Berkholz, Nordström, LICS 2016].
+We also investigate tradeoffs between the dimension and the iteration number of WL,
+and show that d-WL, where d = ⌈ 3(k+1)
+2
+⌉, can simulate the k-WL algorithm using only
+O(k2 · n⌊k/2⌋+1 log n) many iterations, but still requires at least nΩ(k) iterations for any d
+(that is sufficiently smaller than n).
+The number of iterations required by k-WL to distinguish two structures corresponds to
+the quantifier rank of a sentence distinguishing them in the (k + 1)-variable fragment Ck+1
+of first-order logic with counting quantifiers. Hence, our results also imply new upper and
+lower bounds on the quantifier rank required in the logic Ck+1, as well as tradeoffs between
+variable number and quantifier rank.
+1
+Introduction
+The Weisfeiler-Leman (WL) algorithm is a combinatorial algorithm that, given a relational struc-
+ture A (in most applications, this structure is a graph), iteratively computes an isomorphism-
+invariant coloring of tuples of vertices of A. The original algorithm introduced by Weisfeiler
+and Leman [26] is the 2-dimensional version that colors pairs of vertices. Its generalization to
+arbitrary dimension k ≥ 1, independently introduced by Babai and Mathon as well as Immer-
+man and Lander [11] (see also [1] for a historic note), yields for every natural number k the
+k-dimensional WL algorithm (k-WL), which iteratively refines a coloring of vertex k-tuples by
+aggregating local structural information encoded in the colors. More concretely, the k-WL algo-
+rithm initially colors all k-tuples of vertices v = (v1, . . . , vk) of a structure A by the isomorphism
+type of the underlying induced ordered substructure. Afterwards, in each iteration, the coloring
+is refined by taking the colors of all tuples into account that can be obtained from v by replacing
+a single entry of the tuple. This process necessarily stabilizes after a finite number of iterations
+and the resulting coloring can be used to classify k-tuples of vertices.
+The most prominent application of the WL algorithm lies in the context of the graph isomor-
+phism problem. Indeed, since no isomorphism between two structures A and B can map tuples
+of vertices of different colors to each other, the WL algorithm provides a hierarchy of increasingly
+powerful heuristics to the graph isomorphism problem. While there is no dimension k for which
+1
+arXiv:2301.13317v1  [cs.DS]  30 Jan 2023
+
+k-WL serves as a complete isomorphism test [4], the algorithm is still surprisingly powerful.
+For example, Grohe [7] proved that for every non-trivial minor-closed graph class there is some
+k ∈ N such that k-WL computes a different coloring on all non-isomorphic graphs, and thus
+provides a polynomial-time isomorphism test on that class. Moreover, the WL algorithm is also
+regularly used as a subroutine in isomorphism algorithms (see, e.g., [19, 20, 23]) which includes
+Babai’s [1] quasipolynomial-time graph isomorphism test that employs the WL algorithm with
+dimension k = O(log n).
+More recently, the WL algorithm has also received significant attention in the machine
+learning context where it characterizes the expressiveness of graph neural networks [8, 17, 27]
+and, more generally, the colorings computed by WL are used in classification tasks on graph-
+structured data sets (see, e.g., [16, 22]).
+Since the late 1980s, the WL algorithm has played an important role in descriptive complexity
+theory. Indeed, it was independently introduced in the context of descriptive complexity by
+Immerman and Lander [11]. The main reason for this is that k-WL can be seen as an equivalence
+test for the logic Ck+1, the (k +1)-variable fragment of first-order logic with counting quantifiers
+∃≥nx. Through this connection, the algorithm has turned out to be important for studying the
+expressiveness of fixed-point logic with counting [4] and, more generally, for the quest for a logic
+capturing polynomial time [6, 21].
+In this work, we study the iteration number of k-WL, i.e., the number of iterations the algo-
+rithm requires until stabilization. Since the number of color classes increases in each iteration,
+the k-WL algorithm trivially requires at most nk − 1 rounds to stabilize. For k = 1, Kiefer and
+McKay [12] proved that this trivial bound is optimal by providing several infinite families of
+graphs G for which 1-WL requires n − 1 iterations to stabilize (where n denotes the number of
+vertices of G). In contrast, for k = 2, Lichter, Ponomarenko and Schweitzer [15] (improving an
+earlier upper bound by Kiefer and Schweitzer [14]) obtained an upper bound of O(n log n) on
+the iteration number of 2-WL. Beyond that, no improved upper bounds are known for k ≥ 3.
+As our first main contribution, we obtain non-trivial bounds on the iteration number of k-WL
+for all k ≥ 2.
+Theorem 1.1. For all k ≥ 2, the k-dimensional Weisfeiler-Leman algorithm stabilizes after
+O(knk−1 log n) refinement rounds on all relational structures A of arity at most k where n
+denotes the size of the universe.
+For the proof, we extend the algebraic arguments from [15]. Consider a structure A with
+vertex set V of size n and let χ0, . . . , χℓ : V k → C denote the sequence of colorings computed
+by k-WL, i.e., χi is the coloring computed in the i-th iteration. For k = 2, Lichter et al. [15]
+associate with each coloring χi a matrix algebra as follows. For each color c in the image of
+χi, let Mi,c denote the V × V indicator matrix that sets Mi,c(v1, v2) := 1 if χi(v1, v2) = c, and
+Mi,c(v1, v2) := 0 otherwise. The matrices Mi,c, where c ranges over all colors in the image of
+χi, generate a matrix algebra A(i) of V × V matrices over the complex numbers using standard
+matrix multiplication. Using representation-theoretic arguments, it is possible to bound the
+length of the sequence of matrix algebras generated this way which eventually leads to the
+upper bound of O(n log n).
+The proof of Theorem 1.1 follows a similar strategy. For each color in the image of χi, we
+obtain an indicator tensor Mi,c ∈ CV k. Now, the key challenge in generalizing the arguments
+of [15] is to define a suitable multiplication of those tensors that can be “simulated” by a single
+round of k-WL. Given such a multiplication, we then show that the generated algebra A(i) is
+isomorphic to a subalgebra of the nk−1 × nk−1 full matrix algebra (over the complex numbers)
+which then again allows us to use algebraic arguments to obtain the desired upper bound.
+Our arguments actually prove a more general result. Let χ0, . . . , χℓ : V k → C be a sequence
+of finer and finer colorings (i.e., the partition into color classes of χi refines the partition into
+color classes of χi−1 for all i ∈ [ℓ]) where in each step the coloring is refined at least as much as
+2
+
+by a single iteration of k-WL. Then the length of the sequence is bounded by ℓ = O(knk−1 log n).
+As a lower bound to our arguments, we show that, in this more general setting, our upper bound
+is tight up to a factor Ok(log n) (the Ok(·)-notation hides constant factors in k). Here, the key
+insight is that we can find a sequence of finer and finer colorings χ0, . . . , χℓ : V k → C of length
+Ωk(nk−1) that are all stable with respect to k-WL. As such, it provides a lower bound in the
+more general setting explained above (but it does not give any lower bounds on the iteration
+number of k-WL) and implies that new ideas are likely required to obtain further improvements
+on the upper bounds of iteration number of k-WL (see Section 4 for more details).
+Looking for lower bounds on the iteration of k-WL, Fürer [5] provided, for every k ≥ 2, a
+family of graphs on which k-WL requires at least Ω(n) many iterations until stabilization. For k
+sufficiently large, this result was strengthened by Berkholz and Nordström [3] who constructed
+k-ary relational structures A of size n on which k-WL requires at least nΩ(k/ log k) many iterations.
+Answering an open question from [3], our second main contribution is an improved lower bound
+that gets rid of the 1/ log k factor in the exponent. Actually, we prove the following even stronger
+result.
+Theorem 1.2. There are absolute constants k0 ∈ N and α, ε > 0 such that for every d ≥ k ≥ k0
+and every n ≥ αd8k6 there is a is pair of k-ary relational structures A and B of size |V (A)| =
+|V (B)| = n that are distinguished by k-WL, but d-WL does not distinguish A and B after nεk
+refinement rounds.
+We note that, as in the work of Berkholz and Nordström [3], the structures we need to prove
+this theorem are k-ary, that is, have relations of arity k.
+The structures A and B provided by the theorem can be distinguished by k-WL which
+trivially requires at most nk − 1 rounds. The theorem states that, even if we are allowed to
+increase the dimension of the Weisfeiler-Leman algorithm to d, the structures can still not be
+distinguished unless d-WL runs for at least nεk rounds. This result stands in strong contrast to
+several existing results for restricted classes of graphs. For example, k-WL distinguishes between
+all non-isomorphic pairs of graphs of tree-width at most k [13], and increasing the dimension
+to 4k + 3 guarantees that O(log n) iterations suffices to distinguish between all non-isomorphic
+pairs of graphs of tree-width at most k [10]. Similar results are known for planar graphs [9, 25].
+The above theorem rules out such results for general relational structures even if we only wish
+to improve the iteration number to, for example, linear in n.
+By setting d = k, we obtain the following corollary which shows that the upper bound in
+Theorem 1.1 is optimal up to a constant factor (that does not depend on k) in the exponent.
+Corollary 1.3. There are absolute constants k0 ∈ N and α, ε > 0 such that for every k ≥ k0
+and every n ≥ αk14 there is a k-ary structure A of size |V (A)| = n such that the k-dimensional
+Weisfeiler-Leman algorithm does not stabilize within nεk refinement rounds on A.
+For the proof of Theorem 1.2, our main technical contribution is to show that there is a
+k0 ∈ N such that, for all d ≥ k0, there are structures A and B of size n that are distinguished by
+k0-WL, but d-WL still requires Ω(n/d2) many iterations to distinguish A and B. Afterwards,
+we obtain Theorem 1.2 by using a known hardness condensation [3] that reduces the size of the
+structures while roughly preserving the number of iterations required to distinguish them.
+Let us point out that Fürer [5] constructed graphs G and H which are distinguished by
+k0-WL after Ω(n) many rounds. However, as Fürer also shows, his instances are distinguished
+by (3k0)-WL after only O(log n) many rounds which means that we cannot use them for our
+purposes. Berkholz and Nordström [3] provided, for all d ≥ 2, structures A and B of size n that
+are distinguished by 2-WL, but d-WL still requires Ω(n1/(1+log d)) many rounds to distinguish
+them. In combination with the hardness condensation, this leads to the previous lower bound
+of nΩ(k/ log k).
+For the construction of our structures, we introduce the notion of layered expanders whose
+global structure is similar to a (k × n)-grid, but that locally (when looking at O(k) consecutive
+3
+
+columns) behave like an expander graph. We then obtain propositional XOR-formulas from
+layered expanders which can be transformed into relational structures which satisfy the desired
+properties.
+Connection to Logics.
+As pointed out above, k-WL is an equivalence test for the logic Ck+1.
+That is, k-WL distinguishes between two structures A and B if and only if there is a sentence
+ϕ ∈ Ck+1 such that A |= ϕ and B ̸|= ϕ. Additionally, the minimal quantifier rank of such a
+sentence equals (up to an additive error of at most k) the number of iterations k-WL requires to
+distinguish between A and B. With this in mind, Theorem 1.1 can be reformulated as follows.
+Corollary 1.4. Let k ≥ 3. Let A and B be two relational structures of arity at most k that can
+be distinguished by a sentence in Ck. Then there is a sentence ϕ ∈ Ck of quantifier rank at most
+q = O(knk−2 log n) such that A |= ϕ and B ̸|= ϕ.
+Similarly, we can reformulate Theorem 1.2, but here it turns out that we can obtain an even
+stronger result since the structures constructed in the theorem can already be distinguished in
+the logic Lk+1, the k + 1-variable fragment of first-order logic without counting quantifiers.
+Theorem 1.5. There are absolute constants k0 ∈ N and α, ε > 0 such that for every d ≥ k ≥ k0
+and every n ≥ αd8k6 there is a pair of k-ary structures A and B of size |V (A)| = |V (B)| = n
+that can be distinguished by a sentence in k-variable first-order logic Lk, but satisfy the same
+sentences in Ld and Cd up to quantifier rank nεk.
+Hence, we obtain lower bounds for the quantifier rank not only for the logic Ck, but also
+for the logic Lk. Observe that the lower bounds on the quantifier rank remain valid even if we
+arbitrarily increase the number of variables to any number d (as long as d is sufficiently far away
+from the size of the structures). In other words, even if we are allowed to increase the number of
+variables, we cannot in general hope for significant improvements on the quantifier rank required
+to distinguish between two structures.
+Having said that, our final result shows that at least some improvements on the upper bound
+are possible if we are allowed to increase the number of variables by roughly a factor of 3/2.
+Theorem 1.6. Let k ≥ 2. Let A and B be two relational structures of arity at most k such that
+n := |V (A)| = |V (B)|. Also suppose there is a sentence ϕ ∈ Ck+1 such that A |= ϕ and B ̸|= ϕ.
+Let d := ⌈ 3(k+1)
+2
+⌉. Then there is a sentence ψ ∈ C(q)
+d
+of quantifier rank q = O(k2 · n⌊k/2⌋+1 log n)
+such that A |= ψ and B ̸|= ψ.
+Structure of the Paper.
+After introducing the necessary preliminaries in the next section,
+we prove Theorem 1.1 in Section 3. Afterwards, we prove limitations of our approach to obtain
+improved upper bounds on the iteration number in Section 4. In Section 5, we obtain the lower
+bounds on the iteration number of WL and prove Theorems 1.2 and 1.5. Finally, Theorem 1.6
+is proved in Section 6.
+2
+Preliminaries
+We use N = {1, 2, 3, . . . } to denote the positive integers. For n ∈ N we write [n] := {1, . . . , n}
+and [0, n] := {0, . . . , n}.
+Graphs.
+We use standard graph notation. A graph is a pair G = (V (G), E(G)) with finite
+vertex set V (G) and edge set E(G). In this paper, all graphs are simple (i.e., there are no loops
+or multiedges) and undirected. We write vw to denote an edge {v, w} ∈ E(G). The (open)
+neighborhood of a vertex v ∈ V (G) is the set NG(v) := {w ∈ V (G) | vw ∈ E(G)}. The degree
+4
+
+of a vertex, denoted by degG(v), is the size of its neighborhood. For X ⊆ V (G) we define
+NG(X) := (�
+v∈X NG(v)) \ X to denote the neighborhood of X. If the graph G is clear from
+context, we usually omit the index G and simply write N(v), deg(v) and N(X). For X ⊆ V (G)
+we also write G[X] to denote the subgraph of G induced by X.
+Relational Structures.
+In this work, we restrict ourselves to relational vocabularies (sig-
+natures) σ = {R1, . . . , Rm} where each Ri is a relation symbol of a prescribed arity ki ≥ 1.
+We say that σ has arity at most k if ki ≤ k for all i ∈ [m].
+A σ-structure is a tuple
+A = (V (A), RA
+1 , . . . , RA
+m) where V (A) is a finite universe and RA
+i
+⊆ (V (A))ki is a relation
+of arity ki. In the remainder of this work, we usually do not explicitly refer to the vocabulary
+underlying a structure A. With this in mind, we say a structure A = (V (A), RA
+1 , . . . , RA
+m) has
+arity at most k if the underlying vocabulary has arity at most k.
+For X ⊆ V (A) we define A[X] to be the induced substructure of A on X, i.e., A[X] is the
+relational structure with V (A[X]) = X and
+RA[X]
+i
+= RA
+i ∩ Xki
+for all i ∈ [m]. Let B = (V (B), RB
+1 , . . . , RB
+m) be a second structure (over the same vocabulary
+σ). An isomorphism from A to B is a bijection f : V (A) → V (B) such that, for all i ∈ [m] and
+all v1, . . . , vki ∈ V (A), it holds that
+(v1, . . . , vki) ∈ RA
+i
+⇐⇒ (f(v1), . . . , f(vki)) ∈ RB
+i .
+The structures A to B are isomorphic if there is an isomorphism from A to B.
+Logics.
+Next, we cover bounded-variable fragments of first-order logic (with counting quanti-
+fiers). Let σ = {R1, . . . , Rm} be a relational vocabulary and suppose Ri has arity ki ≥ 1. We
+write FO to denote standard first-order logic defined via the grammar
+ϕ ::= x1 = x2 | Ri(x1, . . . , xki) | ϕ ∧ ϕ | ¬ϕ | ∃x1ϕ
+for all i ∈ [m] and all variables xj ∈ V where V is an infinite set of variables.
+We write
+ϕ(x1, . . . , xk) to indicate that the free variables of ϕ are among the variables {x1, . . . , xk}. For
+a structure A = (V (A), RA
+1 , . . . , RA
+m) and v = (v1, . . . , vk) ∈ (V (A))k we write A |= ϕ(v) if A is
+a model of ϕ when xi is interpreted by vi.
+We define the quantifier rank of a formula ϕ ∈ FO inductively via
+• qr(x1 = x2) = qr(Ri(x1, . . . , xki)) := 0 for all i ∈ [m] and all variables xj ∈ V,
+• qr(ϕ ∧ ψ) := max(qr(ϕ), qr(ψ)),
+• qr(¬ϕ) := qr(ϕ), and
+• qr(∃xϕ) := qr(ϕ) + 1 for all x ∈ V.
+We define first-order logic with counting quantifiers C to be the extension of FO by counting
+quantifiers of the form ∃≥jxϕ. The formula ∃≥jxϕ is satisfied over a structure A if there are
+at least j distinct elements v ∈ V (A) that satisfy ϕ. We extend the definition of the quantifier
+rank in the natural way by setting qr(∃≥jxϕ) := qr(ϕ) + 1 for all x ∈ V.
+For k ∈ N we define Lk to be the restriction of FO to formulas over at most k variables, i.e.,
+we restrict ourselves to a set of variables V of size exactly k. Similarly, we define Ck to be the
+restriction of C to formulas over at most k variables.
+Moreover, for q ≥ 0, we define L(q)
+k
+to the restriction of Lk to formulas ϕ of quantifier rank
+qr(ϕ) ≤ q. Similarly, we define C(q)
+k
+to the restriction of Ck to formulas of quantifier rank at
+most q.
+5
+
+The Weisfeiler-Leman Algorithm.
+Next, we describe the k-WL algorithm.
+While it is
+most commonly used as a heuristic to graph isomorphism, the algorithm can be applied to any
+relational structure of arity at most k.
+Let χ1, χ2 : V k → C be colorings of k-tuples over a finite set V where C is some finite set of
+colors. The coloring χ1 refines χ2, denoted χ1 ⪯ χ2, if χ1(v) = χ1(w) implies χ2(v) = χ2(w) for
+all v, w ∈ V k. Observe that χ1 ⪯ χ2 if and only if the partition into color classes of χ1 refines
+the corresponding partition into color classes of χ2. The colorings χ1 and χ2 are equivalent,
+denoted χ1 ≡ χ2, if χ1 ⪯ χ2 and χ2 ⪯ χ1. Also, χ1 strictly refines χ2, denoted χ1 ≺ χ2, if
+χ1 ⪯ χ2 and χ1 ̸≡ χ2.
+Let us fix k ≥ 2 and consider a relational structure A = (V (A), RA
+1 , . . . , RA
+m) of arity at most
+k. Let v = (v1, . . . , vk) ∈ (V (A))k. We define the atomic type of v, denoted by atpA(v), to be
+the isomorphism type of the ordered substructure of A that is induced by {v1, . . . , vk}. More
+concretely, for a second structure B = (V (B), RB
+1 , . . . , RB
+m) and a tuple w = (w1, . . . , wk) ∈
+(V (B))k, it holds that atpA(v) = atpB(w) if the mapping vi �→ wi is an isomorphism from
+A[{v1, . . . , vk}] to B[{w1, . . . , wk}].
+Next, we describe a single refinement step of k-WL. Let V be a finite set and let χ: V k → C
+be a coloring of all k-tuples over V . We define the coloring stepk(χ) by setting
+�
+stepk(χ)
+�
+(v) :=
+�
+χ(v), Mχ(v)
+�
+for all v = (v1, . . . , vk) ∈ V k where
+Mχ(v) :=
+���
+χ(v[w/1]), . . . , χ(v[w/k])
+� ��� w ∈ V
+��
+and v[w/i] := (v1, . . . , vi−1, w, vi+1, . . . , vk) is the tuple obtained from v by replacing the i-th
+entry by w (and {{. . . }} denotes a multiset). Observe that stepk(χ) ⪯ χ. We say the coloring χ
+is k-stable if stepk(χ) ≡ χ.
+We define the initial coloring computed by k-WL on the structure A via χ(0)
+k [A](v) := atpA(v)
+for all v ∈ (V (A))k. For r ≥ 0 we set
+χ(r+1)
+k
+[A] := stepk
+�
+χ(r)
+k [A]
+�
+.
+Since χ(r+1)
+k
+[A] ⪯ χ(r)
+k [A] for all r ≥ 0, there is some minimal r∞ ≤ |V |k − 1 such that
+χ(r∞)
+k
+[A] ≡ χ(r∞+1)
+k
+[A].
+We say that k-WL stabilizes after r∞ rounds on A and define χ(∞)
+k
+[A] := χ(r∞)
+k
+[A] to be the
+output coloring of k-WL. Observe that χ(∞)
+k
+[A] is a k-stable coloring.
+Now, let B = (V (B), RB
+1 , . . . , RB
+m) be a second structure. Let r ≥ 0. We say that k-WL
+distinguishes A and B after r rounds if there is some color c such that
+���
+�
+v ∈ (V (A))k ��� χ(r)
+k [A](v) = c
+���� ̸=
+���
+�
+w ∈ (V (B))k ��� χ(r)
+k [B](w) = c
+����.
+We also say that k-WL distinguishes A and B if there is some integer r ≥ 0 such that k-WL
+distinguishes A and B after r rounds. We write A ≃k B if k-WL does not distinguish A and
+B. Note that, if k-WL distinguishes A and B and k-WL stabilizes after r∞ rounds on A, then
+k-WL distinguishes A and B after (at most) r∞ + 1 rounds.
+The following connections to bounded-variable fragments of first-order logic with counting
+quantifiers are well-known.
+Theorem 2.1 ([4, 11]). Let k ≥ 2. Also let A and B be structures of arity at most k and suppose
+v ∈ V (A)k and w ∈ V (B)k. Then, for every r ≥ 0, it holds that χ(r)
+k [A](v) ̸= χ(r)
+k [B](w) if and
+only if there is some ϕ(x) ∈ C(r)
+k+1 such that A |= ϕ(v) and B ̸|= ϕ(w).
+6
+
+Corollary 2.2. Let k ≥ 2. Also let A and B be structures of arity at most k.
+If there is a sentence ϕ ∈ C(r)
+k+1 such that A |= ϕ and B ̸|= ϕ, then the k-dimensional
+Weisfeiler-Leman algorithm distinguishes A and B after at most r refinement rounds.
+If the k-dimensional Weisfeiler-Leman algorithm distinguishes A and B after r refinement
+rounds, then there is a sentence ϕ ∈ C(r+k)
+k+1
+such that A |= ϕ and B ̸|= ϕ.
+3
+Upper Bounds
+In this section, we prove Theorem 1.1. Actually, we prove a more general result on the maximal
+iteration number of any refinement method that is at least as strong as k-WL.
+For the remainder of this section, let us fix some integer k ≥ 2. Let V be a finite set and let
+P be a partition of V k. We say that P is compatible with equality if for all P ∈ P, all tuples
+(v1, . . . , vk), (v′
+1, . . . , v′
+k) ∈ P, and all i, j ∈ [k] it holds that
+vi = vj ⇐⇒ v′
+i = v′
+j.
+Moreover, the partition P is shufflable if for every function π: [k] → [k] and every P ∈ P it
+holds that
+P π :=
+�
+(vπ(1), . . . , vπ(k))
+�� (v1, . . . , vk) ∈ P
+�
+∈ P.
+Finally, the partition P is permutable if for every permutation π ∈ Sk (we write Sk for the
+symmetric group on the set [k]) and every P ∈ P it holds that P π ∈ P.
+Note that every
+shufflable partition is also permutable.
+We say a coloring χ: V k → C of k-tuples is compatible with equality if the corresponding
+partition P into color classes is compatible with equality. Similarly, χ is permutable (shufflable)
+if P is permutable (shufflable).
+Recall that stepk(χ) denotes the coloring obtained from χ after applying a single refinement
+round of k-WL.
+Theorem 3.1. Let V be a finite set of size n := |V |. Also let χ0, . . . , χℓ : V k → C be a sequence
+of colorings such that
+(I) χt is shufflable and compatible with equality for all t ∈ [0, ℓ],
+(II) stepk(χt−1) ⪰ χt for all t ∈ [ℓ], and
+(III) χt−1 ≻ χt for all t ∈ [ℓ].
+Then ℓ ≤ 2nk−1(⌈k log n⌉ + 1) = O(knk−1 log n).
+Note that Theorem 1.1 immediately follows from Theorem 3.1 by observing that all colorings
+χ(i)
+k [A] obtained from the refinement process of k-WL are shufflable and compatible with equality.
+The proof of Theorem 3.1 relies on algebraic tools. Let V be a finite set of size n := |V |. We
+define a multiplication on the space CV k by
+(a · b)(v1, . . . , vk) :=
+�
+v∈V
+a(v1, . . . , vk−1, v)b(v1, . . . , vk−2, v, vk)
+(1)
+for all a, b ∈ CV k. Note that this multiplication is associative and has a unit 1, defined by
+1(v1, . . . , vk) :=
+�
+1
+if vk−1 = vk,
+0
+otherwise.
+Furthermore, the multiplication is compatible with the vector space structure. Hence, it defines
+an algebra which we denote by A.
+7
+
+With every a ∈ CV k we associate a matrix Ma ∈ CV k−1×V k−1 with entries
+Ma
+�
+(v1, . . . , vk−1), (w1, . . . , wk−1)
+� :=
+�
+a(v1, . . . , vk−2, vk−1, wk−1)
+if vi = wi for all i ∈ [k − 2],
+0
+otherwise.
+It is easy to see that the mapping a �→ Ma is injective and linear. Moreover, it is compatible
+with multiplication:
+Ma · Mb
+�
+(v1, . . . , vk−1), (w1, . . . , wk−1)
+�
+=
+�
+u1,...,uk−1∈V
+Ma
+�
+(v1, . . . , vk−1), (u1, . . . , uk−1)
+�
+Mb
+�
+(u1, . . . , uk−1), (w1, . . . , wk−1)
+�
+=
+��
+u∈V a(v1, . . . , vk−2, vk−1, u)b(v1, . . . , vk−2, u, wk−1)
+if vi = wi for all i ∈ [k − 2],
+0
+otherwise
+=
+�
+(a · b)(v1, . . . , vk−1, wk−1)
+if vi = wi for all i ∈ [k − 2],
+0
+otherwise
+= Ma·b
+�
+(v1, . . . , vk−1), (w1, . . . , wk−1)
+�
+.
+And finally, M1 is the identity matrix. Thus, A is isomorphic to a subalgebra of the nk−1×nk−1-
+dimensional matrix algebra CV k−1×V k−1.
+For every a ∈ A we define a∗ ∈ A by
+a∗(v1, . . . , vk) = a(v1, . . . , vk−2, vk, vk−1)
+(here, c denotes the complex conjugate of a number c ∈ C, i.e., if c = a + bi then c = a − bi).
+Then Ma∗ := (Ma)∗ (the conjugate transpose). Thus, ∗ is an involution on A compatible with
+the algebra structure, which turns A into a ∗-algebra. In particular, as a finite-dimensional
+∗-algebra, A is semisimple.
+Let us denote by Md(C) for the full matrix algebra of all (d × d)-matrices over the complex
+numbers.
+Theorem 3.2 ([15, Theorem 5]). Let A(1) ⊂ · · · ⊂ A(ℓ) ⊆ Md(C) be a sequence of semisimple
+strict subalgebras. Then ℓ ≤ 2d.
+So using the fact that A is isomorphic to a subalgebra of Mnk−1(C), we obtain the following
+corollary.
+Corollary 3.3. Let A(1) ⊂ · · · ⊂ A(ℓ) ⊆ A be a sequence of semisimple strict subalgebras of A.
+Then ℓ ≤ 2nk−1.
+For every subset A ⊆ CV k, we let span(A) be the linear subspace of CV k generated by A, and
+we let ⟨A⟩ be the closure of span(A) under multiplication. If 1 ∈ ⟨A⟩, then ⟨A⟩ is a subalgebra
+of A. We are interested in subalgebras of A generated by partitions of the set V k in the way
+explained next.
+For every subset P ⊆ V k, we define
+cP (v) :=
+�
+1
+if v ∈ P,
+0
+otherwise
+to be the characteristic vector of P. For a partition P of V k, we let CP := {cP | P ∈ P} and
+AP := ⟨CP⟩. If 1 ∈ AP, then AP is a subalgebra of A.
+Lemma 3.4. Let P and Q be partitions of V k such that Q strictly refines P. Then span(CP) ⊂
+span(CQ) and AP ⊆ AQ.
+8
+
+Proof. If P ∈ P is the disjoint union of Q1, . . . , Qm ∈ Q, then cP = �m
+i=1 cQi. Thus, CP ⊆
+span(CQ) and therefore span(CP) ⊆ span(CQ). Moreover, there are P ∈ P, Q ∈ Q such that
+Q ⊂ P. Then cQ ̸∈ span(CP), because all a ∈ span(CP) are constant on P. Hence the inclusion
+is strict.
+The second assertion AP ⊆ AQ follows immediately from the definitions of AP and AQ.
+Observation 3.5. Let P be a partition of V k.
+(1) If P is compatible with equality, then 1 ∈ AP and hence AP is a subalgebra of A.
+(2) If P is permutable, then AP is closed under ∗.
+Corollary 3.6. Let P be a partition of V k that is permutable and compatible with equality. Then
+AP is a ∗-subalgebra of A. In particular, AP is semisimple.
+We say that a ∈ CV k distinguishes v, w ∈ V k if a(v) ̸= a(w), and we say that A ⊆ CV k
+distinguishes v, w if some a ∈ A distinguishes them.
+Lemma 3.7. Let A ⊆ CV k and v, w ∈ V k such that ⟨A⟩ distinguishes v, w. Then there are an
+s ≤ nk and a1, . . . , as ∈ A such that a1 · · · as distinguishes v, w.
+Proof. As a linear subspace of CV k, the space ⟨A⟩ consists of finite linear combinations of
+“monomials” a1 · · · as for ai ∈ A. Since the dimension of the space is at most nk, we only need
+to consider such monomials for s ≤ nk. Hence v, w are distinguished by a linear combination
+m
+�
+i=1
+λiai1 · · · aisi
+with λi ∈ C, aij ∈ A, and si ≤ nk. This immediately implies that v, w are distinguished by
+ai1 · · · aisi for some i ∈ [m].
+With every partition P = {P1, . . . , Pm} we associate a relational structure (V, RP
+1 , . . . , RP
+m)
+whose vocabulary consists of k-ary relation symbols Ri interpreted by RP
+i
+= Pi (to uniquely
+define the associated structure, we fix an arbitrary order on the blocks P1, . . . , Pm). Slightly
+abusing notation, we denote this structure by P as well. We say that a formula ϕ(x) distinguishes
+v, w ∈ V k over P if
+P |= ϕ(v)
+⇐⇒
+P ̸|= ϕ(w).
+Recall that C(q)
+k+1 denotes the fragment of first-order logic with counting consisting of all formulas
+of quantifier rank at most q with at most k + 1 variables.
+Lemma 3.8. Let P be a partition of V k and let v, w ∈ V k such that AP distinguishes v and
+w. Then there is a formula ϕ(x) ∈ C(q)
+k+1 of quantifier rank q ≤ ⌈k log n⌉ that distinguishes v, w
+over P.
+Proof. Suppose that P = {P1, . . . , Pm}, and let ci := cPi. Then AP = ⟨{c1, . . . , cm}⟩. Thus, by
+Lemma 3.7, there is an s ≤ nk and i1, . . . , is ∈ [m] such that ci1 · · · cis distinguishes v, w.
+By induction on s ≥ 1, we prove that if ci1 · · · cis distinguishes v, w, then there is a formula
+ϕ(x) ∈ C(⌈log s⌉)
+k+1
+that distinguishes v, w. The assertion of the lemma follows.
+For the base step s = 1, note that if ci distinguishes v, w, then the atomic formula Ri(x)
+distinguishes v, w.
+For the inductive step, let s ≥ 2.
+Suppose that b = ci1 · · · cis distinguishes v, w.
+Let
+r := ⌈s/2⌉ and note that r ≤ 2⌈log s⌉−1 and therefore
+⌈log r⌉ ≤ ⌈log s⌉ − 1.
+9
+
+Let b1 := ci1 · · · cir and b2 := cir+1 · · · cis. Then b = b1 · b2. Suppose that v = (v1, . . . , vk) and
+w = (w1, . . . , wk). We have
+b(v) =
+�
+u∈V
+b1(v1, . . . , vk−1, u) · b2(v1, . . . , vk−2, u, vk)
+̸= b(w) =
+�
+u∈V
+b1(w1, . . . , wk−1, u) · b2(w1, . . . , wk−2, u, wk).
+Thus, there are b1, b2 ∈ C such that
+p :=
+���
+�
+u ∈ V
+��� b1(v1, . . . , vk−1, u) = b1 and b2(v1, . . . , vk−2, u, vk) = b2
+����
+̸=
+���
+�
+u ∈ V
+��� b1(w1, . . . , wk−1, u) = b1 and b2(w1, . . . , wk−2, u, wk) = b2
+���� =: q.
+It follows from the induction hypothesis that for i = 1, 2 and for all v′, w′ ∈ V k such that bi
+distinguishes v′, w′ there is a formula ψv′,w′
+i
+(x) ∈ C(⌈log r⌉)
+k+1
+that distinguishes v′, w′. Without
+loss of generality,
+P |= ψv′,w′
+i
+(v′)
+and
+P ̸|= ψv′,w′
+i
+(w′),
+otherwise we replace ψv′,w′
+i
+(x) by its negation. Let Vi ⊆ V k be the set of all v′ ∈ V k such that
+bi(v′) = bi and let
+ϕi(x) :=
+�
+v′∈Vi
+�
+w′∈V k\Vi
+ψv′,w′
+i
+(x).
+Then for all v′ ∈ V k we have
+P |= ϕi(v′) ⇐⇒ bi(v′) = bi.
+Without loss of generality we assume that p > q. Then the formula
+ϕ(x1, . . . , xk) := ∃≥pxk+1
+�
+ϕ1(x1, . . . , xk−1, xk+1) ∧ ϕ2(x1, . . . , xk−2, xk+1, xk)
+�
+∈ C(⌈log s⌉)
+k+1
+distinguishes v, w.
+We are now ready to prove Theorem 3.1.
+Proof of Theorem 3.1. For every t ∈ [0, ℓ] let P(t) be the partition of V k into the color classes
+of χt.
+Claim 3.9. Let t, q ≥ 0 such that t + q ≤ ℓ. Suppose that there is a formula ϕ(x) ∈ C(q)
+k+1 that
+distinguishes v, w ∈ V k over P(t). Then v, w belong to different classes of the partition P(t+q).
+Proof. By Condition (I), the partition P(t) is shufflable and compatible with equality.
+This
+implies that χt ≡ χ(0)
+k [P(t)]. Together with Condition (II), we get that χt+q ⪯ χ(q)
+k [P(t)].
+Also, using Theorem 2.1, we get that χ(q)
+k [P(t)](v) ̸= χ(q)
+k [P(t)](w). Overall, it follows that
+v, w belong to different classes of the partition P(t+q).
+⌟
+For every t ∈ [0, ℓ] we define C(t) := CP(t) and A(t) := AP(t). Note that A(t) is a semisimple
+∗-subalgebra of A by Condition (I) and Corollary 3.6. By Lemma 3.4, we have
+A(0) ⊆ A(1) ⊆ . . . ⊆ A(ℓ) ⊆ A.
+(2)
+Claim 3.10. For all t ∈ [0, ℓ − ⌈k log n⌉],
+A(t) ⊆ span(C(t+⌈k log n⌉)).
+10
+
+Proof. Let a ∈ A(t).
+By Lemma 3.8 and Claim 3.9, for all v, w ∈ V k, if a(v) ̸= a(w),
+that is, if a distinguishes v and w, then v and w belong to different classes of the partition
+P(t+⌈k log n⌉). Thus, a is constant on each class of the partition P(t+⌈k log n⌉), which immediately
+implies that a can be written as a linear combination of the characteristic vectors cP of the
+classes P ∈ P(t+⌈k log n⌉). This is the assertion of the claim.
+⌟
+Claim 3.11. For all t ∈ [0, ℓ − ⌈k log n⌉ − 1],
+A(t) ⊂ A(t+⌈k log n⌉+1).
+Proof. By Claim 3.10, we have A(t) ⊆ span(C(t+⌈k log n⌉)). Moreover, by Condition (III), the
+partition P(t+⌈k log n⌉+1) strictly refines the partition P(t+⌈k log n⌉). By Lemma 3.4, this implies
+span(C(t+⌈k log n⌉)) ⊂ span(C(t+⌈k log n⌉+1)). As span(C(t+⌈k log n⌉+1)) ⊆ A(t+⌈k log n⌉+1), the as-
+sertion of the claim follows.
+⌟
+Recall that by Corollary 3.6, the algebras A(t) are semisimple. Thus, by Corollary 3.3, at
+most 2nk−1 of the inclusions in (2) are strict. Then Claim 3.11 implies
+ℓ ≤ 2nk−1(⌈k log n⌉ + 1) = O(knk−1 log n).
+4
+Long Sequences of Stable Colorings
+Next, we prove an almost matching lower bound for Theorem 3.1, i.e., we prove that there are
+sequences of colorings χ0, . . . , χℓ : Vk → C satisfying Condition (I) - (III) of Theorem 3.1 of
+length ℓ = Ω(nk−1). Actually, we prove a slightly stronger result.
+As before, let us fix an integer k ≥ 2. We present a construction for a sequence χ0 ≻ χ1 ≻
+· · · ≻ χℓ of colorings of V k such that χt is k-stable (i.e., the coloring is stable with respect to
+k-WL) for all t ∈ [0, ℓ]. More precisely, the main result of this section is the following theorem.
+Theorem 4.1. Suppose n ≥ 2k2 and let V be a set of size |V | = 2n. Then there is a sequence
+of colorings χ0, . . . , χℓ : V k → C of length ℓ ≥
+� n
+2k
+�k−1 such that
+(I) χt is shufflable and compatible with equality for all t ∈ [0, ℓ],
+(II) χt is k-stable for all t ∈ [0, ℓ], and
+(III) χt−1 ≻ χt for all t ∈ [ℓ].
+Before diving into the proof, let us first discuss some implications of the theorem.
+First of all, Theorem 4.1 implies that the upper bound in Theorem 3.1 is tight up to a factor
+of Ok(log n). This follows from the simple observation that, if χt is k-stable and χt−1 ≻ χt, then
+stepk(χt−1) ≡ χt−1 ⪰ χt, i.e., the sequence of colorings constructed in Theorem 4.1 satisfies the
+requirements of Theorem 3.1.
+On the other hand, since all colorings χt are already k-stable, the theorem does not provide
+any lower bounds on the iteration number of k-WL. However, Theorem 4.1 still provides some
+valuable insights in this setting. Indeed, all existing methods to bound the iteration number
+of k-WL [14, 15] rely on “parallelization arguments”, i.e., it is argued that at some point in the
+refinement process many color classes have to be split at the same time. Theorem 4.1 essentially
+implies that such arguments do not suffice to push the upper bounds on the iteration number
+beyond O(nk−1) since such “parallelization arguments” typically also work in the extended setting
+of Theorem 3.1.
+As a concrete example, Kiefer and Schweitzer [14] prove upper bounds on
+iteration number of 2-WL by bounding the cost of a certain game related to 2-WL. This game
+naturally generalizes to k-WL, but Theorem 4.1 immediately implies that its cost is Ω(nk−1)
+and thus, it is not possible to obtain improved upper bounds by analyzing said game. So overall,
+11
+
+Theorem 4.1 can be interpreted as saying that, in order to obtain improved upper bounds on
+the iteration number of k-WL, we need to rely on arguments that also exploit the possibility of
+stabilization at an early point, and it is not possible to solely rely on “parallelization arguments”.
+Let us now turn to the proof of Theorem 4.1.
+It relies on the following theorem which
+provides a large set family with restricted intersections between its members. Let U be a set of
+size n. A k-uniform set family (over U) is a collection F of k-element subsets of U.
+Theorem 4.2 ([2, Theorem 4.11]). For every n ≥ 2k2 there exists a k-uniform set family F
+over a universe U of n points such that
+1. |E1 ∩ E2| ≤ k − 2 for all distinct E1, E2 ∈ F, and
+2. |F| ≥
+� n
+2k
+�k−1.
+Now, let U be a universe of size n ≥ k and let F be a k-uniform set family over U. We set
+V := U × {0, 1}
+and define a coloring χF : V k → C as follows. Since the actual names of the colors are not
+relevant for our purposes, we only define the color classes, i.e., we specify when two tuples
+receive the same color.
+Let ((u1, a1), . . . , (uk, ak)), ((u′
+1, a′
+1), . . . , (u′
+k, a′
+k)) ∈ V k. We define χF in such a way that
+χF((u1, a1), . . . , (uk, ak)) = χF((u′
+1, a′
+1), . . . , (u′
+k, a′
+k)) if and only if
+(A) ui = u′
+i for all i ∈ [k],
+(B) (ui, ai) = (uj, aj)
+⇔
+(u′
+i, a′
+i) = (u′
+j, a′
+j) for all i, j ∈ [k], and
+(C) if {u1, . . . , uk} ∈ F, then �
+i∈[k] ai ≡ �
+i∈[k] a′
+i mod 2.
+Lemma 4.3. Suppose |E1 ∩ E2| ≤ k − 2 for all distinct E1, E2 ∈ F. Then χF is k-stable.
+Proof. Let v = ((u1, a1), . . . , (uk, ak)), v′ = ((u′
+1, a′
+1), . . . , (u′
+k, a′
+k)) ∈ V k such that
+χF(v) = χF(v′).
+Observe that ui = u′
+i for all i ∈ [k] by Condition (A). We need to show that the two tuples do
+not receive distinct colors after a single refinement step of k-WL, that is, we need to argue that
+���
+χF(v[(u, a)/1]), . . . , χF(v[(u, a)/k])
+� ��� u ∈ U, a ∈ {0, 1}
+��
+=
+���
+χF(v′[(u, a)/1]), . . . , χF(v′[(u, a)/k])
+� ��� u ∈ U, a ∈ {0, 1}
+��
+where v[(u, a)/i]) = ((u1, a1), . . . , (ui−1, ai1), (u, a), (ui+1, ai+1), . . . , (uk, ak)) is the tuple ob-
+tained from v by replacing the i-th entry by (u, a). Towards this end, we actually show the
+stronger statement that
+���
+χF(v[(u, a)/1]), . . . , χF(v[(u, a)/k])
+� ��� a ∈ {0, 1}
+��
+=
+���
+χF(v′[(u, a)/1]), . . . , χF(v′[(u, a)/k])
+� ��� a ∈ {0, 1}
+��
+holds for all u ∈ U.
+Fix some u ∈ U. To see that these two multisets are equal, consider the set
+V ′ := {u1, . . . , uk, u} × {0, 1} ⊆ V
+and the restriction χ′
+F : (V ′)k → C : v �→ χF(v) of χF to the set (V ′)k. Also, let F′ := {E ∈ F |
+E ⊆ V ′}. Since |E1 ∩ E2| ≤ k − 2 for all distinct E1, E2 ∈ F and |V ′| ≤ k + 1, we conclude that
+|F′| ≤ 1.
+12
+
+Claim 4.4. There is a bijection ϕ: V ′ → V ′ such that
+(i) χ′
+F(v) = χ′
+F(ϕ(v)) for all v ∈ (V ′)k, and
+(ii) ϕ(ui, ai) = (u′
+i, a′
+i) for all i ∈ [k].
+Proof. For i ∈ [k] we define ϕ(ui, ai) := (ui, a′
+i) and ϕ(ui, 1 − ai) := (ui, 1 − a′
+i). In particular,
+Condition (ii) is satisfied since ui = u′
+i for all i ∈ [k]. If there is some E′ ∈ F′ such that u ∈ E′,
+then we define
+ϕ(u, a) :=
+�
+(u, a)
+if �
+ui∈E′ ai ≡ �
+ui∈E′ a′
+i mod 2
+(u, 1 − a)
+otherwise
+for both a ∈ {0, 1}. If no such set E′ ∈ F′ exists, then we set ϕ(u, a) := (u, a) for both i ∈ {0, 1}.
+It can be easily verified that χ′
+F(v) = χ′
+F(ϕ(v)) for all v ∈ (V ′)k.
+⌟
+Since the multisets above are defined in an isomorphism-invariant manner over the structure
+induced by (V ′, χ′
+F ), we conclude that they have to be equal.
+Proof of Theorem 4.1. Let F be the set family obtained from Theorem 4.2 and suppose that F =
+{E1, . . . , Eℓ}. Observe that ℓ ≥
+� n
+2k
+�k−1 as desired. For t ∈ [0, ℓ] we define Ft := {E1, . . . , Et} ⊆
+F and χt := χFt. Then χt is k-stable by Lemma 4.3. Also, Ft−1 ⊂ Ft which implies that
+χt−1 ≻ χt by definition of the coloring χt. Finally, it is easy to verify that all colorings are
+shufflable and compatible with equality.
+5
+Lower Bounds on the Iteration Number of WL
+In this section, we obtain improved lower bounds on the iteration number of the Weisfeiler-
+Leman algorithm. More precisely, we prove Theorem 1.2. Our proof strategy is similar to the
+one employed by Berkholz and Nordström in [3]. First, for every sufficiently large ℓhi ≥ ℓlo, we
+construct pairs of structures that can be distinguished by ℓlo-WL, but ℓhi-WL still requires a
+linear number of iterations to distinguish them. Afterwards, we apply a hardness compression
+that reduces the number of vertices in the obtained structures while preserving the iteration
+number of the Weisfeiler-Leman algorithm. Actually, for the second step, we can rely on the
+same tools that are already used by Berkholz and Nordström in [3].
+5.1
+Overview
+The hard instances we construct are based on propositional XOR-formulas that can also be
+viewed as systems of linear equations over the 2-element field F2.
+Let V be a finite set which we interpret as a set of variables that take values in {0, 1}. An
+XOR-constraint (over V ) is a pair (C, a) where C ⊆ V and a ∈ {0, 1}. The reader is encouraged
+to think of such a constraint as the equation x1 + · · · + xk ≡ a mod 2 where C = {x1, . . . , xk}
+is the set of those variables that appear on the left side of the equation. Let C be a set of
+XOR-constraints. We define the arity of C to be the maximum cardinality of C for any pair
+(C, a) ∈ C.
+We can translate a set of XOR-constraints into a pair of relational structures as follows. Let
+C be a set of XOR-constraints over a set V . Also suppose that V = {x1, . . . , xn}. We define
+A = A(C) and B = B(C) as follows. We set V (A) = V (B) := V × {0, 1}, i.e., each element
+of the structures A and B corresponds to an assignment of a single variable. For each i ∈ [n],
+we add a unary relation Xi and set XA
+i = XB
+i
+:= {(xi, 0), (xi, 1)}. Finally, for every constraint
+(C, a) ∈ C with C = {xi1, . . . , xik} we introduce a k-ary relation RC,a and define
+RA
+C,a :=
+�
+�
+�
+�
+(xi1, b1), . . . , (xik, bk)
+�
+������
+b1, . . . , bk ∈ {0, 1},
+k
+�
+j=1
+bj ≡ 0
+mod 2
+�
+�
+�
+13
+
+and
+RB
+C,a :=
+�
+�
+�
+�
+(xi1, b1), . . . , (xik, bk)
+�
+������
+b1, . . . , bk ∈ {0, 1},
+k
+�
+j=1
+bj ≡ a
+mod 2
+�
+�
+�
+Instead of analysing the Weisfeiler-Leman algorithm directly on A(C) and B(C), it turns out
+to more convenient to consider the following game that is directly played on C and is known
+to capture the same information as applying the Weisfeiler-Leman algorithm to the associated
+structures.
+Let C be a set of XOR-constraints over a set V . Let k ∈ N such that C has arity at most
+k. A partial assignment β : X → {0, 1} with X ⊆ V violates an XOR-constraint (C, a) ∈ C if
+C ⊆ X and
+�
+x∈C
+β(x) ̸≡ a
+mod 2.
+(3)
+For a partial assignment β0 : X0 → {0, 1} with |X0| ≤ k the r-round k-pebble game Gr
+k(V, C, β0)
+is played as follows:
+• The game has two players called Verifier and Falsifier.
+• The game is played in rounds with initial position β0.
+• Suppose β : X → {0, 1} is the current position.
+Then the next round consists of the
+following steps:
+– Falsifier chooses x ∈ V \ X and X′ ⊆ X such that |X′ ∪ {x}| ≤ k.
+– Verifier chooses b ∈ {0, 1}.
+– The game moves to position β′ : X′ ∪ {x} → {0, 1} with β′(x′) = β(x′) for x′ ∈ X′
+and β′(x) = b.
+• Falsifier wins a play if within the first r rounds an assignment β violates some XOR-
+constraint (C, a) ∈ C (if r = 0, then Falsifier wins if the initial assignment β0 violates some
+constraint in C).
+• Verifier wins a play if Falsifier does not win within the first r rounds.
+We say Falsifier (respectively Verifier) wins the game Gr
+k(V, C, β0) if Falsifier (respectively
+Verifier) has a winning strategy for the game. The k-pebble game Gk(V, C, β0) is played in the
+same way, but without any restriction on the number of rounds played.
+The following lemma relates the pebble game Gr
+k(V, C, ∅) to bounded-variable fragments of
+first-order logic and thereby, using Corollary 2.2, also to the Weisfeiler-Leman algorithm. (Here,
+we use ∅ to denote the empty assignment, i.e., the domain X0 of the initial partial assignment
+β0 is empty.)
+Lemma 5.1 ([3, Lemma 2.1]). Let k, r ∈ N such that r > 0 and k ≥ 3.
+Let C be a set
+of XOR-constraints over a universe V of arity at most k. Then the following statements are
+equivalent:
+(i) Falsifier wins the r-round k-pebble game Gr
+k(V, C, ∅).
+(ii) There exists a sentence ϕ ∈ L(r)
+k
+such that ϕ |= A(C) and ϕ ̸|= B(C).
+(iii) There exists a sentence ϕ ∈ C(r)
+k
+such that ϕ |= A(C) and ϕ ̸|= B(C).
+14
+
+To obtain a set of XOR-constraints on which Falsifier requires a large number of rounds to
+win the pebble game, we proceed in two steps. First, for every sufficiently large ℓhi ≥ ℓlo, we
+construct a set of XOR-constraints such that Falsifier wins the ℓlo-pebble game, but still requires
+a linear number of rounds to win the ℓhi-pebble game. This is formalized by the next lemma
+which forms the main technical contribution of this section.
+Lemma 5.2. There are absolute constants ℓlo ≥ 2 and δ > 1 such that for every ℓhi ≥ ℓlo
+and every r ≥ 1 there is a set of XOR-constraints C of arity at most ℓlo over a set V of size
+|V | ≤ δ · ℓ2
+hi · r such that Falsifier
+(a) wins the ℓlo-pebble game Gℓlo(V, C, ∅), but
+(b) does not win the r-round ℓhi-pebble game Gr
+ℓhi(V, C, ∅).
+We remark that a similar result has also been obtained in [3], but with weaker guarantees
+on the number of rounds required to win the ℓhi-pebble game. It is exactly this improvement
+that allows us to obtain stronger lower bounds on the iteration number of k-WL in comparison
+to [3].
+Afterwards, we rely on the following hardness compression lemma that reduces the number
+of variables while essentially maintaining the number of rounds that Falsifier requires to win the
+game.
+Lemma 5.3 (Berkholz, Nordström [3, Lemma 3.3]). There is an absolute constant ∆0 ≥ 1 such
+that the following holds. Suppose C is a set of XOR-constraints of arity at most p over a set V
+of size |V | = m. Also assume there are parameters ℓlo > 0, ℓhi ≥ ∆0ℓlo and r > 0 such that
+Falsifier
+(a) wins the ℓlo-pebble game Gℓlo(V, C, ∅), but
+(b) does not win the r-round ℓhi-pebble game Gr
+ℓhi(V, C, ∅).
+Let ∆ be an integer such that ∆0 ≤ ∆ ≤ ℓhi/ℓlo and (2ℓhi∆)2∆ ≤ m. Then there is a set of
+XOR-constraints D of arity at most ∆p over a set W of size |W| = ⌈m3/∆⌉ such that Falsifier
+(A) wins the (∆ℓlo)-pebble game G∆ℓlo(W, D, ∅), but
+(B) does not win the
+r
+2ℓhi -round ℓhi-pebble game Gr/(2ℓhi)
+ℓhi
+(W, D, ∅).
+Combining Lemmas 5.2 and 5.3, we obtain the following corollary.
+Corollary 5.4. There are absolute constants k0 ∈ N and α, ε > 0 such that for every d ≥ k ≥ k0
+and every n ≥ α · d8 · k6 there is a set of XOR-constraints C of arity at most k over a set V of
+size |V | ≤ n such that Falsifier wins the k-pebble game Gk(V, C, ∅), but does not win the r-round
+d-pebble game Gr
+d(V, C, ∅) for all r ≤ nεk.
+Proof. Let ℓlo ≥ 2 and δ > 1 denote the constants from Lemma 5.2. Also, let ∆0 denote the
+constant from Lemma 5.3 and suppose without loss of generality that δ, ∆0 are integers and
+∆0 ≥ 3. We choose
+k0 := max{∆0ℓlo, 6ℓlo}.
+Let d ≥ k ≥ k0. We set p := ℓlo, ℓhi := d and ∆ := ⌊ k
+ℓlo ⌋. We have ℓhi = d ≥ k ≥ k0 ≥ ∆0ℓlo
+and ∆0 ≤ k
+ℓlo . Since ∆0 is an integer, we conclude that ∆0 ≤ ∆.
+We define
+n0 := max{
+�
+δ · ℓ2
+hi · (2ℓhi∆)2∆�3/∆ , 4 · δ · ℓ3
+hi}
+and set α := max{64 · δ · ℓ−6
+lo , 4 · δ}. Let n ≥ α · d8 · k6. Using ∆ ≥ ∆0 ≥ 3, we get that
+�
+δ · ℓ2
+hi · (2ℓhi∆)2∆�3/∆ ≤ δ · ℓ2
+hi · (2ℓhi∆)6 ≤ 64 · δ · ℓ8
+hi ·
+� k
+ℓlo
+�6
+≤ α · d8 · k6
+15
+
+and 4 · δ · ℓ3
+hi ≤ α · d3. So in particular n ≥ n0. Let r be the maximal integer such that
+�
+δ · ℓ2
+hi · r
+�3/∆ ≤ n.
+(4)
+Note that r ≥ (2ℓhi∆)2∆ since n ≥ n0. Let C be the set of XOR-constraints of arity at most ℓlo
+over a set V of size |V | ≤ δ · ℓ2
+hi · r obtained from Lemma 5.2. By adding dummy variables, we
+may assume without loss of generality that m := |V | ≥ r ≥ (2ℓhi∆)2∆.
+By applying Lemma 5.3, we obtain a set of XOR-constraints D of arity at most ∆p over a
+set W of size |W| = ⌈m3/∆⌉ such that Falsifier
+(A) wins the (∆ℓlo)-pebble game G∆ℓlo(W, D, ∅), but
+(B) does not win the
+r
+2ℓhi -round ℓhi-pebble game Gr/(2ℓhi)
+ℓhi
+(W, D, ∅).
+First observe that ∆p = ⌊ k
+ℓlo ⌋ · ℓlo ≤ k and
+|W| =
+�
+m3/∆�
+≤
+��
+δ · ℓ2
+hi · r
+�3/∆�
+≤ n.
+Since ∆ℓlo ≤ k, it holds that Falsifier wins the k-pebble game Gk(W, D, ∅). Moreover, Falsifier
+does not win the
+r
+2ℓhi -round d-pebble game Gr/(2ℓhi)
+d
+(W, D, ∅). We have that
+�
+δ · ℓ2
+hi · 2r
+�3/∆ ≥ n
+since r is the maximal integer to satisfy Equation (4). This implies that
+r
+2ℓhi
+≥
+n∆/3
+4 · δ · ℓ3
+hi
+≥ n
+1
+3 ⌊ k
+ℓlo ⌋−1 ≥ nεk
+for some sufficiently small absolute constant ε > 0.
+With Corollary 5.4 at hand, we are now ready to prove Theorems 1.2 and 1.5.
+Proof of Theorem 1.2. Let k′
+0 ∈ N and α′, ε′ > 0 denote the absolute constants from Corollary
+5.4.
+Let k0 := max{k′
+0, 3}. We set α ≥ 1 and ε > 0 in such a way that for all d ≥ k ≥ k0 and
+n ≥ αd8k6 it holds that
+�n
+2
+�
+≥ α′(d + 1)8(k + 1)6
+and
+�n
+2 − 1
+�ε′k
+− d ≥ nεk.
+Now, let us fix some d ≥ k ≥ k0 and n ≥ αd8k6. Let d′ := d + 1, k′ := k and n′ := ⌊ n
+2 ⌋. We
+apply Corollary 5.4 with parameters d′, k′, n′ and obtain a set of XOR-constraints C of arity at
+most k′ over a set V ′ of size |V ′| ≤ n′ such that Falsifier wins the k′-pebble game Gk′(V ′, C, ∅),
+but does not win the r′-round d′-pebble game Gr′
+d′(V ′, C, ∅) for r′ = (n′)ε′k′.
+Let A := A(C) and B := B(C). Then |V (A)| = |V (B)| = 2|V ′| ≤ 2n′ ≤ n. Note that we can
+easily increase the size of both structures by adding isolated elements that do not participate in
+any relations. Also, note that both structures have arity at most k′ = k.
+By Lemma 5.1 and Corollary 2.2, k-WL distinguishes between A and B. On the other hand,
+again by Lemma 5.1 and Corollary 2.2, d-WL does not distinguish A and B after r := r′ − d
+refinement rounds. We get that
+r = r′ − d = (n′)ε′k′ − d ≥
+�n
+2 − 1
+�ε′k
+− d ≥ nεk.
+Proof of Theorem 1.5. This follows directly from Corollary 5.4 and Lemma 5.1.
+The remainder of this section is devoted to the proof of Lemma 5.2.
+16
+
+5.2
+The Closure of the Constraint Set
+The critical step in the proof of Lemma 5.2 is to argue that Verifier survives a linear number
+of rounds even for a large number of pebbles. Here, we rely on an alternative description of
+winning positions in terms of a closure operator.
+Let k ∈ N. Let V be a finite set and let C be a set of XOR-constraints over V of arity at
+most k. We define the k-attractor
+attrk(C) := C ∪
+�
+(C1 ⊕ C2, a1 + a2 mod 2)
+��� (C1, a1), (C2, a2) ∈ C, |C1 ⊕ C2| ≤ k
+�
+.
+Here, C1 ⊕ C2 denotes the symmetric difference between the two sets, that is, C1 ⊕ C2 :=
+(C1 ∪ C2) \ (C1 ∩ C2).
+Intuitively speaking, if x1 + · · · + xℓ ≡ a1 mod 2 and y1 + · · · + ym ≡ a2 mod 2 are two
+constraints in C, then every satisfying assignment also has to satisfy the equation x1 + · · · +
+xℓ + y1 + · · · + ym ≡ a1 + a2 mod 2. Since all variables appearing in both sets {x1, . . . , xk} and
+{y1, . . . , ym} cancel over F2, we only need to keep those variables appearing in the symmetric
+difference.
+In the case that the resulting number of variables is bounded by k, we add the
+corresponding equation to the k-attractor of the constraint set.
+We define cl(0)
+k (C) := C and cl(r+1)
+k
+(C) := attrk(cl(r)
+k (C)) for all r ≥ 0. Finally, we define the
+k-closure of C to be the set clk(C) := cl(r)
+k (C) for the minimal r ≥ 0 such that cl(r+1)
+k
+(C) = cl(r)
+k (C).
+The following lemma provides the key method to prove that Verifier can survive a certain
+number of rounds.
+Lemma 5.5. Let β : X → {0, 1} be a partial assignment with |X| ≤ k such that β violates no
+XOR-constraint (C, a) ∈ cl(r)
+k (C). Then Verifier wins Gr
+k(V, C, β).
+Proof. We prove the statement by induction on r. For r = 0 the statement is trivial. So suppose
+r ≥ 1 and Spoiler chooses x ∈ V \ X and X′ ⊆ X such that |X′ ∪ {x}| ≤ k in the first round.
+For b ∈ {0, 1} let βb : X′ ∪ {x} → {0, 1} be the partial assignment with βb(x′) = β(x′) for
+x′ ∈ X′ and βb(x) = b. Assume towards a contradiction that, for every b ∈ {0, 1}, there is some
+XOR-constraint (Cb, ab) ∈ cl(r−1)
+k
+(C) violated by βb. Observe that x ∈ Cb for both b ∈ {0, 1}
+(since otherwise β would violate (Cb, ab) contradicting our assumption). Let C := C0 ⊕ C1 ⊆ X
+and a := (a0 + a1) mod 2. Note that |C| ≤ k since C ⊆ X. Then
+�
+y∈C
+β(y) ≡
+�
+y∈C0\{x}
+β(y) +
+�
+y∈C1\{x}
+β(y) ≡ 1 +
+�
+y∈C0
+β0(y) +
+�
+y∈C1
+β1(y) ≡ 1 + a
+mod 2
+and (C, a) ∈ cl(r)
+k (C). Hence, β violates some (C, a) ∈ cl(r)
+k (C) which is a contradiction.
+So there is some b ∈ {0, 1} such that βb violates no XOR-constraint in cl(r−1)
+k
+(C). Verifier
+chooses such a b ∈ {0, 1} and the game moves to position βb which violates no XOR-constraint
+in cl(r−1)
+k
+(C). So Verifier wins Gr−1
+k
+(V, C, βb) by the induction hypothesis which implies that
+Verifier also wins Gr
+k(V, C, β).
+5.3
+Layered Graphs and Expansion
+Next, we discuss the construction of certain expander graphs. Overall, we are aiming to con-
+struct what we refer to as single-neighbor layered expanders. Towards this end, we start with
+constructing standard bipartite expander graphs with an expansion that is close to the minimum
+degree of one side of the bipartite graph. We then define single-neighbor expanders and observe
+that bipartite expanders with large expansion also are single-neighbor expanders (with a slightly
+smaller expansion parameter). Finally, we obtain single-neighbor layered expanders by “stacking
+single-neighbor expanders on top of each other”.
+17
+
+5.3.1
+Expander Graphs
+We start by defining standard bipartite expander graphs.
+Definition 5.6. Let 0 < γ < 1 and α > 1 be constants and let G = (V, W, E) be a bipartite
+graph. We say that G is an (α, γ)-expander if for every ∅ ̸= Y ⊆ W with |Y | ≤ γ|W| it holds
+that
+N(Y ) ≥ α|Y |.
+For more information on expander graphs we refer to [18, 24]. The references also contain
+variants of the following standard argument that guarantees the existence of graphs with good
+expansion properties. For our purposes, the crucial property in the lemma below is that the
+expansion α is relatively close to the degree of the vertices in W.
+Lemma 5.7. There is some number R0 ≥ 2 such that for every r ≥ R0 and every n ≥ 4r there
+is a ( 3
+4r,
+1
+20r)-expander G = (V, W, E) such that |V | = |W| = n and deg(w) = r for all w ∈ W.
+Proof. Suppose r is sufficiently large. Let V, W be two sets with |V | = |W| ≥ 4r. We construct
+a bipartite graph G = (V, W, E) using the following random process: for each w ∈ W we select
+independently and uniformly at random a set of r distinct neighbors from V . We prove that,
+for r sufficiently large, with positive probability the graph G is a ( 3
+4r,
+1
+20r)-expander.
+Let n := |V | = |W|. For X ⊆ V and Y ⊆ W let pX,Y denote the probability that N(Y ) ⊆ X.
+Then
+pX,Y ≤
+�|X|
+n
+�r·|Y |
+.
+Furthermore, let α := 3
+4r and γ :=
+1
+20r. Let p be the probability that G is not a (γ, α)-expander.
+Then, using the inequality
+�n
+k
+�
+≤ (ne/k)k, we get
+p ≤
+�
+Y ⊆W
+|Y |≤γ·n
+�
+X⊆V
+|X|=⌊α|Y |⌋
+pX,Y
+≤
+⌊γ·n⌋
+�
+s=1
+�
+Y ⊆W
+|Y |=s
+�
+X⊆V
+|X|=⌊α|Y |⌋
+�|X|
+n
+�r·|Y |
+≤
+⌊γ·n⌋
+�
+s=1
+�n
+s
+�� n
+⌊αs⌋
+� �αs
+n
+�r·s
+≤
+⌊γ·n⌋
+�
+s=1
+�ne
+s
+�s �ne
+αs
+�α·s �αs
+n
+�r·s
+=
+⌊γ·n⌋
+�
+s=1
+��ne
+s
+� �ne
+αs
+�α �αs
+n
+�r�s
+=
+⌊γ·n⌋
+�
+s=1
+�� s
+n
+�r−α−1
+e1+ααr−α
+�s
+=
+⌊γ·n⌋
+�
+s=1
+�� s
+n
+�r/4−1
+e1+3r/4(3r/4)r/4
+�s
+≤
+⌊γ·n⌋
+�
+s=1
+�
+γr/4−1e1+3r/4(3r/4)r/4�s
+.
+18
+
+Now let x := γr/4−1e1+3r/4(3r/4)r/4. For r sufficiently large we get
+x = (20r)1−r/4e1+3r/4(3r/4)r/4 = 20er
+�3e3
+80
+�r/4
+< 1/10.
+It follows that
+p ≤
+∞
+�
+s=1
+xs =
+x
+1 − x ≤ 1
+9.
+In particular, p < 1 which implies the existence of the desired expander graph.
+Next, we turn to what we call single-neighbor expanders where each sufficiently small set
+Y ⊆ V is required to have a large number of neighbors that additionally satisfy the property
+that they are the neighbor of only a single vertex from Y . Let G = (V, W, E) be a bipartite
+graph. For Y ⊆ W we define
+N∗(Y ) = {v ∈ N(Y ) | |N(v) ∩ Y | = 1}.
+Definition 5.8. Let 0 < γ < 1 and α > 1 be constants and let G = (V, W, E) be a bipartite
+graph.
+We say that G is an (α, γ)-single-neighbor expander if for every ∅ ̸= Y ⊆ W with
+|Y | ≤ γ|W| it holds that
+N∗(Y ) ≥ α|Y |.
+We can obtain single-neighbor expanders from Lemma 5.7 by allowing some loss on the
+expansion parameter α.
+Corollary 5.9. There is some number R0 ≥ 5 such that for every r ≥ R0 and every n ≥ 4r there
+is a ( 1
+4r,
+1
+20r)-single-neighbor expander G = (V, W, E) such that |V | = |W| = n and deg(w) = r
+for all w ∈ W.
+Proof. Choose R0 := max(5, R′
+0) where R′
+0 is the constant from Lemma 5.7 and suppose r ≥ R0
+and n ≥ 4r. By Lemma 5.7, there is a ( 3
+4r,
+1
+20r)-expander G = (V, W, E) such that |V | = |W| = n
+and deg(w) = r for all w ∈ W. We claim that G is a ( 1
+4r,
+1
+20r)-single-neighbor expander. Let
+Y ⊆ W with |Y | ≤
+n
+20r.
+Then |N(Y )| ≥
+3
+4r|Y |.
+Furthermore |N(Y )| = |N∗(Y )| + |{v ∈
+N(Y ) | |N(v) ∩ Y | ≥ 2}| ≤ |N∗(Y )| + 1
+2r|Y | because every vertex in Y has degree r. Thus,
+|N∗(Y )| ≥ 1
+4r|Y |.
+5.3.2
+Layered Graphs
+Now, we turn to the construction of single-neighbor layered expanders which is the main tool
+for constructing the desired constraint sets in the proof of Lemma 5.2. We start by defining a
+certain notion of layered graphs (see also Figure 1).
+Let ℓ, m ∈ N. An (ℓ × m)-layered graph is a bipartite graph G = (V, W, E) for which there
+are partitions V = V0 ⊎ · · · ⊎ Vℓ and W = W1 ⊎ · · · ⊎ Wℓ such that
+1. |Vi| = m for all i ∈ [0, ℓ],
+2. |Wi| = m for all i ∈ [ℓ],
+3. NG(Wi) ⊆ Vi−1 ∪ Vi for all i ∈ [ℓ], and
+4. G[Vi ∪ Wi] is 1-regular (i.e., a matching) for all i ∈ [ℓ].
+With this, we are now ready to define the notion of single-neighbor layered expanders.
+19
+
+V0
+W1
+V1
+W2
+Wℓ
+Vℓ
+...
+...
+. . .
+. . .
+. . .
+. . .
+. . .
+. . .
+Figure 1: Visualization of (ℓ × m)-layered graphs.
+Definition 5.10. Let ℓ, m ≥ 2. Let 0 < γ < 1 and α > 1 be constants and let G = (V, W, E) be
+an (ℓ × m)-layered graph. We say that G is an (α, γ)-single-neighbor (ℓ × m)-layered expander
+if for every ∅ ̸= Y ⊆ W with |Y | ≤ γm it holds that
+N∗(Y ) ≥ α|Y |.
+Note that an (α, γ)-single-neighbor (ℓ × m)-layered expander is not a (α, γ)-single-neighbor
+expander since we are only considering sets Y ⊆ W of size |Y | ≤ γm, i.e., we are only considering
+sets that are smaller (by a factor of γ) than a single layer of a layered graph. In particular, the
+reader is encouraged to think of ℓ being much larger than m. In this case, such a graph is far
+from being a (global) expander, but the key property is that it behaves like an expander when
+only considering a few layers of the graph.
+By again allowing some small loss on the expansion parameter α, we can obtain single-
+neighbor layered expanders by “stacking ℓ copies of a single-neighbor expander on top of each
+other”.
+Corollary 5.11. There is some number R0 ≥ 9 such that for every r ≥ R0, every ℓ ≥ 1, and
+every m ≥ 4r there is a ( 1
+4r − 1,
+1
+20r)-single-neighbor (ℓ × m)-layered expander G = (V, W, E)
+with partitions V = V0 ⊎ · · · ⊎ Vℓ and W = W1 ⊎ · · · ⊎ Wℓ such that NG(w) ∩ Vi−1 = r for all
+w ∈ Wi and all i ∈ [ℓ].
+Proof. Choose R0 := max(9, R′
+0) where R′
+0 is the constant from Corollary 5.9 and suppose
+r ≥ R0, ℓ ≥ 1, and m ≥ 4r. By Corollary 5.9, there is a ( 1
+4r,
+1
+20r)-single-neighbor expander
+G′ = (V ′, W ′, E′) such that |V ′| = |W ′| = m and degG′(w′) = r for all w′ ∈ W ′. Suppose
+V ′ = {v′
+1, . . . , v′
+m} and W ′ = {w′
+1, . . . , w′
+m}.
+We set Vi := {vi,1, . . . , vi,m} for all i ∈ [0, ℓ] and Wi := {wi,1, . . . , wi,m} for all i ∈ [ℓ]. Also,
+we set
+E := {vi−1,jwi,k | i ∈ [ℓ], v′
+jw′
+k ∈ E′} ∪ {vi,jwi,j | i ∈ [ℓ], j ∈ [m]}.
+Clearly, G = (V, W, E) is an (ℓ × m)-layered graph.
+Let α := 1
+4r and γ :=
+1
+20r. Also let Y ⊆ W such that |Y | ≤ γm. We define Yi := Y ∩ Wi
+for all i ∈ [ℓ]. Observe that |Yi| ≤ γm for all i ∈ [ℓ] and Y1, . . . , Yℓ forms a partition of Y . Let
+I := {i ∈ [ℓ] | Yi ̸= ∅}. Since G′ is an (α, γ)-single-neighbor expander, we conclude that
+|N∗(Yi) ∩ Vi−1| ≥ α|Yi|
+20
+
+for all i ∈ I. Moreover, since G[Vi ∪Wi] is 1-regular (i.e., a matching) for all i ∈ [ℓ], we conclude
+that
+|N∗(Y ) ∩ Vi−1| ≥ α|Yi| − |Yi−1|
+for all i ∈ I (we set Y0 := ∅). So overall
+|N∗(Y )| ≥
+�
+i∈I
+α|Yi| − |Yi−1| ≥
+�
+i∈I
+(α − 1)|Yi| = (α − 1)|Y |
+as desired.
+5.4
+Constraint Sets from Layered Expanders
+Now, we turn to the construction of constraint sets from layered graphs. For a bipartite graph
+G = (V, W, E) we define the XOR-constraint set CG := {(N(w), 0) | w ∈ W} over the variable
+set V . Slightly abusing notation, for C ⊆ V , we shall also write C ∈ CG if (C, 0) ∈ CG.
+The basic idea for the construction of the XOR-constraint set C is to take a layered graph
+G = (V, W, E) with partitions V = V0 ⊎ · · · ⊎ Vℓ and W = W1 ⊎ · · · ⊎ Wℓ, and set
+C := CG ∪
+�
+({x}, 0)
+�� x ∈ V0
+�
+∪
+�
+({xℓ}, 1)
+�
+for some arbitrary xℓ ∈ Vℓ. It is not difficult to see that this constraint set is unsatisfiable.
+Indeed, every variable in layer V0 needs to be set to 0, and if all variables in layer Vi−1 are set to
+0, then the constraints obtained from the vertices in Wi enforce that every variable in layer Vi
+needs to be set to 0 as well (using that G[Vi ∪ Wi] is a matching). This inductive argument can
+be easily turned into a winning strategy for Falsifier that requires O(ℓ) many rounds (assuming
+the degree of all vertices in V is bounded by some absolute constant d ≤ k where k denotes the
+number of variables available in the game).
+Now, the central claim is that, if we start with a single-neighbor layered expander, this
+strategy is essentially optimal.
+Let us suppose for the moment that only constraints from
+CG are present and consider the k-closure clk(CG).
+What we need to avoid is that clk(CG)
+contains some constraint that is “non-local”. For example, if clk(CG) would contain a constraint
+({x1, x2, x3}, 0) such that x1, x2 ∈ V0 and x3 ∈ Vℓ, then Falsifier could use such a (derived)
+constraint to immediately conclude that certain variables in the last layer need to be set to 0
+and potentially follow a different strategy to win the game faster. The main point is that, by
+using single-neighbor layered expanders, we ensure that all “relevant” constraints in clk(CG) are
+“local”, i.e., they can only contain variables of O(k) consecutive layers. (Here, the reader may
+note that if |N(v1) ∪ N(v2)| ≤ k then (N(v1) ⊕ N(v2), 0) is always contained in the closure even
+if v1 and v2 are far apart. However, in such a case, N(v1)∩N(v2) = ∅ and the derived constraint
+(N(v1) ∪ N(v2), 0) is not “relevant” since, whenever it is violated by a partial assignment, one
+of the constraints associated with v1 or v2 is also violated.) This way, even when adding all
+constraints from clk(CG) to the initial set, the best that Falsifier can do is essentially to follow
+the above inductive strategy (with the exception that Falsifier may skip up to O(k) layers in
+one step which, however, does not cause any problems for our arguments).
+For technical reasons, the formal arguments slightly deviate from the intuitive ideas described
+above. To start, instead of working with the k-closure clk(CG), it turns out to be more convenient
+to work with the following set.
+Let α > 1 and k ≥ 1. We define the set
+cl∗
+k,α(CG) :=
+���
+D∈D
+D, 0
+� ����� ∅ ̸= D ⊆ CG, |D| ≤ k
+α,
+���
+�
+D∈D
+D
+��� ≤ k
+�
+.
+The next lemma implies that clk(CG) ⊆ cl∗
+k,α(CG) if G is a suitable single-neighbor layered
+expander.
+21
+
+Lemma 5.12. Suppose α > 1 and 0 < γ < 1. Let G = (V, W, E) be an (α, γ)-single-neighbor
+(ℓ × m)-layered expander such that deg(w) ≤ d for all w ∈ W and suppose d ≤ k ≤ 1
+2γm. Then
+attrk
+�
+cl∗
+k,α(CG)
+�
+= cl∗
+k,α(CG).
+Proof. Let C∗ := cl∗
+k,α(CG). Suppose C ∈ attrk(C∗), that is, there are C1, C2 ∈ C∗ such that C :=
+C1 ⊕C2 and |C| ≤ k. By definition, there are integers s, t ≤ k
+α and D1, . . . , Ds, Ds+1, . . . , Ds+t ∈
+CG such that C1 = D1⊕· · ·⊕Ds and C2 = Ds+1⊕· · ·⊕Ds+t. Moreover, D1, . . . , Ds are pairwise
+distinct as well as Ds+1, . . . , Ds+t are pairwise distinct. We have C = D1 ⊕ · · · ⊕ Ds+t. Let
+D := {D1, . . . , Ds} ⊕ {Ds+1, . . . , Ds+t}
+and let Y := {w ∈ W | N(w) ∈ D}. Clearly, C = �
+D∈D D. Suppose towards a contradiction
+that |D| > k
+α. Then |Y | > k
+α and moreover, |Y | ≤ s + t ≤ 2 k
+α ≤ 2k ≤ γm and thus, |N∗(Y )| ≥
+α|Y | > k. But on the other hand N∗(Y ) ⊆ C which implies that |N∗(Y )| ≤ k. This is a
+contradiction. So |D| ≤ k
+α which implies that C ∈ C∗ as desired.
+Lemma 5.13. Suppose α > 1 and 0 < γ < 1. Let G = (V, W, E) be an (α, γ)-single-neighbor
+(ℓ × m)-layered expander such that deg(w) ≤ d for all w ∈ W and suppose d ≤ k ≤ 1
+2γm. Then
+|C| ≥ 2 for all C ∈ cl∗
+k,α(CG).
+Proof. Let C ∈ cl∗
+k,α(CG) and let C1, . . . , Cs ∈ CG such that C = C1⊕· · ·⊕Cs for some s ≤ k
+α ≤ k.
+Furthermore, let Y := {w ∈ W | ∃i ∈ [s]: N(w) = Ci}. Observe that 1 ≤ |Y | ≤ k ≤ γm. Then
+N∗(Y ) ⊆ C and thus, |C| ≥ |N∗(Y )| ≥ α|Y | > 1.
+Next, we prove that Falsifier wins the pebble game if we set all variables in layer V0 to 0,
+and a single variable in the last layer Vℓ to 1. For technical reasons, we do not add ({xℓ}, 1) to
+the constraint set, but rather consider an initial assignment that assigns value 1 to variable xℓ.
+Lemma 5.14. Let G = (V, W, E) be an (ℓ × m)-layered graph with partitions V = V0 ⊎ · · · ⊎ Vℓ
+and W = W1⊎· · ·⊎Wℓ such that deg(w) ≤ k for all w ∈ W. Let xℓ ∈ Vℓ and suppose βℓ : {xℓ} →
+{0, 1} is the partial assignment defined via βℓ(xℓ) = 1. Then Falsifier wins Gk(W, C, βℓ) where
+C := CG ∪
+�
+({x}, 0)
+�� x ∈ V0
+�
+.
+Proof. We prove by induction on i = 0, . . . , ℓ that Falsifier wins Gk(V, C, βi) where βi is any
+partial assignment for which βi(xi) = 1 for some xi ∈ Vi.
+The base case i = 0 is trivial since ({x0}, 0) ∈ C for every x0 ∈ V0.
+For the inductive
+step, suppose i ∈ [ℓ] and consider some partial assignment βi for which there is some xi ∈ Vi
+such that βi(xi) = 1.
+Since G = (V, W, E) is an (ℓ × m)-layered graph, there is a unique
+vertex wi ∈ Wi such that wixi ∈ E. Moreover, NG(wi) ⊆ Vi−1 ∪ Vi. If NG(wi) = {xi}, then
+({xi}, 0) ∈ CG and Falsifier wins immediately.
+So suppose that NG(wi) ∩ Vi−1 ̸= ∅.
+Since
+deg(wi) ≤ k, Falsifier can move to a partial assignment βi−1 : Xi → {0, 1} where Xi = NG(wi)
+and βi−1(xi) = 1. If βi−1 violates the XOR-constraint (Xi, 0), then Falsifier wins immediately.
+Otherwise, �
+y∈Xi βi−1(y) = 0. Together with the fact that βi−1(xi) = 1, this implies that
+there is some xi−1 ∈ Xi ∩ Vi−1 such that βi−1(xi−1) = 1. So Falsifier wins by the induction
+hypothesis.
+The next lemma forms the key technical lemma stating that Falsifier requires a large number
+of rounds to win if the constraint set is obtained from a single-neighbor layered expander.
+Lemma 5.15. Suppose α > 1 and 0 < γ < 1. Let G = (V, W, E) be an (α, γ)-single-neighbor
+(ℓ × m)-layered expander with partitions V = V0 ⊎ · · · ⊎ Vℓ and W = W1 ⊎ · · · ⊎ Wℓ such that
+deg(w) ≤ d for all w ∈ W and suppose d ≤ k ≤ 1
+2γm.
+22
+
+Let xℓ ∈ Vℓ and suppose βℓ : {xℓ} → {0, 1} is the partial assignment defined via βℓ(xℓ) = 1.
+Then Verifier wins Gr−1
+k
+(V, C, βℓ) where
+C := CG ∪
+�
+({x}, 0)
+�� x ∈ V0
+�
+and r := ⌊ℓ/2k⌋.
+Proof. Let
+C∗
+G := cl∗
+k,α(CG) ∪ {(∅, 0)}
+and define C∗ := C∗
+G ∪ {({x}, 0) | x ∈ V0}. We show that Verifier wins Gr−1
+k
+(V, C∗, βℓ) which
+clearly implies the claim since C ⊆ C∗. By Lemma 5.5, it suffices to show that βℓ violates no
+XOR-constraint from the set cl(r−1)
+k
+(C∗), or equivalently ({xℓ}, 0) /∈ cl(r−1)
+k
+(C∗).
+We define
+Vi :=
+2ik
+�
+j=0
+Vj
+for all i ∈ {0, . . . , ⌊ℓ/2k⌋}. Finally, we define
+C∗
+i :=
+�
+C ⊆ V
+��� |C| ≤ k, C = D ⊕ U for some D ∈ C∗
+G, U ⊆ Vi
+�
+for all i ∈ {0, . . . , ⌊ℓ/2k⌋}.
+Claim 5.16. attrk(C∗
+i ) ⊆ C∗
+i+1 for all i ∈ {0, . . . , ⌊ℓ/2k⌋ − 1}.
+Proof. Let C1, C2 ∈ C∗
+i such that |C1 ⊕ C2| ≤ k. Let C := C1 ⊕ C2. For j ∈ {1, 2} pick Dj ∈ C∗
+G
+and Uj ⊆ Vi such that Cj = Dj ⊕Uj. Let U ′ := U1⊕U2. Clearly, U ′ ⊆ Vi and C = D1⊕D2⊕U ′.
+Let Yj ⊆ W, j ∈ {1, 2}, be a set of vertices of size |Yj| ≤ k
+α < k such that Dj = �
+w∈Yj N(w).
+Then there is some λ ∈ {2ik + 1, . . . , 2(i + 1)k} such that Wλ ∩ (Y1 ∪ Y2) = ∅. We define
+Y <λ
+j
+:= Yj ∩ W<λ
+where W<λ := �
+µ<λ Wµ and
+Y >λ
+j
+:= Yj ∩ W>λ
+where W>λ := �
+µ>λ Wµ. Moreover, let
+C>λ
+j
+:=
+�
+w∈Y >λ
+j
+N(w) ⊆ Cj
+for both j ∈ {1, 2} and
+C>λ := C>λ
+1
+⊕ C>λ
+2
+⊆ C.
+Hence, |C>λ
+j
+| ≤ k and |C>λ| ≤ k. So C>λ
+j
+∈ C∗
+G for both j ∈ {1, 2}. It follows that C>λ ∈ C∗
+G by
+Lemma 5.12.
+Now, C = C>λ ⊕ U for some U ⊆ V0 ∪ · · · ∪ Vλ−1 ⊆ Vi+1. It follows that C ∈ C∗
+i+1.
+⌟
+Since C∗ ⊆ C∗
+0 it follows by induction that
+cl(i)
+k (C∗) ⊆ C∗
+i
+(5)
+for all i ∈ {0, . . . , ⌊ℓ/2k⌋} using Claim 5.16. So it only remains the prove the following claim.
+Claim 5.17. {xℓ} /∈ C∗
+r−1.
+23
+
+Proof. Let C ∈ C∗
+r−1 such that C ∩Vℓ ̸= ∅. Also pick D ∈ C∗
+G and U ⊆ Vr−1 such that C = D⊕U
+(which exist by the definition of C∗
+r−1). We have that
+U ⊆ Vr−1 =
+2k(r−1)
+�
+i=0
+Vi ⊆
+ℓ−2k
+�
+i=0
+Vi.
+Let Y ⊆ W such that |Y | ≤ k
+α < k and D = �
+w∈Y N(w). Let λ ∈ [ℓ] be the maximal number
+such that Y ∩ Wλ = ∅. Note that λ > ℓ − k since |Y | < k. Now let D′ := �
+w∈Y ∩W>λ N(w)
+where W>λ := �
+µ>λ Wµ. Then D′ = C ∩ (Vλ ∪ · · · ∪ Vℓ) and hence, |D′| ≤ k. It follows that
+D′ ∈ cl∗
+k,α(CG) which implies that |D′| ≥ 2 using Lemma 5.13. So |C| ≥ 2.
+⌟
+Finally, we require one more technical lemma that allows us to add the XOR-constraint
+({xℓ}, 1) to the final constraint set.
+Lemma 5.18. Let k ≥ 2 and r ≥ 1. Let V be a finite set and let C be a set of XOR-constraints
+over V . Let x0 ∈ V and define β0 : {x0} → {0, 1} via β0(x0) = 1. If Verifier wins Gr
+k(V, C, β0),
+then Verifier also wins Gr
+k−1(V, C ∪ {({x0}, 1)}, ∅).
+Proof. Consider a position β : X → {0, 1} of the game Gr
+k−1(V, C ∪ {({x0}, 1)}, ∅). Throughout
+the game, by following a winning strategy for Gr
+k(V, C, β0), Verifier can maintain the following
+properties after every round ℓ ∈ [0, r]:
+(i) If x0 ∈ X, then β(x0) = 1, and
+(ii) Verifier wins the game Gr−ℓ
+k
+(V, C, β′) where β′ : X ∪ {x0} → {0, 1} is defined via β′(x) :=
+β(x) for all x ∈ X and β(x0) := 1.
+Observe that the condition is satisfied initially since Verifier wins Gr
+k(W, C, β0). All positions
+reached this way clearly satisfy all XOR-constraints in C ∪{({x0}, 1)} which implies that Verifier
+wins Gr
+k−1(W, C ∪ {({x0}, 1)}, ∅).
+With this, we are ready to prove Lemma 5.2.
+Proof of Lemma 5.2. Let R0 ≥ 9 denote the constant from Corollary 5.11 and define ℓlo :=
+R0 + 1. Let d := R0, α := 1
+4d − 1 > 1 and γ :=
+1
+20d. Let ℓhi ≥ ℓlo and r ≥ 1 be given. We define
+k := ℓhi + 1. Also, let m := 2 · k
+γ = 40dk ≥ 4d and ℓ := 2k(r + 1).
+By Corollary 5.11, there is an (α, γ)-single-neighbor (ℓ×m)-layered expander G = (V, W, E)
+such that deg(w) = d + 1 for all w ∈ W. Let V0, . . . , Vℓ and W1, . . . , Wℓ denote the layers of G.
+Also fix some arbitrary element xℓ ∈ Vℓ. We define
+C := CG ∪ {({x}, 0) | x ∈ V0} ∪ {({xℓ}, 1)}.
+Note that C is a set of XOR-constraints over V of arity at most d + 1 = ℓlo.
+To complete the proof, we show that C has the desired properties. First,
+|V | = (ℓ + 1)m = (2k(r + 1) + 1)40dk ≤ 8kr · 40dk = 320R0(ℓhi + 1)2r ≤ δ · ℓ2
+hi · r
+for some suitable absolute constant δ. Moreover, Falsifier wins the ℓlo-pebble game Gℓlo(V, C, ∅)
+by Lemma 5.14. Finally, by Lemma 5.15, Verifier wins Gr
+k(V, C\{({xℓ}, 1)}, βℓ) where βℓ : {xℓ} →
+{0, 1} is the partial assignment defined via βℓ(xℓ) = 1. So Verifier wins the r-round ℓhi-pebble
+game Gr
+ℓhi(V, C, ∅) by Lemma 5.18.
+24
+
+6
+Trading Variable Number for Quantifier Depth
+In this section, we investigate tradeoffs between the number of variables and the quantifier rank
+of formulas used to distinguish relational structures. More concretely, suppose A and B are
+two structures of size n that are distinguished by k-WL. By Corollary 2.2, there is a formula
+ϕ ∈ Ck+1 such that A |= ϕ and B ̸|= ϕ.
+Using Theorem 1.1, we may assume that ϕ has
+quantifier rank at most O(knk−1 log n). In this section, we show that there are sentences ψ
+that distinguish between A and B with smaller quantifier rank if we are allowed to increase the
+number of variables by some function in k. In other words, we can show improved bounds on
+the number of WL-iterations required to distinguish between A and B (compared to Theorem
+1.1) by increasing the dimension of the WL-algorithm.
+Theorem 6.1 (Theorem 1.6 restated). Let k ≥ 2. Let A and B be two relational structures of
+arity at most k such that n := |V (A)| = |V (B)|. Also suppose there is a sentence ϕ ∈ Ck+1 such
+that A |= ϕ and B ̸|= ϕ. Let d := ⌈ 3(k+1)
+2
+⌉. Then there is a sentence ψ ∈ C(q)
+d
+of quantifier rank
+q = O(k2 · n⌊k/2⌋+1 log n) such that A |= ψ and B ̸|= ψ.
+Toward the proof of this theorem, let us fix some k ≥ 2 and suppose that k is odd, i.e.,
+k = 2ℓ − 1 from some integer ℓ ≥ 2 (this is the crucial case). Let A be a relational structure of
+arity at most k. We translate A into a binary structure (i.e., a structure of arity at most two)
+Bin(A) defined as follows. The universe of Bin(A) is set to
+V (Bin(A)) := (V (A))ℓ.
+For every atomic type typ ∈ {atpA(v) | v ∈ (V (A))2ℓ} (on 2ℓ vertices) we introduce a binary
+relation symbol Rtyp and set
+RBin(A)
+typ
+:=
+��
+(v1, . . . , vℓ), (vℓ+1, . . . , v2ℓ)
+� �� atpA(v1, . . . , v2ℓ) = typ
+�
+.
+Now, the key idea behind the proof of Theorem 6.1 is to use d variables to simulate the
+execution of 2-WL on the binary structure Bin(A). We can then obtain the upper bound on the
+quantifier rank by exploiting that 2-WL stabilizes after at most O(n log n) rounds (see Theorem
+1.1).
+The next lemma translates a formula that distinguishes between Bin(A) and Bin(B) into a
+formula distinguishing A and B.
+Lemma 6.2. Let A and B be two relational structures of arity at most k. Suppose there is a
+sentence ϕ ∈ C(q)
+d
+such that Bin(A) |= ϕ and Bin(B) ̸|= ϕ. Then there is a sentence �ϕ ∈ C(q·ℓ)
+d·ℓ
+such that A |= �ϕ and B ̸|= �ϕ.
+The proof of the lemma is a standard syntactic translation (see, e.g., [21, Chapter 1.5]) and
+we omit the details here.
+Lemma 6.3. Let A and B be two relational structures of arity at most k such that Bin(A) ≃2
+Bin(B). Then A ≃k B.
+Proof. Consider an arbitrary structure C and define χ2 := χ(∞)
+2
+[Bin(C)] to be the coloring
+computed by 2-WL on the structure Bin(C). We define a coloring χ: (V (C))k → C by setting
+χ(v1, . . . , vk) := χ2((v1, . . . , vℓ), (vℓ+1, . . . , vk−1, vk, vk)).
+Claim 6.4. Suppose atpC(v1, . . . , vk) ̸= atpC(v′
+1, . . . , v′
+k). Then χ(v1, . . . , vk) ̸= χ(v′
+1, . . . , v′
+k).
+25
+
+Proof. Let typ := atpC(v1, . . . , vk−1, vk, vk). Then ((v1, . . . , vℓ), (vℓ+1, . . . , vk, vk)) ∈ RBin(C)
+typ
+, but
+on the other hand ((v′
+1, . . . , v′
+ℓ), (v′
+ℓ+1, . . . , v′
+k, v′
+k)) /∈ RBin(C)
+typ
+. So
+atpBin(C)((v1, . . . , vℓ), (vℓ+1, . . . , vk, vk)) ̸= atpBin(C)((v′
+1, . . . , v′
+ℓ), (v′
+ℓ+1, . . . , v′
+k, v′
+k))
+which implies that
+χ2((v1, . . . , vℓ), (vℓ+1, . . . , vk, vk)) ̸= χ2((v′
+1, . . . , v′
+ℓ), (v′
+ℓ+1, . . . , v′
+k, v′
+k)).
+This directly implies the claim.
+⌟
+Claim 6.5. χ is k-stable.
+Proof. Let v, v′ ∈ (V (C))k such that χ(v) = χ(v′).
+Suppose v = (v1, . . . , vk) and v′ =
+(v′
+1, . . . , v′
+k). Let us write v1 := (v1, . . . , vℓ) for the “first half” of v, and v2 := (vℓ+1, . . . , vk)
+for the “second half”.
+Note that v2 has only ℓ − 1 entries since k = 2ℓ − 1.
+Similarly, we
+define v′
+1 := (v′
+1, . . . , v′
+ℓ) and v′
+2 := (v′
+ℓ+1, . . . , v′
+k). For w ∈ V (C) we write v2 ◦ w for the tuple
+(vℓ+1, . . . , vk, w) obtained from v2 by appending w. The tuple v′
+2 ◦ w is defined analogously.
+Since χ2 is 2-stable and χ2(v1, v2 ◦ vk) = χ2(v′
+1, v′
+2 ◦ v′
+k), we conclude that
+���
+χ2(v1, w), χ2(w, v2 ◦ vk)
+� ��� w ∈ (V (C))ℓ��
+=
+���
+χ2(v′
+1, w), χ2(w, v′
+2 ◦ v′
+k)
+� ��� w ∈ (V (C))ℓ��
+.
+Using that χ2 refines the coloring by atomic types, it follows that
+���
+χ2(v1, v2 ◦ w), χ2(v2 ◦ w, v2 ◦ vk)
+� ��� w ∈ V (C)
+��
+=
+���
+χ2(v′
+1, v′
+2 ◦ w), χ2(v′
+2 ◦ w, v′
+2 ◦ v′
+k)
+� ��� w ∈ V (C)
+��
+.
+In particular, we get that
+��
+χ2(v1, v2 ◦ w)
+��� w ∈ V (C)
+��
+=
+��
+χ2(v′
+1, v′
+2 ◦ w)
+��� w ∈ V (C)
+��
+.
+Now let w, w′ ∈ V (C) such that χ2(v1, v2 ◦ w) = χ2(v′
+1, v′
+2 ◦ w′). Then
+χ(v[w/i]) = χ(v′[w′/i])
+for all i ∈ [k] using again that χ2 is 2-stable and refines the coloring by atomic types. It follows
+that
+���
+χ(v[w/1]), . . . , χ(v[w/k])
+� ��� w ∈ V (C)
+��
+=
+���
+χ(v′[w/1]), . . . , χ(v′[w/k])
+� ��� w ∈ V (C)
+��
+.
+Overall, this implies that χ is k-stable.
+⌟
+Combining both claims, we obtain that χ ⪯ χ(∞)
+k
+[C]. Now, we complete the proof by setting
+C to the disjoint union of A and B.
+Proof of Theorem 6.1. First suppose that k odd, i.e., k = 2ℓ − 1 for some integer ℓ ≥ 2. Since
+there is a sentence ϕ ∈ Ck+1 such that A |= ϕ and B ̸|= ϕ, we conclude that A ̸≃k B using
+Corollary 2.2. So Bin(A) ̸≃2 Bin(B) by Lemma 6.3. By Theorem 1.1, the 2-WL algorithm
+distinguishes between Bin(A) and Bin(B) after at most r = O(|V (Bin(A))| log |V (Bin(A))|) =
+O(ℓ·nℓ·log n) many refinement rounds. Using Corollary 2.2 again, this means there is a sentence
+ϕ′ ∈ C(r)
+3
+such that Bin(A) |= ϕ′ and Bin(B) ̸|= ϕ′. So there is a sentence ψ ∈ C(r·ℓ)
+3·ℓ
+such that
+A |= ψ and B ̸|= ψ using Lemma 6.2. Note that 3ℓ = 3 · k+1
+2
+= d and r · ℓ = O(ℓ2 · nℓ · log n) =
+O(k2 · n(k+1)/2 log n).
+For k being even, the statement the of theorem follows by applying the first case to k′ =
+k + 1.
+26
+
+7
+Conclusion
+We obtained new upper and lower bounds for the iteration number of the WL algorithm. First,
+we showed that k-WL always stabilizes after at most O(knk−1 log n) rounds for all k ≥ 2, which
+is the first non-trivial upper on the iteration number for k ≥ 3. We complemented this result by
+a lower bound of nΩ(k) which improves over the previously known lower bound of nΩ(k/ log k) [3].
+Finally, we also investigated tradeoffs between the dimension and the iteration number of WL.
+Using known characterizations of WL, our results also imply upper and lower bounds on the
+quantifier rank of formulas in Ck required to distinguish between two structures.
+Still, several questions remain open. The first question concerns the iteration number of
+k-WL on graphs. The structures on which our lower bounds hold are n-element structures of
+arity Θ(k) and size nΘ(k), and the increase in arity is inherent in the hardness condensation
+from [3]. The best known lower bound on the iteration number of k-WL on graphs is Ω(n) due
+to Fürer [5]. As an intermediate question, one can also ask for improved lower bounds in the
+size of the structure (i.e., the sum of the sizes of all relations), i.e., are there structures on which
+the iteration number of k-WL exceeds Ω(m) where m denotes the size of the structure?
+Our next question concerns the quantifier rank of formulas in Lk. While our lower bounds
+extend to the logic Lk (see Theorem 1.5), this is not the case for the upper bounds that crucially
+rely on the availability of counting quantifiers. A non-trivial upper bound of O(n2/ log n) on the
+quantifier rank of formulas in L3 has been obtained in [14]. Can we also obtain improved upper
+bounds on the quantifier rank of formulas in Lk for k ≥ 4?
+Finally, we ask for further results on tradeoffs between the variable number and the quantifier
+rank. Specifically, is there an integer d ≥ 3 such that, for all structures A and B of size n
+distinguished by 3-WL, d-WL distinguishes between A and B in at most �O(n) rounds (where
+�O(·) hides polylogarithmic factors)? We remark that even d = 3 may be a valid choice, but any
+d ≥ 3 is sufficient to obtain further tradeoffs in the spirit of Theorem 1.6.
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+
diff --git a/QtFQT4oBgHgl3EQfZzYk/content/tmp_files/load_file.txt b/QtFQT4oBgHgl3EQfZzYk/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..cd4fa5aef64db28c8ae7c18d32abbdb39b782e70
--- /dev/null
+++ b/QtFQT4oBgHgl3EQfZzYk/content/tmp_files/load_file.txt
@@ -0,0 +1,1722 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf,len=1721
+page_content='The Iteration Number of the Weisfeiler-Leman Algorithm  Martin Grohe  RWTH Aachen University  grohe@informatik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='rwth-aachen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='de  Moritz Lichter  TU Darmstadt  lichter@mathematik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='tu-darmstadt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='de  Daniel Neuen  Simon Fraser University  dneuen@sfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='ca  Abstract  We prove new upper and lower bounds on the number of iterations the k-dimensional  Weisfeiler-Leman algorithm (k-WL) requires until stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For k ≥ 3, we show that  k-WL stabilizes after at most O(knk−1 log n) iterations (where n denotes the number of  vertices of the input structures), obtaining the first improvement over the trivial upper  bound of nk − 1 and extending a previous upper bound of O(n log n) for k = 2 [Lichter et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', LICS 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  We complement our upper bounds by constructing k-ary relational structures on which  k-WL requires at least nΩ(k) iterations to stabilize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' This improves over a previous lower  bound of nΩ(k/ log k) [Berkholz, Nordström, LICS 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  We also investigate tradeoffs between the dimension and the iteration number of WL,  and show that d-WL, where d = ⌈ 3(k+1)  2  ⌉, can simulate the k-WL algorithm using only  O(k2 · n⌊k/2⌋+1 log n) many iterations, but still requires at least nΩ(k) iterations for any d  (that is sufficiently smaller than n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  The number of iterations required by k-WL to distinguish two structures corresponds to  the quantifier rank of a sentence distinguishing them in the (k + 1)-variable fragment Ck+1  of first-order logic with counting quantifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Hence, our results also imply new upper and  lower bounds on the quantifier rank required in the logic Ck+1, as well as tradeoffs between  variable number and quantifier rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  1  Introduction  The Weisfeiler-Leman (WL) algorithm is a combinatorial algorithm that, given a relational struc-  ture A (in most applications, this structure is a graph), iteratively computes an isomorphism-  invariant coloring of tuples of vertices of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' The original algorithm introduced by Weisfeiler  and Leman [26] is the 2-dimensional version that colors pairs of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Its generalization to  arbitrary dimension k ≥ 1, independently introduced by Babai and Mathon as well as Immer-  man and Lander [11] (see also [1] for a historic note), yields for every natural number k the  k-dimensional WL algorithm (k-WL), which iteratively refines a coloring of vertex k-tuples by  aggregating local structural information encoded in the colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' More concretely, the k-WL algo-  rithm initially colors all k-tuples of vertices v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk) of a structure A by the isomorphism  type of the underlying induced ordered substructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Afterwards, in each iteration, the coloring  is refined by taking the colors of all tuples into account that can be obtained from v by replacing  a single entry of the tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' This process necessarily stabilizes after a finite number of iterations  and the resulting coloring can be used to classify k-tuples of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  The most prominent application of the WL algorithm lies in the context of the graph isomor-  phism problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Indeed, since no isomorphism between two structures A and B can map tuples  of vertices of different colors to each other, the WL algorithm provides a hierarchy of increasingly  powerful heuristics to the graph isomorphism problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' While there is no dimension k for which  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='13317v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='DS]  30 Jan 2023  k-WL serves as a complete isomorphism test [4], the algorithm is still surprisingly powerful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  For example, Grohe [7] proved that for every non-trivial minor-closed graph class there is some  k ∈ N such that k-WL computes a different coloring on all non-isomorphic graphs, and thus  provides a polynomial-time isomorphism test on that class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Moreover, the WL algorithm is also  regularly used as a subroutine in isomorphism algorithms (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', [19, 20, 23]) which includes  Babai’s [1] quasipolynomial-time graph isomorphism test that employs the WL algorithm with  dimension k = O(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  More recently, the WL algorithm has also received significant attention in the machine  learning context where it characterizes the expressiveness of graph neural networks [8, 17, 27]  and, more generally, the colorings computed by WL are used in classification tasks on graph-  structured data sets (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', [16, 22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Since the late 1980s, the WL algorithm has played an important role in descriptive complexity  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Indeed, it was independently introduced in the context of descriptive complexity by  Immerman and Lander [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' The main reason for this is that k-WL can be seen as an equivalence  test for the logic Ck+1, the (k +1)-variable fragment of first-order logic with counting quantifiers  ∃≥nx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Through this connection, the algorithm has turned out to be important for studying the  expressiveness of fixed-point logic with counting [4] and, more generally, for the quest for a logic  capturing polynomial time [6, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  In this work, we study the iteration number of k-WL, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', the number of iterations the algo-  rithm requires until stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Since the number of color classes increases in each iteration,  the k-WL algorithm trivially requires at most nk − 1 rounds to stabilize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For k = 1, Kiefer and  McKay [12] proved that this trivial bound is optimal by providing several infinite families of  graphs G for which 1-WL requires n − 1 iterations to stabilize (where n denotes the number of  vertices of G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' In contrast, for k = 2, Lichter, Ponomarenko and Schweitzer [15] (improving an  earlier upper bound by Kiefer and Schweitzer [14]) obtained an upper bound of O(n log n) on  the iteration number of 2-WL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Beyond that, no improved upper bounds are known for k ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  As our first main contribution, we obtain non-trivial bounds on the iteration number of k-WL  for all k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For all k ≥ 2, the k-dimensional Weisfeiler-Leman algorithm stabilizes after  O(knk−1 log n) refinement rounds on all relational structures A of arity at most k where n  denotes the size of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  For the proof, we extend the algebraic arguments from [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Consider a structure A with  vertex set V of size n and let χ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , χℓ : V k → C denote the sequence of colorings computed  by k-WL, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', χi is the coloring computed in the i-th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For k = 2, Lichter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' [15]  associate with each coloring χi a matrix algebra as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For each color c in the image of  χi, let Mi,c denote the V × V indicator matrix that sets Mi,c(v1, v2) := 1 if χi(v1, v2) = c, and  Mi,c(v1, v2) := 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' The matrices Mi,c, where c ranges over all colors in the image of  χi, generate a matrix algebra A(i) of V × V matrices over the complex numbers using standard  matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Using representation-theoretic arguments, it is possible to bound the  length of the sequence of matrix algebras generated this way which eventually leads to the  upper bound of O(n log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1 follows a similar strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For each color in the image of χi, we  obtain an indicator tensor Mi,c ∈ CV k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Now, the key challenge in generalizing the arguments  of [15] is to define a suitable multiplication of those tensors that can be “simulated” by a single  round of k-WL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Given such a multiplication, we then show that the generated algebra A(i) is  isomorphic to a subalgebra of the nk−1 × nk−1 full matrix algebra (over the complex numbers)  which then again allows us to use algebraic arguments to obtain the desired upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Our arguments actually prove a more general result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let χ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , χℓ : V k → C be a sequence  of finer and finer colorings (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', the partition into color classes of χi refines the partition into  color classes of χi−1 for all i ∈ [ℓ]) where in each step the coloring is refined at least as much as  2  by a single iteration of k-WL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then the length of the sequence is bounded by ℓ = O(knk−1 log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  As a lower bound to our arguments, we show that, in this more general setting, our upper bound  is tight up to a factor Ok(log n) (the Ok(·)-notation hides constant factors in k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Here, the key  insight is that we can find a sequence of finer and finer colorings χ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , χℓ : V k → C of length  Ωk(nk−1) that are all stable with respect to k-WL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' As such, it provides a lower bound in the  more general setting explained above (but it does not give any lower bounds on the iteration  number of k-WL) and implies that new ideas are likely required to obtain further improvements  on the upper bounds of iteration number of k-WL (see Section 4 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Looking for lower bounds on the iteration of k-WL, Fürer [5] provided, for every k ≥ 2, a  family of graphs on which k-WL requires at least Ω(n) many iterations until stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For k  sufficiently large, this result was strengthened by Berkholz and Nordström [3] who constructed  k-ary relational structures A of size n on which k-WL requires at least nΩ(k/ log k) many iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Answering an open question from [3], our second main contribution is an improved lower bound  that gets rid of the 1/ log k factor in the exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Actually, we prove the following even stronger  result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' There are absolute constants k0 ∈ N and α, ε > 0 such that for every d ≥ k ≥ k0  and every n ≥ αd8k6 there is a is pair of k-ary relational structures A and B of size |V (A)| =  |V (B)| = n that are distinguished by k-WL, but d-WL does not distinguish A and B after nεk  refinement rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  We note that, as in the work of Berkholz and Nordström [3], the structures we need to prove  this theorem are k-ary, that is, have relations of arity k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  The structures A and B provided by the theorem can be distinguished by k-WL which  trivially requires at most nk − 1 rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' The theorem states that, even if we are allowed to  increase the dimension of the Weisfeiler-Leman algorithm to d, the structures can still not be  distinguished unless d-WL runs for at least nεk rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' This result stands in strong contrast to  several existing results for restricted classes of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For example, k-WL distinguishes between  all non-isomorphic pairs of graphs of tree-width at most k [13], and increasing the dimension  to 4k + 3 guarantees that O(log n) iterations suffices to distinguish between all non-isomorphic  pairs of graphs of tree-width at most k [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Similar results are known for planar graphs [9, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  The above theorem rules out such results for general relational structures even if we only wish  to improve the iteration number to, for example, linear in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  By setting d = k, we obtain the following corollary which shows that the upper bound in  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1 is optimal up to a constant factor (that does not depend on k) in the exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' There are absolute constants k0 ∈ N and α, ε > 0 such that for every k ≥ k0  and every n ≥ αk14 there is a k-ary structure A of size |V (A)| = n such that the k-dimensional  Weisfeiler-Leman algorithm does not stabilize within nεk refinement rounds on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  For the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2, our main technical contribution is to show that there is a  k0 ∈ N such that, for all d ≥ k0, there are structures A and B of size n that are distinguished by  k0-WL, but d-WL still requires Ω(n/d2) many iterations to distinguish A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Afterwards,  we obtain Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2 by using a known hardness condensation [3] that reduces the size of the  structures while roughly preserving the number of iterations required to distinguish them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Let us point out that Fürer [5] constructed graphs G and H which are distinguished by  k0-WL after Ω(n) many rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' However, as Fürer also shows, his instances are distinguished  by (3k0)-WL after only O(log n) many rounds which means that we cannot use them for our  purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Berkholz and Nordström [3] provided, for all d ≥ 2, structures A and B of size n that  are distinguished by 2-WL, but d-WL still requires Ω(n1/(1+log d)) many rounds to distinguish  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' In combination with the hardness condensation, this leads to the previous lower bound  of nΩ(k/ log k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  For the construction of our structures, we introduce the notion of layered expanders whose  global structure is similar to a (k × n)-grid, but that locally (when looking at O(k) consecutive  3  columns) behave like an expander graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We then obtain propositional XOR-formulas from  layered expanders which can be transformed into relational structures which satisfy the desired  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Connection to Logics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  As pointed out above, k-WL is an equivalence test for the logic Ck+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  That is, k-WL distinguishes between two structures A and B if and only if there is a sentence  ϕ ∈ Ck+1 such that A |= ϕ and B ̸|= ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Additionally, the minimal quantifier rank of such a  sentence equals (up to an additive error of at most k) the number of iterations k-WL requires to  distinguish between A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' With this in mind, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1 can be reformulated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let k ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let A and B be two relational structures of arity at most k that can  be distinguished by a sentence in Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then there is a sentence ϕ ∈ Ck of quantifier rank at most  q = O(knk−2 log n) such that A |= ϕ and B ̸|= ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Similarly, we can reformulate Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2, but here it turns out that we can obtain an even  stronger result since the structures constructed in the theorem can already be distinguished in  the logic Lk+1, the k + 1-variable fragment of first-order logic without counting quantifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' There are absolute constants k0 ∈ N and α, ε > 0 such that for every d ≥ k ≥ k0  and every n ≥ αd8k6 there is a pair of k-ary structures A and B of size |V (A)| = |V (B)| = n  that can be distinguished by a sentence in k-variable first-order logic Lk, but satisfy the same  sentences in Ld and Cd up to quantifier rank nεk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Hence, we obtain lower bounds for the quantifier rank not only for the logic Ck, but also  for the logic Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Observe that the lower bounds on the quantifier rank remain valid even if we  arbitrarily increase the number of variables to any number d (as long as d is sufficiently far away  from the size of the structures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' In other words, even if we are allowed to increase the number of  variables, we cannot in general hope for significant improvements on the quantifier rank required  to distinguish between two structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Having said that, our final result shows that at least some improvements on the upper bound  are possible if we are allowed to increase the number of variables by roughly a factor of 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let A and B be two relational structures of arity at most k such that  n := |V (A)| = |V (B)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Also suppose there is a sentence ϕ ∈ Ck+1 such that A |= ϕ and B ̸|= ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Let d := ⌈ 3(k+1)  2  ⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then there is a sentence ψ ∈ C(q)  d  of quantifier rank q = O(k2 · n⌊k/2⌋+1 log n)  such that A |= ψ and B ̸|= ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Structure of the Paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  After introducing the necessary preliminaries in the next section,  we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1 in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Afterwards, we prove limitations of our approach to obtain  improved upper bounds on the iteration number in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' In Section 5, we obtain the lower  bounds on the iteration number of WL and prove Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Finally, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='6  is proved in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  2  Preliminaries  We use N = {1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' } to denote the positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For n ∈ N we write [n] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , n}  and [0, n] := {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  We use standard graph notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' A graph is a pair G = (V (G), E(G)) with finite  vertex set V (G) and edge set E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' In this paper, all graphs are simple (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', there are no loops  or multiedges) and undirected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We write vw to denote an edge {v, w} ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' The (open)  neighborhood of a vertex v ∈ V (G) is the set NG(v) := {w ∈ V (G) | vw ∈ E(G)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' The degree  4  of a vertex, denoted by degG(v), is the size of its neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For X ⊆ V (G) we define  NG(X) := (�  v∈X NG(v)) \\ X to denote the neighborhood of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' If the graph G is clear from  context, we usually omit the index G and simply write N(v), deg(v) and N(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For X ⊆ V (G)  we also write G[X] to denote the subgraph of G induced by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Relational Structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  In this work, we restrict ourselves to relational vocabularies (sig-  natures) σ = {R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , Rm} where each Ri is a relation symbol of a prescribed arity ki ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  We say that σ has arity at most k if ki ≤ k for all i ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  A σ-structure is a tuple  A = (V (A), RA  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , RA  m) where V (A) is a finite universe and RA  i  ⊆ (V (A))ki is a relation  of arity ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' In the remainder of this work, we usually do not explicitly refer to the vocabulary  underlying a structure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' With this in mind, we say a structure A = (V (A), RA  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , RA  m) has  arity at most k if the underlying vocabulary has arity at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  For X ⊆ V (A) we define A[X] to be the induced substructure of A on X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', A[X] is the  relational structure with V (A[X]) = X and  RA[X]  i  = RA  i ∩ Xki  for all i ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let B = (V (B), RB  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , RB  m) be a second structure (over the same vocabulary  σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' An isomorphism from A to B is a bijection f : V (A) → V (B) such that, for all i ∈ [m] and  all v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vki ∈ V (A), it holds that  (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vki) ∈ RA  i  ⇐⇒ (f(v1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , f(vki)) ∈ RB  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  The structures A to B are isomorphic if there is an isomorphism from A to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Logics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Next, we cover bounded-variable fragments of first-order logic (with counting quanti-  fiers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let σ = {R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , Rm} be a relational vocabulary and suppose Ri has arity ki ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We  write FO to denote standard first-order logic defined via the grammar  ϕ ::= x1 = x2 | Ri(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , xki) | ϕ ∧ ϕ | ¬ϕ | ∃x1ϕ  for all i ∈ [m] and all variables xj ∈ V where V is an infinite set of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  We write  ϕ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , xk) to indicate that the free variables of ϕ are among the variables {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , xk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For  a structure A = (V (A), RA  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , RA  m) and v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk) ∈ (V (A))k we write A |= ϕ(v) if A is  a model of ϕ when xi is interpreted by vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  We define the quantifier rank of a formula ϕ ∈ FO inductively via  qr(x1 = x2) = qr(Ri(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , xki)) := 0 for all i ∈ [m] and all variables xj ∈ V,  qr(ϕ ∧ ψ) := max(qr(ϕ), qr(ψ)),  qr(¬ϕ) := qr(ϕ), and  qr(∃xϕ) := qr(ϕ) + 1 for all x ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  We define first-order logic with counting quantifiers C to be the extension of FO by counting  quantifiers of the form ∃≥jxϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' The formula ∃≥jxϕ is satisfied over a structure A if there are  at least j distinct elements v ∈ V (A) that satisfy ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We extend the definition of the quantifier  rank in the natural way by setting qr(∃≥jxϕ) := qr(ϕ) + 1 for all x ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  For k ∈ N we define Lk to be the restriction of FO to formulas over at most k variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=',  we restrict ourselves to a set of variables V of size exactly k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Similarly, we define Ck to be the  restriction of C to formulas over at most k variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Moreover, for q ≥ 0, we define L(q)  k  to the restriction of Lk to formulas ϕ of quantifier rank  qr(ϕ) ≤ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Similarly, we define C(q)  k  to the restriction of Ck to formulas of quantifier rank at  most q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  5  The Weisfeiler-Leman Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Next, we describe the k-WL algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  While it is  most commonly used as a heuristic to graph isomorphism, the algorithm can be applied to any  relational structure of arity at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Let χ1, χ2 : V k → C be colorings of k-tuples over a finite set V where C is some finite set of  colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' The coloring χ1 refines χ2, denoted χ1 ⪯ χ2, if χ1(v) = χ1(w) implies χ2(v) = χ2(w) for  all v, w ∈ V k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Observe that χ1 ⪯ χ2 if and only if the partition into color classes of χ1 refines  the corresponding partition into color classes of χ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' The colorings χ1 and χ2 are equivalent,  denoted χ1 ≡ χ2, if χ1 ⪯ χ2 and χ2 ⪯ χ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Also, χ1 strictly refines χ2, denoted χ1 ≺ χ2, if  χ1 ⪯ χ2 and χ1 ̸≡ χ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Let us fix k ≥ 2 and consider a relational structure A = (V (A), RA  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , RA  m) of arity at most  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk) ∈ (V (A))k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We define the atomic type of v, denoted by atpA(v), to be  the isomorphism type of the ordered substructure of A that is induced by {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' More  concretely, for a second structure B = (V (B), RB  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , RB  m) and a tuple w = (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , wk) ∈  (V (B))k, it holds that atpA(v) = atpB(w) if the mapping vi �→ wi is an isomorphism from  A[{v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk}] to B[{w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , wk}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Next, we describe a single refinement step of k-WL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let V be a finite set and let χ: V k → C  be a coloring of all k-tuples over V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We define the coloring stepk(χ) by setting  �  stepk(χ)  �  (v) :=  �  χ(v), Mχ(v)  �  for all v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk) ∈ V k where  Mχ(v) :=  ���  χ(v[w/1]), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , χ(v[w/k])  � ��� w ∈ V  ��  and v[w/i] := (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vi−1, w, vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk) is the tuple obtained from v by replacing the i-th  entry by w (and {{.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' }} denotes a multiset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Observe that stepk(χ) ⪯ χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We say the coloring χ  is k-stable if stepk(χ) ≡ χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  We define the initial coloring computed by k-WL on the structure A via χ(0)  k [A](v) := atpA(v)  for all v ∈ (V (A))k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For r ≥ 0 we set  χ(r+1)  k  [A] := stepk  �  χ(r)  k [A]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Since χ(r+1)  k  [A] ⪯ χ(r)  k [A] for all r ≥ 0, there is some minimal r∞ ≤ |V |k − 1 such that  χ(r∞)  k  [A] ≡ χ(r∞+1)  k  [A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  We say that k-WL stabilizes after r∞ rounds on A and define χ(∞)  k  [A] := χ(r∞)  k  [A] to be the  output coloring of k-WL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Observe that χ(∞)  k  [A] is a k-stable coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Now, let B = (V (B), RB  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , RB  m) be a second structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We say that k-WL  distinguishes A and B after r rounds if there is some color c such that  ���  �  v ∈ (V (A))k ��� χ(r)  k [A](v) = c  ���� ̸=  ���  �  w ∈ (V (B))k ��� χ(r)  k [B](w) = c  ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  We also say that k-WL distinguishes A and B if there is some integer r ≥ 0 such that k-WL  distinguishes A and B after r rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We write A ≃k B if k-WL does not distinguish A and  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Note that, if k-WL distinguishes A and B and k-WL stabilizes after r∞ rounds on A, then  k-WL distinguishes A and B after (at most) r∞ + 1 rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  The following connections to bounded-variable fragments of first-order logic with counting  quantifiers are well-known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1 ([4, 11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Also let A and B be structures of arity at most k and suppose  v ∈ V (A)k and w ∈ V (B)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then, for every r ≥ 0, it holds that χ(r)  k [A](v) ̸= χ(r)  k [B](w) if and  only if there is some ϕ(x) ∈ C(r)  k+1 such that A |= ϕ(v) and B ̸|= ϕ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  6  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Also let A and B be structures of arity at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  If there is a sentence ϕ ∈ C(r)  k+1 such that A |= ϕ and B ̸|= ϕ, then the k-dimensional  Weisfeiler-Leman algorithm distinguishes A and B after at most r refinement rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  If the k-dimensional Weisfeiler-Leman algorithm distinguishes A and B after r refinement  rounds, then there is a sentence ϕ ∈ C(r+k)  k+1  such that A |= ϕ and B ̸|= ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  3  Upper Bounds  In this section, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Actually, we prove a more general result on the maximal  iteration number of any refinement method that is at least as strong as k-WL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  For the remainder of this section, let us fix some integer k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let V be a finite set and let  P be a partition of V k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We say that P is compatible with equality if for all P ∈ P, all tuples  (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk), (v′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , v′  k) ∈ P, and all i, j ∈ [k] it holds that  vi = vj ⇐⇒ v′  i = v′  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Moreover, the partition P is shufflable if for every function π: [k] → [k] and every P ∈ P it  holds that  P π :=  �  (vπ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vπ(k))  �� (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk) ∈ P  �  ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Finally, the partition P is permutable if for every permutation π ∈ Sk (we write Sk for the  symmetric group on the set [k]) and every P ∈ P it holds that P π ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Note that every  shufflable partition is also permutable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  We say a coloring χ: V k → C of k-tuples is compatible with equality if the corresponding  partition P into color classes is compatible with equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Similarly, χ is permutable (shufflable)  if P is permutable (shufflable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Recall that stepk(χ) denotes the coloring obtained from χ after applying a single refinement  round of k-WL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let V be a finite set of size n := |V |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Also let χ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , χℓ : V k → C be a sequence  of colorings such that  (I) χt is shufflable and compatible with equality for all t ∈ [0, ℓ],  (II) stepk(χt−1) ⪰ χt for all t ∈ [ℓ], and  (III) χt−1 ≻ χt for all t ∈ [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Then ℓ ≤ 2nk−1(⌈k log n⌉ + 1) = O(knk−1 log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Note that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1 immediately follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1 by observing that all colorings  χ(i)  k [A] obtained from the refinement process of k-WL are shufflable and compatible with equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1 relies on algebraic tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let V be a finite set of size n := |V |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We  define a multiplication on the space CV k by  (a · b)(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk) :=  �  v∈V  a(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk−1, v)b(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk−2, v, vk)  (1)  for all a, b ∈ CV k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Note that this multiplication is associative and has a unit 1, defined by  1(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk) :=  �  1  if vk−1 = vk,  0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Furthermore, the multiplication is compatible with the vector space structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Hence, it defines  an algebra which we denote by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  7  With every a ∈ CV k we associate a matrix Ma ∈ CV k−1×V k−1 with entries  Ma  �  (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk−1), (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , wk−1)  � :=  �  a(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk−2, vk−1, wk−1)  if vi = wi for all i ∈ [k − 2],  0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  It is easy to see that the mapping a �→ Ma is injective and linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Moreover, it is compatible  with multiplication:  Ma · Mb  �  (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk−1), (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , wk−1)  �  =  �  u1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=',uk−1∈V  Ma  �  (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk−1), (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , uk−1)  �  Mb  �  (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , uk−1), (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , wk−1)  �  =  ��  u∈V a(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk−2, vk−1, u)b(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk−2, u, wk−1)  if vi = wi for all i ∈ [k − 2],  0  otherwise  =  �  (a · b)(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk−1, wk−1)  if vi = wi for all i ∈ [k − 2],  0  otherwise  = Ma·b  �  (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk−1), (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , wk−1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  And finally, M1 is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Thus, A is isomorphic to a subalgebra of the nk−1×nk−1-  dimensional matrix algebra CV k−1×V k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  For every a ∈ A we define a∗ ∈ A by  a∗(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk) = a(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk−2, vk, vk−1)  (here, c denotes the complex conjugate of a number c ∈ C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', if c = a + bi then c = a − bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Then Ma∗ := (Ma)∗ (the conjugate transpose).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Thus, ∗ is an involution on A compatible with  the algebra structure, which turns A into a ∗-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' In particular, as a finite-dimensional  ∗-algebra, A is semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Let us denote by Md(C) for the full matrix algebra of all (d × d)-matrices over the complex  numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2 ([15, Theorem 5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let A(1) ⊂ · · · ⊂ A(ℓ) ⊆ Md(C) be a sequence of semisimple  strict subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then ℓ ≤ 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  So using the fact that A is isomorphic to a subalgebra of Mnk−1(C), we obtain the following  corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let A(1) ⊂ · · · ⊂ A(ℓ) ⊆ A be a sequence of semisimple strict subalgebras of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Then ℓ ≤ 2nk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  For every subset A ⊆ CV k, we let span(A) be the linear subspace of CV k generated by A, and  we let ⟨A⟩ be the closure of span(A) under multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' If 1 ∈ ⟨A⟩, then ⟨A⟩ is a subalgebra  of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We are interested in subalgebras of A generated by partitions of the set V k in the way  explained next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  For every subset P ⊆ V k, we define  cP (v) :=  �  1  if v ∈ P,  0  otherwise  to be the characteristic vector of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For a partition P of V k, we let CP := {cP | P ∈ P} and  AP := ⟨CP⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' If 1 ∈ AP, then AP is a subalgebra of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let P and Q be partitions of V k such that Q strictly refines P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then span(CP) ⊂  span(CQ) and AP ⊆ AQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  8  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' If P ∈ P is the disjoint union of Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , Qm ∈ Q, then cP = �m  i=1 cQi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Thus, CP ⊆  span(CQ) and therefore span(CP) ⊆ span(CQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Moreover, there are P ∈ P, Q ∈ Q such that  Q ⊂ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then cQ ̸∈ span(CP), because all a ∈ span(CP) are constant on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Hence the inclusion  is strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  The second assertion AP ⊆ AQ follows immediately from the definitions of AP and AQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let P be a partition of V k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  (1) If P is compatible with equality, then 1 ∈ AP and hence AP is a subalgebra of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  (2) If P is permutable, then AP is closed under ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let P be a partition of V k that is permutable and compatible with equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then  AP is a ∗-subalgebra of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' In particular, AP is semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  We say that a ∈ CV k distinguishes v, w ∈ V k if a(v) ̸= a(w), and we say that A ⊆ CV k  distinguishes v, w if some a ∈ A distinguishes them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let A ⊆ CV k and v, w ∈ V k such that ⟨A⟩ distinguishes v, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then there are an  s ≤ nk and a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , as ∈ A such that a1 · · · as distinguishes v, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' As a linear subspace of CV k, the space ⟨A⟩ consists of finite linear combinations of  “monomials” a1 · · · as for ai ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Since the dimension of the space is at most nk, we only need  to consider such monomials for s ≤ nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Hence v, w are distinguished by a linear combination  m  �  i=1  λiai1 · · · aisi  with λi ∈ C, aij ∈ A, and si ≤ nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' This immediately implies that v, w are distinguished by  ai1 · · · aisi for some i ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  With every partition P = {P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , Pm} we associate a relational structure (V, RP  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , RP  m)  whose vocabulary consists of k-ary relation symbols Ri interpreted by RP  i  = Pi (to uniquely  define the associated structure, we fix an arbitrary order on the blocks P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , Pm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Slightly  abusing notation, we denote this structure by P as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We say that a formula ϕ(x) distinguishes  v, w ∈ V k over P if  P |= ϕ(v)  ⇐⇒  P ̸|= ϕ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Recall that C(q)  k+1 denotes the fragment of first-order logic with counting consisting of all formulas  of quantifier rank at most q with at most k + 1 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let P be a partition of V k and let v, w ∈ V k such that AP distinguishes v and  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then there is a formula ϕ(x) ∈ C(q)  k+1 of quantifier rank q ≤ ⌈k log n⌉ that distinguishes v, w  over P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Suppose that P = {P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , Pm}, and let ci := cPi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then AP = ⟨{c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , cm}⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Thus, by  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='7, there is an s ≤ nk and i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , is ∈ [m] such that ci1 · · · cis distinguishes v, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  By induction on s ≥ 1, we prove that if ci1 · · · cis distinguishes v, w, then there is a formula  ϕ(x) ∈ C(⌈log s⌉)  k+1  that distinguishes v, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' The assertion of the lemma follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  For the base step s = 1, note that if ci distinguishes v, w, then the atomic formula Ri(x)  distinguishes v, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  For the inductive step, let s ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Suppose that b = ci1 · · · cis distinguishes v, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Let  r := ⌈s/2⌉ and note that r ≤ 2⌈log s⌉−1 and therefore  ⌈log r⌉ ≤ ⌈log s⌉ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  9  Let b1 := ci1 · · · cir and b2 := cir+1 · · · cis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then b = b1 · b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Suppose that v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk) and  w = (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , wk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We have  b(v) =  �  u∈V  b1(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk−1, u) · b2(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk−2, u, vk)  ̸= b(w) =  �  u∈V  b1(w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , wk−1, u) · b2(w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , wk−2, u, wk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Thus, there are b1, b2 ∈ C such that  p :=  ���  �  u ∈ V  ��� b1(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk−1, u) = b1 and b2(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk−2, u, vk) = b2  ����  ̸=  ���  �  u ∈ V  ��� b1(w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , wk−1, u) = b1 and b2(w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , wk−2, u, wk) = b2  ���� =: q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  It follows from the induction hypothesis that for i = 1, 2 and for all v′, w′ ∈ V k such that bi  distinguishes v′, w′ there is a formula ψv′,w′  i  (x) ∈ C(⌈log r⌉)  k+1  that distinguishes v′, w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Without  loss of generality,  P |= ψv′,w′  i  (v′)  and  P ̸|= ψv′,w′  i  (w′),  otherwise we replace ψv′,w′  i  (x) by its negation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let Vi ⊆ V k be the set of all v′ ∈ V k such that  bi(v′) = bi and let  ϕi(x) :=  �  v′∈Vi  �  w′∈V k\\Vi  ψv′,w′  i  (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Then for all v′ ∈ V k we have  P |= ϕi(v′) ⇐⇒ bi(v′) = bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Without loss of generality we assume that p > q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then the formula  ϕ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , xk) := ∃≥pxk+1  �  ϕ1(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , xk−1, xk+1) ∧ ϕ2(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , xk−2, xk+1, xk)  �  ∈ C(⌈log s⌉)  k+1  distinguishes v, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  We are now ready to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For every t ∈ [0, ℓ] let P(t) be the partition of V k into the color classes  of χt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let t, q ≥ 0 such that t + q ≤ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Suppose that there is a formula ϕ(x) ∈ C(q)  k+1 that  distinguishes v, w ∈ V k over P(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then v, w belong to different classes of the partition P(t+q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' By Condition (I), the partition P(t) is shufflable and compatible with equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  This  implies that χt ≡ χ(0)  k [P(t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Together with Condition (II), we get that χt+q ⪯ χ(q)  k [P(t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Also, using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1, we get that χ(q)  k [P(t)](v) ̸= χ(q)  k [P(t)](w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Overall, it follows that  v, w belong to different classes of the partition P(t+q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  ⌟  For every t ∈ [0, ℓ] we define C(t) := CP(t) and A(t) := AP(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Note that A(t) is a semisimple  ∗-subalgebra of A by Condition (I) and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='4, we have  A(0) ⊆ A(1) ⊆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' ⊆ A(ℓ) ⊆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  (2)  Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For all t ∈ [0, ℓ − ⌈k log n⌉],  A(t) ⊆ span(C(t+⌈k log n⌉)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  10  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let a ∈ A(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='8 and Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='9, for all v, w ∈ V k, if a(v) ̸= a(w),  that is, if a distinguishes v and w, then v and w belong to different classes of the partition  P(t+⌈k log n⌉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Thus, a is constant on each class of the partition P(t+⌈k log n⌉), which immediately  implies that a can be written as a linear combination of the characteristic vectors cP of the  classes P ∈ P(t+⌈k log n⌉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' This is the assertion of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  ⌟  Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For all t ∈ [0, ℓ − ⌈k log n⌉ − 1],  A(t) ⊂ A(t+⌈k log n⌉+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' By Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='10, we have A(t) ⊆ span(C(t+⌈k log n⌉)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Moreover, by Condition (III), the  partition P(t+⌈k log n⌉+1) strictly refines the partition P(t+⌈k log n⌉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='4, this implies  span(C(t+⌈k log n⌉)) ⊂ span(C(t+⌈k log n⌉+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' As span(C(t+⌈k log n⌉+1)) ⊆ A(t+⌈k log n⌉+1), the as-  sertion of the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  ⌟  Recall that by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='6, the algebras A(t) are semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Thus, by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='3, at  most 2nk−1 of the inclusions in (2) are strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='11 implies  ℓ ≤ 2nk−1(⌈k log n⌉ + 1) = O(knk−1 log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  4  Long Sequences of Stable Colorings  Next, we prove an almost matching lower bound for Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', we prove that there are  sequences of colorings χ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , χℓ : Vk → C satisfying Condition (I) - (III) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1 of  length ℓ = Ω(nk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Actually, we prove a slightly stronger result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  As before, let us fix an integer k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We present a construction for a sequence χ0 ≻ χ1 ≻  · · ≻ χℓ of colorings of V k such that χt is k-stable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', the coloring is stable with respect to  k-WL) for all t ∈ [0, ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' More precisely, the main result of this section is the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Suppose n ≥ 2k2 and let V be a set of size |V | = 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then there is a sequence  of colorings χ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , χℓ : V k → C of length ℓ ≥  � n  2k  �k−1 such that  (I) χt is shufflable and compatible with equality for all t ∈ [0, ℓ],  (II) χt is k-stable for all t ∈ [0, ℓ], and  (III) χt−1 ≻ χt for all t ∈ [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Before diving into the proof, let us first discuss some implications of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  First of all, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1 implies that the upper bound in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1 is tight up to a factor  of Ok(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' This follows from the simple observation that, if χt is k-stable and χt−1 ≻ χt, then  stepk(χt−1) ≡ χt−1 ⪰ χt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', the sequence of colorings constructed in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1 satisfies the  requirements of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  On the other hand, since all colorings χt are already k-stable, the theorem does not provide  any lower bounds on the iteration number of k-WL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' However, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1 still provides some  valuable insights in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Indeed, all existing methods to bound the iteration number  of k-WL [14, 15] rely on “parallelization arguments”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', it is argued that at some point in the  refinement process many color classes have to be split at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1 essentially  implies that such arguments do not suffice to push the upper bounds on the iteration number  beyond O(nk−1) since such “parallelization arguments” typically also work in the extended setting  of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  As a concrete example, Kiefer and Schweitzer [14] prove upper bounds on  iteration number of 2-WL by bounding the cost of a certain game related to 2-WL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' This game  naturally generalizes to k-WL, but Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1 immediately implies that its cost is Ω(nk−1)  and thus, it is not possible to obtain improved upper bounds by analyzing said game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' So overall,  11  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1 can be interpreted as saying that, in order to obtain improved upper bounds on  the iteration number of k-WL, we need to rely on arguments that also exploit the possibility of  stabilization at an early point, and it is not possible to solely rely on “parallelization arguments”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Let us now turn to the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  It relies on the following theorem which  provides a large set family with restricted intersections between its members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let U be a set of  size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' A k-uniform set family (over U) is a collection F of k-element subsets of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2 ([2, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For every n ≥ 2k2 there exists a k-uniform set family F  over a universe U of n points such that  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' |E1 ∩ E2| ≤ k − 2 for all distinct E1, E2 ∈ F, and  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' |F| ≥  � n  2k  �k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Now, let U be a universe of size n ≥ k and let F be a k-uniform set family over U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We set  V := U × {0, 1}  and define a coloring χF : V k → C as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Since the actual names of the colors are not  relevant for our purposes, we only define the color classes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', we specify when two tuples  receive the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Let ((u1, a1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , (uk, ak)), ((u′  1, a′  1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , (u′  k, a′  k)) ∈ V k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We define χF in such a way that  χF((u1, a1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , (uk, ak)) = χF((u′  1, a′  1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , (u′  k, a′  k)) if and only if  (A) ui = u′  i for all i ∈ [k],  (B) (ui, ai) = (uj, aj)  ⇔  (u′  i, a′  i) = (u′  j, a′  j) for all i, j ∈ [k], and  (C) if {u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , uk} ∈ F, then �  i∈[k] ai ≡ �  i∈[k] a′  i mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Suppose |E1 ∩ E2| ≤ k − 2 for all distinct E1, E2 ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then χF is k-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let v = ((u1, a1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , (uk, ak)), v′ = ((u′  1, a′  1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , (u′  k, a′  k)) ∈ V k such that  χF(v) = χF(v′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Observe that ui = u′  i for all i ∈ [k] by Condition (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We need to show that the two tuples do  not receive distinct colors after a single refinement step of k-WL, that is, we need to argue that  ���  χF(v[(u, a)/1]), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , χF(v[(u, a)/k])  � ��� u ∈ U, a ∈ {0, 1}  ��  =  ���  χF(v′[(u, a)/1]), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , χF(v′[(u, a)/k])  � ��� u ∈ U, a ∈ {0, 1}  ��  where v[(u, a)/i]) = ((u1, a1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , (ui−1, ai1), (u, a), (ui+1, ai+1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , (uk, ak)) is the tuple ob-  tained from v by replacing the i-th entry by (u, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Towards this end, we actually show the  stronger statement that  ���  χF(v[(u, a)/1]), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , χF(v[(u, a)/k])  � ��� a ∈ {0, 1}  ��  =  ���  χF(v′[(u, a)/1]), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , χF(v′[(u, a)/k])  � ��� a ∈ {0, 1}  ��  holds for all u ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Fix some u ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' To see that these two multisets are equal, consider the set  V ′ := {u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , uk, u} × {0, 1} ⊆ V  and the restriction χ′  F : (V ′)k → C : v �→ χF(v) of χF to the set (V ′)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Also, let F′ := {E ∈ F |  E ⊆ V ′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Since |E1 ∩ E2| ≤ k − 2 for all distinct E1, E2 ∈ F and |V ′| ≤ k + 1, we conclude that  |F′| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  12  Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' There is a bijection ϕ: V ′ → V ′ such that  (i) χ′  F(v) = χ′  F(ϕ(v)) for all v ∈ (V ′)k, and  (ii) ϕ(ui, ai) = (u′  i, a′  i) for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For i ∈ [k] we define ϕ(ui, ai) := (ui, a′  i) and ϕ(ui, 1 − ai) := (ui, 1 − a′  i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' In particular,  Condition (ii) is satisfied since ui = u′  i for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' If there is some E′ ∈ F′ such that u ∈ E′,  then we define  ϕ(u, a) :=  �  (u, a)  if �  ui∈E′ ai ≡ �  ui∈E′ a′  i mod 2  (u, 1 − a)  otherwise  for both a ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' If no such set E′ ∈ F′ exists, then we set ϕ(u, a) := (u, a) for both i ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  It can be easily verified that χ′  F(v) = χ′  F(ϕ(v)) for all v ∈ (V ′)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  ⌟  Since the multisets above are defined in an isomorphism-invariant manner over the structure  induced by (V ′, χ′  F ), we conclude that they have to be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let F be the set family obtained from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2 and suppose that F =  {E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , Eℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Observe that ℓ ≥  � n  2k  �k−1 as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For t ∈ [0, ℓ] we define Ft := {E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , Et} ⊆  F and χt := χFt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then χt is k-stable by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Also, Ft−1 ⊂ Ft which implies that  χt−1 ≻ χt by definition of the coloring χt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Finally, it is easy to verify that all colorings are  shufflable and compatible with equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  5  Lower Bounds on the Iteration Number of WL  In this section, we obtain improved lower bounds on the iteration number of the Weisfeiler-  Leman algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' More precisely, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Our proof strategy is similar to the  one employed by Berkholz and Nordström in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' First, for every sufficiently large ℓhi ≥ ℓlo, we  construct pairs of structures that can be distinguished by ℓlo-WL, but ℓhi-WL still requires a  linear number of iterations to distinguish them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Afterwards, we apply a hardness compression  that reduces the number of vertices in the obtained structures while preserving the iteration  number of the Weisfeiler-Leman algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Actually, for the second step, we can rely on the  same tools that are already used by Berkholz and Nordström in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1  Overview  The hard instances we construct are based on propositional XOR-formulas that can also be  viewed as systems of linear equations over the 2-element field F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Let V be a finite set which we interpret as a set of variables that take values in {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' An  XOR-constraint (over V ) is a pair (C, a) where C ⊆ V and a ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' The reader is encouraged  to think of such a constraint as the equation x1 + · · · + xk ≡ a mod 2 where C = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , xk}  is the set of those variables that appear on the left side of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let C be a set of  XOR-constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We define the arity of C to be the maximum cardinality of C for any pair  (C, a) ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  We can translate a set of XOR-constraints into a pair of relational structures as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let  C be a set of XOR-constraints over a set V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Also suppose that V = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , xn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We define  A = A(C) and B = B(C) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We set V (A) = V (B) := V × {0, 1}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', each element  of the structures A and B corresponds to an assignment of a single variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For each i ∈ [n],  we add a unary relation Xi and set XA  i = XB  i  := {(xi, 0), (xi, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Finally, for every constraint  (C, a) ∈ C with C = {xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , xik} we introduce a k-ary relation RC,a and define  RA  C,a :=  �  �  �  �  (xi1, b1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , (xik, bk)  �  ������  b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , bk ∈ {0, 1},  k  �  j=1  bj ≡ 0  mod 2  �  �  �  13  and  RB  C,a :=  �  �  �  �  (xi1, b1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , (xik, bk)  �  ������  b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , bk ∈ {0, 1},  k  �  j=1  bj ≡ a  mod 2  �  �  �  Instead of analysing the Weisfeiler-Leman algorithm directly on A(C) and B(C), it turns out  to more convenient to consider the following game that is directly played on C and is known  to capture the same information as applying the Weisfeiler-Leman algorithm to the associated  structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Let C be a set of XOR-constraints over a set V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let k ∈ N such that C has arity at most  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' A partial assignment β : X → {0, 1} with X ⊆ V violates an XOR-constraint (C, a) ∈ C if  C ⊆ X and  �  x∈C  β(x) ̸≡ a  mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  (3)  For a partial assignment β0 : X0 → {0, 1} with |X0| ≤ k the r-round k-pebble game Gr  k(V, C, β0)  is played as follows:  The game has two players called Verifier and Falsifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  The game is played in rounds with initial position β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Suppose β : X → {0, 1} is the current position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Then the next round consists of the  following steps:  – Falsifier chooses x ∈ V \\ X and X′ ⊆ X such that |X′ ∪ {x}| ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  – Verifier chooses b ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  – The game moves to position β′ : X′ ∪ {x} → {0, 1} with β′(x′) = β(x′) for x′ ∈ X′  and β′(x) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Falsifier wins a play if within the first r rounds an assignment β violates some XOR-  constraint (C, a) ∈ C (if r = 0, then Falsifier wins if the initial assignment β0 violates some  constraint in C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Verifier wins a play if Falsifier does not win within the first r rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  We say Falsifier (respectively Verifier) wins the game Gr  k(V, C, β0) if Falsifier (respectively  Verifier) has a winning strategy for the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' The k-pebble game Gk(V, C, β0) is played in the  same way, but without any restriction on the number of rounds played.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  The following lemma relates the pebble game Gr  k(V, C, ∅) to bounded-variable fragments of  first-order logic and thereby, using Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2, also to the Weisfeiler-Leman algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' (Here,  we use ∅ to denote the empty assignment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', the domain X0 of the initial partial assignment  β0 is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=')  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1 ([3, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let k, r ∈ N such that r > 0 and k ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Let C be a set  of XOR-constraints over a universe V of arity at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then the following statements are  equivalent:  (i) Falsifier wins the r-round k-pebble game Gr  k(V, C, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  (ii) There exists a sentence ϕ ∈ L(r)  k  such that ϕ |= A(C) and ϕ ̸|= B(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  (iii) There exists a sentence ϕ ∈ C(r)  k  such that ϕ |= A(C) and ϕ ̸|= B(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  14  To obtain a set of XOR-constraints on which Falsifier requires a large number of rounds to  win the pebble game, we proceed in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' First, for every sufficiently large ℓhi ≥ ℓlo, we  construct a set of XOR-constraints such that Falsifier wins the ℓlo-pebble game, but still requires  a linear number of rounds to win the ℓhi-pebble game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' This is formalized by the next lemma  which forms the main technical contribution of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' There are absolute constants ℓlo ≥ 2 and δ > 1 such that for every ℓhi ≥ ℓlo  and every r ≥ 1 there is a set of XOR-constraints C of arity at most ℓlo over a set V of size  |V | ≤ δ · ℓ2  hi · r such that Falsifier  (a) wins the ℓlo-pebble game Gℓlo(V, C, ∅), but  (b) does not win the r-round ℓhi-pebble game Gr  ℓhi(V, C, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  We remark that a similar result has also been obtained in [3], but with weaker guarantees  on the number of rounds required to win the ℓhi-pebble game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' It is exactly this improvement  that allows us to obtain stronger lower bounds on the iteration number of k-WL in comparison  to [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Afterwards, we rely on the following hardness compression lemma that reduces the number  of variables while essentially maintaining the number of rounds that Falsifier requires to win the  game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='3 (Berkholz, Nordström [3, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' There is an absolute constant ∆0 ≥ 1 such  that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Suppose C is a set of XOR-constraints of arity at most p over a set V  of size |V | = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Also assume there are parameters ℓlo > 0, ℓhi ≥ ∆0ℓlo and r > 0 such that  Falsifier  (a) wins the ℓlo-pebble game Gℓlo(V, C, ∅), but  (b) does not win the r-round ℓhi-pebble game Gr  ℓhi(V, C, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Let ∆ be an integer such that ∆0 ≤ ∆ ≤ ℓhi/ℓlo and (2ℓhi∆)2∆ ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then there is a set of  XOR-constraints D of arity at most ∆p over a set W of size |W| = ⌈m3/∆⌉ such that Falsifier  (A) wins the (∆ℓlo)-pebble game G∆ℓlo(W, D, ∅), but  (B) does not win the  r  2ℓhi -round ℓhi-pebble game Gr/(2ℓhi)  ℓhi  (W, D, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Combining Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='3, we obtain the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' There are absolute constants k0 ∈ N and α, ε > 0 such that for every d ≥ k ≥ k0  and every n ≥ α · d8 · k6 there is a set of XOR-constraints C of arity at most k over a set V of  size |V | ≤ n such that Falsifier wins the k-pebble game Gk(V, C, ∅), but does not win the r-round  d-pebble game Gr  d(V, C, ∅) for all r ≤ nεk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let ℓlo ≥ 2 and δ > 1 denote the constants from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Also, let ∆0 denote the  constant from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='3 and suppose without loss of generality that δ, ∆0 are integers and  ∆0 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We choose  k0 := max{∆0ℓlo, 6ℓlo}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Let d ≥ k ≥ k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We set p := ℓlo, ℓhi := d and ∆ := ⌊ k  ℓlo ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We have ℓhi = d ≥ k ≥ k0 ≥ ∆0ℓlo  and ∆0 ≤ k  ℓlo .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Since ∆0 is an integer, we conclude that ∆0 ≤ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  We define  n0 := max{  �  δ · ℓ2  hi · (2ℓhi∆)2∆�3/∆ , 4 · δ · ℓ3  hi}  and set α := max{64 · δ · ℓ−6  lo , 4 · δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let n ≥ α · d8 · k6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Using ∆ ≥ ∆0 ≥ 3, we get that  �  δ · ℓ2  hi · (2ℓhi∆)2∆�3/∆ ≤ δ · ℓ2  hi · (2ℓhi∆)6 ≤ 64 · δ · ℓ8  hi ·  � k  ℓlo  �6  ≤ α · d8 · k6  15  and 4 · δ · ℓ3  hi ≤ α · d3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' So in particular n ≥ n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let r be the maximal integer such that  �  δ · ℓ2  hi · r  �3/∆ ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  (4)  Note that r ≥ (2ℓhi∆)2∆ since n ≥ n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let C be the set of XOR-constraints of arity at most ℓlo  over a set V of size |V | ≤ δ · ℓ2  hi · r obtained from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' By adding dummy variables, we  may assume without loss of generality that m := |V | ≥ r ≥ (2ℓhi∆)2∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  By applying Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='3, we obtain a set of XOR-constraints D of arity at most ∆p over a  set W of size |W| = ⌈m3/∆⌉ such that Falsifier  (A) wins the (∆ℓlo)-pebble game G∆ℓlo(W, D, ∅), but  (B) does not win the  r  2ℓhi -round ℓhi-pebble game Gr/(2ℓhi)  ℓhi  (W, D, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  First observe that ∆p = ⌊ k  ℓlo ⌋ · ℓlo ≤ k and  |W| =  �  m3/∆�  ≤  ��  δ · ℓ2  hi · r  �3/∆�  ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Since ∆ℓlo ≤ k, it holds that Falsifier wins the k-pebble game Gk(W, D, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Moreover, Falsifier  does not win the  r  2ℓhi -round d-pebble game Gr/(2ℓhi)  d  (W, D, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We have that  �  δ · ℓ2  hi · 2r  �3/∆ ≥ n  since r is the maximal integer to satisfy Equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' This implies that  r  2ℓhi  ≥  n∆/3  4 · δ · ℓ3  hi  ≥ n  1  3 ⌊ k  ℓlo ⌋−1 ≥ nεk  for some sufficiently small absolute constant ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  With Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='4 at hand, we are now ready to prove Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let k′  0 ∈ N and α′, ε′ > 0 denote the absolute constants from Corollary  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Let k0 := max{k′  0, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We set α ≥ 1 and ε > 0 in such a way that for all d ≥ k ≥ k0 and  n ≥ αd8k6 it holds that  �n  2  �  ≥ α′(d + 1)8(k + 1)6  and  �n  2 − 1  �ε′k  − d ≥ nεk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Now, let us fix some d ≥ k ≥ k0 and n ≥ αd8k6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let d′ := d + 1, k′ := k and n′ := ⌊ n  2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We  apply Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='4 with parameters d′, k′, n′ and obtain a set of XOR-constraints C of arity at  most k′ over a set V ′ of size |V ′| ≤ n′ such that Falsifier wins the k′-pebble game Gk′(V ′, C, ∅),  but does not win the r′-round d′-pebble game Gr′  d′(V ′, C, ∅) for r′ = (n′)ε′k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Let A := A(C) and B := B(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then |V (A)| = |V (B)| = 2|V ′| ≤ 2n′ ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Note that we can  easily increase the size of both structures by adding isolated elements that do not participate in  any relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Also, note that both structures have arity at most k′ = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2, k-WL distinguishes between A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' On the other hand,  again by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2, d-WL does not distinguish A and B after r := r′ − d  refinement rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We get that  r = r′ − d = (n′)ε′k′ − d ≥  �n  2 − 1  �ε′k  − d ≥ nεk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' This follows directly from Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='4 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  The remainder of this section is devoted to the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  16  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2  The Closure of the Constraint Set  The critical step in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2 is to argue that Verifier survives a linear number  of rounds even for a large number of pebbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Here, we rely on an alternative description of  winning positions in terms of a closure operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Let k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let V be a finite set and let C be a set of XOR-constraints over V of arity at  most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We define the k-attractor  attrk(C) := C ∪  �  (C1 ⊕ C2, a1 + a2 mod 2)  ��� (C1, a1), (C2, a2) ∈ C, |C1 ⊕ C2| ≤ k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Here, C1 ⊕ C2 denotes the symmetric difference between the two sets, that is, C1 ⊕ C2 :=  (C1 ∪ C2) \\ (C1 ∩ C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Intuitively speaking, if x1 + · · · + xℓ ≡ a1 mod 2 and y1 + · · · + ym ≡ a2 mod 2 are two  constraints in C, then every satisfying assignment also has to satisfy the equation x1 + · · · +  xℓ + y1 + · · · + ym ≡ a1 + a2 mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Since all variables appearing in both sets {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , xk} and  {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , ym} cancel over F2, we only need to keep those variables appearing in the symmetric  difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  In the case that the resulting number of variables is bounded by k, we add the  corresponding equation to the k-attractor of the constraint set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  We define cl(0)  k (C) := C and cl(r+1)  k  (C) := attrk(cl(r)  k (C)) for all r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Finally, we define the  k-closure of C to be the set clk(C) := cl(r)  k (C) for the minimal r ≥ 0 such that cl(r+1)  k  (C) = cl(r)  k (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  The following lemma provides the key method to prove that Verifier can survive a certain  number of rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let β : X → {0, 1} be a partial assignment with |X| ≤ k such that β violates no  XOR-constraint (C, a) ∈ cl(r)  k (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then Verifier wins Gr  k(V, C, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We prove the statement by induction on r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For r = 0 the statement is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' So suppose  r ≥ 1 and Spoiler chooses x ∈ V \\ X and X′ ⊆ X such that |X′ ∪ {x}| ≤ k in the first round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  For b ∈ {0, 1} let βb : X′ ∪ {x} → {0, 1} be the partial assignment with βb(x′) = β(x′) for  x′ ∈ X′ and βb(x) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Assume towards a contradiction that, for every b ∈ {0, 1}, there is some  XOR-constraint (Cb, ab) ∈ cl(r−1)  k  (C) violated by βb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Observe that x ∈ Cb for both b ∈ {0, 1}  (since otherwise β would violate (Cb, ab) contradicting our assumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let C := C0 ⊕ C1 ⊆ X  and a := (a0 + a1) mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Note that |C| ≤ k since C ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then  �  y∈C  β(y) ≡  �  y∈C0\\{x}  β(y) +  �  y∈C1\\{x}  β(y) ≡ 1 +  �  y∈C0  β0(y) +  �  y∈C1  β1(y) ≡ 1 + a  mod 2  and (C, a) ∈ cl(r)  k (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Hence, β violates some (C, a) ∈ cl(r)  k (C) which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  So there is some b ∈ {0, 1} such that βb violates no XOR-constraint in cl(r−1)  k  (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Verifier  chooses such a b ∈ {0, 1} and the game moves to position βb which violates no XOR-constraint  in cl(r−1)  k  (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' So Verifier wins Gr−1  k  (V, C, βb) by the induction hypothesis which implies that  Verifier also wins Gr  k(V, C, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='3  Layered Graphs and Expansion  Next, we discuss the construction of certain expander graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Overall, we are aiming to con-  struct what we refer to as single-neighbor layered expanders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Towards this end, we start with  constructing standard bipartite expander graphs with an expansion that is close to the minimum  degree of one side of the bipartite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We then define single-neighbor expanders and observe  that bipartite expanders with large expansion also are single-neighbor expanders (with a slightly  smaller expansion parameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Finally, we obtain single-neighbor layered expanders by “stacking  single-neighbor expanders on top of each other”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  17  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1  Expander Graphs  We start by defining standard bipartite expander graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let 0 < γ < 1 and α > 1 be constants and let G = (V, W, E) be a bipartite  graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We say that G is an (α, γ)-expander if for every ∅ ̸= Y ⊆ W with |Y | ≤ γ|W| it holds  that  N(Y ) ≥ α|Y |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  For more information on expander graphs we refer to [18, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' The references also contain  variants of the following standard argument that guarantees the existence of graphs with good  expansion properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For our purposes, the crucial property in the lemma below is that the  expansion α is relatively close to the degree of the vertices in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' There is some number R0 ≥ 2 such that for every r ≥ R0 and every n ≥ 4r there  is a ( 3  4r,  1  20r)-expander G = (V, W, E) such that |V | = |W| = n and deg(w) = r for all w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Suppose r is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let V, W be two sets with |V | = |W| ≥ 4r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We construct  a bipartite graph G = (V, W, E) using the following random process: for each w ∈ W we select  independently and uniformly at random a set of r distinct neighbors from V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We prove that,  for r sufficiently large, with positive probability the graph G is a ( 3  4r,  1  20r)-expander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Let n := |V | = |W|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For X ⊆ V and Y ⊆ W let pX,Y denote the probability that N(Y ) ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Then  pX,Y ≤  �|X|  n  �r·|Y |  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Furthermore, let α := 3  4r and γ :=  1  20r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let p be the probability that G is not a (γ, α)-expander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' using the inequality  �n  k  �  ≤ (ne/k)k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' we get  p ≤  �  Y ⊆W  |Y |≤γ·n  �  X⊆V  |X|=⌊α|Y |⌋  pX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='Y  ≤  ⌊γ·n⌋  �  s=1  �  Y ⊆W  |Y |=s  �  X⊆V  |X|=⌊α|Y |⌋  �|X|  n  �r·|Y |  ≤  ⌊γ·n⌋  �  s=1  �n  s  �� n  ⌊αs⌋  � �αs  n  �r·s  ≤  ⌊γ·n⌋  �  s=1  �ne  s  �s �ne  αs  �α·s �αs  n  �r·s  =  ⌊γ·n⌋  �  s=1  ��ne  s  � �ne  αs  �α �αs  n  �r�s  =  ⌊γ·n⌋  �  s=1  �� s  n  �r−α−1  e1+ααr−α  �s  =  ⌊γ·n⌋  �  s=1  �� s  n  �r/4−1  e1+3r/4(3r/4)r/4  �s  ≤  ⌊γ·n⌋  �  s=1  �  γr/4−1e1+3r/4(3r/4)r/4�s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  18  Now let x := γr/4−1e1+3r/4(3r/4)r/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For r sufficiently large we get  x = (20r)1−r/4e1+3r/4(3r/4)r/4 = 20er  �3e3  80  �r/4  < 1/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  It follows that  p ≤  ∞  �  s=1  xs =  x  1 − x ≤ 1  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  In particular, p < 1 which implies the existence of the desired expander graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Next, we turn to what we call single-neighbor expanders where each sufficiently small set  Y ⊆ V is required to have a large number of neighbors that additionally satisfy the property  that they are the neighbor of only a single vertex from Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let G = (V, W, E) be a bipartite  graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For Y ⊆ W we define  N∗(Y ) = {v ∈ N(Y ) | |N(v) ∩ Y | = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let 0 < γ < 1 and α > 1 be constants and let G = (V, W, E) be a bipartite  graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  We say that G is an (α, γ)-single-neighbor expander if for every ∅ ̸= Y ⊆ W with  |Y | ≤ γ|W| it holds that  N∗(Y ) ≥ α|Y |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  We can obtain single-neighbor expanders from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='7 by allowing some loss on the  expansion parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' There is some number R0 ≥ 5 such that for every r ≥ R0 and every n ≥ 4r there  is a ( 1  4r,  1  20r)-single-neighbor expander G = (V, W, E) such that |V | = |W| = n and deg(w) = r  for all w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Choose R0 := max(5, R′  0) where R′  0 is the constant from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='7 and suppose r ≥ R0  and n ≥ 4r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='7, there is a ( 3  4r,  1  20r)-expander G = (V, W, E) such that |V | = |W| = n  and deg(w) = r for all w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We claim that G is a ( 1  4r,  1  20r)-single-neighbor expander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let  Y ⊆ W with |Y | ≤  n  20r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Then |N(Y )| ≥  3  4r|Y |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Furthermore |N(Y )| = |N∗(Y )| + |{v ∈  N(Y ) | |N(v) ∩ Y | ≥ 2}| ≤ |N∗(Y )| + 1  2r|Y | because every vertex in Y has degree r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Thus,  |N∗(Y )| ≥ 1  4r|Y |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2  Layered Graphs  Now, we turn to the construction of single-neighbor layered expanders which is the main tool  for constructing the desired constraint sets in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We start by defining a  certain notion of layered graphs (see also Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Let ℓ, m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' An (ℓ × m)-layered graph is a bipartite graph G = (V, W, E) for which there  are partitions V = V0 ⊎ · · · ⊎ Vℓ and W = W1 ⊎ · · · ⊎ Wℓ such that  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' |Vi| = m for all i ∈ [0, ℓ],  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' |Wi| = m for all i ∈ [ℓ],  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' NG(Wi) ⊆ Vi−1 ∪ Vi for all i ∈ [ℓ], and  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' G[Vi ∪ Wi] is 1-regular (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', a matching) for all i ∈ [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  With this, we are now ready to define the notion of single-neighbor layered expanders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  19  V0  W1  V1  W2  Wℓ  Vℓ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Figure 1: Visualization of (ℓ × m)-layered graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let ℓ, m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let 0 < γ < 1 and α > 1 be constants and let G = (V, W, E) be  an (ℓ × m)-layered graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We say that G is an (α, γ)-single-neighbor (ℓ × m)-layered expander  if for every ∅ ̸= Y ⊆ W with |Y | ≤ γm it holds that  N∗(Y ) ≥ α|Y |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Note that an (α, γ)-single-neighbor (ℓ × m)-layered expander is not a (α, γ)-single-neighbor  expander since we are only considering sets Y ⊆ W of size |Y | ≤ γm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', we are only considering  sets that are smaller (by a factor of γ) than a single layer of a layered graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' In particular, the  reader is encouraged to think of ℓ being much larger than m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' In this case, such a graph is far  from being a (global) expander, but the key property is that it behaves like an expander when  only considering a few layers of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  By again allowing some small loss on the expansion parameter α, we can obtain single-  neighbor layered expanders by “stacking ℓ copies of a single-neighbor expander on top of each  other”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' There is some number R0 ≥ 9 such that for every r ≥ R0, every ℓ ≥ 1, and  every m ≥ 4r there is a ( 1  4r − 1,  1  20r)-single-neighbor (ℓ × m)-layered expander G = (V, W, E)  with partitions V = V0 ⊎ · · · ⊎ Vℓ and W = W1 ⊎ · · · ⊎ Wℓ such that NG(w) ∩ Vi−1 = r for all  w ∈ Wi and all i ∈ [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Choose R0 := max(9, R′  0) where R′  0 is the constant from Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='9 and suppose  r ≥ R0, ℓ ≥ 1, and m ≥ 4r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='9, there is a ( 1  4r,  1  20r)-single-neighbor expander  G′ = (V ′, W ′, E′) such that |V ′| = |W ′| = m and degG′(w′) = r for all w′ ∈ W ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Suppose  V ′ = {v′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , v′  m} and W ′ = {w′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , w′  m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  We set Vi := {vi,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vi,m} for all i ∈ [0, ℓ] and Wi := {wi,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , wi,m} for all i ∈ [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Also,  we set  E := {vi−1,jwi,k | i ∈ [ℓ], v′  jw′  k ∈ E′} ∪ {vi,jwi,j | i ∈ [ℓ], j ∈ [m]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Clearly, G = (V, W, E) is an (ℓ × m)-layered graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Let α := 1  4r and γ :=  1  20r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Also let Y ⊆ W such that |Y | ≤ γm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We define Yi := Y ∩ Wi  for all i ∈ [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Observe that |Yi| ≤ γm for all i ∈ [ℓ] and Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , Yℓ forms a partition of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let  I := {i ∈ [ℓ] | Yi ̸= ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Since G′ is an (α, γ)-single-neighbor expander, we conclude that  |N∗(Yi) ∩ Vi−1| ≥ α|Yi|  20  for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Moreover, since G[Vi ∪Wi] is 1-regular (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', a matching) for all i ∈ [ℓ], we conclude  that  |N∗(Y ) ∩ Vi−1| ≥ α|Yi| − |Yi−1|  for all i ∈ I (we set Y0 := ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' So overall  |N∗(Y )| ≥  �  i∈I  α|Yi| − |Yi−1| ≥  �  i∈I  (α − 1)|Yi| = (α − 1)|Y |  as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='4  Constraint Sets from Layered Expanders  Now, we turn to the construction of constraint sets from layered graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For a bipartite graph  G = (V, W, E) we define the XOR-constraint set CG := {(N(w), 0) | w ∈ W} over the variable  set V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Slightly abusing notation, for C ⊆ V , we shall also write C ∈ CG if (C, 0) ∈ CG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  The basic idea for the construction of the XOR-constraint set C is to take a layered graph  G = (V, W, E) with partitions V = V0 ⊎ · · · ⊎ Vℓ and W = W1 ⊎ · · · ⊎ Wℓ, and set  C := CG ∪  �  ({x}, 0)  �� x ∈ V0  �  ∪  �  ({xℓ}, 1)  �  for some arbitrary xℓ ∈ Vℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' It is not difficult to see that this constraint set is unsatisfiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Indeed, every variable in layer V0 needs to be set to 0, and if all variables in layer Vi−1 are set to  0, then the constraints obtained from the vertices in Wi enforce that every variable in layer Vi  needs to be set to 0 as well (using that G[Vi ∪ Wi] is a matching).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' This inductive argument can  be easily turned into a winning strategy for Falsifier that requires O(ℓ) many rounds (assuming  the degree of all vertices in V is bounded by some absolute constant d ≤ k where k denotes the  number of variables available in the game).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Now, the central claim is that, if we start with a single-neighbor layered expander, this  strategy is essentially optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Let us suppose for the moment that only constraints from  CG are present and consider the k-closure clk(CG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  What we need to avoid is that clk(CG)  contains some constraint that is “non-local”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For example, if clk(CG) would contain a constraint  ({x1, x2, x3}, 0) such that x1, x2 ∈ V0 and x3 ∈ Vℓ, then Falsifier could use such a (derived)  constraint to immediately conclude that certain variables in the last layer need to be set to 0  and potentially follow a different strategy to win the game faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' The main point is that, by  using single-neighbor layered expanders, we ensure that all “relevant” constraints in clk(CG) are  “local”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', they can only contain variables of O(k) consecutive layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' (Here, the reader may  note that if |N(v1) ∪ N(v2)| ≤ k then (N(v1) ⊕ N(v2), 0) is always contained in the closure even  if v1 and v2 are far apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' However, in such a case, N(v1)∩N(v2) = ∅ and the derived constraint  (N(v1) ∪ N(v2), 0) is not “relevant” since, whenever it is violated by a partial assignment, one  of the constraints associated with v1 or v2 is also violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=') This way, even when adding all  constraints from clk(CG) to the initial set, the best that Falsifier can do is essentially to follow  the above inductive strategy (with the exception that Falsifier may skip up to O(k) layers in  one step which, however, does not cause any problems for our arguments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  For technical reasons, the formal arguments slightly deviate from the intuitive ideas described  above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' To start, instead of working with the k-closure clk(CG), it turns out to be more convenient  to work with the following set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Let α > 1 and k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We define the set  cl∗  k,α(CG) :=  ���  D∈D  D, 0  � ����� ∅ ̸= D ⊆ CG, |D| ≤ k  α,  ���  �  D∈D  D  ��� ≤ k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  The next lemma implies that clk(CG) ⊆ cl∗  k,α(CG) if G is a suitable single-neighbor layered  expander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  21  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Suppose α > 1 and 0 < γ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let G = (V, W, E) be an (α, γ)-single-neighbor  (ℓ × m)-layered expander such that deg(w) ≤ d for all w ∈ W and suppose d ≤ k ≤ 1  2γm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then  attrk  �  cl∗  k,α(CG)  �  = cl∗  k,α(CG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let C∗ := cl∗  k,α(CG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Suppose C ∈ attrk(C∗), that is, there are C1, C2 ∈ C∗ such that C :=  C1 ⊕C2 and |C| ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' By definition, there are integers s, t ≤ k  α and D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , Ds, Ds+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , Ds+t ∈  CG such that C1 = D1⊕· · ·⊕Ds and C2 = Ds+1⊕· · ·⊕Ds+t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Moreover, D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , Ds are pairwise  distinct as well as Ds+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , Ds+t are pairwise distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We have C = D1 ⊕ · · · ⊕ Ds+t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let  D := {D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , Ds} ⊕ {Ds+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , Ds+t}  and let Y := {w ∈ W | N(w) ∈ D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Clearly, C = �  D∈D D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Suppose towards a contradiction  that |D| > k  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then |Y | > k  α and moreover, |Y | ≤ s + t ≤ 2 k  α ≤ 2k ≤ γm and thus, |N∗(Y )| ≥  α|Y | > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' But on the other hand N∗(Y ) ⊆ C which implies that |N∗(Y )| ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' This is a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' So |D| ≤ k  α which implies that C ∈ C∗ as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Suppose α > 1 and 0 < γ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let G = (V, W, E) be an (α, γ)-single-neighbor  (ℓ × m)-layered expander such that deg(w) ≤ d for all w ∈ W and suppose d ≤ k ≤ 1  2γm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then  |C| ≥ 2 for all C ∈ cl∗  k,α(CG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let C ∈ cl∗  k,α(CG) and let C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , Cs ∈ CG such that C = C1⊕· · ·⊕Cs for some s ≤ k  α ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Furthermore, let Y := {w ∈ W | ∃i ∈ [s]: N(w) = Ci}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Observe that 1 ≤ |Y | ≤ k ≤ γm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then  N∗(Y ) ⊆ C and thus, |C| ≥ |N∗(Y )| ≥ α|Y | > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Next, we prove that Falsifier wins the pebble game if we set all variables in layer V0 to 0,  and a single variable in the last layer Vℓ to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For technical reasons, we do not add ({xℓ}, 1) to  the constraint set, but rather consider an initial assignment that assigns value 1 to variable xℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let G = (V, W, E) be an (ℓ × m)-layered graph with partitions V = V0 ⊎ · · · ⊎ Vℓ  and W = W1⊎· · ·⊎Wℓ such that deg(w) ≤ k for all w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let xℓ ∈ Vℓ and suppose βℓ : {xℓ} →  {0, 1} is the partial assignment defined via βℓ(xℓ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then Falsifier wins Gk(W, C, βℓ) where  C := CG ∪  �  ({x}, 0)  �� x ∈ V0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We prove by induction on i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , ℓ that Falsifier wins Gk(V, C, βi) where βi is any  partial assignment for which βi(xi) = 1 for some xi ∈ Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  The base case i = 0 is trivial since ({x0}, 0) ∈ C for every x0 ∈ V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  For the inductive  step, suppose i ∈ [ℓ] and consider some partial assignment βi for which there is some xi ∈ Vi  such that βi(xi) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Since G = (V, W, E) is an (ℓ × m)-layered graph, there is a unique  vertex wi ∈ Wi such that wixi ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Moreover, NG(wi) ⊆ Vi−1 ∪ Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' If NG(wi) = {xi}, then  ({xi}, 0) ∈ CG and Falsifier wins immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  So suppose that NG(wi) ∩ Vi−1 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Since  deg(wi) ≤ k, Falsifier can move to a partial assignment βi−1 : Xi → {0, 1} where Xi = NG(wi)  and βi−1(xi) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' If βi−1 violates the XOR-constraint (Xi, 0), then Falsifier wins immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Otherwise, �  y∈Xi βi−1(y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Together with the fact that βi−1(xi) = 1, this implies that  there is some xi−1 ∈ Xi ∩ Vi−1 such that βi−1(xi−1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' So Falsifier wins by the induction  hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  The next lemma forms the key technical lemma stating that Falsifier requires a large number  of rounds to win if the constraint set is obtained from a single-neighbor layered expander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Suppose α > 1 and 0 < γ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let G = (V, W, E) be an (α, γ)-single-neighbor  (ℓ × m)-layered expander with partitions V = V0 ⊎ · · · ⊎ Vℓ and W = W1 ⊎ · · · ⊎ Wℓ such that  deg(w) ≤ d for all w ∈ W and suppose d ≤ k ≤ 1  2γm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  22  Let xℓ ∈ Vℓ and suppose βℓ : {xℓ} → {0, 1} is the partial assignment defined via βℓ(xℓ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Then Verifier wins Gr−1  k  (V, C, βℓ) where  C := CG ∪  �  ({x}, 0)  �� x ∈ V0  �  and r := ⌊ℓ/2k⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let  C∗  G := cl∗  k,α(CG) ∪ {(∅, 0)}  and define C∗ := C∗  G ∪ {({x}, 0) | x ∈ V0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We show that Verifier wins Gr−1  k  (V, C∗, βℓ) which  clearly implies the claim since C ⊆ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='5, it suffices to show that βℓ violates no  XOR-constraint from the set cl(r−1)  k  (C∗), or equivalently ({xℓ}, 0) /∈ cl(r−1)  k  (C∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  We define  Vi :=  2ik  �  j=0  Vj  for all i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , ⌊ℓ/2k⌋}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Finally, we define  C∗  i :=  �  C ⊆ V  ��� |C| ≤ k, C = D ⊕ U for some D ∈ C∗  G, U ⊆ Vi  �  for all i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , ⌊ℓ/2k⌋}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' attrk(C∗  i ) ⊆ C∗  i+1 for all i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , ⌊ℓ/2k⌋ − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let C1, C2 ∈ C∗  i such that |C1 ⊕ C2| ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let C := C1 ⊕ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For j ∈ {1, 2} pick Dj ∈ C∗  G  and Uj ⊆ Vi such that Cj = Dj ⊕Uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let U ′ := U1⊕U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Clearly, U ′ ⊆ Vi and C = D1⊕D2⊕U ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Let Yj ⊆ W, j ∈ {1, 2}, be a set of vertices of size |Yj| ≤ k  α < k such that Dj = �  w∈Yj N(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Then there is some λ ∈ {2ik + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , 2(i + 1)k} such that Wλ ∩ (Y1 ∪ Y2) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We define  Y <λ  j  := Yj ∩ W<λ  where W<λ := �  µ<λ Wµ and  Y >λ  j  := Yj ∩ W>λ  where W>λ := �  µ>λ Wµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Moreover, let  C>λ  j  :=  �  w∈Y >λ  j  N(w) ⊆ Cj  for both j ∈ {1, 2} and  C>λ := C>λ  1  ⊕ C>λ  2  ⊆ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Hence, |C>λ  j  | ≤ k and |C>λ| ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' So C>λ  j  ∈ C∗  G for both j ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' It follows that C>λ ∈ C∗  G by  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Now, C = C>λ ⊕ U for some U ⊆ V0 ∪ · · · ∪ Vλ−1 ⊆ Vi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' It follows that C ∈ C∗  i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  ⌟  Since C∗ ⊆ C∗  0 it follows by induction that  cl(i)  k (C∗) ⊆ C∗  i  (5)  for all i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , ⌊ℓ/2k⌋} using Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' So it only remains the prove the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' {xℓ} /∈ C∗  r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  23  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let C ∈ C∗  r−1 such that C ∩Vℓ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Also pick D ∈ C∗  G and U ⊆ Vr−1 such that C = D⊕U  (which exist by the definition of C∗  r−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We have that  U ⊆ Vr−1 =  2k(r−1)  �  i=0  Vi ⊆  ℓ−2k  �  i=0  Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Let Y ⊆ W such that |Y | ≤ k  α < k and D = �  w∈Y N(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let λ ∈ [ℓ] be the maximal number  such that Y ∩ Wλ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Note that λ > ℓ − k since |Y | < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Now let D′ := �  w∈Y ∩W>λ N(w)  where W>λ := �  µ>λ Wµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then D′ = C ∩ (Vλ ∪ · · · ∪ Vℓ) and hence, |D′| ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' It follows that  D′ ∈ cl∗  k,α(CG) which implies that |D′| ≥ 2 using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' So |C| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  ⌟  Finally, we require one more technical lemma that allows us to add the XOR-constraint  ({xℓ}, 1) to the final constraint set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let k ≥ 2 and r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let V be a finite set and let C be a set of XOR-constraints  over V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let x0 ∈ V and define β0 : {x0} → {0, 1} via β0(x0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' If Verifier wins Gr  k(V, C, β0),  then Verifier also wins Gr  k−1(V, C ∪ {({x0}, 1)}, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Consider a position β : X → {0, 1} of the game Gr  k−1(V, C ∪ {({x0}, 1)}, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Throughout  the game, by following a winning strategy for Gr  k(V, C, β0), Verifier can maintain the following  properties after every round ℓ ∈ [0, r]:  (i) If x0 ∈ X, then β(x0) = 1, and  (ii) Verifier wins the game Gr−ℓ  k  (V, C, β′) where β′ : X ∪ {x0} → {0, 1} is defined via β′(x) :=  β(x) for all x ∈ X and β(x0) := 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Observe that the condition is satisfied initially since Verifier wins Gr  k(W, C, β0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' All positions  reached this way clearly satisfy all XOR-constraints in C ∪{({x0}, 1)} which implies that Verifier  wins Gr  k−1(W, C ∪ {({x0}, 1)}, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  With this, we are ready to prove Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let R0 ≥ 9 denote the constant from Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='11 and define ℓlo :=  R0 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let d := R0, α := 1  4d − 1 > 1 and γ :=  1  20d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let ℓhi ≥ ℓlo and r ≥ 1 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We define  k := ℓhi + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Also, let m := 2 · k  γ = 40dk ≥ 4d and ℓ := 2k(r + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='11, there is an (α, γ)-single-neighbor (ℓ×m)-layered expander G = (V, W, E)  such that deg(w) = d + 1 for all w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let V0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , Vℓ and W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , Wℓ denote the layers of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Also fix some arbitrary element xℓ ∈ Vℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We define  C := CG ∪ {({x}, 0) | x ∈ V0} ∪ {({xℓ}, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Note that C is a set of XOR-constraints over V of arity at most d + 1 = ℓlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  To complete the proof, we show that C has the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' First,  |V | = (ℓ + 1)m = (2k(r + 1) + 1)40dk ≤ 8kr · 40dk = 320R0(ℓhi + 1)2r ≤ δ · ℓ2  hi · r  for some suitable absolute constant δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Moreover, Falsifier wins the ℓlo-pebble game Gℓlo(V, C, ∅)  by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Finally, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='15, Verifier wins Gr  k(V, C\\{({xℓ}, 1)}, βℓ) where βℓ : {xℓ} →  {0, 1} is the partial assignment defined via βℓ(xℓ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' So Verifier wins the r-round ℓhi-pebble  game Gr  ℓhi(V, C, ∅) by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  24  6  Trading Variable Number for Quantifier Depth  In this section, we investigate tradeoffs between the number of variables and the quantifier rank  of formulas used to distinguish relational structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' More concretely, suppose A and B are  two structures of size n that are distinguished by k-WL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2, there is a formula  ϕ ∈ Ck+1 such that A |= ϕ and B ̸|= ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Using Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1, we may assume that ϕ has  quantifier rank at most O(knk−1 log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' In this section, we show that there are sentences ψ  that distinguish between A and B with smaller quantifier rank if we are allowed to increase the  number of variables by some function in k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' In other words, we can show improved bounds on  the number of WL-iterations required to distinguish between A and B (compared to Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1) by increasing the dimension of the WL-algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1 (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='6 restated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let A and B be two relational structures of  arity at most k such that n := |V (A)| = |V (B)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Also suppose there is a sentence ϕ ∈ Ck+1 such  that A |= ϕ and B ̸|= ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let d := ⌈ 3(k+1)  2  ⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then there is a sentence ψ ∈ C(q)  d  of quantifier rank  q = O(k2 · n⌊k/2⌋+1 log n) such that A |= ψ and B ̸|= ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Toward the proof of this theorem, let us fix some k ≥ 2 and suppose that k is odd, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=',  k = 2ℓ − 1 from some integer ℓ ≥ 2 (this is the crucial case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let A be a relational structure of  arity at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We translate A into a binary structure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', a structure of arity at most two)  Bin(A) defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' The universe of Bin(A) is set to  V (Bin(A)) := (V (A))ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  For every atomic type typ ∈ {atpA(v) | v ∈ (V (A))2ℓ} (on 2ℓ vertices) we introduce a binary  relation symbol Rtyp and set  RBin(A)  typ  :=  ��  (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vℓ), (vℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , v2ℓ)  � �� atpA(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , v2ℓ) = typ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Now, the key idea behind the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1 is to use d variables to simulate the  execution of 2-WL on the binary structure Bin(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We can then obtain the upper bound on the  quantifier rank by exploiting that 2-WL stabilizes after at most O(n log n) rounds (see Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  The next lemma translates a formula that distinguishes between Bin(A) and Bin(B) into a  formula distinguishing A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let A and B be two relational structures of arity at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Suppose there is a  sentence ϕ ∈ C(q)  d  such that Bin(A) |= ϕ and Bin(B) ̸|= ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then there is a sentence �ϕ ∈ C(q·ℓ)  d·ℓ  such that A |= �ϕ and B ̸|= �ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  The proof of the lemma is a standard syntactic translation (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', [21, Chapter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='5]) and  we omit the details here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let A and B be two relational structures of arity at most k such that Bin(A) ≃2  Bin(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then A ≃k B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Consider an arbitrary structure C and define χ2 := χ(∞)  2  [Bin(C)] to be the coloring  computed by 2-WL on the structure Bin(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We define a coloring χ: (V (C))k → C by setting  χ(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk) := χ2((v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vℓ), (vℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk−1, vk, vk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Suppose atpC(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk) ̸= atpC(v′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , v′  k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then χ(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk) ̸= χ(v′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , v′  k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  25  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let typ := atpC(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk−1, vk, vk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then ((v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vℓ), (vℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk, vk)) ∈ RBin(C)  typ  , but  on the other hand ((v′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , v′  ℓ), (v′  ℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , v′  k, v′  k)) /∈ RBin(C)  typ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' So  atpBin(C)((v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vℓ), (vℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk, vk)) ̸= atpBin(C)((v′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , v′  ℓ), (v′  ℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , v′  k, v′  k))  which implies that  χ2((v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vℓ), (vℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk, vk)) ̸= χ2((v′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , v′  ℓ), (v′  ℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , v′  k, v′  k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  This directly implies the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  ⌟  Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' χ is k-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let v, v′ ∈ (V (C))k such that χ(v) = χ(v′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Suppose v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk) and v′ =  (v′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , v′  k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Let us write v1 := (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vℓ) for the “first half” of v, and v2 := (vℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk)  for the “second half”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Note that v2 has only ℓ − 1 entries since k = 2ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Similarly, we  define v′  1 := (v′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , v′  ℓ) and v′  2 := (v′  ℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , v′  k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' For w ∈ V (C) we write v2 ◦ w for the tuple  (vℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , vk, w) obtained from v2 by appending w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' The tuple v′  2 ◦ w is defined analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Since χ2 is 2-stable and χ2(v1, v2 ◦ vk) = χ2(v′  1, v′  2 ◦ v′  k), we conclude that  ���  χ2(v1, w), χ2(w, v2 ◦ vk)  � ��� w ∈ (V (C))ℓ��  =  ���  χ2(v′  1, w), χ2(w, v′  2 ◦ v′  k)  � ��� w ∈ (V (C))ℓ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Using that χ2 refines the coloring by atomic types, it follows that  ���  χ2(v1, v2 ◦ w), χ2(v2 ◦ w, v2 ◦ vk)  � ��� w ∈ V (C)  ��  =  ���  χ2(v′  1, v′  2 ◦ w), χ2(v′  2 ◦ w, v′  2 ◦ v′  k)  � ��� w ∈ V (C)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  In particular, we get that  ��  χ2(v1, v2 ◦ w)  ��� w ∈ V (C)  ��  =  ��  χ2(v′  1, v′  2 ◦ w)  ��� w ∈ V (C)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Now let w, w′ ∈ V (C) such that χ2(v1, v2 ◦ w) = χ2(v′  1, v′  2 ◦ w′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Then  χ(v[w/i]) = χ(v′[w′/i])  for all i ∈ [k] using again that χ2 is 2-stable and refines the coloring by atomic types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' It follows  that  ���  χ(v[w/1]), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , χ(v[w/k])  � ��� w ∈ V (C)  ��  =  ���  χ(v′[w/1]), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' , χ(v′[w/k])  � ��� w ∈ V (C)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Overall, this implies that χ is k-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  ⌟  Combining both claims, we obtain that χ ⪯ χ(∞)  k  [C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Now, we complete the proof by setting  C to the disjoint union of A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' First suppose that k odd, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', k = 2ℓ − 1 for some integer ℓ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Since  there is a sentence ϕ ∈ Ck+1 such that A |= ϕ and B ̸|= ϕ, we conclude that A ̸≃k B using  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' So Bin(A) ̸≃2 Bin(B) by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1, the 2-WL algorithm  distinguishes between Bin(A) and Bin(B) after at most r = O(|V (Bin(A))| log |V (Bin(A))|) =  O(ℓ·nℓ·log n) many refinement rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Using Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2 again, this means there is a sentence  ϕ′ ∈ C(r)  3  such that Bin(A) |= ϕ′ and Bin(B) ̸|= ϕ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' So there is a sentence ψ ∈ C(r·ℓ)  3·ℓ  such that  A |= ψ and B ̸|= ψ using Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Note that 3ℓ = 3 · k+1  2  = d and r · ℓ = O(ℓ2 · nℓ · log n) =  O(k2 · n(k+1)/2 log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  For k being even, the statement the of theorem follows by applying the first case to k′ =  k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  26  7  Conclusion  We obtained new upper and lower bounds for the iteration number of the WL algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' First,  we showed that k-WL always stabilizes after at most O(knk−1 log n) rounds for all k ≥ 2, which  is the first non-trivial upper on the iteration number for k ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We complemented this result by  a lower bound of nΩ(k) which improves over the previously known lower bound of nΩ(k/ log k) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Finally, we also investigated tradeoffs between the dimension and the iteration number of WL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Using known characterizations of WL, our results also imply upper and lower bounds on the  quantifier rank of formulas in Ck required to distinguish between two structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Still, several questions remain open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' The first question concerns the iteration number of  k-WL on graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' The structures on which our lower bounds hold are n-element structures of  arity Θ(k) and size nΘ(k), and the increase in arity is inherent in the hardness condensation  from [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' The best known lower bound on the iteration number of k-WL on graphs is Ω(n) due  to Fürer [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' As an intermediate question, one can also ask for improved lower bounds in the  size of the structure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', the sum of the sizes of all relations), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=', are there structures on which  the iteration number of k-WL exceeds Ω(m) where m denotes the size of the structure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Our next question concerns the quantifier rank of formulas in Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' While our lower bounds  extend to the logic Lk (see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='5), this is not the case for the upper bounds that crucially  rely on the availability of counting quantifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' A non-trivial upper bound of O(n2/ log n) on the  quantifier rank of formulas in L3 has been obtained in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Can we also obtain improved upper  bounds on the quantifier rank of formulas in Lk for k ≥ 4?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Finally, we ask for further results on tradeoffs between the variable number and the quantifier  rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Specifically, is there an integer d ≥ 3 such that, for all structures A and B of size n  distinguished by 3-WL, d-WL distinguishes between A and B in at most �O(n) rounds (where  �O(·) hides polylogarithmic factors)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' We remark that even d = 3 may be a valid choice, but any  d ≥ 3 is sufficient to obtain further tradeoffs in the spirit of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  References  [1] László Babai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Graph isomorphism in quasipolynomial time [extended abstract].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' In Daniel Wichs  and Yishay Mansour, editors, Proceedings of the 48th Annual ACM SIGACT Symposium on Theory  of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 684–697.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' ACM, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1145/2897518.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='2897542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  [2] László Babai and Péter Frankl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' Linear algebra methods in combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' University of Chicago,  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  [3] Christoph Berkholz and Jakob Nordström.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Near-optimal lower bounds on quantifier depth and  Weisfeiler-Leman refinement steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' In Martin Grohe, Eric Koskinen, and Natarajan Shankar, edi-  tors, Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS  ’16, New York, NY, USA, July 5-8, 2016, pages 267–276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' ACM, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='1145/2933575.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  2934560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
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+page_content=' In  Fernando Orejas, Paul G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
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+page_content=' Springer, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
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+page_content=' The logic of graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' In 36th Annual ACM/IEEE Symposium on  Logic in Computer Science, LICS 2021, Rome, Italy, June 29 - July 2, 2021, pages 1–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
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+page_content=' In 34th Annual ACM/IEEE Symposium on  Logic in Computer Science, LICS 2019, Vancouver, BC, Canada, June 24-27, 2019, pages 1–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
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+page_content=' Cambridge University Press,  1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
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+page_content=' Isomorphism testing parameterized by genus and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' In Petra Mutzel, Rasmus  Pagh, and Grzegorz Herman, editors, 29th Annual European Symposium on Algorithms, ESA 2021,  September 6-8, 2021, Lisbon, Portugal (Virtual Conference), volume 204 of LIPIcs, pages 72:1–72:18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
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+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  [20] Daniel Neuen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  Isomorphism testing for graphs excluding small topological subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  In  Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium  on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12,  2022, pages 1411–1434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
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+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
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+page_content='59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  [21] Martin Otto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
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+page_content=' Cambridge University Press, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
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+page_content='acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
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+page_content='2746617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
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+page_content='  [25] Oleg Verbitsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
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+page_content=' In Wolfgang  Thomas and Pascal Weil, editors, STACS 2007, 24th Annual Symposium on Theoretical Aspects of  Computer Science, Aachen, Germany, February 22-24, 2007, Proceedings, volume 4393 of Lecture  Notes in Computer Science, pages 682–693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
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+page_content=' The reduction of a graph to canonical form and the algebra  which appears therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' NTI, Series 2, 1968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
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+page_content='iti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='zcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='cz/wl2018/pdf/wl_paper_translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
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+page_content='  How powerful are graph neural  networks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' In 7th International Conference on Learning Representations, ICLR 2019, New Orleans,  LA, USA, May 6-9, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content=' OpenReview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='net, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='  URL: https://openreview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
+page_content='net/forum?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
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+page_content='  29' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFQT4oBgHgl3EQfZzYk/content/2301.13317v1.pdf'}
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+Quantifying Multiphase Flow of Aqueous Acid and Gas CO2 in De-
+forming Porous Media Subject to Dissolution†
+Rafid Musabbir Rahman, Carson Kocmick, Colin Shaw and Yaofa Li∗
+Mineral dissolution in porous media coupled with single- or multi-phase flows is pervasive in nat-
+ural and engineering systems including carbon capture and sequestration (CCS) and acid stimu-
+lation in reservoir engineering. Dissolution of minerals occurs as chemicals in the solid phase is
+transformed into ions in the aqueous phase, effectively modifying the physical, hydrological and
+geochemical properties of the solid matrix, and thus leading to the strong coupling between local
+dissolution rate and pore-scale flow. However, our fundamental understanding of this coupling
+effect at the pore level is still limited. In this work, mineral dissolution is studied using novel
+calcite-based porous micromodels under single- and multiphase conditions, with a focus on the in-
+teractions of mineral dissolution with pore flow. The microfluidic devices used in the experiments
+were fabricated in calcite using photolithography and wet etching. These surrogate porous media
+offer precise control over the structures and chemical properties and facilitate unobstructed and
+unaberrated optical access to the pore flow with µPIV methods. The preliminary results provide
+a unique view of the flow dynamics during mineral dissolution. Based on these pore-scale mea-
+surements, correlations between pore-scale flow and dissolution rates can be developed for several
+representative conditions.
+1
+Introduction
+Reactive dissolution of minerals in porous media coupled with single- or multi-phase flows is per-
+vasive in natural and engineering systems1–16. In subsurface environments, the solid porous matrix
+is composed of various types of minerals, through which subsurface water flows. Dissolution of
+minerals occurs as chemicals in the solid phase is transformed into ions in the aqueous phase, ef-
+fectively modifying the physical, hydrological and geochemical properties of the solid matrix as
+well as the chemistry in the aqueous phase. These processes play a defining role in a broad range
+of applications including: (i) carbon capture and sequestration (CCS)11,16; (ii) acid stimulation in
+reservoir engineering1,2; (iii) soil formation and contaminant transport in soil4,5; (iv) development
+of underground cave systems17; and (v) global carbon and nutrient cycling18,19. For instance, CCS
+is considered as a viable technology to reduce carbon emissions to the atmosphere, thus effectively
+mitigating global climate change. However, injection of CO2 into geologic formations leads to disso-
+lution of minerals comprising reservoirs rocks, potentially creating leakage pathways that threaten
+the safety and security of CO2 storage11,16. In agriculture, mineral dissolution supplies nutrients
+to soil19–22, but at the same time it can also cause the release of heavy metals leading to soil con-
+tamination4,5. In reservoir stimulation, reactive fluids such as hydrochloric acid (HCl) are injected
+into damaged wellbores to increase its permeability and thus the well productivity1. Therefore, a
+Mechanical & Industrial Engineering Department, Montana State University, Bozeman, Montana, USA. Fax: 406-994-6292; Tel: 406-994-6773; E-mail:
+yaofa.li@montana.edu
+∗ Corresponding author.
+1
+1–12 | 1
+arXiv:2301.00378v1  [physics.flu-dyn]  1 Jan 2023
+
+comprehensive understanding of mineral dissolution is crucial to successfully modeling, predicting,
+controlling and optimizing many those processes.
+However, a full understanding of reactive dissolution in porous media is still lacking due to a
+number of complications. The porous materials are often of high heterogeneity structurally and
+chemically, causing complex flow and reaction patterns. The ranges of the underlying spatial and
+temporal scales for practical problems are enormous, and processes at fine (pore) scales are known
+to meaningfully impact dissolution at much larger (reservoir) scales.
+In fact, one of the most
+quoted challenges in the literature is that dissolution rates in porous media measured in the lab
+are typically orders of magnitude higher than those observed in the field, often referred to as the
+“lab–field discrepancy"3,6,8,9,23–30. The mismatch not only poses strong challenges in developing
+accurate predictive models based on the rate laws developed in laboratory, but also highlights a
+lack of fundamental understanding of mineral dissolution in porous media.
+Dissolution of minerals in porous media is subject to chemical reaction, advection and diffusion.
+Field-based dissolution rates are usually estimated by mass balance approaches in soil profiles and
+watersheds, which are often subject to transport limitations6,31, whereas the laboratory reaction
+rates are often measured in well-mixed systems using freshly crushed mineral suspensions, where
+mass transport limitations in the aqueous phase are intentionally eliminated through forced mix-
+ing32. As such, several factors have been proposed or identified to be collectively responsible for
+this discrepancy, which fall into two groups: extrinsic ones including the density of reactive sites
+and secondary mineralization, and intrinsic ones including heterogeneous pore flow and microbial
+processes12,23. While the exact reason(s) and their relative contributions to the enormous dis-
+crepancy is still under debate, it is well accepted that that the laboratory measurement based on
+well-mixed systems (i.e., no concentration gradient) does not appropriately represent the scenario
+in real fields, because: (a) in fields, pore-level concentration gradient can develop depending on the
+relative strengths of reaction and transport; (b) accessibility of reactive surfaces depends strongly
+on the local flow conditions. In fact, several studies have conjectured the lab-field discrepancy is pri-
+marily due to the concentration gradients resulting from incomplete mixing within individual pores
+that are in turn subject to heterogeneous flow fields at the microscopic scale9,23–25,28,30. Additionally,
+it has been shown that multiphase flow is possible in mineral dissolution, which significantly affects
+the accessibility of reactive surfaces11,13,15,33–37.
+To further our understanding, quantifying pore flow is highly needed. Pore-scale processes in
+mineral dissolution are coupled through a non-linear feedback loop linking fluid flow, reactant
+transport, surface reaction and pore evolution (cf. Fig. 2)9. A positive feedback mechanism is
+well known to exist between local fluid flow and reactive dissolution, which is responsible for the
+creation of wormholes during acid stimulation1,38,39. Additionally, increasing evidences reinforce
+that pore flow (i.e., advective transport) has very subtle effects on the development of concentration
+gradient and thus local dissolution rate24,25. Pore flow is especially important when assessing and
+upscaling the pore-scale effects to a much larger domain, considering that for a given distance l the
+required time would scale with l for advective transport, but with l2 for diffusive transport.
+Moreover, previous studies on reactive dissolution in porous media have been almost exclusively
+focused on single-phase flow conditions, although abundant evidences have shown that multiphase
+can play an important role11,13,35–38. For instance, during acid stimulation of carbonate reservoirs
+2
+1–12 | 2
+
+Inlet
+Outlet
+Inlet
+Outlet
+SIDE VIEW
+TOP VIEW
+Calcite
+Glass
+Glass
+Porous
+Structure
+Fig.
+1 Schematic diagram illustrating the design of the micromodel for the top [top] and side [bottom] views.
+The micromodel consists of 2 layers of glass with a thin calcite layer sandwiched in between, and the porous
+section is formed by regularly arranged circular posts.
+under practical injection conditions, produced CO2 can emerge as a separate phase, creating a
+multiphase environment for subsequent dissolution processes15,33,34. An immiscible phase such
+as oil or gas can even be present prior to acid injection36,40. Multiphase flow introduces effects
+like capillarity and flow instability and its complex phase configuration can alter the abundance
+of reactive surface areas thus affecting dissolution rate. Accounting for multiphase flow effects
+through innovative pore flow measurement is strongly needed to reshape our understanding of
+mineral dissolution.
+To this end, single- and multiphase flows in porous media subject to simultaneous mineral
+dissolution are investigated in novel calcite-based porous micromodels. The microfluidic devices
+used in the experiments were fabricated in calcite using photolithography and wet etching. These
+surrogate porous media offer precise control over the structures and chemical properties and facil-
+itate unobstructed and unaberrated optical access to the pore flow with µPIV methods. White field
+microscopy and the fluorescent micro-PIV method were simultaneously employed by seeding the
+water phase with fluorescent particles, in order to achieve simultaneous measurement of the veloc-
+ity fields of the aqueous and structure evolution of the solid phase. Specifically, the objective of the
+study is to: 1) design and fabricate calcite-based micromodels using microfabrication techniques;
+2) evaluate how local dissolution rate is affect by local flow field and multiphase flow.
+2
+Experimental Methods
+2.1
+Micromodel
+The micromodel consist of an inlet, an outlet, and a porous section, as shown in Fig. 1 [top]. While
+the inlet and outlet serve for fluid delivery, the porous section is the region of interest for flow
+and dissolution quantification. To mimic a porous media, the porous section is formed by regularly
+arranged circular posts with an overall size of 3.2 mm × 4.8 mm, and an porosity of approximately
+45%.
+From the side view (Fig. 1 [bottom]), the micromodel is constructed with three layers,
+with two glass layers sandwiching a calcite layer in between. The glass layers provide a structural
+support for the calcite layer, and also facilitate unobstructed and unaberrated optical access to the
+pore flow within the porous structure. The calcite layer presents an opportunity for us to effectively
+3
+1–12 | 3
+
+Fig.
+2 Flowchart illustrating the major steps required to fabricate a calcite-based micromodel. A bulk calcite
+crystal was sectioned, double side polished, and grounded to 100 µm. By photolithography, the porous pattern
+was transferred to the calcite wafer, which was then wet-etched in 0.4% HCl solution. Two through holes were
+drilled. The wafer was then be cleaned and adhesively bonded to another glass wafer on the top to form closed
+flow channels.
+simulate the geochemistry and pore-scale fluid–rock interactions in real time.
+The fabrication procedures are illustrated in Fig. 2. First the calcite crystal was cleaned with
+acetone and attached to a reference glass using an epoxy glue. The calcite crystal was then sec-
+tioned using a thin-sectioning saw (Pelcon Automatic Thin Section Machine). The top surface was
+ground with three steel grinding rollers, which were coated with nickel bonded diamonds. The
+grinding process begins with the coarse roller embedded with 64 mum diamonds, continues on the
+intermediate roller with 32 mum diamonds and ends on the fine roller with 16 umm diamonds. Fol-
+lowing the grinding process, the top surface was attached to a clear microscopic slide using UV
+glue which was cured with UV light for 10 minutes while the sample was securely clamped. Using
+the same thin-sectioning machine, the reference glass was removed. The new open surface was
+then ground following the same procedures with the same 3 rollers. Finally, a polishing step was
+carried out using 6 µm grit for 2 minutes at 150 rpm. The resulting thickness of the calcite sample
+is ∼31 µm with an of uncertainty of 5.5 µm.
+The calcite sample attached to the glass substrate was then ready for photolithography to trans-
+fer the designed pattern (Fig. 1) to the sample. First, a spin coater (Laurell) was used to spin coat
+a 9 mum thick photoresist (AZ10XT) at 2000 rpm for 45 s. The coated calcite wafer was soft baked
+for 6 minutes at 115 ◦C, following which the sample was allowed to rest for 1 hour to equilibrate
+with the environment. Second, the AB-M contact aligner was used to expose the photoresist at
+15 mW/cm2 for 25 s, which was then developed in AZ300 MIF developer for 15 minutes. The com-
+plete the photolithograpy, the sample was rinsed with DI water and dried with N2 stream. Finally,
+the porous structure was etched by immersing the sample in 0.4% hydrochloric acid (HCl) for 8
+minutes. To complete the etching process, the calcite was again rinsed with DI water and then dried
+with N2 gas (Fig. 4). Following the etching process, a second microscope slide was bonded to form
+a closed micromodel. Two nanoports (IDEX Health & Science) were attached to the micromodel
+for fluid delivery41
+4
+1–12 | 4
+
+ Start
+Mask Printing
+UV Exposure
+Calcite Wafer
+Photoresist (PR
+Complete
+Drilling/Bonding
+Wet Etching
+PR Stripping/Cleaning
+PR Developing
+Calcite
+Glass
+PhotoresistFig.
+3 (a) A photo of the sample after photolithography illustrating the pattern; (b)
+An SEM photo of the
+resulting porous structure after etching.
+2.2
+Experimental Procedure
+The working fluid used in this study was HCl with a concentration of 0.8% to ensure an effective but
+controlled reaction with the solid calcite. To facilitate µPIV measurement in the aqueous phase, the
+HCl solution was seeded with fluorescent polystyrene particles of 1 mum in diameter (Invitrogen
+F8819), at a nominal volume fraction of 3 × 10−4. The volume fraction of the solid particles is
+low so as to not appreciably change the aqueous phase properties and their density matched that of
+water so as to behave in a neutrally buoyant manner. Prior to each experiment, a new micromodel is
+cleaned and dried using a Nitrogen stream, and then the nanoports were connected to the plumbing
+tubings. The flow of the aqueous phase was then initiated, which was controlled by syringe pump
+(Harvard Apparatus PHD 2000).
+2.3
+Image Acquisition and Processing
+In µPIV, two consecutive images of the particles are recorded with a prescribed time delay, ∆t. Each
+image is then subdivided into small areas, called interrogation windows. The average displacement
+of the tracer particles ∆∆∆x within each interrogation window is determined statistically from cross-
+correlation analysis of consecutive images. The velocity vector uuu for each interrogation window
+can then be obtained by dividing the displacement ∆∆∆x with the prescribed ∆t 42. To perform µPIV
+measurement, a green LED (Thorlabs Solis-525c) with a dominant wavelength of 525 nm was used
+to excite the fluorescent particles in the aqueous phase. The fluorescence emitted from the tracer
+particles was passed through a λ = 575 ± 13 nm bandpass filter and focused by an Olympus IX-
+71 microscope equipped with an objective lens with 4x magnification and 0.1 numerical aperture
+(NA), onto the detector of a high-speed scientific CMOS camera (Phantom VEO440). The camera
+was run at 100 frame per second, providing sufficient temporal resolution to capture the evolution
+of dynamic pore-scale phenomena.
+5
+1–12 | 5
+
+aOwing to the multi-phase nature of some of the flows under study, the gas phase of the flow
+where no tracer particle is present needes to be masked out prior to image interrogation for quan-
+tification of the water velocity distribution43–45. This masking was performed based on a simple
+thresholding procedure complemented by manual adjustment when necessary41. For details re-
+garding the image segmentation and image mask creation, the reader is referred to Refs.41,45. To
+calculate the water velocity field, both frames in an acquired frame pair were first masked using
+the mask created in the previous steps and then interrogated in an in-house MATLAB code using
+a multi-pass, two-frame cross-correlation method45. Unless otherwise stated, the size of the inter-
+rogation windows was 32×32 pixels, with 50% overlap, which yielded a velocity vector spacing of
+40 µm and a spatial resolution of 80 µm.
+3
+Preliminary Results
+While the micromodel fabrication and experiments are still undergoing, herein in this paper we
+present some preliminary results to showcase our study. It is worth noting that, the micromodel
+used in this preliminary study was fabricated in a slightly different way from what was described
+above. Specifically, to create the porous structures, instead of wet-etching, the micro-structure
+was micro-milled directly into a polished calcite crystal based on the patterns as shown in Fig. 4.
+Nevertheless, the preliminary results provide a unique view of the flow dynamics during mineral
+dissolution.
+Figures 5a and 5b show the sample velocity fields of the pore-scale flow under single and
+multiphase flow conditions, respectively. When HCl concentration is low but the pore flow rate is
+relatively high, the reaction product (i.e., CO2) is instantaneously dissolved in the aqueous phase
+leading to a single phase flow. However, when HCl concentration is high such that the produced
+CO2 cannot be instantaneously dissolved, it will emerge as a separate phase, leading to a multiphase
+flow.
+While the flow field with single-phase flow is relatively simple, that flow is significantly
+modified in the multiphase flow case due to the presence CO2 bubbles that are generated in-situ
+as a result of chemical reaction between the liquid and solid phases. The separate CO2 phase is
+expected to not only divert the HCl flow, but also shied the solid surfaces from further reaction,
+thus significantly modifying the local dissolution pattern and rate.
+The velocity fields generated by this technique also enables computations of various scalar
+fields, such as Péclet (Pe) and Damköhler number(Da) fields, all involved in the mechanisms con-
+trolling the reactive transport and used as characterizating metrics of the relative importance of
+reaction, diffusion and advection. The Péclet number (Pe) and the Damköhler number are (Da)
+defined as46,47:
+Pe = Vl/D
+(1)
+Da = kl/V
+(2)
+Here, V is the fluid velocity, l is the characteristic length scale (e.g., pore diameter), D is the
+diffusion coefficient, and k is the reaction rate constant. Physically, Pe defines the ratio of advective
+to diffusive transport rates, and Da defines the ratio of the overall chemical reaction rate to the
+advective mass transport rate. Fig. 6 shows the Pe and Da fields calculated with Eqs.
+1 and 2
+6
+1–12 | 6
+
+a
+b
+Inlet
+Outlet
+Fig. 4 (a) A photo of the preliminary micromodel fabricated by micro-milling a piece of polished calcite crystal.
+The top layer is the polished calcite crystal, which is bonded to a glass slide, whereas the yellow parts are the
+nanoports used for fluid delivery. (b) A schematic showing the machined porous structure in the preliminary
+micromodel.
+based on the velocity field shown in Fig. 5B, highlighting the spatial variability of these fields,
+which will be valuable in understanding transport and reaction potential30,48.
+Another key attribute of the this flow is the pore-scale dissolution rate of grain. Fig. 7 depicts
+the dissolution process of a particular grain subject to a flow of 0.8% HCl solution at 0.2 ml/min.
+Fig. 7A acquired with the white-field microscopy shows at an instance (t = 0 s) the geometry of
+the grain with a CO2 bubble attached to its bottom. Bubbles are observed to preferentially grow
+at this particular location, presumably due to the relatively sharp corner therein, which serves as
+nucleation site. Fig. 7B illustrates the evolution of the solid boundaries with time. These plots
+clearly indicate a much faster erosion of the grain at its front than its back due to stronger convec-
+tive transport in the upstream. It also shows that the dissolution rate at the bottom of the grain
+corresponding to the location of the bubble in Fig. 7A is very low, suggesting that the presence of
+CO2 significantly impeded dissolution therein and the neighboring region adjacent to it. Fig. 7C
+plots the temporal variation of the total grain volume and the overall dissolution rate as a func-
+tion of time, whereas Fig. 7D plots the spatial variation of the dissolution rate around the grain
+at a certain moment (t = 24 s). While both the total grain volume and the overall dissolution rate
+decrease with time in a quite smooth manner, the local dissolution rate displays a large variability
+owing to the complex flow fields and disturbance of CO2 bubble around the grain. This behavior
+7
+1–12 | 7
+
+Fig.
+5 Sample velocity fields of a single-phase flow of water (A) and a multiphase flow of water and air (B)
+through the calcite micromodel at the same flow rate. Contours and arrows indicate the velocity magnitude and
+direction, respectively, in the aqueous phase; and gray and yellow regions represent solid grains and gas bubbles,
+respectively.
+again reinforces the importance of the characterization of local dissolution and pore flow, which
+has never been achieved before.
+The spatially- and temporally-resolved velocity data also enables the computation of flow statis-
+tics, providing insight into the integrated effects of pore-scale process thus can be particularly useful
+to develop constitutive correlations necessary for large-scale predictive models. Fig. 8 presents the
+probability density functions (PDF) of normalized velocity components, u and v in a single- and
+multiphase flow, respectively. The PDFs reveal that a second phase renders the flow much more
+random with broader distributions for both u and v. These results also highlight the flow alteration
+induced by multiphase flow, justifying the need for systematic studies of the subtle effects of mul-
+tiphase flow on mineral dissolution. In fact, statistical measures, specifically PDFs, have also been
+used as a great tool to characterize pore structures in order to reveal underlying physics that is not
+apparent via local observations48,49. Additionally, PDFs also provide another convenient basis for
+experimental validation of computer codes for multiphase fluid flow. As pointed out by Meakin et
+al.50, due to fabrication error, microscale roughness, trace impurities and other experimental and
+numerical uncertainties, pore by pore comparison is virtually impractical, and thus comparison of
+statistical measures such as PDFs is often desired.
+8
+1–12 | 8
+
+A
+300 μm
+B
+Velocity Magnitude
+[mm/s]
+0
+0.4
+0.8Fig. 6 Pe (A) and Da (B) fields calculated based on the velocity field in Fig. 5B and Equations 1 and 2.
+4
+Conclusions
+In this study, a calcite-based micromodel was successfully designed and fabricated using micro-
+fabrication techniques. The µPIV technique was employed that captured spatially and temporally
+resolved dynamics of single- and multiphase flows of CO2 and diluted HCl in 2D calcite-based
+porous media. While the flow field with single-phase flow is relatively simple, that flow is signif-
+icantly modified in the multiphase flow case due to the presence CO2 bubbles that are generated
+in-situ as a result of chemical reaction between the liquid and solid phases. The separate CO2 phase
+is expected to not only divert the HCl flow, but also shied the solid surfaces from further reaction,
+thus significantly modifying the local dissolution pattern and rate. The preliminary results not
+only provide a unique view of the flow dynamics during multiphase mineral dissolution, but also
+demonstrate great potential of this methodology for future more in-depth studies.
+Author Contributions
+Conflicts of interest
+There are no conflicts to declare.
+9
+1–12 | 9
+
+A
+300 μm
+Pe
+0
+4
+8
+B
+Da
+0
+3 × 10-3
+6 × 10-3Fig.
+7 Dissolution of a particular grain: (A) grain at t = 0 s, (B) evolution of grain boundaries with time, (C)
+temporal variation of total grain volume and overall dissolution rate and (D) spatial variation of local dissolution
+rate around the grain.
+Acknowledgements
+This work was performed in part at the Montana Nanotechnology Facility, an NNCI facility sup-
+ported by NSF Grant ECCS-1542210, and with support by the Murdock Charitable Trust. The work
+was partially supported by the Petroleum Research Fund, ACS (62687-DNI9YL) with Dr. Thomas
+Clancy serving as the program officer. YL thanks the Norm Asbjornson College of Engineering and
+the Center for Faculty Excellence at Montana State University for their support through the Fac-
+ulty Excellence Grants. RMR thanks the Norm Asbjornson College of Engineering for their support
+through the Benjamin Fellowship.
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+and H. S. Viswanathan, Geophysical Research Letters, 2020, 47, e2020GL087163.
+49 Y. Al-Khulaifi, Q. Lin, M. J. Blunt and B. Bijeljic, Environmental science & technology, 2017, 51,
+4108–4116.
+50 P. Meakin and A. M. Tartakovsky, Reviews of Geophysics, 2009, 47, RG3002.
+12
+1–12 | 12
+
diff --git a/U9AyT4oBgHgl3EQfhfiE/content/tmp_files/load_file.txt b/U9AyT4oBgHgl3EQfhfiE/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf,len=511
+page_content='Quantifying Multiphase Flow of Aqueous Acid and Gas CO2 in De-  forming Porous Media Subject to Dissolution†  Rafid Musabbir Rahman, Carson Kocmick, Colin Shaw and Yaofa Li∗  Mineral dissolution in porous media coupled with single- or multi-phase flows is pervasive in nat-  ural and engineering systems including carbon capture and sequestration (CCS) and acid stimu-  lation in reservoir engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Dissolution of minerals occurs as chemicals in the solid phase is  transformed into ions in the aqueous phase, effectively modifying the physical, hydrological and  geochemical properties of the solid matrix, and thus leading to the strong coupling between local  dissolution rate and pore-scale flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' However, our fundamental understanding of this coupling  effect at the pore level is still limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' In this work, mineral dissolution is studied using novel  calcite-based porous micromodels under single- and multiphase conditions, with a focus on the in-  teractions of mineral dissolution with pore flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The microfluidic devices used in the experiments  were fabricated in calcite using photolithography and wet etching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' These surrogate porous media  offer precise control over the structures and chemical properties and facilitate unobstructed and  unaberrated optical access to the pore flow with µPIV methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The preliminary results provide  a unique view of the flow dynamics during mineral dissolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Based on these pore-scale mea-  surements, correlations between pore-scale flow and dissolution rates can be developed for several  representative conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  1  Introduction  Reactive dissolution of minerals in porous media coupled with single- or multi-phase flows is per-  vasive in natural and engineering systems1–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' In subsurface environments, the solid porous matrix  is composed of various types of minerals, through which subsurface water flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Dissolution of  minerals occurs as chemicals in the solid phase is transformed into ions in the aqueous phase, ef-  fectively modifying the physical, hydrological and geochemical properties of the solid matrix as  well as the chemistry in the aqueous phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' These processes play a defining role in a broad range  of applications including: (i) carbon capture and sequestration (CCS)11,16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' (ii) acid stimulation in  reservoir engineering1,2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' (iii) soil formation and contaminant transport in soil4,5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' (iv) development  of underground cave systems17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' and (v) global carbon and nutrient cycling18,19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' For instance, CCS  is considered as a viable technology to reduce carbon emissions to the atmosphere, thus effectively  mitigating global climate change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' However, injection of CO2 into geologic formations leads to disso-  lution of minerals comprising reservoirs rocks, potentially creating leakage pathways that threaten  the safety and security of CO2 storage11,16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' In agriculture, mineral dissolution supplies nutrients  to soil19–22, but at the same time it can also cause the release of heavy metals leading to soil con-  tamination4,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' In reservoir stimulation, reactive fluids such as hydrochloric acid (HCl) are injected  into damaged wellbores to increase its permeability and thus the well productivity1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Therefore, a  Mechanical & Industrial Engineering Department, Montana State University, Bozeman, Montana, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Fax: 406-994-6292;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Tel: 406-994-6773;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' E-mail:  yaofa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='li@montana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='edu  ∗ Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  1  1–12 | 1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='00378v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='flu-dyn]  1 Jan 2023  comprehensive understanding of mineral dissolution is crucial to successfully modeling, predicting,  controlling and optimizing many those processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  However, a full understanding of reactive dissolution in porous media is still lacking due to a  number of complications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The porous materials are often of high heterogeneity structurally and  chemically, causing complex flow and reaction patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The ranges of the underlying spatial and  temporal scales for practical problems are enormous, and processes at fine (pore) scales are known  to meaningfully impact dissolution at much larger (reservoir) scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  In fact, one of the most  quoted challenges in the literature is that dissolution rates in porous media measured in the lab  are typically orders of magnitude higher than those observed in the field, often referred to as the  “lab–field discrepancy"3,6,8,9,23–30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The mismatch not only poses strong challenges in developing  accurate predictive models based on the rate laws developed in laboratory, but also highlights a  lack of fundamental understanding of mineral dissolution in porous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  Dissolution of minerals in porous media is subject to chemical reaction, advection and diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  Field-based dissolution rates are usually estimated by mass balance approaches in soil profiles and  watersheds, which are often subject to transport limitations6,31, whereas the laboratory reaction  rates are often measured in well-mixed systems using freshly crushed mineral suspensions, where  mass transport limitations in the aqueous phase are intentionally eliminated through forced mix-  ing32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' As such, several factors have been proposed or identified to be collectively responsible for  this discrepancy, which fall into two groups: extrinsic ones including the density of reactive sites  and secondary mineralization, and intrinsic ones including heterogeneous pore flow and microbial  processes12,23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' While the exact reason(s) and their relative contributions to the enormous dis-  crepancy is still under debate, it is well accepted that that the laboratory measurement based on  well-mixed systems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=', no concentration gradient) does not appropriately represent the scenario  in real fields, because: (a) in fields, pore-level concentration gradient can develop depending on the  relative strengths of reaction and transport;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' (b) accessibility of reactive surfaces depends strongly  on the local flow conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' In fact, several studies have conjectured the lab-field discrepancy is pri-  marily due to the concentration gradients resulting from incomplete mixing within individual pores  that are in turn subject to heterogeneous flow fields at the microscopic scale9,23–25,28,30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Additionally,  it has been shown that multiphase flow is possible in mineral dissolution, which significantly affects  the accessibility of reactive surfaces11,13,15,33–37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  To further our understanding, quantifying pore flow is highly needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Pore-scale processes in  mineral dissolution are coupled through a non-linear feedback loop linking fluid flow, reactant  transport, surface reaction and pore evolution (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' 2)9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' A positive feedback mechanism is  well known to exist between local fluid flow and reactive dissolution, which is responsible for the  creation of wormholes during acid stimulation1,38,39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Additionally, increasing evidences reinforce  that pore flow (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=', advective transport) has very subtle effects on the development of concentration  gradient and thus local dissolution rate24,25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Pore flow is especially important when assessing and  upscaling the pore-scale effects to a much larger domain, considering that for a given distance l the  required time would scale with l for advective transport, but with l2 for diffusive transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  Moreover, previous studies on reactive dissolution in porous media have been almost exclusively  focused on single-phase flow conditions, although abundant evidences have shown that multiphase  can play an important role11,13,35–38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' For instance, during acid stimulation of carbonate reservoirs  2  1–12 | 2  Inlet  Outlet  Inlet  Outlet  SIDE VIEW  TOP VIEW  Calcite  Glass  Glass  Porous  Structure  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  1 Schematic diagram illustrating the design of the micromodel for the top [top] and side [bottom] views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  The micromodel consists of 2 layers of glass with a thin calcite layer sandwiched in between, and the porous  section is formed by regularly arranged circular posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  under practical injection conditions, produced CO2 can emerge as a separate phase, creating a  multiphase environment for subsequent dissolution processes15,33,34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' An immiscible phase such  as oil or gas can even be present prior to acid injection36,40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Multiphase flow introduces effects  like capillarity and flow instability and its complex phase configuration can alter the abundance  of reactive surface areas thus affecting dissolution rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Accounting for multiphase flow effects  through innovative pore flow measurement is strongly needed to reshape our understanding of  mineral dissolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  To this end, single- and multiphase flows in porous media subject to simultaneous mineral  dissolution are investigated in novel calcite-based porous micromodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The microfluidic devices  used in the experiments were fabricated in calcite using photolithography and wet etching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' These  surrogate porous media offer precise control over the structures and chemical properties and facil-  itate unobstructed and unaberrated optical access to the pore flow with µPIV methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' White field  microscopy and the fluorescent micro-PIV method were simultaneously employed by seeding the  water phase with fluorescent particles, in order to achieve simultaneous measurement of the veloc-  ity fields of the aqueous and structure evolution of the solid phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Specifically, the objective of the  study is to: 1) design and fabricate calcite-based micromodels using microfabrication techniques;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  2) evaluate how local dissolution rate is affect by local flow field and multiphase flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  2  Experimental Methods  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='1  Micromodel  The micromodel consist of an inlet, an outlet, and a porous section, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' 1 [top].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' While  the inlet and outlet serve for fluid delivery, the porous section is the region of interest for flow  and dissolution quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' To mimic a porous media, the porous section is formed by regularly  arranged circular posts with an overall size of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='2 mm × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='8 mm, and an porosity of approximately  45%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  From the side view (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' 1 [bottom]), the micromodel is constructed with three layers,  with two glass layers sandwiching a calcite layer in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The glass layers provide a structural  support for the calcite layer, and also facilitate unobstructed and unaberrated optical access to the  pore flow within the porous structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The calcite layer presents an opportunity for us to effectively  3  1–12 | 3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  2 Flowchart illustrating the major steps required to fabricate a calcite-based micromodel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' A bulk calcite  crystal was sectioned, double side polished, and grounded to 100 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' By photolithography, the porous pattern  was transferred to the calcite wafer, which was then wet-etched in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='4% HCl solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Two through holes were  drilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The wafer was then be cleaned and adhesively bonded to another glass wafer on the top to form closed  flow channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  simulate the geochemistry and pore-scale fluid–rock interactions in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  The fabrication procedures are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' First the calcite crystal was cleaned with  acetone and attached to a reference glass using an epoxy glue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The calcite crystal was then sec-  tioned using a thin-sectioning saw (Pelcon Automatic Thin Section Machine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The top surface was  ground with three steel grinding rollers, which were coated with nickel bonded diamonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The  grinding process begins with the coarse roller embedded with 64 mum diamonds, continues on the  intermediate roller with 32 mum diamonds and ends on the fine roller with 16 umm diamonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Fol-  lowing the grinding process, the top surface was attached to a clear microscopic slide using UV  glue which was cured with UV light for 10 minutes while the sample was securely clamped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Using  the same thin-sectioning machine, the reference glass was removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The new open surface was  then ground following the same procedures with the same 3 rollers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Finally, a polishing step was  carried out using 6 µm grit for 2 minutes at 150 rpm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The resulting thickness of the calcite sample  is ∼31 µm with an of uncertainty of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  The calcite sample attached to the glass substrate was then ready for photolithography to trans-  fer the designed pattern (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' 1) to the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' First, a spin coater (Laurell) was used to spin coat  a 9 mum thick photoresist (AZ10XT) at 2000 rpm for 45 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The coated calcite wafer was soft baked  for 6 minutes at 115 ◦C, following which the sample was allowed to rest for 1 hour to equilibrate  with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Second, the AB-M contact aligner was used to expose the photoresist at  15 mW/cm2 for 25 s, which was then developed in AZ300 MIF developer for 15 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The com-  plete the photolithograpy, the sample was rinsed with DI water and dried with N2 stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Finally,  the porous structure was etched by immersing the sample in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='4% hydrochloric acid (HCl) for 8  minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' To complete the etching process, the calcite was again rinsed with DI water and then dried  with N2 gas (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Following the etching process, a second microscope slide was bonded to form  a closed micromodel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Two nanoports (IDEX Health & Science) were attached to the micromodel  for fluid delivery41  4  1–12 | 4  Start  Mask Printing  UV Exposure  Calcite Wafer  Photoresist (PR  Complete  Drilling/Bonding  Wet Etching  PR Stripping/Cleaning  PR Developing  Calcite  Glass  PhotoresistFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  3 (a) A photo of the sample after photolithography illustrating the pattern;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' (b)  An SEM photo of the  resulting porous structure after etching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='2  Experimental Procedure  The working fluid used in this study was HCl with a concentration of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='8% to ensure an effective but  controlled reaction with the solid calcite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' To facilitate µPIV measurement in the aqueous phase, the  HCl solution was seeded with fluorescent polystyrene particles of 1 mum in diameter (Invitrogen  F8819), at a nominal volume fraction of 3 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The volume fraction of the solid particles is  low so as to not appreciably change the aqueous phase properties and their density matched that of  water so as to behave in a neutrally buoyant manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Prior to each experiment, a new micromodel is  cleaned and dried using a Nitrogen stream, and then the nanoports were connected to the plumbing  tubings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The flow of the aqueous phase was then initiated, which was controlled by syringe pump  (Harvard Apparatus PHD 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='3  Image Acquisition and Processing  In µPIV, two consecutive images of the particles are recorded with a prescribed time delay, ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Each  image is then subdivided into small areas, called interrogation windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The average displacement  of the tracer particles ∆∆∆x within each interrogation window is determined statistically from cross-  correlation analysis of consecutive images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The velocity vector uuu for each interrogation window  can then be obtained by dividing the displacement ∆∆∆x with the prescribed ∆t 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' To perform µPIV  measurement, a green LED (Thorlabs Solis-525c) with a dominant wavelength of 525 nm was used  to excite the fluorescent particles in the aqueous phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The fluorescence emitted from the tracer  particles was passed through a λ = 575 ± 13 nm bandpass filter and focused by an Olympus IX-  71 microscope equipped with an objective lens with 4x magnification and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='1 numerical aperture  (NA), onto the detector of a high-speed scientific CMOS camera (Phantom VEO440).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The camera  was run at 100 frame per second, providing sufficient temporal resolution to capture the evolution  of dynamic pore-scale phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  5  1–12 | 5  aOwing to the multi-phase nature of some of the flows under study, the gas phase of the flow  where no tracer particle is present needes to be masked out prior to image interrogation for quan-  tification of the water velocity distribution43–45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' This masking was performed based on a simple  thresholding procedure complemented by manual adjustment when necessary41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' For details re-  garding the image segmentation and image mask creation, the reader is referred to Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='41,45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' To  calculate the water velocity field, both frames in an acquired frame pair were first masked using  the mask created in the previous steps and then interrogated in an in-house MATLAB code using  a multi-pass, two-frame cross-correlation method45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Unless otherwise stated, the size of the inter-  rogation windows was 32×32 pixels, with 50% overlap, which yielded a velocity vector spacing of  40 µm and a spatial resolution of 80 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  3  Preliminary Results  While the micromodel fabrication and experiments are still undergoing, herein in this paper we  present some preliminary results to showcase our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' It is worth noting that, the micromodel  used in this preliminary study was fabricated in a slightly different way from what was described  above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Specifically, to create the porous structures, instead of wet-etching, the micro-structure  was micro-milled directly into a polished calcite crystal based on the patterns as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  Nevertheless, the preliminary results provide a unique view of the flow dynamics during mineral  dissolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  Figures 5a and 5b show the sample velocity fields of the pore-scale flow under single and  multiphase flow conditions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' When HCl concentration is low but the pore flow rate is  relatively high, the reaction product (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=', CO2) is instantaneously dissolved in the aqueous phase  leading to a single phase flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' However, when HCl concentration is high such that the produced  CO2 cannot be instantaneously dissolved, it will emerge as a separate phase, leading to a multiphase  flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  While the flow field with single-phase flow is relatively simple, that flow is significantly  modified in the multiphase flow case due to the presence CO2 bubbles that are generated in-situ  as a result of chemical reaction between the liquid and solid phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The separate CO2 phase is  expected to not only divert the HCl flow, but also shied the solid surfaces from further reaction,  thus significantly modifying the local dissolution pattern and rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  The velocity fields generated by this technique also enables computations of various scalar  fields, such as Péclet (Pe) and Damköhler number(Da) fields, all involved in the mechanisms con-  trolling the reactive transport and used as characterizating metrics of the relative importance of  reaction, diffusion and advection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The Péclet number (Pe) and the Damköhler number are (Da)  defined as46,47:  Pe = Vl/D  (1)  Da = kl/V  (2)  Here, V is the fluid velocity, l is the characteristic length scale (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=', pore diameter), D is the  diffusion coefficient, and k is the reaction rate constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Physically, Pe defines the ratio of advective  to diffusive transport rates, and Da defines the ratio of the overall chemical reaction rate to the  advective mass transport rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' 6 shows the Pe and Da fields calculated with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  1 and 2  6  1–12 | 6  a  b  Inlet  Outlet  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' 4 (a) A photo of the preliminary micromodel fabricated by micro-milling a piece of polished calcite crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  The top layer is the polished calcite crystal, which is bonded to a glass slide, whereas the yellow parts are the  nanoports used for fluid delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' (b) A schematic showing the machined porous structure in the preliminary  micromodel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  based on the velocity field shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' 5B, highlighting the spatial variability of these fields,  which will be valuable in understanding transport and reaction potential30,48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  Another key attribute of the this flow is the pore-scale dissolution rate of grain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' 7 depicts  the dissolution process of a particular grain subject to a flow of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='8% HCl solution at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='2 ml/min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' 7A acquired with the white-field microscopy shows at an instance (t = 0 s) the geometry of  the grain with a CO2 bubble attached to its bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Bubbles are observed to preferentially grow  at this particular location, presumably due to the relatively sharp corner therein, which serves as  nucleation site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' 7B illustrates the evolution of the solid boundaries with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' These plots  clearly indicate a much faster erosion of the grain at its front than its back due to stronger convec-  tive transport in the upstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' It also shows that the dissolution rate at the bottom of the grain  corresponding to the location of the bubble in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' 7A is very low, suggesting that the presence of  CO2 significantly impeded dissolution therein and the neighboring region adjacent to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' 7C  plots the temporal variation of the total grain volume and the overall dissolution rate as a func-  tion of time, whereas Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' 7D plots the spatial variation of the dissolution rate around the grain  at a certain moment (t = 24 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' While both the total grain volume and the overall dissolution rate  decrease with time in a quite smooth manner, the local dissolution rate displays a large variability  owing to the complex flow fields and disturbance of CO2 bubble around the grain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' This behavior  7  1–12 | 7  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  5 Sample velocity fields of a single-phase flow of water (A) and a multiphase flow of water and air (B)  through the calcite micromodel at the same flow rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Contours and arrows indicate the velocity magnitude and  direction, respectively, in the aqueous phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' and gray and yellow regions represent solid grains and gas bubbles,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  again reinforces the importance of the characterization of local dissolution and pore flow, which  has never been achieved before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  The spatially- and temporally-resolved velocity data also enables the computation of flow statis-  tics, providing insight into the integrated effects of pore-scale process thus can be particularly useful  to develop constitutive correlations necessary for large-scale predictive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' 8 presents the  probability density functions (PDF) of normalized velocity components, u and v in a single- and  multiphase flow, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The PDFs reveal that a second phase renders the flow much more  random with broader distributions for both u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' These results also highlight the flow alteration  induced by multiphase flow, justifying the need for systematic studies of the subtle effects of mul-  tiphase flow on mineral dissolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' In fact, statistical measures, specifically PDFs, have also been  used as a great tool to characterize pore structures in order to reveal underlying physics that is not  apparent via local observations48,49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Additionally, PDFs also provide another convenient basis for  experimental validation of computer codes for multiphase fluid flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' As pointed out by Meakin et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='50, due to fabrication error, microscale roughness, trace impurities and other experimental and  numerical uncertainties, pore by pore comparison is virtually impractical, and thus comparison of  statistical measures such as PDFs is often desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  8  1–12 | 8  A  300 μm  B  Velocity Magnitude  [mm/s]  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='8Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' 6 Pe (A) and Da (B) fields calculated based on the velocity field in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' 5B and Equations 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  4  Conclusions  In this study, a calcite-based micromodel was successfully designed and fabricated using micro-  fabrication techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The µPIV technique was employed that captured spatially and temporally  resolved dynamics of single- and multiphase flows of CO2 and diluted HCl in 2D calcite-based  porous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' While the flow field with single-phase flow is relatively simple, that flow is signif-  icantly modified in the multiphase flow case due to the presence CO2 bubbles that are generated  in-situ as a result of chemical reaction between the liquid and solid phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The separate CO2 phase  is expected to not only divert the HCl flow, but also shied the solid surfaces from further reaction,  thus significantly modifying the local dissolution pattern and rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The preliminary results not  only provide a unique view of the flow dynamics during multiphase mineral dissolution, but also  demonstrate great potential of this methodology for future more in-depth studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  Author Contributions  Conflicts of interest  There are no conflicts to declare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  9  1–12 | 9  A  300 μm  Pe  0  4  8  B  Da  0  3 × 10-3  6 × 10-3Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  7 Dissolution of a particular grain: (A) grain at t = 0 s, (B) evolution of grain boundaries with time, (C)  temporal variation of total grain volume and overall dissolution rate and (D) spatial variation of local dissolution  rate around the grain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  Acknowledgements  This work was performed in part at the Montana Nanotechnology Facility, an NNCI facility sup-  ported by NSF Grant ECCS-1542210, and with support by the Murdock Charitable Trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' The work  was partially supported by the Petroleum Research Fund, ACS (62687-DNI9YL) with Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Thomas  Clancy serving as the program officer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' YL thanks the Norm Asbjornson College of Engineering and  the Center for Faculty Excellence at Montana State University for their support through the Fac-  ulty Excellence Grants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' RMR thanks the Norm Asbjornson College of Engineering for their support  through the Benjamin Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  Notes and references  1 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Fogler, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' McCune and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Ault, Chemical Engineering Science, 1975, 30, 825–835.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  2 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Daccord, Physical review letters, 1987, 58, 479.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
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+page_content=' Clow, Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
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+page_content=' Bianco, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
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+page_content=' Spycher, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Sonnenthal and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
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+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Lin, Geochimica et Cosmochimica Acta, 2004, 68, 4629–  4648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  10  1–12 | 10  B  A  Calcite Grain  0>0  0<0  Time  14  27  [s]  0  Gas Bubble  Diss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Rate x103 [mm3/s]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='3  Ero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' Speed [μm/s]  5  10  15  20  0  4  0  4  Time (s)  0 (Rad)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='  8 PDFs: (A) horizontal component, u and (B) vertical component, v for the single and multiphase flows  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
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+page_content=' Shah, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
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+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfhfiE/content/2301.00378v1.pdf'}
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+1 
+ 
+ 
+Collective Coordinates Method in Relativistic 
+Field Theory 
+A.V. Shurgaia 
+Department of theoretical physics 
+A. Razmadze Mathematical Institute 
+at Ivane. Javakhishvili State University 
+2 Merab Aleksidze 2nd Lane  
+0193 Tbilisi Georgia 
+ Abstract 
+A field theoretical perturbation theory in inverse powers of coupling constant is developed which is 
+manifestly covariant in every order of the expansion. A dilatation operator serves as an evolution 
+dynamical one in a scale non-invariant theory. 
+ 
+1. Introduction 
+ 
+The quantization of interacting fields with nontrivial solutions of the classical equations of 
+motion requires a knowledge of the correct wave function of the ground state of the overall system. 
+Symmetry principles can be a great help to solve this problem. These principles constitute the basis 
+of the strong coupling method (or the method of collective coordinates, as it is now called), which 
+was first proposed by G. WENTZEL, W. PAULI, and S. DANCOFF [l] in an investigation of the 
+fixed source model in the strong coupling limit. A mathematically rigorous and consistent 
+formulation of this method was given by N. N. BOGOLUBOV [2] in 1950. More recently this 
+method attracted attention in studies of nonlinear classical field theories, and in the papers of many 
+authors, it was developed for interacting systems with arbitrary symmetries and applied to various 
+quantum field theoretical models with either the canonical method or the path integral formulation 
+of the field theory [3-71. An alternative procedure was proposed in ref. [S] in which the collective 
+variables are introduced in the quantum Yang-Feldman equations. One should note, that the authors 
+of these papers have used nonrelativistic symmetry group parameters as the collective coordinates. 
+In principle, the corresponding consideration of relativistic symmetry parameters is impossible in 
+ordinary quantum field theory. Indeed, it is well known, that from a relativistic point of view, all 
+physical quantities can be subdivided into dynamical i. e. interaction-dependent quantities and 
+kinematical quantities [9]. The strong coupling method makes it possible to construct a perturbation 
+theory for the dynamical quantities in such a way, that their eigenvalue spectrum can be described 
+by an exact representation of the symmetry group. In standard field theory, constructed on the 𝑡 =
+ 𝑐𝑜𝑛𝑠𝑡. surface, only the boost and energy are dynamical and we can describe them by the invariants 
+of the three-dimensional Euclidean motion group (and an internal symmetry group, if present). Thus 
+it is impossible to introduce Lorentz group parameters as collective variables into ordinary field 
+theory in the sense mentioned above. To solve this problem we have to drop the definition of 
+absolute time and construct the field theory on a surface, on which the Lorentz transformations carry 
+the kinematical information. The generalized Hamiltonian formalism, developed by DIRAC [l0], 
+opens, we think, the possibility to construct such a theory. The connection between the strong 
+coupling method and the generalized Hamiltonian approach was first pointed out in ref. [ll]. In ref. 
+[12] the general theory of the strong coupling method is developed for a nonrelativistic particle, 
+
+2 
+ 
+interacting with the quantum field and for an arbitrary symmetry. A similar procedure is adopted in 
+ref. [13] in a study of relativistic symmetry with the help of the strong coupling method. With this 
+aim in mind, the author has constructed the theory on an arbitrary space-like surface, choosing the 
+gauge at the flat surface, thereby losing the exact covariance. The relativistic symmetry has been 
+investigated in ref. [14], in particular, the Yang-Feldman equation is studied in the two-dimensional 
+scalar theory. The results the authors obtained correspond to the quantization on the flat surface. 
+The covariant generalization of the strong coupling method with the application has been given in 
+ref. [l5]. The strong coupling transformations are defined in such a way, that the field operators 
+become at the same time operators in the space of the Poincare group generator algebra. These 
+generators are then equated to the corresponding physical quantities constructed with the Noether 
+theorem. Then Dirac-type constraints arise. In ref. [16] an alternative procedure is proposed that 
+treats Lorentz invariance as an exact symmetry. In particular, the field theory is constructed on the 
+surface, on which the Lorentz transformations carry the kinematical information, and the 
+perturbation theory is developed, with exact covariance in each order. In what follows, we present 
+a revised version of this approach for arbitrary strongly interacting fields.  
+ 
+2. Generalized Hamiltonian Dynamics  
+ 
+In this section, we recall some basic concepts of generalized Hamiltonian dynamics on an 
+arbitrary space-like surface [l0]. Any three-dimensional surface is defined by specifying the four 
+coordinates at any point on it as functions of three parameters 𝑢𝑟: 𝑥𝜇 = 𝑥𝜇(𝑢𝑟), 𝑟 =  1, 2, 3. The 
+space-likeness of the surface implies the existence of the normal time-like vector 𝜂𝜇(𝑢𝑟) at every 
+point on it: 𝜂𝜇𝜂𝜇 = 1, 𝜂𝜇
+𝜕
+𝜕𝑢𝑟 𝜂𝜇 = 0.  The surface metric is given by the relation:  
+𝑑𝑠2 = 𝑑𝑥𝜇𝑑𝑥𝜇 = 𝛾𝑟𝑠𝑢𝑟𝑢𝑠, 
+with   
+𝛾𝑟𝑠 = 𝜕𝑥𝜇
+𝜕𝑢𝑟
+𝜕𝑥𝜇
+𝑢𝑠
+. 
+is the metric tensor on it. Since 𝑑𝑠2 < 0, det𝛾𝑟𝑠 ≡ −𝛤2 < 0. An inverse tensor is defined by the 
+equality  𝛾𝑟𝑠𝛾𝜎𝑡 = 𝛿   𝑡
+𝑟 . Any four-vector 𝑎𝜇 can be decomposed with respect to the coordinate 
+system into normal and contravariant parts: 𝑎⊥ = 𝑎𝜇𝜂𝜇 and 𝑎𝑟 = 𝑎𝜇
+𝜕𝜂𝜇
+𝜕𝑢𝑟. respectively. A covariant 
+component is defined by the relation 𝑎𝑟 = 𝛾𝑟𝑠𝑎𝑠.   
+    We can consider the quantities 𝑥𝜇(𝑢𝑟), defined in this way, as dynamical variables, which can 
+also be unphysical. We next introduce a parameter 𝜏 (which is monotonically increasing with t), t), 
+which varies from one surface to the other.   
+𝑉𝜇 = 𝜋𝜇(𝜏, 𝑢𝑟) + 𝐾𝜇 ≈ 0, 
+where 𝜋𝜇(𝜏, 𝑢𝑟) are the canonical momenta conjugate to 𝑥𝜇(𝜏, 𝑢𝑟) and the functions 𝐾𝜇 do not 
+contain them. The Lagrangian becomes a homogeneous function of the velocities, and the canonical 
+Hamiltonian vanishes. The Hamiltonian now is a linear combination of the constraints, i.e.  
+𝐻′ = ∫ 𝑑3𝑢𝑐𝜇(𝜏, 𝑢𝑟)𝑉𝜇. 
+ 
+The equations of motion for an arbitrary function of the canonical variables take the form: 
+ 
+ 𝑓̇ =
+𝜕𝑓
+𝜕𝜏 + ∫ 𝛿3𝑢𝑐𝜇(𝜏, 𝑢𝑟){𝑓, 𝑉𝜇}, 
+
+3 
+ 
+where 
+𝑥̇𝜇(𝜏, 𝑢𝑟) = 𝑐𝜇(𝜏, 𝑢𝑟). 
+ 
+Thus, the quantities 𝑐𝜇(𝜏, 𝑢𝑟) turn out to be the velocities 𝑥̇𝜇(𝜏, 𝑢𝑟) and, consequently, a choice of 
+𝑐𝜇(𝜏, 𝑢𝑟) defines the variation of the surface variation depending on 𝜏. Suppose now the 𝑐𝜇(𝜏, 𝑢𝑟) 
+are 𝜏-independent functions. Then we get the following representation for the 𝑥𝜇(𝜏, 𝑢𝑟): 
+ 
+𝑥𝜇(𝜏, 𝑢𝑟) = 𝑥0𝜇(𝜏, 𝑢𝑟) + 𝑐𝜇(𝑢𝑟)𝜏. 
+ 
+The equations of motion for z,(u, z) then assume the form:  
+ 
+𝜋̇𝜇(𝜏, 𝑢𝑟) = −𝑐𝜈(𝑢𝑟) 𝜕𝐾𝜈
+𝜕𝑥𝑚. 
+ 
+Via 𝐾𝜇, these equations involve the interaction terms and, consequently, depend on them in some 
+complicated manner. Turning now to the field variables we can conclude, that the invariance of the 
+Euler-Lagrange equations under space-time translations makes it possible to define the following 
+parametrization for the solutions :  
+ 
+𝜑(𝜏, 𝑢𝑟) = 𝜑(𝑥𝜇(𝜏, 𝑢𝑟) − 𝑐𝜇(𝑢𝑟)𝜏. 
+ 
+so that all the interaction effects are included in c,(u). It implies, that the energy-momentum vector 
+contains the interaction part of the Lagrangian and, hence, is a dynamical quantity. Since 𝑐𝜇(𝑢𝑟) is 
+a velocity, it is directed along the surface normal. In this way, we shall seek a choice of 𝑐𝜇(𝑢𝑟) for 
+which the Lorentz transformations are kinematical. Such a choice of c,(u) is equivalent to the 
+following gauge fixing:  
+ 
+𝑈𝜇 = 𝑥𝜇(𝜏, 𝑢𝑟) − 𝜏𝜂𝜇(𝑢) ≈ 0.  
+Introducing then according to DIRAC [l0] new brackets for the arbitrary functions of dynamical 
+variables one can convince oneself, that brackets for 𝑥𝜇(𝜏, 𝑢𝑟) and 𝜋𝜇(𝜏, 𝑢𝑟) become zero. These 
+are therefore unphysical. The phase space is now defined by the conditions 𝑈𝜇 = 𝑉𝜇 = 0, which 
+allows us to introduce in this gauge the Hamiltonian of the system.  
+ 
+3. Transformation of variables  
+ 
+We consider below the theory of the classical N-component field 𝜑𝑖(𝑥), i == 1, ... , N, with action:                                                    
+𝑠 = ∫ 𝑑4𝑥ℒ (𝜑𝑖(𝑥), 𝜕𝜇𝜑𝑖(𝑥)).                                                         (1) 
+This action is a relativistically invariant functional. Noether’s theorem implies the following 
+conserved quantities (cf. also the Appendix for notational conventions):     
+ 
+                                                              𝑃𝜇 = ∫ 𝑑𝜎𝜈 𝑇𝜇𝜈                                                                  (2) 
+𝑀𝛼𝛽 = ∫ 𝑑𝜎𝜈 𝜉𝛼𝛽
+     𝜇𝑇𝜇𝜈 − (𝐼𝛼𝛽)𝑖𝑗𝜑𝑗
+𝜕ℒ
+𝜕𝜕𝜈𝜑𝑖(𝑥),                                          (3) 
+in which  
+
+4 
+ 
+𝑇𝜇𝜈 =
+𝜕ℒ
+𝜕𝜕𝜈𝜑𝑖(𝑥)
+𝜕𝜑𝑖
+𝜕𝑥𝜇 − 𝑔𝜇𝜈ℒ 
+ 
+is the energy-momentum tensor. In relations (2) and (3) we have preserved the integration over the 
+arbitrary surface to be free to choose a surface on which the Lorentz transformations possess 
+kinematical character. We define the transformation of our field 𝜑𝑖(𝑥) as follows:  
+ 
+𝜑𝑖(𝑥) = 𝑆𝑖𝑘(𝜗(𝜏))𝜑̃𝜅(𝛬̅(𝜗(𝜏))(𝑥(𝜏, 𝑢) − 𝑞(𝜏)))                                         (4) 
+ 
+and make a change of the integration variables in (1): 
+ 𝑥𝜇 = 𝛬𝜇
+  𝜈(𝛬̅(𝜗(𝜏)) {𝜏𝛼𝜈(𝑢) − 𝑞𝜈(𝜏)}.                                                  (5) 
+where the unit 4-vector 𝛼𝜈(𝑢) is time-like. The field quantities 𝜑̃𝜅(𝑢, 𝜏) together with the scale 
+deformation parameter 𝑐(𝜏) and the Poincare group parameters 𝜗𝛼𝛽(𝜏) = −𝜗𝛽𝛼(𝜏) and 
+𝑞𝜈(𝜏) provide the new set of dynamical variables. The surface metric tensor is given by the equation: 
+𝛾𝑟𝑠 = 𝜕𝛼𝜈(𝑢)
+𝜕𝑢𝑟
+𝜕𝛼𝜈
+𝜕𝑢𝑠
+. 
+ 
+A four-volume element in x-space is equal to the product of the 3-dimensional elementary volume 
+in the (𝑢1, 𝑢2, 𝑢3)-space, given on the surface, and the elementary distance along the surface normal: 
+ 
+𝑑4𝑥 = 𝑐̇(𝜏)𝑐(𝜏)𝑑3𝑢𝑑𝜏. 
+The action can then be written as follows:  
+ 
+𝑆 = ∫ 𝑑𝜏𝐿,       𝐿 = ∫ 𝑐̇(𝜏)𝑐(𝜏)𝑑3𝑢𝑑𝜏ℒ𝜑𝑖(𝑥), 𝜕𝜇𝜑𝑖(𝑥)|
+ 
+𝜑=𝑆𝜑̃                                  (6) 
+ 
+where 𝐿 is the new Lagrangian of our system. To reexpress it in the new  representation we need to 
+calculate the derivatives 𝜕𝜇𝜑𝑖(𝑥), i. e. 
+ 
+𝜕𝜑𝑖
+𝜕𝑥𝜇
+= (𝜕𝑆𝑖𝑘
+𝜕𝜏 𝜑̃𝑘 + 𝑆𝑖𝑘
+𝜕𝜑̃𝑘
+𝜕𝜏 ) 𝜕𝜏
+𝜕𝑥𝜇
++ 𝑆𝑖𝑘
+𝜕𝜑̃𝑘
+𝜕𝑢𝑟
+𝜕𝑢𝑟
+𝜕𝑥𝜇
+.                                                (7) 
+ 
+Differentiating equation (5), one obtains the following expressions for 
+𝜕𝜏
+𝜕𝑥𝜇 and 
+𝜕𝑢𝑟
+𝜕𝑥𝜇 :   
+ 
+Differentiating equation (5), one obtains the following expressions for 
+𝜕𝜏
+𝜕𝑥𝜇 and 
+𝜕𝑢𝑟
+𝜕𝑥𝜇 :  
+𝜕𝜏
+𝜕𝑥𝜇
+= 𝛼𝜈𝛬̅𝜇
+  𝜈
+𝛼𝜎𝛺𝜎, 
+𝜕𝑢𝑟
+𝜕𝑥𝜇
+= 𝛬̅𝜈
+  𝜇 {𝑐(𝜏) 𝜕𝛼𝜈(𝑢)
+𝜕𝑢𝑟
++ 𝛼𝜈(𝑢)
+𝛼𝜎𝛺𝜎
+𝜕𝛼𝜌(𝑢)
+𝜕𝑢𝑟
+𝛺𝜌} 
+ 
+where the following notation is used for reasons of brevity: 
+ 
+𝛺𝜎 = 𝑐̇(𝜏)𝛼𝜎 + 𝛬̅𝜎
+  𝜈𝑞̇(𝜏) − 𝜗̇𝛼𝛽𝐵𝛼𝛽
+  𝛾𝛿(𝜗)𝜉𝛾𝛿,𝜎(𝛼). 
+
+5 
+ 
+ 
+The derivative 𝑆̇𝑖𝑘 is of the form:  
+ 
+𝑆̇𝑖𝑘 = 𝑆𝑖𝑛𝜗̇𝛼𝛽𝐵𝛼𝛽
+  𝛾𝛿(𝜗)(𝐼𝛾𝛿)𝑛𝑘. 
+ 
+We can now pass over to the Hamiltonian approach. Since the number of variables of the theory has 
+been extended there are constraints. Indeed, by calculating the conjugate momenta, we obtain: 
+ 
+𝜋̃(𝜏, 𝑢) =
+𝜕𝐿
+𝜕𝜑̇ (𝜏, 𝑢) = 𝑐̇(𝜏)𝑐(𝜏)𝑆𝑚𝑛𝛬̅𝜈
+  𝜇
+𝜕ℒ
+𝜕𝜕𝜇𝜑|
+𝜑=𝑆𝜑̃
+𝛼𝜈(𝑢),                         (8) 
+ 
+𝑃𝛼𝛽 =
+𝜕𝐿
+𝜕𝜗̇𝛼𝛽(𝜏, 𝑢) 𝐵𝛼𝛽
+  𝛾𝛿(𝜗)∫ 𝑑3𝑢 {𝜉̅𝛾𝛿
+   𝜎(𝛼) 𝜕𝛼𝜎
+𝜕𝑢𝑟
+𝜕𝜑̃𝑚
+𝜕𝑢𝑟 𝜋̃𝑚 − (𝐼̅𝛾𝛿)𝑚𝑛𝜑̃𝑛(𝜏, 𝑢)𝜋̃𝑚 (𝜏, 𝑢)}             (9) 
+ 
+𝑃𝝁 =
+𝜕𝐿
+𝜕𝑞̇ 𝜇(𝜏) = −𝛬̅𝜈
+  𝜇∫ 𝑑3𝑢 {𝛼𝜈(𝑢)ℋ + 𝑐(𝜏) 𝜕𝛼𝜈
+𝜕𝑢𝑟
+𝒫𝑟}                                          (10) 
+ 
+                                                                         𝑃𝑐 = −∫ 𝑑3𝑢ℋ                                                                 (11) 
+ 
+where 
+ℋ = 𝜋̃𝑚 𝜑̃𝑛𝑙 − 𝑐̇(𝜏)𝑐(𝜏)ℒ, 
+ 
+𝒫𝑟 = 𝜋̃𝑚 (𝜏, 𝑢) 𝜕𝜑̃𝑚(𝜏, 𝑢)
+𝜕𝑢𝑟
+, 
+                                                                𝜑̃𝑛𝑙 = 𝛬̅𝜈
+  𝜇𝑆𝑛𝑚
+𝜕𝜑𝑚
+𝜕𝑥𝜇 𝛼𝜈.                                                           (12 )    
+ 
+It is now easy to find the following constraints: 
+ 
+𝜓𝜇 = 𝑃𝜇 + 𝛬̅𝜈𝜇∫ 𝑑3𝑢𝛼𝜈ℋ + 𝑐(𝜏) 𝜕𝛼𝜈
+𝜕𝑢𝑟
+𝒫𝑟,                                                                (13 ) 
+ 
+𝜓𝛼𝛽 = 𝐿̅𝛼𝛽 − ∫ 𝑑3𝑢Σ𝛼𝛽,𝑚(𝜑̃(𝜏, 𝑢))𝜋̃𝑚 (𝜏, 𝑢),                                                           (14 ) 
+ 
+𝜓𝑐 = 𝑃𝑐 + 𝑑3𝑢ℋ.                                                                                                            (15) 
+ 
+Here we used the notation: 
+ 
+Σ𝛼𝛽,𝑚(𝜑̃(𝜏, 𝑢)) = 𝜉̅𝛾𝛿
+   𝜎(𝛼)𝑐2(𝜏) 𝜕𝛼𝜎
+𝜕𝑢𝑟
+𝜕𝜑̃𝑚
+𝜕𝑢𝑟 − (𝐼̅𝛾𝛿)𝑚𝑛𝜑̃𝑚(𝜏, 𝑢). 
+ 
+One should note that in every specific model 𝜑̃𝑚 has to be reexpressed in terms of the canonical 
+variables. 
+ 
+
+6 
+ 
+4. The Conservation Laws 
+ 
+We should now express all physical quantities in the new representation and verify the validity of 
+the conservation laws. We consider expressions (2) and (3). The surface element can be expressed 
+in terms of the 4-volume element as follows: 
+ 
+𝑑𝜎𝜈 = 𝜕𝐹(𝑥)
+𝜕𝑥𝜈
+𝛿(𝐹(𝑥) − 𝜏0)𝑑4𝑥, 
+in which the function 𝐹(𝑥) determines the form of the surface in space-time. Substituting the 
+transformations (4), (5) into the expressions of physical quantities and assuming 
+ 
+𝐹(𝑥) = 𝑥𝜇(𝜏, 𝑢)𝛬𝜈
+  𝜇𝛼𝜈 
+ 
+we obtain after some manipulations: 
+ 
+𝑃𝜇 = 𝛬̅𝜈𝜇∫ 𝑑3𝑢𝛼𝜈ℋ + 𝑐(𝜏) 𝜕𝛼𝜈
+𝜕𝑢𝑟
+𝒫𝑟,                                                        (16) 
+ 
+𝑀𝛼𝛽 = −𝐿𝛼𝛽 − (𝐽𝛼𝛽)𝜇𝜈𝑞𝜇𝑃𝜈.                                                                    (17) 
+ 
+It is known from the theory of Lie groups, that there exists a simple connection between the 
+quantities 𝐿𝛼𝛽, and 𝐿̅𝛼𝛽, indeed 
+ 
+𝐿𝛼𝛽 = 𝛬𝛼
+  𝜎𝛬𝛽
+   𝜌𝐿̅𝜎𝜌.                                                                   (18) 
+ 
+Based on this relationship together with the constraint (14), we conclude, that the 4-angular-
+momenta is kinematical, whereas the 4-energy-momentum vector is a dynamical one. Taking the 
+conservation laws into account one should note that because of the constraints the momenta  
+𝜋̃𝑚 (𝜏, 𝑢) contain transverse and longitudinal parts concerning some direction defined by the 
+solutions of the classical equations. The longitudinal part  turns out to be unphysical and one can  
+ 
+𝜑̃𝑚(𝜏, 𝑢) = 𝑔𝜑̃0𝑚(𝜏, 𝑢) + Φ𝑚(𝜏, 𝑢),      𝜋̃𝑚 (𝜏, 𝑢) = 𝜋̃𝑚𝑡 (𝜏, 𝑢) + 𝜋̃𝑚𝑙 (𝜏, 𝑢).     
+ 
+where 𝜑̃0𝑚(𝜏, 𝑢) are the classical solutions, and Φ𝑚(𝜏, 𝑢) together with the transverse momenta 
+Φ𝑚(𝜏, 𝑢) describe some fluctuations around them. The 𝜋̃𝑚𝑡 (𝜏, 𝑢) satisfy the following 
+orthogonality conditions: 
+ 
+∫ 𝑑3𝑢Σ𝛼𝛽,𝑚(𝜑̃(𝜏, 𝑢))𝜋̃𝑚𝑡 (𝜏, 𝑢) ≈ 0.                                                         (19) 
+ 
+They are, in fact, transversality conditions and will later be seen to play the role of constraints. For 
+the longitudinal part 𝜋̃𝑚𝑙 (𝜏, 𝑢) we then obtain the following expression:  
+ 
+𝜋̃𝑛𝑙 (𝜏, 𝑢) = 1
+𝑔 𝑁    𝑛
+𝛼𝛽(𝜏, 𝑢) (1 + 1
+𝑔 𝐹)
+𝛼𝛽,𝛾𝛿
+−1
+{𝐿̅𝛾𝛿(𝜗(𝜏)) − ∫ 𝑑3𝑢Σ    𝑚  
+𝛾𝛿 (𝜑̃(𝜏, 𝑢))𝜋̃𝑚𝑡 (𝜏, 𝑢)}, (20) 
+
+7 
+ 
+where the arbitrary functions 𝑁    𝑛
+𝛼𝛽(𝜏, 𝑢) are normalized as follows: 
+ 
+ ∫ 𝑑3𝑢𝑁    𝑛
+𝛼𝛽(𝜏, 𝑢)Σ    𝑚  
+𝛾𝛿 (𝜑0𝑚(𝜏, 𝑢)) = 𝑔𝛼𝛾𝑔𝛽𝛿.                                              (21) 
+Besides, 
+ 
+ 𝐹𝛼𝛽,𝛾𝛿 = ∫ 𝑑3𝑢Σ𝛼𝛽,𝑛(Φ(𝜏, 𝑢) 𝑁𝛾𝛿(𝜏, 𝑢).                                                       (22) 
+ 
+Thus the momenta 𝜋̃𝑚𝑡 (𝜏, 𝑢) and, consequently, the energy-momentum vector depends on the 
+Lorentz group parameters through the quantities 𝐿̅𝜎𝜌.. At the same time the 4-angular-momenta 
+𝑀𝛼𝛽, coincide with the generators, but their Poisson brackets with the 𝐿̅𝜎𝜌., is zero. This is just what  
+we need because as a result of this and expression (15), we can develop the perturbation theory 
+in inverse powers of 𝑔, which will be manifestly Lorentz-covariant in every order of the expansion. 
+ 
+5. Transition to the Quantum Theory 
+ 
+The considerations of the previous section allow us to pass to the Hamiltonian approach. 
+It is easy to verify that the canonical Hamiltonian: 
+ 
+𝐻 = 𝑐̇(𝜏)𝑃𝑐(𝜏) + 𝜗̇𝛼𝛽(𝜏)𝑃𝛼𝛽(𝜏) + 𝑞̇(𝜏)𝑃𝑐 + ∫ 𝑑3𝑢𝜑̃̇𝑚 (𝜏, 𝑢)𝜋̃𝑚 (𝜏, 𝑢) − 𝐿 
+ 
+is identically zero upon insertion of the constraints. Following Dirac’s theory, we take the 
+Hamiltonian as a linear combination of constraints. i.e. 
+ 
+𝐻′ = 𝑎(𝜏)Ψ𝑐 + 𝑎𝜇(𝜏)Ψ𝜇 + 𝑏𝛼𝛽Ψ𝛼𝛽.                                                       (23)  
+ 
+The equation of motion for any particular dynamical variable has the form 
+𝑓(𝜏)
+̇
+= 𝜕𝑓
+𝜕𝑡 + {𝑓, 𝐻′}                                                                             (24) 
+and lead for the constraints (they must be time independent) to the following consistency conditions; 
+  
+𝑎(𝜏){Ψ𝑐, Ψ𝑐} + 𝑎𝜇(𝜏){Ψ𝑐, Ψ𝜇} + 𝑏𝛼𝛽(𝜏){Ψ𝑐, Ψ𝛼𝛽} ≈ 0, 
+ 
+𝑎(𝜏){Ψ𝜈, Ψ𝑐 + 𝑎𝜇(𝜏){Ψ𝜈, Ψ𝜇} + 𝑏𝛼𝛽(𝜏){Ψ𝜈, Ψ𝛼𝛽} ≈ 0, 
+ 
+𝑎(𝜏){Ψ𝜌𝜎, Ψ𝑐} + 𝑎𝜇(𝜏){Ψ𝜌𝜎, Ψ𝜇} + 𝑏𝛼𝛽(𝜏){Ψ𝜌𝜎, Ψ𝛼𝛽} ≈ 0. 
+ 
+One can verify, that the constraints form a closed algebra with constant structure coefficients so that 
+the consistency conditions become the identities: 0 = 0. This means that our theory does not contain 
+constraints except (13)-(15). At the same time, there are the arbitrary coefficients 𝑎(𝜏),   
+ 𝑎𝜇(𝜏) and 𝑏𝛼𝛽(𝜏) in the theory, which can be connected with the velocities of the parameters, 
+introduced by transformations (4) and (5). Indeed from the equation of motion, we obtain: 
+ 
+𝑎(𝜏) = 𝑐̇(𝜏),
+𝑎𝜇(𝜏) = 𝑞̇𝜇(𝜏),   𝑏𝛼𝛽(𝜏) = 𝛣̅𝛼𝛽
+  𝛾𝛿(𝜗)𝜗̇𝛾𝛿(𝜏).             
+
+8 
+ 
+ 
+The arbitrariness can be avoided by introducing the gauge conditions. Before we fix these conditions 
+we rewrite the constraints: 
+ 
+Ψ𝑐 = 𝑃𝑐 + ∫ 𝑑3𝑢ℋ,                                                                                 (25) 
+ 
+Ψ𝜇 = 𝛬𝜇
+𝜈𝑃𝜈 + ∫ 𝑑3𝑢 (𝛼𝜇ℋ + 𝑐(𝜏) 𝜕𝛼𝜇
+𝜕𝑢𝑟
+𝒫𝑟),                                                        (26) 
+ 
+Ψ𝛼𝛽 = ∫ 𝑑3𝑢Σ    𝑚  
+𝛾𝛿 (𝜑0𝑚(𝜏, 𝑢))𝜋𝑛𝑡(𝜏, 𝑢)                                                           (27) 
+ 
+We recall that the variables 
+ 
+𝑐(𝜏),   𝑃𝑐(𝜏),   𝑞𝜇(𝜏),   𝑃𝜇(𝜏),  𝜗𝛼𝛽(𝜏),   𝑃𝛼𝛽(𝜏), Φ𝑘(𝜏, 𝑢), 𝜋𝑛𝑡(𝜏, 𝑢)                                (28)    
+ 
+constitute the phase space of theory. We next introduce the following gauge conditions: 
+ 
+𝜒𝑐 = 𝑐(𝜏) − 𝜏,                                                                      (29) 
+ 
+𝜒𝜇 = 𝛬𝜇
+  𝜈𝑞𝜈(𝜏) − 𝜏𝑃𝜇(𝜏),                                                                 (30) 
+ 
+𝜒𝛼𝛽 = ∫ 𝑑3𝑢𝑁𝛼𝛽,𝑛(𝜏, 𝑢)Φ𝑛(𝜏, 𝑢).                                                            (31) 
+ 
+These conditions are chosen in such a way, that their Poisson brackets with each other become zero 
+(such a choice is convenient, but not necessary). The second requirement we impose on the gauge 
+conditions is that the matrix composed of Poisson brackets of constraints with gauge conditions 
+ should be nonsingular. We now modify the Hamiltonian dynamics with the help of the brackets 
+introduced by DIRAC [l0]. We introduce the notation 𝐴𝑎𝑏 = {Ψ𝑎, 𝜒𝑏}, where Ψ𝑎 and 𝜒𝑏 are the sets  
+of constraints and gauge conditions respectively. The new brackets for any dynamical quantity are 
+now defined as follows: 
+ 
+{𝑓, ℎ}𝐷 = {𝑓, ℎ} − {𝑓, Ψ𝑎}𝐴𝑏𝑎
+−1{𝜒𝑏,ℎ} − {𝑓, 𝜒𝑎}𝐴𝑎𝑏
+−1{Ψ𝑏, ℎ} 
+ 
+where the nonsingular matrix A can be represented in the form 
+ 
+𝐴 = (
+𝑔
+{Ψ, 𝜒𝑐}
+{Ψ, 𝜒𝛼𝛽}
+0
+−1
+{ Ψ𝑐 , 𝜒𝛼𝛽}
+0
+0
+−𝐼
+) 
+where g is the metric tensor of 4-space-time, Ψ = Ψ𝜇  , and 𝐼 is a unit 6 x6 matrix. The  inverse 
+matrix 𝐴−1 has the following form: 
+ 
+𝐴−1 = (
+−𝑔
+−{Ψ, 𝜒𝑐}
+{Ψ, 𝜒𝛼𝛽} + {Ψ, 𝜒𝑐} ∙ { Ψ𝑐, 𝜒𝛼𝛽}
+0
+−1
+𝑔{ Ψ𝑐, 𝜒𝛼𝛽}
+0
+0
+−𝐼
+). 
+
+9 
+ 
+ 
+Evaluating the Dirac brackets for our canonical variables, we find that 𝑐(𝜏), 𝑃𝑐(𝜏),  𝑞𝜇(𝜏),  𝑃𝜇(𝜏) 
+become unphysical; indeed: 
+ 
+{𝑐(𝜏), 𝑃𝑐(𝜏)}𝐷 =  {𝑞𝜇(𝜏),  𝑃𝜈(𝜏)𝐷 = 0, 
+ 
+whereas  the Dirac brackets for the other canonical pairs have the following form: 
+ 
+{𝜗𝛼𝛽(𝜏), 𝑃𝛾𝛿(𝜏)}𝐷 = 𝑔𝛼𝛾𝑔𝛽𝛿,  𝛼 > 𝛾, 𝛽 > 𝛿,                                              (32) 
+ 
+{Φ𝑚(𝜏, 𝑢), 𝜋𝑛𝑡(𝜏, 𝑢′)}𝐷 = 𝛿𝑚𝑛𝛿3(𝑢 − 𝑢′) − Σ𝛼𝛽,𝑛(𝜑0(𝜏, 𝑢))𝑁      𝑚
+𝛼𝛽 (𝜏, 𝑢′) ≡ 𝐴𝑚𝑛(𝑢, 𝑢′).       (33) 
+ 
+We have to choose a quantity, which generates the evolution of the system. One can verify, that the 
+brackets of dynamical variables (or physical quantities) with 𝑃𝑐 provide the evolution in 𝜏. For 
+example 
+ 
+{𝑃𝑐, Φ𝑚(𝜏, 𝑢)}𝐷 = ∫ 𝑑3𝑣𝐴𝑚𝑛(𝑢, 𝑣)
+𝛿
+𝛿𝜋𝑛𝑡(𝜏, 𝑣) ∫ 𝑑3𝑢′ℋ,                           (34) 
+ 
+{𝑃𝑐, 𝜋𝑛𝑡(𝜏, 𝑢)}𝐷 =
+𝛿
+𝛿𝜋𝑛𝑡(𝜏, 𝑣) ∫ 𝑑3𝑢′ℋ.                                                            (35) 
+ 
+Due to constraints we can eliminate unphysical degrees of freedom and define the Hamiltonian - 
+which generates the evolution of the system - as follows: 
+ 
+𝐻 = ∫ 𝑑3𝑢ℋ. 
+In general, we have to add to this Hamiltonian the linear combination of remaining primary 
+constraints Ψ𝛼𝛽, but their Dirac brackets with dynamical quantities become zero, so we can get 
+these constraints in a strong sense. Therefore the equations of motion have the form: 
+ 
+𝑓̇ = 𝜕𝑓
+𝜕𝜏 + {𝑓, 𝐻}𝐷.                                                                   (36) 
+The transition to quantum theory is achieved with the help of the new brackets. In particular, the 
+commutator of two operators 𝑓, 𝑔 is defined by the relation: 
+ 
+[𝑓, 𝑔] = 𝑖{𝑓, 𝐻}𝐷.                                                                      (37) 
+ 
+From conditions  (32) and (33)  we can obtain the algebra of the operators, realized in the Hilbert 
+space, in which the variables 𝑃𝛼𝛽 and 𝜋𝑛𝑡(𝜏, 𝑣) are acting as the following differentiation operators: 
+ 
+𝑃𝛼𝛽 = −𝑖
+𝜕
+𝜕𝜗𝛼𝛽 ,
+𝜋𝑚𝑡(𝜏, 𝑣) = −𝑖 ∫ 𝑑3𝑣𝐴𝑚𝑛(𝑢, 𝑣)
+𝛿
+𝛿Φ𝑛𝑡(𝜏, 𝑣),                        (38)  
+ 
+he space is a direct sum of two spaces, realizing the representation (38).  
+
+10 
+ 
+Returning to expression (20), we can conclude that the state vector is an eigenvector of the two 
+commuting operators, 𝐻 and 𝐿̅𝛼𝛽 because [𝐻, 𝐿̅𝛼𝛽] = 0. Hence we can separate the 𝜗𝛼𝛽-dependence 
+of the state vector: 
+ 
+𝜓(𝜗, Φ(𝜏, 𝑢) = 𝐷(𝜗)𝜓′(Φ(𝜏, 𝑢)).                                                     (39) 
+ 
+Hence, the quantum equations for the remaining fluctuation operators  i.e. 
+ 
+𝑖 𝜕Φ𝑚(𝜏, 𝑢)
+𝜕𝜏
+= [Φ𝑚(𝜏, 𝑢), 𝐻],    𝑖 𝜕𝜋𝑚𝑡(𝜏, 𝑣)
+𝜕𝜏
+= −[𝜋𝑚𝑡(𝜏, 𝑣), 𝐻].                              (40) 
+ 
+are explicitly covariant. Expanding the momenta 𝜋𝑚𝑡(𝜏, 𝑣) (and correspondingly 𝐻)) in a series of 
+inverse powers of the coupling constant 𝑔, we can construct the exact covariant perturbation theory  
+for the equations (40) to every order of the expansion. Before expanding it is necessary to transform 
+the state vector as follows: 
+ 
+𝜓′ ⇒ exp (𝑖𝑔∫ 𝑑3𝑢𝑠𝑚(𝜏, 𝑢)Φ𝑚))𝜓′                                                    (41) 
+provided that 
+ 
+  ∫ 𝑑3𝑢Σ𝛼𝛽,𝑚(𝜑0(𝜏, 𝑢))𝑠𝑚(𝜏, 𝑢) = 0.                                                  (42)   
+ 
+If the 𝑠𝑚(𝜏, 𝑢) do not satisfy (42), we can define another 𝑠𝑚′(𝜏, 𝑢) which would satisfy this 
+requirement, indeed 
+ 
+𝑠𝑚
+′(𝜏,𝑢) = 𝑠𝑚(𝜏, 𝑢) − 𝑁𝛼𝛽,𝑚(𝜏, 𝑢)∫ 𝑑3𝑣Σ𝛼𝛽,𝑚(𝜑0(𝜏, 𝑣))𝑠𝑚(𝜏, 𝑣). 
+ 
+The transformation (41) implies, that the quantum operator 𝜋𝑚𝑡(𝜏, 𝑣) has to be replaced by 
+ 
+𝜋𝑚𝑡(𝜏, 𝑣) ⇒ 𝑔𝑠𝑚(𝜏, 𝑢) + 𝜋𝑚𝑡(𝜏, 𝑣).  
+ 
+We then expand the momenta 𝜋𝑚𝑡(𝜏, 𝑣) and the Hamiltonian 𝐻 in a series of inverse powers of 𝑔, 
+i.e. 
+ 
+𝜋𝑚𝑡(𝜏, 𝑣) = 𝑔𝜋0𝑚𝑡(𝜏, 𝑣)+𝜋1𝑚𝑡(𝜏, 𝑣) + 𝑔−1𝜋2𝑚𝑡(𝜏, 𝑣) + ⋯                                      (43) 
+ 
+𝐻 = 𝑔2𝐻0 + 𝑔𝐻1 + 𝐻2 + 𝑔−1𝐻3 + ⋯                                                                   (44) 
+ 
+The dominant terms of these series are purely classical. The terms that follow are  linear (~𝑔) and 
+quadratic (~𝑔0) in the fluctuation variables. Thus, the quantum equations up to the one-loop 
+approximation are: 
+ 
+𝑖 𝜕𝜑0𝑛(𝜏, 𝑢)
+𝜕𝜏
+= [Φ𝑛(𝜏, 𝑢), 𝛨1], 
+ 
+
+11 
+ 
+𝑖 𝜕𝜋0𝑛𝑡(𝜏, 𝑢)
+𝜕𝜏
+= [𝜋1𝑛𝑡(𝜏, 𝑢), 𝛨1], 
+ 
+ 
+𝑖 𝜕𝜋0𝑛𝑡(𝜏, 𝑢)
+𝜕𝜏
+= [𝜋1𝑛𝑡(𝜏, 𝑢), 𝛨1], 
+ 
+𝑖 𝜕Φ𝑛(𝜏, 𝑢)
+𝜕𝜏
+= [Φ𝑛(𝜏, 𝑢), 𝛨1], 
+ 
+𝑖 𝜕𝜋1𝑛𝑡(𝜏, 𝑢)
+𝜕𝜏
+= [𝜋1𝑛𝑡(𝜏, 𝑢), 𝛨2] + [𝜋2𝑛𝑡(𝜏, 𝑢), 𝛨1]. 
+ 
+We see, that the first line reproduces the classical equations of motion whereas the others coincide 
+with one-loop approximation equations. It should be noted that one has to be careful in manipulating  
+quantum variables, but up to the one-loop approximations no operator ordering problem arises, and 
+hence this question does not concern us here. 
+ 
+Appendix  
+ 
+In this appendix we summarize some properties of the Lorentz group which we use in the text. The 
+Lorentz transformation matrix satisfies the following orthogonality and product conditions : 
+ 
+𝛬𝛼
+  𝛽𝛬̅𝛽
+  𝛾 = 𝛿𝛼
+ 𝛾,           𝛬(𝜗1)𝛬(𝜗2) = 𝛬(𝜗3).                                                    (A. 1) 
+ 
+From the latter condition, we deduce that 
+ 
+𝜗2
+𝛾𝛿 = 𝜑𝛾𝛿(𝜃1, 𝜃2). 
+ 
+The matrices satisfy the equations: 
+ 
+𝜕𝛬(𝜗)
+𝜕𝜗𝛼𝛽 = 𝐵𝛼𝛽
+  𝛾𝛿(𝜃)𝐽𝛾𝛿𝛬(𝜗), 
+ 
+𝜕𝛬̅(𝜗)
+𝜕𝜗𝛼𝛽 = 𝐵̅𝛼𝛽
+  𝛾𝛿(𝜃)𝐽̅𝛾𝛿𝛬(𝜗).                                                     (Α. 2) 
+ 
+Where 
+𝐵𝛼𝛽
+  𝛾𝛿(𝜗2) = 𝜕𝜑𝛾𝛿(𝜃1, 𝜃2)
+𝜕𝜗1
+𝛼𝛽
+|
+𝜗1=𝜗2−1
+, 
+𝐵̅𝛼𝛽
+  𝛾𝛿(𝜗1) = 𝜕𝜑𝛾𝛿(𝜃1, 𝜃2)
+𝜕𝜗2
+𝛼𝛽
+|
+𝜗2=𝜗1−1
+                                             (Α. 3) 
+ 
+which become unit matrixes for zero arguments. Then 
+
+12 
+ 
+𝐽𝛼𝛽 = 𝜕𝛬(𝜗
+𝜕𝜗𝛼𝛽|
+𝜗=0
+, 
+ 
+𝐽̅𝛼𝛽 = 𝜕𝛬̅(𝜗
+𝜕𝜗𝛼𝛽|
+𝜗=0
+                                                                  (Α. 4) 
+ 
+We define matrices 𝐴, 𝐴̅ which are inverse to 𝐵,  𝐵̅  respectively by  equations: 
+ 
+𝐴𝐵 = I ,     𝐴̅𝐵̅ = I ,                                                                       (Α. 5)  
+ 
+I am a unit matrix. With the help of these matrices, we can construct the following quantities: 
+ 
+𝐿𝛼𝛽 = 𝐵𝛼𝛽
+  𝛾𝛿𝑃𝛾𝛿,      𝐿̅𝛼𝛽 = 𝐵̅𝛼𝛽
+  𝛾𝛿𝑃𝛾𝛿.                                                            (Α. 6) 
+ 
+They obey the usual Poisson brackets for the Lorentz group parameters¨ 
+ 
+{𝐿𝛼𝛽, 𝐿𝛾𝛿} = 𝐶𝛼𝛽,𝛾𝛿
+        𝜎𝜌𝐿𝜎𝜌,          {𝐿̅𝛼𝛽, 𝐿̅𝛾𝛿} = −𝐶𝛼𝛽,𝛾𝛿
+        𝜎𝜌𝐿̅𝜎𝜌,    {𝐿𝛼𝛽, 𝐿̅𝛾𝛿} = 0.                        (Α. 7) 
+ 
+where 𝐶𝛼𝛽,𝛾𝛿
+        𝜎𝜌 are well-known structure constants. 
+We are interested in the Minkowsky space coordinate transformations : 
+ 
+𝑥𝜇 ⇒ 𝑥𝜇
+′ = 𝛬𝜇
+  𝜈𝑥𝜈 ≡ 𝑓𝜇(𝑥, 𝜗).                                                                      (Α. 8) 
+ 
+From the product law we can obtain the following equations: 
+  
+𝜕𝑥𝜇′
+𝜕𝜗𝛼𝛽 = 𝛣𝛼𝛽
+  𝛾𝛿(𝜗)𝜉𝛼𝛽,𝜇(𝑥′)                                                                                   (Α. 9) 
+ 
+with 
+ 
+𝜉𝛼𝛽,𝜇(𝑥′) = 𝜕𝑓𝜇(𝑥, 𝜗)
+𝜕𝜗𝛼𝛽
+|
+𝜗=0
+.                                                                                   (Α. 10)  
+ 
+Inverting the transformations we obtain the equations¨ 
+ 
+𝜕𝑥𝜇
+𝜕𝜗𝛼𝛽 = 𝛣̅𝛼𝛽
+  𝛾𝛿(𝜗)𝜉̅𝛼𝛽,𝜇(𝑥)                                                                                   (Α. 9) 
+with 
+ 
+𝜉̅𝛼𝛽,𝜇(𝑥) = 𝜕𝑓̅𝜇(𝑥, 𝜗)
+𝜕𝜗𝛼𝛽
+|
+𝜗=0
+.                                                                                   (Α. 10)  
+ 
+The quantities  
+
+13 
+ 
+𝑋𝛼𝛽 = 𝜉𝛼𝛽,𝜇(𝑥)𝑃𝛼𝛽,   𝑋̅𝛼𝛽 = 𝜉̅𝛼𝛽,𝜇(𝑥)𝑃𝛼𝛽                                                               (Α. 11) 
+  
+with {𝑥𝜇, 𝑃𝜎} = 𝑔𝜇𝜎 satisfy the conditions similar to (A.7). These formulae give two different 
+representations of the homogeneous Lorentz group. The matrix 𝑆𝑖𝑘(𝜗), determines the 
+transformation properties of the field variables 𝜑𝜄(x).  They obey the equation 
+ 
+𝜕𝑆𝑖𝑘(𝜗)
+𝜕𝜗𝛼𝛽
+= 𝐵𝛼𝛽
+  𝛾𝛿(𝜃)(𝛪𝛾𝛿)𝑖𝑛𝑆𝑛𝑘(𝜗),        𝛪𝛾𝛿 = 𝜕𝑆(𝜗)
+𝜕𝜗𝛾𝛿 |
+𝜗=0
+                                       (Α. 12) 
+ 
+If 𝑆̅ are inverse to 𝑆, then 
+ 
+𝜕𝑆̅𝑖𝑘(𝜗)
+𝜕𝜗𝛼𝛽
+= 𝐵̅𝛼𝛽
+  𝛾𝛿(𝜃)(𝛪̅𝛾𝛿)𝑖𝑛𝑆𝑛𝑘(𝜗),     𝛪̅𝛾𝛿 = 𝜕𝑆̅(𝜗)
+𝜕𝜗𝛾𝛿 |
+𝜗=0
+    .                                 (Α. 13) 
+ 
+         References 
+ 
+[l] G. Wentzel, Helv. Phys. Acta 13 (1940) 269;  
+    W. Pauli and S. Dancoff, Phys. Rev. 62 (1942) 85.  
+[2] N. N. Bogolubov, Ukr. Math. J. 2 (1950) 3; Selected Topics, Vol. 2, p. 499, Naukov Dumka,  
+[3] E. P. Sslodovnikova,  A. N. Tavkhelidze and O. A. Khrustalev, Teor. Mat. Fiz. 11 (1972) 317  
+[4] E. P. Sslodovnikova, A. N. Tavkhelidze and O. A. Khrustalev, Teor. Mat. Fiz. Fiz. 90 (1972)  
+     168 
+N. E. Turin and A. V. Shurgaia, Teor. Mat. Fiz. 16 (1973) 197;  
+A. A. Arkhipov and N. E. Turin, Teor. Mat. Fiz. 17 (1973) 57;  
+S. V. Semenoff, Teor. Mat. Fiz. 18 (1974) 353;  
+A. V. Shurgaia, Teor. Mat. Fiz. 28 (1976) 223;  
+     A. V. Razumov and O. A. KHRUSTALOV, Teor. Mat. Fiz. 29 (1976) 300.  
+[5] N. Christ and T. D. Lee, Phys. Rev. D12 (1975) 1606. 
+161 J. L. Gervais, A. Jevicki and B. Sakita, Phys. Rev. D12 (1975) 1038;  
+       J. L. Gervais and  A. Jevicki, Nucl. Phys. BllO (1976) 113;  
+       J. L. Gervais and B. Sakita,, Phys. Rev. D16 (1977) 3507.  
+[7] H. J. W. Muller-Kirsten  and A. Widemann, Fortschr. Phys. 32 (1984) 21;  
+      H. J. W. Muller-Kirsten  and A. Widemann, Nuovo Cimento A78 (1983) 61; 
+      L. Maharan and H. J. W. Muller-Kirsten, Nuovo Cimento A83 (1984) 229.  
+      [S] H. Matsumoto, G. Oberlechner, H. Umezava and M. Umezava, J. Math. Phys. 20 (1979) 
+      2088. 
+[9] P. A. M. Dirac, Rev. Mod. Phys. 21 (1949) 391.  
+[10] P. A. M. Dirac, Can. J. Math. Phys. 2 (1950) 129; Can. J. Math. Phys. 3 (1951) 1.  
+[11] E. Tomboulis and G. Woo, Ann. Phys. (N.Y.) 98 (1976) 1,  
+[12] A. V. Surgaia, Theor. Math. Fiz. 45 (1980) 46. 
+[13] K. Ishikawa, Nucl. Phys. B107 (1976) 238, 253. 
+[14] H. Matsumoto, N. J. apastamatian and H. Umazava, Phys. Rev. D24 (1981) 406;  
+       G. Semenoff, M. Matzumoto and M. Umezava, Phys. Rev. D25 (1982) 1054.  
+[I5] K. A. Sveshnikov, Teor. Mat. Fiz. 55 (1983) 553. 
+[16] A. V. Shurgaia, Teor. Mat. Fiz. 57 (1983) 392. 
+
diff --git a/_dAzT4oBgHgl3EQf_f7E/content/tmp_files/load_file.txt b/_dAzT4oBgHgl3EQf_f7E/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,416 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf,len=415
+page_content='1  Collective Coordinates Method in Relativistic Field Theory A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Shurgaia Department of theoretical physics A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Razmadze Mathematical Institute at Ivane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Javakhishvili State University 2 Merab Aleksidze 2nd Lane 0193 Tbilisi Georgia Abstract A field theoretical perturbation theory in inverse powers of coupling constant is developed which is manifestly covariant in every order of the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' A dilatation operator serves as an evolution dynamical one in a scale non-invariant theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Introduction  The quantization of interacting fields with nontrivial solutions of the classical equations of motion requires a knowledge of the correct wave function of the ground state of the overall system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Symmetry principles can be a great help to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' These principles constitute the basis of the strong coupling method (or the method of collective coordinates, as it is now called), which was first proposed by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' WENTZEL, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' PAULI, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' DANCOFF [l] in an investigation of the fixed source model in the strong coupling limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' A mathematically rigorous and consistent formulation of this method was given by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' BOGOLUBOV [2] in 1950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' More recently this method attracted attention in studies of nonlinear classical field theories, and in the papers of many authors, it was developed for interacting systems with arbitrary symmetries and applied to various quantum field theoretical models with either the canonical method or the path integral formulation of the field theory [3-71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' An alternative procedure was proposed in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' [S] in which the collective variables are introduced in the quantum Yang-Feldman equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' One should note, that the authors of these papers have used nonrelativistic symmetry group parameters as the collective coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' In principle, the corresponding consideration of relativistic symmetry parameters is impossible in ordinary quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  Indeed, it is well known, that from a relativistic point of view, all physical quantities can be subdivided into dynamical i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' interaction-dependent quantities and kinematical quantities [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The strong coupling method makes it possible to construct a perturbation theory for the dynamical quantities in such a way, that their eigenvalue spectrum can be described by an exact representation of the symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' In standard field theory, constructed on the 𝑡 = 𝑐𝑜𝑛𝑠𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' surface, only the boost and energy are dynamical and we can describe them by the invariants of the three-dimensional Euclidean motion group (and an internal symmetry group, if present).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Thus it is impossible to introduce Lorentz group parameters as collective variables into ordinary field theory in the sense mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' To solve this problem we have to drop the definition of absolute time and construct the field theory on a surface, on which the Lorentz transformations carry the kinematical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The generalized Hamiltonian formalism, developed by DIRAC [l0], opens, we think, the possibility to construct such a theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The connection between the strong coupling method and the generalized Hamiltonian approach was first pointed out in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' [ll].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' In ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' [12] the general theory of the strong coupling method is developed for a nonrelativistic particle,  2  interacting with the quantum field and for an arbitrary symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' A similar procedure is adopted in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' [13] in a study of relativistic symmetry with the help of the strong coupling method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' With this aim in mind, the author has constructed the theory on an arbitrary space-like surface, choosing the gauge at the flat surface, thereby losing the exact covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The relativistic symmetry has been investigated in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' [14], in particular, the Yang-Feldman equation is studied in the two-dimensional scalar theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The results the authors obtained correspond to the quantization on the flat surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The covariant generalization of the strong coupling method with the application has been given in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' [l5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The strong coupling transformations are defined in such a way, that the field operators become at the same time operators in the space of the Poincare group generator algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' These generators are then equated to the corresponding physical quantities constructed with the Noether theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Then Dirac-type constraints arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' In ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' [16] an alternative procedure is proposed that treats Lorentz invariance as an exact symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' In particular, the field theory is constructed on the surface, on which the Lorentz transformations carry the kinematical information, and the perturbation theory is developed, with exact covariance in each order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' In what follows, we present a revised version of this approach for arbitrary strongly interacting fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Generalized Hamiltonian Dynamics  In this section, we recall some basic concepts of generalized Hamiltonian dynamics on an arbitrary space-like surface [l0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Any three-dimensional surface is defined by specifying the four coordinates at any point on it as functions of three parameters 𝑢𝑟: 𝑥𝜇 = 𝑥𝜇(𝑢𝑟), 𝑟 =  1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The space-likeness of the surface implies the existence of the normal time-like vector 𝜂𝜇(𝑢𝑟) at every point on it: 𝜂𝜇𝜂𝜇 = 1, 𝜂𝜇 𝜕 𝜕𝑢𝑟 𝜂𝜇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The surface metric is given by the relation: 𝑑𝑠2 = 𝑑𝑥𝜇𝑑𝑥𝜇 = 𝛾𝑟𝑠𝑢𝑟𝑢𝑠, with 𝛾𝑟𝑠 = 𝜕𝑥𝜇 𝜕𝑢𝑟 𝜕𝑥𝜇 𝑢𝑠 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' is the metric tensor on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Since 𝑑𝑠2 < 0, det𝛾𝑟𝑠 ≡ −𝛤2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' An inverse tensor is defined by the equality  𝛾𝑟𝑠𝛾𝜎𝑡 = 𝛿   𝑡 𝑟 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Any four-vector 𝑎𝜇 can be decomposed with respect to the coordinate system into normal and contravariant parts: 𝑎⊥ = 𝑎𝜇𝜂𝜇 and 𝑎𝑟 = 𝑎𝜇 𝜕𝜂𝜇 𝜕𝑢𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' A covariant component is defined by the relation 𝑎𝑟 = 𝛾𝑟𝑠𝑎𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' We can consider the quantities 𝑥𝜇(𝑢𝑟), defined in this way, as dynamical variables, which can also be unphysical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' We next introduce a parameter 𝜏 (which is monotonically increasing with t), t), which varies from one surface to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 𝑉𝜇 = 𝜋𝜇(𝜏, 𝑢𝑟) + 𝐾𝜇 ≈ 0, where 𝜋𝜇(𝜏, 𝑢𝑟) are the canonical momenta conjugate to 𝑥𝜇(𝜏, 𝑢𝑟) and the functions 𝐾𝜇 do not contain them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The Lagrangian becomes a homogeneous function of the velocities, and the canonical Hamiltonian vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The Hamiltonian now is a linear combination of the constraints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 𝐻′ = ∫ 𝑑3𝑢𝑐𝜇(𝜏, 𝑢𝑟)𝑉𝜇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  The equations of motion for an arbitrary function of the canonical variables take the form:  𝑓̇ = 𝜕𝑓 𝜕𝜏 + ∫ 𝛿3𝑢𝑐𝜇(𝜏, 𝑢𝑟){𝑓, 𝑉𝜇},  3  where 𝑥̇𝜇(𝜏, 𝑢𝑟) = 𝑐𝜇(𝜏, 𝑢𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  Thus, the quantities 𝑐𝜇(𝜏, 𝑢𝑟) turn out to be the velocities 𝑥̇𝜇(𝜏, 𝑢𝑟) and, consequently, a choice of 𝑐𝜇(𝜏, 𝑢𝑟) defines the variation of the surface variation depending on 𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Suppose now the 𝑐𝜇(𝜏, 𝑢𝑟) are 𝜏-independent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Then we get the following representation for the 𝑥𝜇(𝜏, 𝑢𝑟):  𝑥𝜇(𝜏, 𝑢𝑟) = 𝑥0𝜇(𝜏, 𝑢𝑟) + 𝑐𝜇(𝑢𝑟)𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  The equations of motion for z,(u, z) then assume the form:  𝜋̇𝜇(𝜏, 𝑢𝑟) = −𝑐𝜈(𝑢𝑟) 𝜕𝐾𝜈 𝜕𝑥𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  Via 𝐾𝜇, these equations involve the interaction terms and, consequently, depend on them in some complicated manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Turning now to the field variables we can conclude, that the invariance of the Euler-Lagrange equations under space-time translations makes it possible to define the following parametrization for the solutions :  𝜑(𝜏, 𝑢𝑟) = 𝜑(𝑥𝜇(𝜏, 𝑢𝑟) − 𝑐𝜇(𝑢𝑟)𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  so that all the interaction effects are included in c,(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' It implies, that the energy-momentum vector contains the interaction part of the Lagrangian and, hence, is a dynamical quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Since 𝑐𝜇(𝑢𝑟) is a velocity, it is directed along the surface normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' In this way, we shall seek a choice of 𝑐𝜇(𝑢𝑟) for which the Lorentz transformations are kinematical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Such a choice of c,(u) is equivalent to the following gauge fixing:  𝑈𝜇 = 𝑥𝜇(𝜏, 𝑢𝑟) − 𝜏𝜂𝜇(𝑢) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Introducing then according to DIRAC [l0] new brackets for the arbitrary functions of dynamical variables one can convince oneself, that brackets for 𝑥𝜇(𝜏, 𝑢𝑟) and 𝜋𝜇(𝜏, 𝑢𝑟) become zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' These are therefore unphysical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The phase space is now defined by the conditions 𝑈𝜇 = 𝑉𝜇 = 0, which allows us to introduce in this gauge the Hamiltonian of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Transformation of variables  We consider below the theory of the classical N-component field 𝜑𝑖(𝑥), i == 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' , N, with action: 𝑠 = ∫ 𝑑4𝑥ℒ (𝜑𝑖(𝑥), 𝜕𝜇𝜑𝑖(𝑥)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (1) This action is a relativistically invariant functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Noether’s theorem implies the following conserved quantities (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' also the Appendix for notational conventions):  𝑃𝜇 = ∫ 𝑑𝜎𝜈 𝑇𝜇𝜈                                                                  (2) 𝑀𝛼𝛽 = ∫ 𝑑𝜎𝜈 𝜉𝛼𝛽 𝜇𝑇𝜇𝜈 − (𝐼𝛼𝛽)𝑖𝑗𝜑𝑗 𝜕ℒ 𝜕𝜕𝜈𝜑𝑖(𝑥),                                          (3) in which  4  𝑇𝜇𝜈 =  𝜕ℒ  𝜕𝜕𝜈𝜑𝑖(𝑥)  𝜕𝜑𝑖  𝜕𝑥𝜇 − 𝑔𝜇𝜈ℒ  is the energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' In relations (2) and (3) we have preserved the integration over the arbitrary surface to be free to choose a surface on which the Lorentz transformations possess kinematical character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' We define the transformation of our field 𝜑𝑖(𝑥) as follows:  𝜑𝑖(𝑥) = 𝑆𝑖𝑘(𝜗(𝜏))𝜑̃𝜅(𝛬̅(𝜗(𝜏))(𝑥(𝜏, 𝑢) − 𝑞(𝜏)))                                         (4)  and make a change of the integration variables in (1): 𝑥𝜇 = 𝛬𝜇 𝜈(𝛬̅(𝜗(𝜏)) {𝜏𝛼𝜈(𝑢) − 𝑞𝜈(𝜏)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (5) where the unit 4-vector 𝛼𝜈(𝑢) is time-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The field quantities 𝜑̃𝜅(𝑢, 𝜏) together with the scale deformation parameter 𝑐(𝜏) and the Poincare group parameters 𝜗𝛼𝛽(𝜏) = −𝜗𝛽𝛼(𝜏) and 𝑞𝜈(𝜏) provide the new set of dynamical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The surface metric tensor is given by the equation: 𝛾𝑟𝑠 = 𝜕𝛼𝜈(𝑢) 𝜕𝑢𝑟 𝜕𝛼𝜈 𝜕𝑢𝑠 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  A four-volume element in x-space is equal to the product of the 3-dimensional elementary volume in the (𝑢1, 𝑢2, 𝑢3)-space, given on the surface, and the elementary distance along the surface normal:  𝑑4𝑥 = 𝑐̇(𝜏)𝑐(𝜏)𝑑3𝑢𝑑𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The action can then be written as follows:  𝑆 = ∫ 𝑑𝜏𝐿,       𝐿 = ∫ 𝑐̇(𝜏)𝑐(𝜏)𝑑3𝑢𝑑𝜏ℒ𝜑𝑖(𝑥), 𝜕𝜇𝜑𝑖(𝑥)|  𝜑=𝑆𝜑̃                                  (6)  where 𝐿 is the new Lagrangian of our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' To reexpress it in the new  representation we need to calculate the derivatives 𝜕𝜇𝜑𝑖(𝑥), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  𝜕𝜑𝑖 𝜕𝑥𝜇 = (𝜕𝑆𝑖𝑘 𝜕𝜏 𝜑̃𝑘 + 𝑆𝑖𝑘 𝜕𝜑̃𝑘 𝜕𝜏 ) 𝜕𝜏 𝜕𝑥𝜇 + 𝑆𝑖𝑘 𝜕𝜑̃𝑘 𝜕𝑢𝑟 𝜕𝑢𝑟 𝜕𝑥𝜇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (7)  Differentiating equation (5), one obtains the following expressions for 𝜕𝜏 𝜕𝑥𝜇 and 𝜕𝑢𝑟 𝜕𝑥𝜇 :  Differentiating equation (5), one obtains the following expressions for 𝜕𝜏 𝜕𝑥𝜇 and 𝜕𝑢𝑟 𝜕𝑥𝜇 : 𝜕𝜏 𝜕𝑥𝜇 = 𝛼𝜈𝛬̅𝜇 𝜈 𝛼𝜎𝛺𝜎, 𝜕𝑢𝑟 𝜕𝑥𝜇 = 𝛬̅𝜈 𝜇 {𝑐(𝜏) 𝜕𝛼𝜈(𝑢) 𝜕𝑢𝑟 + 𝛼𝜈(𝑢) 𝛼𝜎𝛺𝜎 𝜕𝛼𝜌(𝑢) 𝜕𝑢𝑟 𝛺𝜌}  where the following notation is used for reasons of brevity:  𝛺𝜎 = 𝑐̇(𝜏)𝛼𝜎 + 𝛬̅𝜎 𝜈𝑞̇(𝜏) − 𝜗̇𝛼𝛽𝐵𝛼𝛽 𝛾𝛿(𝜗)𝜉𝛾𝛿,𝜎(𝛼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  5  The derivative 𝑆̇𝑖𝑘 is of the form:  𝑆̇𝑖𝑘 = 𝑆𝑖𝑛𝜗̇𝛼𝛽𝐵𝛼𝛽  𝛾𝛿(𝜗)(𝐼𝛾𝛿)𝑛𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  We can now pass over to the Hamiltonian approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Since the number of variables of the theory has been extended there are constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' by calculating the conjugate momenta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' we obtain:  𝜋̃(𝜏,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 𝑢) = 𝜕𝐿 𝜕𝜑̇ (𝜏,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 𝑢) = 𝑐̇(𝜏)𝑐(𝜏)𝑆𝑚𝑛𝛬̅𝜈 𝜇 𝜕ℒ 𝜕𝜕𝜇𝜑| 𝜑=𝑆𝜑̃ 𝛼𝜈(𝑢),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='                         (8)  𝑃𝛼𝛽 = 𝜕𝐿 𝜕𝜗̇𝛼𝛽(𝜏,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 𝑢) 𝐵𝛼𝛽 𝛾𝛿(𝜗)∫ 𝑑3𝑢 {𝜉̅𝛾𝛿 𝜎(𝛼) 𝜕𝛼𝜎 𝜕𝑢𝑟 𝜕𝜑̃𝑚 𝜕𝑢𝑟 𝜋̃𝑚 − (𝐼̅𝛾𝛿)𝑚𝑛𝜑̃𝑛(𝜏,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 𝑢)𝜋̃𝑚 (𝜏,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 𝑢)}             (9)  𝑃𝝁 = 𝜕𝐿 𝜕𝑞̇ 𝜇(𝜏) = −𝛬̅𝜈 𝜇∫ 𝑑3𝑢 {𝛼𝜈(𝑢)ℋ + 𝑐(𝜏) 𝜕𝛼𝜈 𝜕𝑢𝑟 𝒫𝑟}                                          (10)  𝑃𝑐 = −∫ 𝑑3𝑢ℋ                                                                 (11)  where ℋ = 𝜋̃𝑚 𝜑̃𝑛𝑙 − 𝑐̇(𝜏)𝑐(𝜏)ℒ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  𝒫𝑟 = 𝜋̃𝑚 (𝜏,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 𝑢) 𝜕𝜑̃𝑚(𝜏,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 𝑢) 𝜕𝑢𝑟 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 𝜑̃𝑛𝑙 = 𝛬̅𝜈 𝜇𝑆𝑛𝑚 𝜕𝜑𝑚 𝜕𝑥𝜇 𝛼𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (12 )  It is now easy to find the following constraints:  𝜓𝜇 = 𝑃𝜇 + 𝛬̅𝜈𝜇∫ 𝑑3𝑢𝛼𝜈ℋ + 𝑐(𝜏) 𝜕𝛼𝜈 𝜕𝑢𝑟 𝒫𝑟,                                                                (13 )  𝜓𝛼𝛽 = 𝐿̅𝛼𝛽 − ∫ 𝑑3𝑢Σ𝛼𝛽,𝑚(𝜑̃(𝜏, 𝑢))𝜋̃𝑚 (𝜏, 𝑢),                                                           (14 )  𝜓𝑐 = 𝑃𝑐 + 𝑑3𝑢ℋ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (15)  Here we used the notation:  Σ𝛼𝛽,𝑚(𝜑̃(𝜏, 𝑢)) = 𝜉̅𝛾𝛿  𝜎(𝛼)𝑐2(𝜏) 𝜕𝛼𝜎  𝜕𝑢𝑟  𝜕𝜑̃𝑚  𝜕𝑢𝑟 − (𝐼̅𝛾𝛿)𝑚𝑛𝜑̃𝑚(𝜏, 𝑢).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  One should note that in every specific model 𝜑̃𝑚 has to be reexpressed in terms of the canonical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The Conservation Laws  We should now express all physical quantities in the new representation and verify the validity of the conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' We consider expressions (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The surface element can be expressed in terms of the 4-volume element as follows:  𝑑𝜎𝜈 = 𝜕𝐹(𝑥) 𝜕𝑥𝜈 𝛿(𝐹(𝑥) − 𝜏0)𝑑4𝑥, in which the function 𝐹(𝑥) determines the form of the surface in space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Substituting the transformations (4), (5) into the expressions of physical quantities and assuming  𝐹(𝑥) = 𝑥𝜇(𝜏, 𝑢)𝛬𝜈  𝜇𝛼𝜈  we obtain after some manipulations:  𝑃𝜇 = 𝛬̅𝜈𝜇∫ 𝑑3𝑢𝛼𝜈ℋ + 𝑐(𝜏) 𝜕𝛼𝜈 𝜕𝑢𝑟 𝒫𝑟,                                                        (16)  𝑀𝛼𝛽 = −𝐿𝛼𝛽 − (𝐽𝛼𝛽)𝜇𝜈𝑞𝜇𝑃𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (17)  It is known from the theory of Lie groups, that there exists a simple connection between the quantities 𝐿𝛼𝛽, and 𝐿̅𝛼𝛽, indeed  𝐿𝛼𝛽 = 𝛬𝛼 𝜎𝛬𝛽 𝜌𝐿̅𝜎𝜌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (18)  Based on this relationship together with the constraint (14), we conclude, that the 4-angular- momenta is kinematical, whereas the 4-energy-momentum vector is a dynamical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Taking the conservation laws into account one should note that because of the constraints the momenta 𝜋̃𝑚 (𝜏, 𝑢) contain transverse and longitudinal parts concerning some direction defined by the solutions of the classical equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The longitudinal part  turns out to be unphysical and one can  𝜑̃𝑚(𝜏, 𝑢) = 𝑔𝜑̃0𝑚(𝜏, 𝑢) + Φ𝑚(𝜏, 𝑢),      𝜋̃𝑚 (𝜏, 𝑢) = 𝜋̃𝑚𝑡 (𝜏, 𝑢) + 𝜋̃𝑚𝑙 (𝜏, 𝑢).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  where 𝜑̃0𝑚(𝜏, 𝑢) are the classical solutions, and Φ𝑚(𝜏, 𝑢) together with the transverse momenta Φ𝑚(𝜏, 𝑢) describe some fluctuations around them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The 𝜋̃𝑚𝑡 (𝜏, 𝑢) satisfy the following orthogonality conditions:  ∫ 𝑑3𝑢Σ𝛼𝛽,𝑚(𝜑̃(𝜏, 𝑢))𝜋̃𝑚𝑡 (𝜏, 𝑢) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (19)  They are, in fact, transversality conditions and will later be seen to play the role of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' For the longitudinal part 𝜋̃𝑚𝑙 (𝜏, 𝑢) we then obtain the following expression:  𝜋̃𝑛𝑙 (𝜏, 𝑢) = 1 𝑔 𝑁    𝑛 𝛼𝛽(𝜏, 𝑢) (1 + 1 𝑔 𝐹) 𝛼𝛽,𝛾𝛿 −1 {𝐿̅𝛾𝛿(𝜗(𝜏)) − ∫ 𝑑3𝑢Σ    𝑚 𝛾𝛿 (𝜑̃(𝜏, 𝑢))𝜋̃𝑚𝑡 (𝜏, 𝑢)}, (20)  7  where the arbitrary functions 𝑁    𝑛 𝛼𝛽(𝜏, 𝑢) are normalized as follows:  ∫ 𝑑3𝑢𝑁    𝑛 𝛼𝛽(𝜏, 𝑢)Σ    𝑚 𝛾𝛿 (𝜑0𝑚(𝜏, 𝑢)) = 𝑔𝛼𝛾𝑔𝛽𝛿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (21) Besides,  𝐹𝛼𝛽,𝛾𝛿 = ∫ 𝑑3𝑢Σ𝛼𝛽,𝑛(Φ(𝜏, 𝑢) 𝑁𝛾𝛿(𝜏, 𝑢).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (22)  Thus the momenta 𝜋̃𝑚𝑡 (𝜏, 𝑢) and, consequently, the energy-momentum vector depends on the Lorentz group parameters through the quantities 𝐿̅𝜎𝜌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='. At the same time the 4-angular-momenta 𝑀𝛼𝛽, coincide with the generators, but their Poisson brackets with the 𝐿̅𝜎𝜌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=', is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' This is just what we need because as a result of this and expression (15), we can develop the perturbation theory in inverse powers of 𝑔, which will be manifestly Lorentz-covariant in every order of the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Transition to the Quantum Theory  The considerations of the previous section allow us to pass to the Hamiltonian approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' It is easy to verify that the canonical Hamiltonian:  𝐻 = 𝑐̇(𝜏)𝑃𝑐(𝜏) + 𝜗̇𝛼𝛽(𝜏)𝑃𝛼𝛽(𝜏) + 𝑞̇(𝜏)𝑃𝑐 + ∫ 𝑑3𝑢𝜑̃̇𝑚 (𝜏, 𝑢)𝜋̃𝑚 (𝜏, 𝑢) − 𝐿  is identically zero upon insertion of the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Following Dirac’s theory, we take the Hamiltonian as a linear combination of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  𝐻′ = 𝑎(𝜏)Ψ𝑐 + 𝑎𝜇(𝜏)Ψ𝜇 + 𝑏𝛼𝛽Ψ𝛼𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (23)  The equation of motion for any particular dynamical variable has the form 𝑓(𝜏) ̇ = 𝜕𝑓 𝜕𝑡 + {𝑓, 𝐻′}                                                                             (24) and lead for the constraints (they must be time independent) to the following consistency conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  𝑎(𝜏){Ψ𝑐, Ψ𝑐} + 𝑎𝜇(𝜏){Ψ𝑐, Ψ𝜇} + 𝑏𝛼𝛽(𝜏){Ψ𝑐, Ψ𝛼𝛽} ≈ 0,  𝑎(𝜏){Ψ𝜈, Ψ𝑐 + 𝑎𝜇(𝜏){Ψ𝜈, Ψ𝜇} + 𝑏𝛼𝛽(𝜏){Ψ𝜈, Ψ𝛼𝛽} ≈ 0,  𝑎(𝜏){Ψ𝜌𝜎, Ψ𝑐} + 𝑎𝜇(𝜏){Ψ𝜌𝜎, Ψ𝜇} + 𝑏𝛼𝛽(𝜏){Ψ𝜌𝜎, Ψ𝛼𝛽} ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  One can verify, that the constraints form a closed algebra with constant structure coefficients so that the consistency conditions become the identities: 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' This means that our theory does not contain constraints except (13)-(15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' At the same time, there are the arbitrary coefficients 𝑎(𝜏), 𝑎𝜇(𝜏) and 𝑏𝛼𝛽(𝜏) in the theory, which can be connected with the velocities of the parameters, introduced by transformations (4) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Indeed from the equation of motion, we obtain:  𝑎(𝜏) = 𝑐̇(𝜏), 𝑎𝜇(𝜏) = 𝑞̇𝜇(𝜏),   𝑏𝛼𝛽(𝜏) = 𝛣̅𝛼𝛽 𝛾𝛿(𝜗)𝜗̇𝛾𝛿(𝜏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  8  The arbitrariness can be avoided by introducing the gauge conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Before we fix these conditions we rewrite the constraints:  Ψ𝑐 = 𝑃𝑐 + ∫ 𝑑3𝑢ℋ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='                                                                                 (25)  Ψ𝜇 = 𝛬𝜇 𝜈𝑃𝜈 + ∫ 𝑑3𝑢 (𝛼𝜇ℋ + 𝑐(𝜏) 𝜕𝛼𝜇 𝜕𝑢𝑟 𝒫𝑟),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='                                                        (26)  Ψ𝛼𝛽 = ∫ 𝑑3𝑢Σ    𝑚 𝛾𝛿 (𝜑0𝑚(𝜏,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 𝑢))𝜋𝑛𝑡(𝜏,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 𝑢)                                                           (27)  We recall that the variables  𝑐(𝜏),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='   𝑃𝑐(𝜏),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='   𝑞𝜇(𝜏),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='   𝑃𝜇(𝜏),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  𝜗𝛼𝛽(𝜏),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='   𝑃𝛼𝛽(𝜏),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Φ𝑘(𝜏,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 𝑢),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 𝜋𝑛𝑡(𝜏,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 𝑢)                                (28)  constitute the phase space of theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' We next introduce the following gauge conditions:  𝜒𝑐 = 𝑐(𝜏) − 𝜏,                                                                      (29)  𝜒𝜇 = 𝛬𝜇 𝜈𝑞𝜈(𝜏) − 𝜏𝑃𝜇(𝜏),                                                                 (30)  𝜒𝛼𝛽 = ∫ 𝑑3𝑢𝑁𝛼𝛽,𝑛(𝜏, 𝑢)Φ𝑛(𝜏, 𝑢).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (31)  These conditions are chosen in such a way, that their Poisson brackets with each other become zero (such a choice is convenient, but not necessary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The second requirement we impose on the gauge conditions is that the matrix composed of Poisson brackets of constraints with gauge conditions should be nonsingular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' We now modify the Hamiltonian dynamics with the help of the brackets introduced by DIRAC [l0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' We introduce the notation 𝐴𝑎𝑏 = {Ψ𝑎, 𝜒𝑏}, where Ψ𝑎 and 𝜒𝑏 are the sets of constraints and gauge conditions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The new brackets for any dynamical quantity are now defined as follows:  {𝑓, ℎ}𝐷 = {𝑓, ℎ} − {𝑓, Ψ𝑎}𝐴𝑏𝑎 −1{𝜒𝑏,ℎ} − {𝑓, 𝜒𝑎}𝐴𝑎𝑏 −1{Ψ𝑏, ℎ}  where the nonsingular matrix A can be represented in the form  𝐴 = ( 𝑔 {Ψ, 𝜒𝑐} {Ψ, 𝜒𝛼𝛽} 0 −1 { Ψ𝑐 , 𝜒𝛼𝛽} 0 0 −𝐼 ) where g is the metric tensor of 4-space-time, Ψ = Ψ𝜇  , and 𝐼 is a unit 6 x6 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The  inverse matrix 𝐴−1 has the following form:  𝐴−1 = ( −𝑔 −{Ψ, 𝜒𝑐} {Ψ, 𝜒𝛼𝛽} + {Ψ, 𝜒𝑐} ∙ { Ψ𝑐, 𝜒𝛼𝛽} 0 −1 𝑔{ Ψ𝑐, 𝜒𝛼𝛽} 0 0 −𝐼 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  9  Evaluating the Dirac brackets for our canonical variables, we find that 𝑐(𝜏), 𝑃𝑐(𝜏),  𝑞𝜇(𝜏),  𝑃𝜇(𝜏) become unphysical;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' indeed:  {𝑐(𝜏), 𝑃𝑐(𝜏)}𝐷 =  {𝑞𝜇(𝜏),  𝑃𝜈(𝜏)𝐷 = 0,  whereas  the Dirac brackets for the other canonical pairs have the following form:  {𝜗𝛼𝛽(𝜏), 𝑃𝛾𝛿(𝜏)}𝐷 = 𝑔𝛼𝛾𝑔𝛽𝛿,  𝛼 > 𝛾, 𝛽 > 𝛿,                                              (32)  {Φ𝑚(𝜏, 𝑢), 𝜋𝑛𝑡(𝜏, 𝑢′)}𝐷 = 𝛿𝑚𝑛𝛿3(𝑢 − 𝑢′) − Σ𝛼𝛽,𝑛(𝜑0(𝜏, 𝑢))𝑁      𝑚 𝛼𝛽 (𝜏, 𝑢′) ≡ 𝐴𝑚𝑛(𝑢, 𝑢′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (33)  We have to choose a quantity, which generates the evolution of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' One can verify, that the brackets of dynamical variables (or physical quantities) with 𝑃𝑐 provide the evolution in 𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' For example  {𝑃𝑐, Φ𝑚(𝜏, 𝑢)}𝐷 = ∫ 𝑑3𝑣𝐴𝑚𝑛(𝑢, 𝑣) 𝛿 𝛿𝜋𝑛𝑡(𝜏, 𝑣) ∫ 𝑑3𝑢′ℋ,                           (34)  {𝑃𝑐, 𝜋𝑛𝑡(𝜏, 𝑢)}𝐷 = 𝛿 𝛿𝜋𝑛𝑡(𝜏, 𝑣) ∫ 𝑑3𝑢′ℋ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (35)  Due to constraints we can eliminate unphysical degrees of freedom and define the Hamiltonian - which generates the evolution of the system - as follows:  𝐻 = ∫ 𝑑3𝑢ℋ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' In general, we have to add to this Hamiltonian the linear combination of remaining primary constraints Ψ𝛼𝛽, but their Dirac brackets with dynamical quantities become zero, so we can get these constraints in a strong sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Therefore the equations of motion have the form:  𝑓̇ = 𝜕𝑓 𝜕𝜏 + {𝑓, 𝐻}𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (36) The transition to quantum theory is achieved with the help of the new brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' In particular, the commutator of two operators 𝑓, 𝑔 is defined by the relation:  [𝑓, 𝑔] = 𝑖{𝑓, 𝐻}𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (37)  From conditions  (32) and (33)  we can obtain the algebra of the operators, realized in the Hilbert space, in which the variables 𝑃𝛼𝛽 and 𝜋𝑛𝑡(𝜏, 𝑣) are acting as the following differentiation operators:  𝑃𝛼𝛽 = −𝑖 𝜕 𝜕𝜗𝛼𝛽 , 𝜋𝑚𝑡(𝜏, 𝑣) = −𝑖 ∫ 𝑑3𝑣𝐴𝑚𝑛(𝑢, 𝑣) 𝛿 𝛿Φ𝑛𝑡(𝜏, 𝑣),                        (38)  he space is a direct sum of two spaces, realizing the representation (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  10  Returning to expression (20), we can conclude that the state vector is an eigenvector of the two commuting operators, 𝐻 and 𝐿̅𝛼𝛽 because [𝐻, 𝐿̅𝛼𝛽] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Hence we can separate the 𝜗𝛼𝛽-dependence of the state vector:  𝜓(𝜗, Φ(𝜏, 𝑢) = 𝐷(𝜗)𝜓′(Φ(𝜏, 𝑢)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (39)  Hence, the quantum equations for the remaining fluctuation operators  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  𝑖 𝜕Φ𝑚(𝜏, 𝑢) 𝜕𝜏 = [Φ𝑚(𝜏, 𝑢), 𝐻],    𝑖 𝜕𝜋𝑚𝑡(𝜏, 𝑣) 𝜕𝜏 = −[𝜋𝑚𝑡(𝜏, 𝑣), 𝐻].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (40)  are explicitly covariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Expanding the momenta 𝜋𝑚𝑡(𝜏, 𝑣) (and correspondingly 𝐻)) in a series of inverse powers of the coupling constant 𝑔, we can construct the exact covariant perturbation theory for the equations (40) to every order of the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Before expanding it is necessary to transform the state vector as follows:  𝜓′ ⇒ exp (𝑖𝑔∫ 𝑑3𝑢𝑠𝑚(𝜏, 𝑢)Φ𝑚))𝜓′                                                    (41) provided that  ∫ 𝑑3𝑢Σ𝛼𝛽,𝑚(𝜑0(𝜏, 𝑢))𝑠𝑚(𝜏, 𝑢) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (42)  If the 𝑠𝑚(𝜏, 𝑢) do not satisfy (42), we can define another 𝑠𝑚′(𝜏, 𝑢) which would satisfy this requirement, indeed  𝑠𝑚 ′(𝜏,𝑢) = 𝑠𝑚(𝜏, 𝑢) − 𝑁𝛼𝛽,𝑚(𝜏, 𝑢)∫ 𝑑3𝑣Σ𝛼𝛽,𝑚(𝜑0(𝜏, 𝑣))𝑠𝑚(𝜏, 𝑣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  The transformation (41) implies, that the quantum operator 𝜋𝑚𝑡(𝜏, 𝑣) has to be replaced by  𝜋𝑚𝑡(𝜏, 𝑣) ⇒ 𝑔𝑠𝑚(𝜏, 𝑢) + 𝜋𝑚𝑡(𝜏, 𝑣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  We then expand the momenta 𝜋𝑚𝑡(𝜏, 𝑣) and the Hamiltonian 𝐻 in a series of inverse powers of 𝑔, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  𝜋𝑚𝑡(𝜏, 𝑣) = 𝑔𝜋0𝑚𝑡(𝜏, 𝑣)+𝜋1𝑚𝑡(𝜏, 𝑣) + 𝑔−1𝜋2𝑚𝑡(𝜏, 𝑣) + ⋯                                      (43)  𝐻 = 𝑔2𝐻0 + 𝑔𝐻1 + 𝐻2 + 𝑔−1𝐻3 + ⋯                                                                   (44)  The dominant terms of these series are purely classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The terms that follow are  linear (~𝑔) and quadratic (~𝑔0) in the fluctuation variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Thus, the quantum equations up to the one-loop approximation are:  𝑖 𝜕𝜑0𝑛(𝜏, 𝑢)  𝜕𝜏  = [Φ𝑛(𝜏, 𝑢), 𝛨1],  11  𝑖 𝜕𝜋0𝑛𝑡(𝜏, 𝑢)  𝜕𝜏  = [𝜋1𝑛𝑡(𝜏, 𝑢), 𝛨1],  𝑖 𝜕𝜋0𝑛𝑡(𝜏, 𝑢)  𝜕𝜏  = [𝜋1𝑛𝑡(𝜏, 𝑢), 𝛨1],  𝑖 𝜕Φ𝑛(𝜏, 𝑢)  𝜕𝜏  = [Φ𝑛(𝜏, 𝑢), 𝛨1],  𝑖 𝜕𝜋1𝑛𝑡(𝜏, 𝑢) 𝜕𝜏 = [𝜋1𝑛𝑡(𝜏, 𝑢), 𝛨2] + [𝜋2𝑛𝑡(𝜏, 𝑢), 𝛨1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  We see, that the first line reproduces the classical equations of motion whereas the others coincide with one-loop approximation equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' It should be noted that one has to be careful in manipulating quantum variables, but up to the one-loop approximations no operator ordering problem arises, and hence this question does not concern us here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  Appendix  In this appendix we summarize some properties of the Lorentz group which we use in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The Lorentz transformation matrix satisfies the following orthogonality and product conditions :  𝛬𝛼 𝛽𝛬̅𝛽 𝛾 = 𝛿𝛼 𝛾,           𝛬(𝜗1)𝛬(𝜗2) = 𝛬(𝜗3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 1)  From the latter condition, we deduce that  𝜗2  𝛾𝛿 = 𝜑𝛾𝛿(𝜃1, 𝜃2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='  The matrices satisfy the equations:  𝜕𝛬(𝜗)  𝜕𝜗𝛼𝛽 = 𝐵𝛼𝛽  𝛾𝛿(𝜃)𝐽𝛾𝛿𝛬(𝜗),  𝜕𝛬̅(𝜗) 𝜕𝜗𝛼𝛽 = 𝐵̅𝛼𝛽 𝛾𝛿(𝜃)𝐽̅𝛾𝛿𝛬(𝜗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (Α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 2)  Where  𝐵𝛼𝛽  𝛾𝛿(𝜗2) = 𝜕𝜑𝛾𝛿(𝜃1, 𝜃2)  𝜕𝜗1  𝛼𝛽  |  𝜗1=𝜗2−1  ,  𝐵̅𝛼𝛽  𝛾𝛿(𝜗1) = 𝜕𝜑𝛾𝛿(𝜃1, 𝜃2)  𝜕𝜗2  𝛼𝛽  |  𝜗2=𝜗1−1  (Α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 3)  which become unit matrixes for zero arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Then  12  𝐽𝛼𝛽 = 𝜕𝛬(𝜗  𝜕𝜗𝛼𝛽|  𝜗=0  ,  𝐽̅𝛼𝛽 = 𝜕𝛬̅(𝜗  𝜕𝜗𝛼𝛽|  𝜗=0  (Α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 4)  We define matrices 𝐴, 𝐴̅ which are inverse to 𝐵,  𝐵̅  respectively by  equations:  𝐴𝐵 = I ,     𝐴̅𝐵̅ = I ,                                                                       (Α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 5)  I am a unit matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' With the help of these matrices, we can construct the following quantities:  𝐿𝛼𝛽 = 𝐵𝛼𝛽 𝛾𝛿𝑃𝛾𝛿,      𝐿̅𝛼𝛽 = 𝐵̅𝛼𝛽 𝛾𝛿𝑃𝛾𝛿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (Α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 6)  They obey the usual Poisson brackets for the Lorentz group parameters¨  {𝐿𝛼𝛽, 𝐿𝛾𝛿} = 𝐶𝛼𝛽,𝛾𝛿 𝜎𝜌𝐿𝜎𝜌,          {𝐿̅𝛼𝛽, 𝐿̅𝛾𝛿} = −𝐶𝛼𝛽,𝛾𝛿 𝜎𝜌𝐿̅𝜎𝜌,    {𝐿𝛼𝛽, 𝐿̅𝛾𝛿} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (Α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 7)  where 𝐶𝛼𝛽,𝛾𝛿 𝜎𝜌 are well-known structure constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' We are interested in the Minkowsky space coordinate transformations :  𝑥𝜇 ⇒ 𝑥𝜇 ′ = 𝛬𝜇 𝜈𝑥𝜈 ≡ 𝑓𝜇(𝑥, 𝜗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (Α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 8)  From the product law we can obtain the following equations:  𝜕𝑥𝜇′ 𝜕𝜗𝛼𝛽 = 𝛣𝛼𝛽 𝛾𝛿(𝜗)𝜉𝛼𝛽,𝜇(𝑥′)                                                                                   (Α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 9)  with  𝜉𝛼𝛽,𝜇(𝑥′) = 𝜕𝑓𝜇(𝑥, 𝜗) 𝜕𝜗𝛼𝛽 | 𝜗=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (Α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 10)  Inverting the transformations we obtain the equations¨  𝜕𝑥𝜇 𝜕𝜗𝛼𝛽 = 𝛣̅𝛼𝛽 𝛾𝛿(𝜗)𝜉̅𝛼𝛽,𝜇(𝑥)                                                                                   (Α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 9) with  𝜉̅𝛼𝛽,𝜇(𝑥) = 𝜕𝑓̅𝜇(𝑥, 𝜗) 𝜕𝜗𝛼𝛽 | 𝜗=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (Α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 10)  The quantities  13  𝑋𝛼𝛽 = 𝜉𝛼𝛽,𝜇(𝑥)𝑃𝛼𝛽,   𝑋̅𝛼𝛽 = 𝜉̅𝛼𝛽,𝜇(𝑥)𝑃𝛼𝛽                                                               (Α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 11)  with {𝑥𝜇, 𝑃𝜎} = 𝑔𝜇𝜎 satisfy the conditions similar to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' These formulae give two different representations of the homogeneous Lorentz group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' The matrix 𝑆𝑖𝑘(𝜗), determines the transformation properties of the field variables 𝜑𝜄(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' They obey the equation  𝜕𝑆𝑖𝑘(𝜗) 𝜕𝜗𝛼𝛽 = 𝐵𝛼𝛽 𝛾𝛿(𝜃)(𝛪𝛾𝛿)𝑖𝑛𝑆𝑛𝑘(𝜗),        𝛪𝛾𝛿 = 𝜕𝑆(𝜗) 𝜕𝜗𝛾𝛿 | 𝜗=0 (Α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 12)  If 𝑆̅ are inverse to 𝑆, then  𝜕𝑆̅𝑖𝑘(𝜗) 𝜕𝜗𝛼𝛽 = 𝐵̅𝛼𝛽 𝛾𝛿(𝜃)(𝛪̅𝛾𝛿)𝑖𝑛𝑆𝑛𝑘(𝜗),     𝛪̅𝛾𝛿 = 𝜕𝑆̅(𝜗) 𝜕𝜗𝛾𝛿 | 𝜗=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' (Α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 13)  References  [l] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Wentzel, Helv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Acta 13 (1940) 269;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Pauli and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Dancoff, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 62 (1942) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' [2] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Bogolubov, Ukr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 2 (1950) 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Selected Topics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 499, Naukov Dumka, [3] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Sslodovnikova,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Tavkhelidze and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Khrustalev, Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 11 (1972) 317 [4] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Sslodovnikova, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Tavkhelidze and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Khrustalev, Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 90 (1972) 168 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Turin and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Shurgaia, Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 16 (1973) 197;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Arkhipov and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Turin, Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 17 (1973) 57;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Semenoff, Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 18 (1974) 353;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Shurgaia, Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 28 (1976) 223;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Razumov and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' KHRUSTALOV, Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 29 (1976) 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' [5] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Christ and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Lee, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' D12 (1975) 1606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 161 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Gervais, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Jevicki and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Sakita, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' D12 (1975) 1038;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Gervais and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Jevicki, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' BllO (1976) 113;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Gervais and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Sakita,, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' D16 (1977) 3507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' [7] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Muller-Kirsten  and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Widemann, Fortschr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 32 (1984) 21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Muller-Kirsten  and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Widemann, Nuovo Cimento A78 (1983) 61;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Maharan and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Muller-Kirsten, Nuovo Cimento A83 (1984) 229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' [S] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Matsumoto, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Oberlechner, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Umezava and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Umezava, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 20 (1979) 2088.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' [9] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Dirac, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 21 (1949) 391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' [10] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Dirac, Can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' 2 (1950) 129;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
+page_content=' Can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQf_f7E/content/2301.01950v1.pdf'}
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+Continuous Newton-like Methods featuring Inertia and
+Variable Mass
+Camille Castera∗
+Department of Mathematics
+University of T¨ubingen
+Germany
+Hedy Attouch
+IMAG
+Universit´e Montpellier
+CNRS, France
+Jalal Fadili
+ENSICAEN
+Normandie Universit´e
+CNRS, GREYC, France
+Peter Ochs
+Department of Mathematics
+University of T¨ubingen
+Germany
+ABSTRACT
+We introduce a new dynamical system, at the interface between second-order dynamics
+with inertia and Newton’s method. This system extends the class of inertial Newton-like
+dynamics by featuring a time-dependent parameter in front of the acceleration, called
+variable mass. For strongly convex optimization, we provide guarantees on how the
+Newtonian and inertial behaviors of the system can be non-asymptotically controlled by
+means of this variable mass. A connection with the Levenberg–Marquardt (or regular-
+ized Newton’s) method is also made. We then show the effect of the variable mass on the
+asymptotic rate of convergence of the dynamics, and in particular, how it can turn the lat-
+ter into an accelerated Newton method. We provide numerical experiments supporting
+our findings. This work represents a significant step towards designing new algorithms
+that benefit from the best of both first- and second-order optimization methods.
+1
+Introduction
+1.1
+Problem Statement
+A major challenge in modern unconstrained convex optimization consists in building fast algorithms
+while maintaining low computational cost and memory footprint. This plays a central role in many key
+applications such as large-scale machine learning problems or data processing. The problems we are
+aiming to study are of the form
+min
+x∈Rn f(x).
+∗Corresponding author: camille.castera@protonmail.com
+arXiv:2301.08726v1  [math.OC]  20 Jan 2023
+
+Large values of n demand for algorithms at the interface of first- and second-order optimization. Lim-
+ited computational capabilities explain why gradient-based (first-order) algorithms remain prominent
+in practice. Unfortunately, they often require many iterations which is true even for the provably best
+algorithms for certain classes of optimization problems; for example that of convex and strongly con-
+vex functions with Lipschitz continuous gradient [37, 33, 34]. On the other hand, algorithms using
+second-order information (the Hessian of f)—with Newton’s method as prototype—adapt locally to
+the geometry of the objective, allowing them to progress much faster towards a solution. However,
+each iteration comes with high computational and memory costs, which highlights a challenging trade-
+off. It is therefore essential to develop algorithms that take the best of both worlds. They are commonly
+referred to as (limited-memory) quasi-Newton methods. Several quasi-Newton algorithms partly ad-
+dress this issue, for example BFGS methods [18, 23, 25, 39, 31], yet, in very large-scale applications,
+first-order algorithms often remain the preferred choice.
+In order to reach a new level of efficiency, deep insights into the mechanism and relations between algo-
+rithms are required. To that aim, an insightful approach is to see optimization algorithms as discretiza-
+tion of ordinary differential equations (ODEs): for small-enough step-sizes, iterates can be modeled
+by a continuous-time trajectory [32, 13]. Obtaining a fast algorithm following this strategy depends
+on two ingredients: choosing an ODE for which rapid convergence to a solution can be proved, and
+discretizing it with an appropriate scheme that preserves the favorable properties of the ODE.
+Both steps are highly challenging, our work focuses on the ODE matter. We study the following
+second-order dynamical system in a general setting:
+ε(t)¨x(t) + α(t) ˙x(t) + β∇2f(x(t)) ˙x(t) + ∇f(x(t)) = 0,
+t ≥ 0,
+(VM-DIN-AVD)
+where f : Rn → R is a smooth convex twice continuously differentiable function, with gradient ∇f and
+Hessian ∇2f defined on Rn equipped with scalar product ⟨·, ·⟩, and induced norm ∥·∥. Additionally, f is
+assumed to be coercive, and strongly convex on bounded subsets of RP. The functions ε, α: R+ → R+
+(where R+ = [0, +∞[) are differentiable, non-increasing, and ε(t) > 0 for all t ≥ 0. Together with
+β > 0, they are control parameters that define the type of dynamics that drives the trajectory (or
+solution) x: R+ → Rn, whose first- and second-order derivatives are denoted ˙x and ¨x respectively.
+We call the above dynamics (VM-DIN-AVD), which stands for “Variable Mass Dynamical Inertial
+Newton-like system with Asymptotically Vanishing Damping” since it generalizes a broad class of
+ODEs whose original member is DIN [2], where ε and α were constant. DIN was then extended to
+the case of non-constant asymptotically vanishing dampings (AVD) α [9]. In this work we introduce
+the non-constant parameter ε called variable mass (VM) in front of the acceleration ¨x, in the same
+way that α is called (viscous) damping by analogy with classical mechanics. A key feature of these
+ODEs, that positions them at the interface of first- and second-order optimization, is that they possess
+equivalent forms involving only ∇f but not ∇2f, significantly reducing computational costs, hence
+enabling the design of practical algorithms, see e.g., [20, 10, 21]. The key idea behind this is the
+relation ∇2f(x(t)) ˙x(t) = d
+dt∇f(x(t)), see Section 2 for an equivalent formulation of (VM-DIN-AVD)
+exploiting this.
+This paper emphasizes the relation between (VM-DIN-AVD) and well-studied special cases. Indeed,
+taking ε = α = 0, one obtains1 the Continuous Newton (CN) method [24]
+β∇2f(xN(t)) ˙xN(t) + ∇f(xN(t)) = 0,
+t ≥ 0,
+(CN)
+1CN is usually considered with β = 1, we put β in the system to ease the discussions below.
+2
+
+= 0
+(t)
+0
+> 0 constant
+Viscous damping 
+= 0
+(t)
+0
+> 0 const.
+Variable mass 
+0
+0
+0
+(t)
+(t)
+max(
+(t),
+(t))
+max(
+,
+)
+(Alvarez et al.)
+Initialization x0
+Continuous Newton
+Continuous gradient descent
+DIN-AVD (no variable mass)
+Minimizer x  of f
+Level lines of f
+fast decay
+slow decay
+Decay speed of (t)
+Figure 1: Left: phase diagram on distances from (VM-DIN-AVD) to (CN) and (LM) (see Section 3).
+For each patch, the color indicates which of the distances ∥xN(t) − x(t)∥ and ∥xLM(t) − x(t)∥ is
+considered, the scaling of a corresponding upper-bound on this distance is written; in white for prior
+work and in black for our contributions. The green line separates the cases ε ≥ α (above) and ε ≤
+α (below). Right: 2D illustration of the trajectories of (VM-DIN-AVD) for several choices of ε on
+a quadratic function. Using fast-vanishing ε(t) (dark-blue solid curves), one can bring the solution
+of (VM-DIN-AVD) close to that of (CN), making it, for example, more robust to bad conditioning
+compared to first-order dynamics (such as gradient descent).
+known notably for being invariant to affine transformations and yielding fast vanishing of the gradient
+(see Section 3). In fact, this observation shows that (VM-DIN-AVD) is a singular perturbation of (CN),
+which also justifies the terminology “Newton-like” in DIN. When α ̸= 0 but ε = 0, we recover the
+Levenberg–Marquardt (LM) method,
+α(t) ˙xLM(t) + β∇2f(xLM(t)) ˙xLM(t) + ∇f(xLM(t)) = 0,
+t ≥ 0,
+(LM)
+also known as regularized Newton method since it stabilizes (CN). In the rest of the paper, the solutions
+of (CN) and (LM) will always be denoted by xN and xLM respectively. Alvarez et al. [2] showed that
+for α = 0, β = 1, and ε constant and small, (VM-DIN-AVD) is a “perturbed” Newton method since
+the distance between the solutions of (VM-DIN-AVD) and (CN) is at most proportional to √ε at all
+time. Yet, despite the benefits of this class of ODEs, such as stabilization properties, see e.g., [9, 10],
+no improvement2 of the rate of convergence (in values) has been shown compared to inertial first-order
+dynamics [37, 41]. This raises the question:
+“are these ODEs really of Newton type?”,
+which is crucial in view of designing faster algorithms from them.
+2DIN-like systems were thought to yield faster vanishing of the gradient compared to inertial first-order dynamics, until
+recently [4].
+3
+
+xn(t) - x(t)II
+Both
+[XLM(t) 一 x(t) 
+Frontier ε(t) = α(t)Table 1: Informal summary of Section 4. Comparison of (VM-DIN-AVD) with other dynamics
+Parameters of (VM-DIN-AVD)
+Speed of convergence
+Dominant parameter Integrability in +∞
+w.r.t. (CN)
+w.r.t. (LM)
+variable mass ε
+yes
+as fast
+as fast
+no
+faster
+faster
+viscous damping α
+yes
+as fast
+only depends
+on ε
+no
+slower
+1.2
+Main Contributions
+We show that the answer to this question is partially positive, and closely related to the choices of ε
+and α. We provide general results on the role played by these two control parameters and how they
+can be chosen to control (VM-DIN-AVD), and make it close to (CN) for all time, as illustrated on the
+right-hand side of Figure 1, but also to obtain fast convergence. This represents a first step towards
+building new fast practical algorithms. Our main contributions are the following:
+– We provide a first-order equivalent formulation of (VM-DIN-AVD), and address the questions of
+existence and uniqueness of the solutions of (VM-DIN-AVD) under mild assumptions.
+– We generalize the perturbed Newtonian property discussed above to non-constant and possibly van-
+ishing variable masses ε, and “not too large” positive dampings α, and derive bounds that (formally)
+take the form ∥x(t) − xN(t)∥ = O(
+�
+ε(t)). We then extend these results to larger dampings α and
+make the connection between (VM-DIN-AVD) and (LM). This contribution is summarized in the phase
+diagram of Figure 1.
+– Using quadratic functions as a model for strongly convex functions, we shed light on techniques to
+efficiently approximate solutions of (VM-DIN-AVD). We then show how ε and α affect the speed of
+convergence. Depending on their setting, the solutions of (VM-DIN-AVD) may either converge as fast
+as that of (CN), faster, or rather have a (LM) nature, as summarized in Table 1.
+– We provide numerical experiments supporting our theoretical findings.
+1.3
+Related work
+The system (VM-DIN-AVD) belongs to the class of inertial systems with viscous and geometric
+(“Hessian-driven”) dampings, initially introduced with constant ε = 1 and constant α in [2] and called
+DIN (for Dynamical Inertial Newton-like system). Except in a few cases [20, 19], most of the follow-up
+work then considered extensions of DIN with non-constant AVD α, with in particular the DIN-AVD
+system with α(t) = α0/t as introduced in [9]. The reason for this popular choice for α is its link with
+Nesterov’s method [41]. Non-constant choices for β have been considered [10, 1, 29, 11]. We keep
+it constant here, and rather focus on non-constant ε, unlike prior work that used constant ε = 1. The
+mass ε was only considered in the original work [2], but only for fixed ε, β = 1 and constant α = 0.
+VM-DIN-AVD is however closely related to the IGS system considered in [11] as it is actually equiv-
+alent to the latter after dividing both sides of (VM-DIN-AVD) by ε(t). Our approach—which consists
+in studying the connections with other second-order dynamics as ε vanishes asymptotically—is how-
+ever different from the one followed in [11], which is of independent interest. The literature on DIN
+is rich, let us mention further connections with Nesterov’s method [40, 1], extensions with Tikhonov
+4
+
+regularization [16] and closed-loop damping [12, 29]. The non-smooth and possibly non-convex cases
+have been considered in [5, 6, 20]. Finally, avoidance of strict saddle points in smooth non-convex
+optimization has been shown in [19].
+The influence of the damping α on the (LM) dynamics has been studied in [7, 8]. Interestingly, the
+conditions enforced on α in these papers (formally a sub-exponential decay) are very similar to those
+we make on ε and α for (VM-DIN-AVD) (see Assumptions 1 and 2).
+Regarding the second part of our analysis, which deals with the case where f is quadratic, Attouch
+et al. [10] provided closed-form solutions to (VM-DIN-AVD) for ε ≡ 1 and special choices of α. Our
+work rather deals with approximate solutions which allows considering a wide class of functions ε and
+α. We rely on the Liouville–Green (LG) method [30, 26] presented in Section 4. Generalizations of
+LG are also often referred to as WKB methods [44, 28, 17] and seem to be mostly used in physics so
+far. To the best of the authors’ knowledge, the current work seem to be one of the first to use the LG
+method in optimization, and the first for DIN-like systems.
+1.4
+Organization
+The paper is organized as follows. We discuss the existence of solutions in Section 2. Our main results,
+from a non-asymptotic control perspective, are then presented in Section 3. An analysis of the role
+played asymptotically by ε and α is then carried out on quadratic functions in Section 4. Finally,
+numerical experiments are presented in Section 5, and some conclusions are then drawn.
+2
+Existence and Uniqueness of Solutions
+In the sequel, we fix x0 ∈ Rn and ˙x0 ∈ Rn, such that, unless stated otherwise, (VM-DIN-AVD) is
+always considered with initial condition (x(0), ˙x(0)) = (x0, ˙x0), and (CN) and (LM) with initial con-
+dition xN(0) = xLM(0) = x0. We also fix initial values for the control parameters ε(0) = ε0 > 0,
+ε′(0) = ε′
+0 ≤ 0 and α(0) = α0 ≥ 0. In addition to the definitions of ε and α in Section 1.1, we assume
+that ε is twice differentiable, with bounded second derivative. In order to use the Cauchy–Lipschitz
+Theorem, we reformulate (VM-DIN-AVD) into a first-order (in time) system by introducing an auxil-
+iary variable y : R+ → Rn. Notably, our reformulation does not involve ∇2f, in the same fashion as
+[2, 9]. For all t, defining ν(t) = α(t) − ε′(t) − 1
+βε(t), we show in Appendix A that (VM-DIN-AVD) is
+equivalent to
+�
+ε(t) ˙x(t) + β∇f(x(t)) + ν(t)x(t) + y(t)
+= 0
+˙y(t) + ν′(t)x(t) + ν(t)
+β x(t) + 1
+βy(t)
+= 0
+(gVM-DIN-AVD)
+with initial conditions (x(0), y(0)) =
+�
+x0, −ε0 ˙x0 − β∇f(x0) − (α0 − ε′
+0 − 1
+βε0)x0
+�
+. One can notice
+that in the special case where ε is taken constant and equal to 1 (that is when (VM-DIN-AVD) is simply
+the DIN-AVD system [9]), we recover the same first-order formulation as that in [9].
+We can then apply the Cauchy–Lipschitz Theorem. For all t ≥ 0 and (u, v) ∈ Rn × Rn, define the
+mapping
+G (t, (u, v)) =
+� 1
+ε(t)(−β∇f(u) − ν(t)u − v)
+−ν′(t)u − ν(t)
+β u − 1
+βv
+�
+,
+5
+
+so that (gVM-DIN-AVD) rewrites ( ˙x(t), ˙y(t)) = G (t, (x(t), y(t))) for all t ≥ 0. Since f is twice
+continuously differentiable, one can see that G is continuously differentiable w.r.t. its second argument
+(u, v). Consequently G is locally Lipschitz continuous w.r.t. (u, v) and by the Cauchy–Lipschitz The-
+orem, for each initial condition, there exists a unique local solution to (gVM-DIN-AVD) and thus to
+(VM-DIN-AVD). We then show that this solution is actually global (in Appendix A) by proving the
+boundedness of (x, y). We omit the existence and uniqueness of the solutions of (CN) and (LM) since
+these are standard results, obtained with similar arguments.
+3
+Non-asymptotic Control of (VM-DIN-AVD)
+The purpose of this section is to understand how close x might be to xN and xLM, as a function of α
+and ε. Since f is coercive and strongly convex on bounded sets, it has a unique minimizer x⋆ ∈ Rn.
+Consequently, any two trajectories that converge to x⋆ will eventually be arbitrarily close to each other.
+Thus, asymptotic results of the form ∥x(t) − xN(t)∥ −−−−→
+t→+∞ 0 are not precise enough to claim, for
+example, that x has a “Newtonian behavior”. Instead, we will derive upper bounds on the distance
+between trajectories that hold for all time t ≥ 0, and which typically depend on ε and/or α. We first
+present the case where α is small relative to ε and then generalize.
+3.1
+Comparison with (CN) under Moderate Viscous Dampings
+When the viscous damping α remains moderate w.r.t. the variable mass ε, one can expect the solutions
+of (VM-DIN-AVD) to be close to that of (CN). We make the following assumptions.
+Assumption 1. There exists c1, c2 ≥ 0 such that for all t ≥ 0, |ε′(t)| ≤ c1ε(t) and α(t) ≤ c2ε(t).
+The assumption states that α must decrease at least as fast as ε (up to a constant).3 The reason for
+assuming |ε′(t)| ≤ c1ε(t) is technical and will appear more clearly in the proofs below. It formally
+means that ε can decrease at most exponentially fast.4 This is a relatively mild assumption that holds,
+for example, for any polynomial decay ε0/(t + 1)a, a ∈ N.
+We start with the main result of this section.
+Theorem 3.1. Let xN be the solution of (CN), and let c1, c2 ≥ 0. There exist C0, C1, C2 ≥ 0, de-
+pending on c1, c2, such that for all (ε, α) for which Assumption 1 holds with constants c1 and c2, the
+corresponding solution x of (VM-DIN-AVD) is such that for all t ≥ 0,
+∥x(t) − xN(t)∥ ≤ C0e− t
+β ε0∥ ˙x0∥ + C1
+�
+ε(t) + C2
+� t
+s=0
+e
+1
+β (s−t)�
+ε(s) ds.
+(1)
+This extends a previous result from [2, Proposition 3.1] which states a similar bound for constant ε,
+α ≡ 0 and β = 1. Theorem 3.1 corresponds to the blue parts in the phase diagram of Figure 1 (see also
+Corollary 3.6 below).
+Remark 3.2. The strength of the above result comes from the fact that the constants C0, C1, C2 do not
+depend on ε and α, and that the result is non-asymptotic. This allows in particular choosing (ε, α) to
+control the distance from x to xN, for all time t ≥ 0.
+3Assumption 1 can actually hold only after some large-enough t0 ≥ 0, we take t0 = 0 for the sake of simplicity.
+4This is a consequence of Gronwall’s lemma, see e.g., [22].
+6
+
+Remark 3.3. Under Assumption 1, the dynamics (VM-DIN-AVD) is dominated by the variable mass.
+The damping α does not appear in Theorem 3.1.
+The above theorem and remarks emphasize the “Newtonian nature” of (VM-DIN-AVD). We present
+two lemmas before proving Theorem 3.1, and then state a simpler bound than (1), see Corollary 3.6.
+Lemma 3.4. Let (ε, α), and let x be the corresponding solution of (VM-DIN-AVD). For all t ≥ 0,
+define the function,
+U(t) = ε(t)
+2 ∥ ˙x(t)∥2 + f(x(t)) − f(x⋆).
+Then U is differentiable and for all t > 0,
+dU
+dt (t) = ε′(t)
+2 ∥ ˙x(t)∥2 − α(t)∥ ˙x(t)∥2 − β⟨∇2f(x(t)) ˙x(t), ˙x(t)⟩ ≤ 0.
+Therefore, in particular, U is non-increasing.
+Proof. Let t ≥ 0, since x is twice differentiable, U is differentiable and,
+dU
+dt (t) = ε′(t)
+2 ∥ ˙x(t)∥2 + ε(t)⟨ ˙x(t), ¨x(t)⟩ + ⟨ ˙x(t), ∇f(x(t))⟩.
+We use the fact that x is solution of (VM-DIN-AVD), to substitute ε(t)¨x(t) by its expression,
+dU
+dt (t) = ε′(t)
+2 ∥ ˙x(t)∥2 − α(t)∥ ˙x(t)∥2 − β⟨∇2f(x(t)) ˙x(t), ˙x(t)⟩.
+By assumption ε is non-increasing so for all t > 0, ε′(t) ≤ 0.
+Furthermore f is convex so
+⟨∇2f(x(t)) ˙x(t), ˙x(t)⟩ ≥ 0. Hence U is non-increasing.
+We then state the following bound.
+Lemma 3.5. There exists C ≥ 0 such that for any (ε, α) and the corresponding solution x of
+(VM-DIN-AVD), for all t ≥ 0 it holds that,
+ε(t)∥ ˙x(t)∥ ≤ C
+�
+ε(t).
+Proof. Let t ≥ 0, according to Lemma 3.4, U is non-increasing so U(t) ≤ U(0), or equivalently,
+ε(t)
+2 ∥ ˙x(t)∥2 + f(x(t)) − f(x⋆) ≤ ε0
+2 ∥ ˙x0∥2 + f(x0) − f(x⋆).
+This implies in particular that,
+ε(t)∥ ˙x(t)∥2 ≤ ε0∥ ˙x0∥2 + 2(f(x0) − f(x⋆)),
+and hence by multiplying both sides by ε(t) and composing with the square-root we obtain that,
+ε(t)∥ ˙x(t)∥ ≤ C
+�
+ε(t),
+where C =
+�
+ε0∥ ˙x0∥2 + 2(f(x0) − f(x⋆)).
+7
+
+Proof of Theorem 3.1. Let (ε, α) as defined in Sections 1.1 and 2, and let x be the corresponding solu-
+tion of (VM-DIN-AVD). Then, according to Lemma 3.4, for all t ≥ 0, U(t) ≤ U(0), so in particular
+f(x(t)) ≤ f(x0) + ε0
+2 ∥ ˙x0∥2.
+Denoting M0 = f(x0) + ε0
+2 ∥ ˙x0∥2, the set K0 = {y ∈ Rn | f(y) ≤ M0} is bounded, since f is coercive
+(lim∥y∥→+∞ f(y) = +∞). So for all t ≥ 0, x(t) ∈ K0. Since M0 (and hence K0) depends only on ε0,
+x0 and ˙x0, we have proved that for any choice (ε, α), the corresponding solution x of (VM-DIN-AVD)
+is inside K0 at all time. Let xN be the solution of (CN). One can see similarly that for all t ≥ 0,
+f(xN(t)) ≤ f(xN(0)) = f(x0) ≤ M0. So we also have xN(t) ∈ K0 for all t ≥ 0.
+Now, fix c1, c2 > 0, and let (ε, α) such that Assumption 1 is satisfied with constants c1, c2. Let x be the
+corresponding solution of (VM-DIN-AVD). Since f is strongly convex on bounded sets, it is strongly
+convex on K0. We denote µ > 0 the strong-convexity parameter of f on K0. Equivalently, we have that
+∇f is strongly monotone on K0, that is, ∀y1, y2 ∈ K0,
+⟨∇f(y1) − ∇f(y2), y1 − y2⟩ ≥ µ∥y1 − y2∥2.
+(2)
+Let t ≥ 0, since x(t) ∈ K0 and xN(t) ∈ K0, by combining (2) with the Cauchy–Schwarz inequality, we
+deduce that
+∥x(t) − xN(t)∥ ≤ 1
+µ∥∇f(x(t)) − ∇f(xN(t))∥.
+(3)
+Therefore, it is sufficient to bound the difference of gradients in order to bound ∥x(t) − xN(t)∥. First,
+remark that (CN) can be rewritten as follows:
+d
+dt∇f(xN(t)) + 1
+β∇f(xN(t)) = 0. So we can integrate,
+for all t ≥ 0,
+∇f(xN(t)) = e− t
+β ∇f(x0).
+(4)
+We now turn our attention to ∇f(x(t)), for which we cannot find a closed-form solution in general.
+We rewrite (VM-DIN-AVD) in the following equivalent form
+d
+dt [ε(t) ˙x(t) + β∇f(x(t))] + 1
+β ε(t) ˙x(t) + ∇f(x(t)) =
+� 1
+β ε(t) + ε′(t) − α(t)
+�
+˙x(t).
+Introducing the variable ω(t) = ε(t) ˙x(t) + β∇f(x(t)), the latter is thus solution to
+�
+˙ω(t) + 1
+βω(t) =
+�
+1
+βε(t) + ε′(t) − α(t)
+�
+˙x(t),
+t ≥ 0,
+ω(0) = ε0 ˙x0 + β∇f(x0).
+This is a non-homogeneous first-order ODE in ω, whose solution can be expressed using the integrating
+factor
+ω(t) = e− t
+β (ε0 ˙x0 + β∇f(x0)) + e− t
+β
+� t
+0
+e
+s
+β
+� 1
+β ε(s) + ε′(s) − α(s)
+�
+˙x(s) ds.
+We thus have the following expression for ∇f(x), for all t ≥ 0,
+β∇f(x(t)) = βe− t
+β ∇f(x0)+e− t
+β ε0 ˙x0−ε(t) ˙x(t)+e− t
+β
+� t
+0
+e
+s
+β
+� 1
+β ε(s) + ε′(s) − α(s)
+�
+˙x(s) ds. (5)
+8
+
+We can now use (4) and (5) in (3) to get
+∥x(t) − xN(t)∥ ≤ 1
+βµ
+����e− t
+β ε0 ˙x0 − ε(t) ˙x(t) + e− t
+β
+� t
+0
+e
+s
+β
+� 1
+β ε(s) + ε′(s) − α(s)
+�
+˙x(s) ds
+���� .
+Using the triangle inequality, we obtain,
+∥x(t) − xN(t)∥ ≤ ε0∥ ˙x0∥
+βµ
+e− t
+β + ε(t)∥ ˙x(t)∥
+βµ
++ 1
+βµ
+� t
+0
+e
+1
+β (s−t)
+����
+1
+β ε(s) + ε′(s) − α(s)
+���� ∥ ˙x(s)∥ ds. (6)
+The first term in (6) corresponds to the first one in (1) with C0 = 1/(βµ). As for the second-one, by
+direct application of Lemma 3.5, there exists C > 0 such that for all t ≥ 0, ε(t)∥ ˙x(t)∥
+βµ
+≤ C
+�
+ε(t), so we
+set C1 = C/(βµ). Regarding the last term in (6), using Assumption 1 and again Lemma 3.5, it holds
+that, for all s ≥ 0,
+����
+1
+β ε(s) + ε′(s) − α(s)
+���� ∥ ˙x(s)∥ ≤
+� 1
+β + c1 + c2
+�
+ε(s)∥ ˙x(s)∥ ≤
+� 1
+β + c1 + c2
+�
+C
+�
+ε(s).
+This proves the theorem with C2 = C
+βµ
+�
+1
+β + c1 + c2
+�
+.
+Let us analyze the bound in Theorem 3.1. The first term in (1) decays exponentially fast and can even
+be zero if the initial speed is ˙x0 = 0, the second-one decays like
+�
+ε(t), however, the rate at which the
+last term decreases is less obvious. The following corollary gives a less-tight but easier-to-understand
+estimate.
+Corollary 3.6. Consider the same assumptions and variables as in Theorem 3.1. If furthermore c1 < 2
+β,
+then there exists C3 > 0 such that for all t ≥ 0,
+∥x(t) − xN(t)∥ ≤ C0e− t
+β ε0∥ ˙x0∥ + C3
+�
+ε(t).
+Proof of Corollary 3.6. For all t ≥ 0, define J(t) =
+� t
+0 e
+s
+β �
+ε(s) ds. An integration by parts yields
+J(t) =
+�
+βe
+s
+β �
+ε(s)
+�t
+s=0 −
+� t
+s=0
+βe
+s
+β
+ε′(s)
+2
+�
+ε(s)
+ds = βe
+t
+β �
+ε(t)−βε0 +
+� t
+s=0
+βe
+s
+β −ε′(s)
+2ε(s)
+�
+ε(s) ds. (7)
+By assumption, 0 ≤ c1 < 2
+β such that for all s > 0, |ε′(s)| ≤ c1ε(s), which in our setting is equivalent
+to −ε′(s)
+ε(s) ≤ c1. So we deduce from (7) that
+J(t) ≤ βe
+t
+β �
+ε(t) + c1
+β
+2
+� t
+s=0
+e
+s
+β �
+ε(s) ds = βe
+t
+β �
+ε(t) + c1
+β
+2 J(t).
+So,
+�
+1 − c1
+β
+2
+�
+J(t) ≤ βet�
+ε(t). By assumption 1 − c1
+β
+2 > 0, therefore, J(t) ≤
+2
+2−c1βe
+t
+β �
+ε(t).
+Finally, using this in (1) and setting C3 = C1 + C2
+2
+2−c1β, we obtain the result.
+Remark 3.7. The above proofs suggest that an extension to the case where f is non-smooth but strongly
+convex is possible using regularization techniques. This is left for future work.
+So far our results only cover the case where α is “not too large” w.r.t. ε, and do not study (LM). We
+now state a more general result that covers these cases.
+9
+
+3.2
+Generalization to Arbitrary Viscous Dampings with Sub-exponential Decay
+This time we do not assume a link between ε and α but only sub-exponential decays.
+Assumption 2. There exists c1, c2 ≥ 0 such that for all t ≥ 0, |ε′(t)| ≤ c1ε(t) and |α′(t)| ≤ c2α(t).
+We are now in position to state the main result of this section.
+Theorem 3.8. Let xN and xLM be the solution of (CN) and (LM) respectively, and let c1, c2 ≥ 0.
+There exist constants C, ˜C ≥ 0, depending on c1, c2, such that for all ε and α for which Assumption 2
+holds with c1 and c2, the corresponding solution x of (VM-DIN-AVD) is such that for all t ≥ 0,
+∥x(t) − xN(t)∥ ≤ C
+�
+e− t
+β +
+�
+ε(t) + α(t) +
+� t
+s=0
+e
+1
+β (s−t)(
+�
+ε(s) + α(s)) ds
+�
+,
+(8)
+and,
+∥x(t) − xLM(t)∥ ≤ ˜C
+�
+e− t
+β +
+�
+ε(t) + α(t) +
+� t
+s=0
+e
+1
+β (s−t)(
+�
+ε(s) + α(s)) ds
+�
+.
+(9)
+The proof is postponed to Appendix B. Although it follows a similar reasoning as that of Theorem 3.1,
+more involved estimates are needed.
+Let us comment on these results. The bound (8) generalizes Theorem 3.1, although the constant in-
+volved will, in general, be larger than those in (1) (see the proof of Theorem 3.8 in appendix). Theo-
+rem 3.8 allows for far more flexibility in the choice of ε and α in order to control x and make it possibly
+close to xN. The bound in (9) is the same as that in (8) (up to a constant), but this time w.r.t. xLM, thus
+connecting (VM-DIN-AVD) to (LM). We make the following two important remarks. First (9) involves
+α, suggesting that making x close to xLM requires not only ε but also α to vanish asymptotically. Ad-
+ditionally, Theorem 3.8 does not state to which of xN and xLM the solution of (VM-DIN-AVD) is the
+closest. It remains an open question to know whether one can make (9) independent of α, and to state
+to which trajectory x is the closest. Yet, the numerical experiments in Section 5 suggest that neither are
+possible. Indeed, we observe that for some functions f, x is sometimes closer to xN than to xLM, even
+when ε(t) ≤ α(t).
+Nevertheless, Theorem 3.8 answers the question asked in the introduction: yes, (VM-DIN-AVD) is
+really of second-order nature since it can be brought close to the second-order dynamics (CN) and
+(LM). Doing so, it benefits from the good properties of these methods, such as the robustness to bad
+conditioning, as previously illustrated on the right of Figure 1. This concludes the analysis from a
+control perspective. We will now derive an approximation of the solution x in order to study the impact
+that ε and α have on the speed of convergence of x to x⋆ compared to the speeds of convergence of xN
+and xLM.
+4
+Approximate Solutions and Asymptotic Analysis on Quadratic Functions
+We consider the particular case where f is a strongly-convex quadratic function in order to study the
+asymptotic behavior of (VM-DIN-AVD) w.r.t. (CN) and (LM). Quadratic functions are the prototypical
+example of strongly-convex functions. In particular, any strongly-convex function can be locally ap-
+proximated by a quadratic one around its minimizer, making the latter a good model for understanding
+the local behavior of dynamics. In this section, f is quadratic: f(y) = 1
+2∥Ay − b∥2
+2 for all y ∈ Rn,
+10
+
+where A ∈ Rn×n is symmetric positive definite and b ∈ Rn. Without loss of generality, we take b = 0,
+so that the unique minimum is x⋆ = 0.
+4.1
+Setting: the Special Case of Quadratic Functions
+Quadratic functions are particularly interesting in our setting since DIN-like ODEs take a simpler form
+(as observed in [10, 40]). Indeed, ∀y ∈ Rn, ∇f(y) = ATAy and ∇2f(y) = ATA. Since ∇2f(y) is
+independent of y we can rewrite (VM-DIN-AVD) in an eigenspace5 of ATA. That is, we can study the
+ODE coordinate-wise by looking at one-dimensional problems of the form
+ε(t)¨x(t) + (α(t) + βλ) ˙x(t) + λx(t) = 0,
+t ≥ 0.
+(Q1-VM-DIN-AVD)
+Here (and throughout what follows) λ > 0 denotes any eigenvalue of ATA and x: R+ → R now
+denotes the corresponding coordinate (function) of the solution of (VM-DIN-AVD) in an eigenspace
+of ATA. The dynamics (Q1-VM-DIN-AVD) is a linear second-order ODE in x with non-constant
+coefficients. Similarly, (LM) can be rewritten coordinate-wise as
+(α(t) + βλ) ˙xLM(t) + λxLM(t) = 0,
+t ≥ 0,
+(Q1-LM)
+where xLM : R+ → R, and (CN) becomes
+β ˙xN(t) + xN(t) = 0,
+t ≥ 0,
+(Q1-CN)
+where again, xN : R+ → R is one-dimensional. Observe in particular that (CN) and (LM) are now
+first-order linear ODEs, whose solutions have the closed forms, for all t ≥ 0,
+xN(t) = x0e− t
+β
+and
+xLM(t) = x0 exp
+�
+−
+� t
+0
+λ
+α(s) + βλ ds
+�
+.
+(10)
+Since the minimizer is x⋆ = 0, we directly see that xN converges exponentially fast to x⋆, with a rate
+independent of λ. The rate of xLM depends however on λ and how fast α vanishes.
+Unfortunately, except for some special choices of ε and α (see [10]), one cannot solve the second-order
+linear ODE (Q1-VM-DIN-AVD) in closed form in general. Additionally, it is hopeless to circumvent
+the difficulty by finding a closed form for ∇f(x), accordingly to what we did in Section 3, since here
+∇f(x) = λx. In order to study the speed of convergence of x despite not having access to a closed
+form, we will approximate it with a controlled error, via a method that we now present.
+4.2
+The Liouville–Green Method
+In what follows, we rely on the Liouville–Green method [30, 26], a technique for obtaining non-
+asymptotic approximations to solutions of linear second-order ODEs with non-constant coefficients.
+First, we give the intuition behind the method, following the presentation of [35]. Consider the differ-
+ential equation
+¨z(t) − r(t)z(t) = 0,
+t ≥ 0,
+(11)
+5This can be generalized to the case where AT A is only semi-definite by considering orthogonal projections on an
+eigenspace spanned by the positive eigenvalues of AT A.
+11
+
+where r is real-valued, positive, and twice continuously differentiable. Any linear second-order ODE
+can be reformulated in the form (11), see Lemma 4.5 below. Since for all t ≥ 0, r(t) ̸= 0, we can use
+the changes of variables τ =
+� t
+0
+�
+r(s) ds and w = r1/4z and show that w is solution to
+¨w(τ) − (1 + ψ(τ))w(τ) = 0,
+t ≥ 0,
+(12)
+where6 ψ(τ) = 4r(t)r′′(t)−5r′(t)2
+16r(t)3
+. The LG method consists in neglecting the term ψ(τ) in (12), which
+simply yields two approximate solutions ˆw1(τ) = eτ and ˆw2(τ) = e−τ. Expressing this in terms of z
+and t, we obtain
+ˆz1(t) = r(t)−1/4 exp
+�� t
+0
+�
+r(s) ds
+�
+and
+ˆz2(t) = r(t)−1/4 exp
+�� t
+0
+−
+�
+r(s) ds
+�
+.
+(13)
+Those are the LG approximations of the solutions of (11). They are formally valid on any interval
+[0, T], T > 0 when ψ is “not too large”, provided that √r is integrable on [0, T].
+Remark 4.1. There exists other (but less intuitive) ways to derive the approximations above, which
+allow for generalization to higher-order linear ODEs, see e.g., [14, Chapter 10].
+The advantage of this approach is the possibility to estimate the error made using (13) w.r.t. the true
+solutions of (11). This is expressed in the following theorem which gathers results from [15, 36, 42].
+Theorem 4.2 (Olver [35]). Let r: R+ → R be a real, positive, twice continuously differentiable func-
+tion, and define ϕ(t) = 4r(t)r′′(t)−5r′(t)2
+16r(t)5/2
+for all t ≥ 0. Then for any T > 0, the differential equation,
+¨z(t) − r(t)z(t) = 0,
+t ∈ [0, T],
+(14)
+has two real and twice continuously differentiable solutions defined for all t ∈ [0, T] by,
+z1(t) = 1 + δ1(t)
+r(t)1/4 exp
+�� t
+0
+�
+r(s) ds
+�
+and
+z2(t) = 1 + δ2(t)
+r(t)1/4 exp
+�
+−
+� t
+0
+�
+r(s) ds
+�
+,
+where |δ1(t)| ≤ exp
+�1
+2
+� t
+0
+|ϕ(s)| ds
+�
+− 1 and |δ2(t)| ≤ exp
+�
+−1
+2
+� T
+t
+|ϕ(s)| ds
+�
+− 1. If in addition
+� +∞
+0
+|ϕ(s)| ds < +∞, then the results above also hold for T = +∞.
+Remark 4.3. We make the following remarks regarding the above result.
+– Note that z1 and z2 in Theorem 4.2 are exact solutions to (14). The LG approximations ˆz1 and ˆz2
+are obtained by neglecting the unknown functions δ1 and δ2 in z1 and z2. The theorem gives a non-
+asymptotic bound for the errors |z1(t) − ˆz1(t)| and |z2(t) − ˆz2(t)|, t ≥ 0.
+– Since we assumed r to be twice continuously differentiable and positive, ϕ is continuous, so it is
+integrable except maybe for t → +∞.
+– For the sake of simplicity, the formulation of Theorem 4.2 slightly differs from that in [35], the original
+formulation can be recovered by a change of variable.
+6We express ψ(τ) using t for the sake of readability, using the one-to-one correspondence between τ and t.
+12
+
+4.3
+Liouville–Green Approximation of (VM-DIN-AVD)
+We now proceed to make use of the LG method for approximating the solutions of (Q1-VM-DIN-AVD).
+The reader only interested in the result can jump directly to the Section 4.4. We first make the following
+assumption.
+Assumption 3. The functions α and ε are three times continuously differentiable, and ε0 is such that
+∀t ≥ 0, ε0 <
+(βλ)2
+2|α′(t)|+4λ.
+Remark 4.4. The condition on ε0 in Assumption 3 is only technical, so that r defined below is positive.
+It can be easily satisfied since |α′(t)| is uniformly bounded. Indeed, α is non-increasing and non-
+negative (see Section 1.1), from which one can deduce that
+� +∞
+0
+|α′(s)| ds ≤ α0.
+We now rewrite (Q1-VM-DIN-AVD) in the form (14).
+Lemma 4.5. Suppose that Assumption 3 holds, and let x be the solution of (Q1-VM-DIN-AVD). For
+all t ≥ 0, define
+p(t) = α(t) + βλ
+ε(t)
+and
+r(t) = p(t)2
+4
++ p′(t)
+2
+−
+λ
+ε(t).
+(15)
+Then, p and r are twice continuously differentiable, r is positive and the function y defined for all t ≥ 0
+by y(t) = x(t) exp
+�� t
+0
+p(s)
+2 ds
+�
+is a solution to
+¨y(t) − r(t)y(t) = 0,
+t ≥ 0,
+(16)
+with initial condition (y(0), ˙y(0)) = (x0, ˙x0 + p(0)
+2 x0).
+Proof. We first check that for all t ≥ 0, r(t) is positive. Let t > 0,
+r(t) > 0 ⇐⇒ (α(t) + βλ)2
+4ε(t)2
++ α′(t)
+2ε(t) − (α(t) + βλ)ε′(t)
+ε(t)2
+−
+λ
+ε(t) > 0.
+(17)
+Since ε′(t) ≤ 0 and α′(t) ≤ 0, one can check that a sufficient condition for (17) to hold is,
+r(t) > 0 ⇐= (α(t) + βλ)2
+4
+>
+�|α′(t)|
+2
++ λ
+�
+ε(t) ⇐=
+(βλ)2
+2|α′(t)| + 4λ > ε0.
+So under Assumption 3, for all t ≥ 0, r(t) > 0. We then check that y is indeed solution to (16). Let
+t > 0,
+˙y(t) = p(t)
+2 x(t) exp
+�� t
+0
+p(s)
+2
+ds
+�
++ ˙x(t) exp
+�� t
+0
+p(s)
+2
+ds
+�
+,
+and
+¨y(t) = exp
+�� t
+0
+p(s)
+2
+ds
+� ��p(t)2
+4
++ p′(t)
+2
+�
+x(t) + p(t) ˙x(t) + ¨x(t)
+�
+.
+Since x solves (Q1-VM-DIN-AVD), it holds that ¨x(t) = −p(t) ˙x(t) −
+λ
+ε(t)x(t), so,
+¨y(t) = exp
+�� t
+0
+p(s)
+2
+ds
+� �p(t)2
+4
++ p′(t)
+2
+−
+λ
+ε(t)
+�
+x(t)
+=
+�p(t)2
+4
++ p′(t)
+2
+−
+λ
+ε(t)
+�
+y(t) = r(t)y(t).
+13
+
+Lemma 4.5 gives a reformulation of (Q1-VM-DIN-AVD) suited to apply Theorem 4.2. To use the
+theorem for all t ≥ 0, we need to ensure that ϕ(t) = 4r(t)r′′(t)−5r′(t)2
+16r(t)5/2
+is integrable. To this aim we make
+the following assumption.
+Assumption 4. The functions ε and α have first, second and third-order derivatives that are integrable
+on [0, +∞[. In addition, lim
+t→∞ ε(t) = 0 and ε′(t)2/ε(t) is integrable on [0, +∞[.
+Remark 4.6. Assumption 4 holds for most decays used in practice, with in particular any polynomial
+decay of the form
+ε0
+(t+1)a and
+α0
+(t+1)b, a ∈ N \ {0} and b ∈ N. Note that ε and α need not be integrable
+and α can even be constant.
+The next lemma states the integrability of ϕ on [0, +∞[.
+Lemma 4.7. Under Assumption 3 and 4,
+� +∞
+0
+|ϕ(s)| ds < +∞.
+The proof of this lemma, relies on relatively simple arguments but involves long computations and is
+thus postponed to Appendix C. We can now use Theorem 4.2 to obtain an exact form for the solution
+of (Q1-VM-DIN-AVD) based on the LG approximations.
+Theorem 4.8. Suppose that Assumptions 3 and 4 hold. There exists A, B ∈ R such that x(0) = x0,
+˙x(0) = ˙x0 and for all t ≥ 0, the solution of (Q1-VM-DIN-AVD) is
+x(t) = A1 + δ1(t)
+r(t)1/4
+�
+α(t) + βλ
+�
+ε(t)
+exp
+�� t
+0
+−
+λ
+α(s) + βλ −
+λ2ε(s)
+(α(s) + βλ)3 + o(ε(s)) ds
+�
++B 1 + δ2(t)
+r(t)1/4
+�
+ε(t)
+�
+α(t) + βλ
+exp
+�� t
+0
+−α(s) + βλ
+ε(s)
++
+λ
+α(s) + βλ +
+λ2ε(s)
+(α(s) + βλ)3 + o(ε(s)) ds
+�
+,
+(18)
+where for all t ≥ 0,
+|δ1(t)| ≤ exp
+�1
+2
+� t
+0
+|ϕ(s)| ds
+�
+− 1
+and
+|δ2(t)| ≤ exp
+�
+−1
+2
+� +∞
+t
+|ϕ(s)| ds
+�
+− 1.
+(19)
+Thanks to the bounds (19), we now have an approximation of x. We will use it in particular to compare
+x asymptotically to the solutions of (Q1-LM) and (Q1-CN). Before this, we prove Theorem 4.8.
+Proof of Theorem 4.8. Let x be the solution of (Q1-VM-DIN-AVD) define p, r as in (15). Let us also
+define y(t)
+def
+= x(t) exp
+�� t
+0
+p(s)
+2 ds
+�
+. According to Lemma 4.5, r is positive and y is solution to (16).
+Then, from Lemma 4.7,
+� T
+t |ϕ(s)| ds < +∞, so we can apply Theorem 4.2 to y on [0, +∞[. Therefore,
+there exists A, B ∈ R, such that ∀t ≥ 0,
+y(t) = A1 + δ1(t)
+r(t)1/4 exp
+�� t
+0
+�
+r(s) ds
+�
++ B 1 + δ2(t)
+r(t)1/4 exp
+�� t
+0
+−
+�
+r(s) ds
+�
+,
+where A and B are determined by the initial conditions, and δ1, δ2 are such that (19) holds.
+Going back to x(t) = y(t) exp
+�� t
+0 − p(s)
+2 ds
+�
+, we obtain that for all t ≥ 0,
+x(t) = A1 + δ1(t)
+r(t)1/4 exp
+�� t
+0
+−p(s)
+2
++
+�
+r(s) ds
+�
++ B 1 + δ2(t)
+r(t)1/4 exp
+�� t
+0
+−p(s)
+2
+−
+�
+r(s) ds
+�
+.
+(20)
+14
+
+It now remains to expand the terms in the two exponentials in (20) in order to obtain (18). To this aim,
+we approximate
+�
+r(s), let s ≥ 0,
+�
+r(s) = p(s)
+2
+�
+1 + 2p′(s)
+p(s)2 −
+4λ
+ε(s)p(s)2
+= p(s)
+2
+�
+1 + p′(s)
+p(s)2 −
+2λ
+ε(s)p(s)2 − 1
+8
+�2p′(s)
+p(s)2 −
+4λ
+ε(s)p(s)2
+�2
++ o(ε(s)2)
+�
+= p(s)
+2
++ p′(s)
+2p(s) −
+λ
+ε(s)p(s) − 1
+16
+� 2p′(s)
+p(s)3/2 −
+4λ
+ε(s)p(s)3/2
+�2
++ o(ε(s))
+= p(s)
+2
++
+p′(s)ε(s)
+2(α(s) + βλ) −
+λ
+α(s) + βλ − 1
+16
+�
+2p′(s)
+p(s)3/2 −
+4λ
+�
+ε(s)
+(α(s) + βλ)3/2
+�2
++ o(ε(s))
+= p(s)
+2
++
+α′(s)
+2(α(s) + βλ) − ε′(s)
+2ε(s) −
+λ
+α(s) + βλ − 1
+16
+�
+2p′(s)
+p(s)3/2 −
+4λ
+�
+ε(s)
+(α(s) + βλ)3/2
+�2
++ o(ε(s))
+(21)
+Focusing on the first exponential term in (20), we deduce from (21) that for all t ≥ 0,
+exp
+�� t
+0
+−p(s)
+2
++
+�
+r(s) ds
+�
+= exp
+�
+�
+� t
+0
+α′(s)/2
+α(s) + βλ − ε′(s)
+2ε(s) −
+λ
+α(s) + βλ − 1
+16
+�
+2p′(s)
+p(s)3/2 −
+4λ
+�
+ε(s)
+(α(s) + βλ)3/2
+�2
++ o(ε(s)) ds
+�
+�
+=
+�
+α(t) + βλ)
+√α0 + βλ
+√ε0
+�
+ε(t)
+exp
+�
+�
+� t
+0
+−λ
+α(s) + βλ − 1
+16
+�
+2p′(s)
+p(s)3/2 −
+4λ
+�
+ε(s)
+(α(s) + βλ)3/2
+�2
++ o(ε(s)) ds
+�
+�
+=
+�
+α(t) + βλ)
+√α0 + βλ
+√ε0
+�
+ε(t)
+exp
+�� t
+0
+−λ
+α(s) + βλ −
+λ2ε(s)
+(α(s) + βλ)3 + o(ε(s)) ds
+�
+,
+where the last line relies on further computations postponed to Lemma C.1 in Appendix C. Performing
+the exact same type of computations on exp
+�� t
+0 − p(s)
+2 −
+�
+r(s) ds
+�
+, and up to redefining A and B so
+as to encompass all the constants, we obtain (18) and the result is proved.
+4.4
+Comparison of x with xLM and xN
+We now have an expression for x which is almost explicit: we do not know δ1 and δ2 in closed form,
+but they are uniformly bounded (by Lemma 4.7). We will now compare the asymptotic behavior of (18)
+with those of the solutions of (Q1-LM) and (Q1-CN) that we denoted xLM and xN respectively. Our
+main result of Section 4 is the following, where ∼+∞ denotes the asymptotic equivalence7 between two
+functions as t → ∞.
+7Two real-valued functions g1 and g2 are asymptotically equivalent in +∞ if and only if limt→∞
+g1(t)
+g2(t) = 1.
+15
+
+Theorem 4.9. Let x be the solution of (Q1-VM-DIN-AVD), given in (18), and xLM and xN whose
+closed forms are stated in (10). Under Assumptions 3 and 4, there exists C > 0 such that the following
+asymptotic equivalences hold:
+x(t) ∼+∞ xLM(t)C exp
+�� t
+0
+−
+λ2ε(s)
+(α(s) + βλ)3 + o(ε(s)) ds
+�
+,
+and
+x(t) ∼+∞ xN(t)C exp
+�� t
+0
+α(s)
+β(α(s) + βλ) −
+λ2ε(s)
+(α(s) + βλ)3 + o(ε(s)) ds
+�
+.
+(22)
+As a consequence, the convergence of x to x⋆ is:
+(i) Faster than that of xLM if ε is non-integrable and as fast otherwise.
+(ii) Slower than that of xN if α is non-integrable and as fast if α is integrable, in the case where
+∀t ≥ 0, α(t) > ε(t).
+(iii) Faster than that of xN if ε is non-integrable and as fast if ε is integrable, in the case where
+∀t ≥ 0, α(t) < ε(t).
+While the results of Section 3 were related to the closeness of (VM-DIN-AVD) w.r.t. (CN) and (LM)
+from a control perspective, Theorem 4.9 provides a different type of insight. First, the results are
+asymptotic, so they only allow to control (VM-DIN-AVD) for large t. They provide however a clear
+understanding of the nature of the solutions of (VM-DIN-AVD) and their convergence. The conclusions
+(summarized in Table 1) are in accordance with what we would expect: when the viscous damping is
+larger than the variable mass, (VM-DIN-AVD) behaves more like the Levenberg–Marquardt method
+than the Newton one, but it actually becomes an accelerated Levenberg–Marquardt dynamics when ε
+is non-integrable but vanishing. However, when the variable mass ε is larger than α, the dynamics is
+closer to the one of the Newton method, and can actually be an accelerated Newton dynamics, again for
+non-integrable ε. This is analogous to the necessary condition that α must be non-integrable in order
+to accelerate first-order methods in convex optimization (see [3]). We conclude this section by proving
+Theorem 4.9.
+Proof of Theorem 4.9. Thanks to Assumptions 3 and 4, Theorem 4.8 tells us that x has the form (18).
+We now analyze the two terms in (18).
+First, we know from Theorem 4.8 that δ1(0) = 0 and limt→+∞ δ2(t) = 0. In addition, by Lemma 4.7,
+δ1 and δ2 are uniformly bounded by some positive constant. Then r(t)−1/4 decays asymptotically like
+�
+ε(t) and α is bounded. So A 1+δ1(t)
+r(t)1/4
+√
+α(t)+βλ
+√
+ε(t)
+is asymptotically equivalent to some constant c1 ∈ R as
+t → +∞. Similarly, B 1+δ2(t)
+r(t)1/4
+√
+ε(t)
+√
+α(t)+βλ is equivalent to c2ε(t), with c2 ∈ R.
+We now analyze the “exponential factors” in (18). On the one hand,
+λ
+α(s)+βλ +
+λ2ε(s)
+(α(s)+βλ)3 + o(ε(s))
+converges to 1
+β as s → ∞, while α(s)+βλ
+ε(s)
+diverges to +∞. Therefore, we deduce that,
+exp
+�� t
+0
+−α(s) + βλ
+ε(s)
++
+λ
+α(s) + βλ +
+λ2ε(s)
+(α(s) + βλ)3 + o(ε(s)) ds
+�
+= o
+�
+exp
+�� t
+0
+−
+λ
+α(s) + βλ −
+λ2ε(s)
+(α(s) + βλ)3 + o(ε(s)) ds
+��
+.
+16
+
+As a consequence, the second term in (18) will decrease to 0 faster than the first-one (let alone the
+additional ε(t) decay that we have just discussed). The asymptotic behavior of x will thus be governed
+by the first term in (18).
+Let us now focus on the first term in (18). Observe that exp
+�� t
+0 −
+λ
+α(s)+βλ ds
+�
+is exactly the decay
+of xLM in (10). Thus, we have proved that there exists C > 0, such that the following asymptotic
+equivalence holds,
+A1 + δ1(t)
+r(t)1/4
+�
+α(t) + βλ
+�
+ε(t)
+exp
+�� t
+0
+−
+λ
+α(s) + βλ −
+λ2ε(s)
+(α(s) + βλ)3 + o(ε(s)) ds
+�
+∼+∞ xLM(t)C exp
+�� t
+0
+−
+λ2ε(s)
+(α(s) + βλ)3 + o(ε(s)) ds
+�
+,
+which proves the first part of (22). The second equivalence in (22) is obtained using the following
+identity,
+� t
+0
+−
+λ
+α(s) + βλ ds =
+� t
+0
+− 1
+β +
+α(s)
+β(α(s) + βλ) ds = − t
+β +
+� t
+0
+α(s)
+β(α(s) + βλ) ds
+(23)
+and e−t/β is precisely the rate at which xN decreases. So (22) holds.
+It finally remains to deduce the conclusions of the theorem from (22).
+– Regarding the comparison with xLM, the integral
+� t
+0 −
+λ2ε(s)
+(α(s)+βλ)3 + o(ε(s)) ds converges if and only
+if ε is integrable on [0, +∞[, and diverges to −∞ when ε is not. So x converges to 0 at least as fast as
+xLM and faster when ε is not integrable.
+– As for the comparison with xN, if α(s) > ε(s) ≥ 0 for all s ≥ 0, then the integral
+� t
+0
+α(s)
+β(α(s)+βλ) −
+λ2ε(s)
+(α(s)+βλ)3 + o(ε(s)) ds is convergent in +∞ if and only if α is integrable and diverges to +∞ when
+α is non-integrable. So when α is integrable, the speed of convergence of x is the same as that of
+xN. When α is not integrable, the convergence to 0 is slower but still holds. Indeed, for all s ≥ 0
+α(s)
+β(α(s)+ 1
+β ) <
+α(s)
+βα(s) = 1
+β. Thus for all t > 0, − t
+β +
+� t
+0
+α(s)
+β(α(s)+βλ) ds < 0.
+– Finally, the comparison with xN in the case ε(s) > α(s) is exactly the same as the comparison with
+xLM using (23).
+5
+Numerical Experiments
+We present two set of experiments that illustrate our main results from Sections 3 and 4. We first detail
+the general methodology.
+5.1
+Methodology
+We compare the solutions of (CN), (LM) and (VM-DIN-AVD) obtained for strongly-convex functions
+in dimension n = 100. Since closed-form solutions are not available, they are estimated via discretiza-
+tion schemes with small step-sizes γ = 10−1. We used Euler semi-explicit schemes, where a linear
+system is solved at each iteration, for the sake of stability. The resulting algorithms are detailed in
+Appendix D.
+17
+
+0
+10
+20
+30
+40
+50
+t
+10
+20
+10
+17
+10
+14
+10
+11
+10
+8
+10
+5
+10
+2
+101
+Distance to CN
+(t) =
+0/log t,
+(t) =
+0/t
+(t) =
+0/ t,
+(t) =
+0/t
+(t) =
+0/t,
+(t) =
+0/t
+(t) =
+0/t2,
+(t) =
+0/t
+(t) =
+0e
+t/ ,
+(t) =
+0/t
+0
+10
+20
+30
+40
+50
+t
+Distance to LM
+0
+10
+20
+30
+40
+50
+t
+Speed of convergence
+Levenberg-Marquardt
+Continuous Newton
+0
+10
+20
+30
+40
+50
+t
+10
+21
+10
+17
+10
+13
+10
+9
+10
+5
+10
+1
+Distance to CN
+(t) =
+0/log t,
+(t) =
+0/t2
+(t) =
+0/ t,
+(t) =
+0/t2
+(t) =
+0/t,
+(t) =
+0/t2
+(t) =
+0/t2,
+(t) =
+0/t2
+(t) =
+0e
+t/ ,
+(t) =
+0/t2
+0
+10
+20
+30
+40
+50
+t
+Distance to LM
+0
+10
+20
+30
+40
+50
+t
+Speed of convergence
+Levenberg-Marquardt
+Continuous Newton
+Figure 2:
+Comparison of the solutions xN, xLM and x of (CN), (LM) and (VM-DIN-AVD) respec-
+tively, for a strongly-convex function of the form f(x) = e−∥x∥2 + 1
+2∥Ax∥2. Left figures: distance
+∥x(t) − xN(t)∥ versus time t, each curve corresponds to a different choice of ε; middle figures: dis-
+tance ∥x(t) − xLM(t)∥, again for several ε. Right figures: distance to the optimum x⋆ for reference,
+xN and xLM are in dotted and dashed lines, other curves correspond to (VM-DIN-AVD) for several
+choices of ε. The brown curve is often hidden behind the purple (and sometimes the pink) curve. Top
+and bottom rows show results respectively for non-integrable and integrable viscous dampings α. The
+theoretical bounds from Theorem 3.8 are only displayed on Figure 4 below, for the sake of readability.
+5.2
+First Experiment: Distance between Trajectories
+We begin with an empirical validation of the results of Section 3 on the distance between x, xLM
+and xN. Each of Figures 2, 3 and 4 (as well as Figures 6 in Appendix D) corresponds to a different
+strongly-convex function, specified below its corresponding figure. In order to ensure strong convexity,
+each function contains a quadratic term of the form ∥Ax∥2, where A is symmetric positive definite.
+Several observations can be made from the numerical results, but we first note on the right plots of
+Figures 2 to 4 that xN always converges asymptotically linearly (i.e., exponentially fast). This is also
+the case for x and xLM in some (but not all) cases. This is important because ∥x(t) − xN(t)∥ ≤
+∥x(t) − x⋆∥ + ∥xN(t) − x⋆∥, so if both x and xN converge linearly, then the bounds of Theorems 3.1
+and 3.8 need not be tight asymptotically. That being said, the strength of these results is to be non-
+asymptotic and this is highlighted by the experiments as we now explain.
+Looking at the left and middle plots of Figures 2, 3 and 4, we observe that Theorems 3.1 and 3.8 seem
+empirically validated, since the distances ∥x(t) − xN(t)∥ and ∥x(t) − xLM(t)∥ decrease relatively fast
+to zero. Again, when x converges rapidly to x⋆ this is not very insightful, however, the main interest
+18
+
+0
+20
+40
+60
+80
+100
+120
+140
+t
+10
+64
+10
+55
+10
+46
+10
+37
+10
+28
+10
+19
+10
+10
+10
+1
+Distance to CN
+(t) =
+0/log t,
+(t) =
+0/t
+(t) =
+0/ t,
+(t) =
+0/t
+(t) =
+0/t,
+(t) =
+0/t
+(t) =
+0/t2,
+(t) =
+0/t
+(t) =
+0e
+t/ ,
+(t) =
+0/t
+0
+20
+40
+60
+80
+100
+120
+140
+t
+Distance to LM
+0
+20
+40
+60
+80
+100
+120
+140
+t
+Speed of convergence
+Levenberg-Marquardt
+Continuous Newton
+0
+20
+40
+60
+80
+100
+120
+140
+t
+10
+64
+10
+55
+10
+46
+10
+37
+10
+28
+10
+19
+10
+10
+10
+1
+Distance to CN
+(t) =
+0/log t,
+(t) =
+0/t2
+(t) =
+0/ t,
+(t) =
+0/t2
+(t) =
+0/t,
+(t) =
+0/t2
+(t) =
+0/t2,
+(t) =
+0/t2
+(t) =
+0e
+t/ ,
+(t) =
+0/t2
+0
+20
+40
+60
+80
+100
+120
+140
+t
+Distance to LM
+0
+20
+40
+60
+80
+100
+120
+140
+t
+Speed of convergence
+Levenberg-Marquardt
+Continuous Newton
+Figure 3: Similar experiment and figures as those described in Figure 2, but for the function f(x) =
+log (�n
+i=1 exi + e−xi) + 1
+2∥Ax∥2.
+of our theorems appears on the left of Figures 3 and 4: the blue and green curves, corresponding to
+slowly decaying choices of ε, converge more slowly than other trajectories. However, when taking
+faster decays, we recover fast convergence and closeness to xN (this is particularly true for the purple
+curve). Very similar observations are made w.r.t. xLM on the middle plots. Despite not being stated in
+the theorems of Section 3, the experiments match the intuition that when ε > α, x may be closer to
+xN and when ε ≤ α, x would rather be closer to xLM. This is more noticeable on the top rows of the
+figures, where α is not integrable.
+Figure 4 suggests that the bounds in Theorem 3.8 are rather tight for small t, since, for example, the
+blue and green curves on the left show a relatively slow vanishing of ∥x(t)−xN(t)∥ for slowly decaying
+ε. The experiments show that the bounds seem however often too pessimistic for large t, for which the
+second part of our study provides better insights (see Section 4 and below). Interestingly, slow decays
+of ε may sometimes result in faster convergence for x than fast decays (and also faster convergence
+than xLM), notably on Figure 2. We also note that ε(t) = ε0/t combined either with α(t) = α0/t or
+α(t) = α0/t2 seems to always yield fast convergence on these experiments (and sometimes the fastest
+of all dynamics).
+5.3
+Second Experiment: Empirical Validation of Theorem 4.9
+We now turn our attention to the solutions x, xN and xLM for a quadratic function of the form f(y) =
+1
+2∥Ay∥2, y ∈ Rn, and for several choices of ε and α. The results in Figure 5 exactly match the expected
+behavior summarized in Table 1. Indeed, looking first at the right-hand side of Figure 5, x is as fast
+19
+
+0
+20
+40
+60
+80
+100
+120
+140
+t
+10
+46
+10
+39
+10
+32
+10
+25
+10
+18
+10
+11
+10
+4
+103
+Distance to CN
+(t) =
+0/log t,
+(t) =
+0/t
+(t) =
+0/ t,
+(t) =
+0/t
+(t) =
+0/t,
+(t) =
+0/t
+(t) =
+0/t2,
+(t) =
+0/t
+(t) =
+0e
+t/ ,
+(t) =
+0/t
+0
+20
+40
+60
+80
+100
+120
+140
+t
+Distance to LM
+0
+20
+40
+60
+80
+100
+120
+140
+t
+Speed of convergence
+Levenberg-Marquardt
+Continuous Newton
+0
+20
+40
+60
+80
+100
+120
+140
+t
+10
+59
+10
+50
+10
+41
+10
+32
+10
+23
+10
+14
+10
+5
+104
+Distance to CN
+(t) =
+0/log t,
+(t) =
+0/t2
+(t) =
+0/ t,
+(t) =
+0/t2
+(t) =
+0/t,
+(t) =
+0/t2
+(t) =
+0/t2,
+(t) =
+0/t2
+(t) =
+0e
+t/ ,
+(t) =
+0/t2
+0
+20
+40
+60
+80
+100
+120
+140
+t
+Distance to LM
+0
+20
+40
+60
+80
+100
+120
+140
+t
+Speed of convergence
+Levenberg-Marquardt
+Continuous Newton
+Figure 4: Similar experiment and figures as those described in Figure 2, but for the function f(x) =
+∥x∥50 + 1
+2∥Ax∥2. The thin “dash dotted” curves represent the theoretical bounds from Theorem 3.8 for
+each choice of (ε, α) considered.
+as the corresponding8 xLM when ε is not integrable and regardless of α, and x is faster when ε is non-
+integrable. Then on the left-hand side, when comparing to xN, x is slower in settings where α is larger
+than ε and non-integrable (red curves), or almost as fast when α is integrable (pink curve). However,
+acceleration w.r.t. to xN is indeed achieved for non-integrable ε regardless of α (first-two blue curves),
+and the rate is the same as that of xN when ε is integrable (third blue curve).
+6
+Conclusions and Perspectives
+We introduced a general ODE (VM-DIN-AVD) featuring variable mass, and provided a deep under-
+standing on the behavior of its solutions w.r.t. time dependent control parameters ε and α, both, asymp-
+totically and non-asymptotically. We can conclude that (VM-DIN-AVD) is indeed of (regularized)
+Newton type, since it can be controlled to be close to both (CN) and (LM). Yet we also showed
+that (VM-DIN-AVD) fundamentally differs from the other two dynamics in its nature. In particular,
+Theorem 4.9 and the numerical experiments emphasized that ε and α can accelerate (or slow down)
+(VM-DIN-AVD) w.r.t. (CN) and (LM). We also note that our bounds in Theorems 3.1 and 3.8 seem
+relatively tight, in particular for functions with large gradients (see Figure 4). Our contribution yields
+a complete and satisfying picture on the relation between the three systems, which was only partially
+understood. We believe that our results build a strong foundation for the development of algorithms
+that combine the best properties of first- and second-order optimization methods.
+8That is, the solution of (LM) for the same α as that considered for (VM-DIN-AVD).
+20
+
+10
+20
+30
+40
+50
+t
+10
+21
+10
+18
+10
+15
+10
+12
+10
+9
+10
+6
+10
+3
+100
+x(t)
+x
+Continuous Newton
+(t) =
+0/ t,
+(t) =
+0/t
+(t) =
+0/t,
+(t) =
+0/t2
+(t) =
+0/t3/2,
+(t) =
+0/t2
+(t) =
+0/t,
+(t) =
+0/ t
+(t) =
+0/t2,
+(t) =
+0/t
+(t) =
+0/t2,
+(t) =
+0/t3/2
+10
+20
+30
+40
+50
+t
+10
+21
+10
+18
+10
+15
+10
+12
+10
+9
+10
+6
+10
+3
+100
+x(t)
+x
+LM: 
+0/t
+(t) =
+0/ t,
+(t) =
+0/t
+(t) =
+0/t2,
+(t) =
+0/t
+LM: 
+0/t2
+(t) =
+0/ t,
+(t) =
+0/t2
+(t) =
+0/t2,
+(t) =
+0/t2
+Figure 5: Numerical validation of Theorem 4.9: distance to the optimum x⋆ as a function of time. on
+a quadratic function f(x) = 1
+2∥Ax∥2. Left: speed comparison w.r.t. (CN) for several choices of ε and
+α. Right: Comparison with LM in for α integrable or not and several choices of ε. Shades of blue
+represent cases where ε(t) > α(t) while shades of red represent the opposite setting.
+As for future work, we showed that (VM-DIN-AVD) is promising from an optimization perspective.
+So far we approximated solutions of (VM-DIN-AVD) via schemes that required solving a linear system
+at each iteration (this is also true for (CN) and (LM)). Our new understanding on (ε, α) paves the way
+towards designing new Newton-like algorithms with a significantly reduced computational cost, which
+is crucial for large-scale optimization. Another open question is whether it is possible to preserve the
+properties evidenced in this work when ε is defined in a closed-loop manner (formally depending on
+x rather than on t). Finally, it would be worth investigating how the current work can be extended to
+general convex and/or non-smooth functions.
+Acknowledgment
+C. Castera, J. Fadili and P. Ochs are supported by the ANR-DFG joint project TRINOM-DS under
+number ANR-20-CE92-0037-01. The numerical experiments were made thanks to the development
+teams of the following libraries: Python [38], Numpy [43] and Matplotlib [27].
+21
+
+Appendices
+A
+Equivalent First-order System and Global Existence of Solutions
+A.1
+First-order Equivalent Formulation
+We reformulate (VM-DIN-AVD) as a system of ODE involving only first-order time derivatives and
+the gradient of f. For this purpose, notice that for all t > 0 (VM-DIN-AVD) can be rewritten as,
+d
+dt [ε(t) ˙x(t)] + β d
+dt∇f(x(t)) + α(t) ˙x(t) − ε′(t) ˙x(t) + ∇f(x(t)) = 0,
+t ≥ 0.
+(24)
+We then integrate (24) for all t ≥ 0,
+ε(t) ˙x(t) + β∇f(x(t)) − ε0 ˙x0 − β∇f(x0) +
+� t
+0
+(α(s) − ε′(s)) ˙x(s) + ∇f(x(s)) ds = 0.
+(25)
+For all t ≥ 0, we define the variable,
+z(t) =
+� t
+0
+(α(s) − ε′(s)) ˙x(s) + ∇f(x(s)) ds − ε0 ˙x0 − β∇f(x0).
+We differentiate z, for all t > 0, ˙z(t) = (α(t) − ε′(t)) ˙x(t) + ∇f(x(t)), so that we can rewrite (25) as,
+�
+ε(t) ˙x(t) + β∇f(x(t)) + z(t) = 0
+˙z(t) − (α(t) − ε′(t)) ˙x(t) − ∇f(x(t)) = 0
+,
+t ≥ 0.
+We substitute the first line in the second-one,
+�
+ε(t) ˙x(t) + β∇f(x(t)) + z(t) = 0
+β ˙z(t) − β(α(t) − ε′(t) − 1
+βε(t)) ˙x(t) + z(t) = 0
+,
+t ≥ 0.
+(26)
+To ease the readability, we recall the notation ν(t) = α(t) − ε′(t) − 1
+βε(t) from Section 2. Then define
+for all t ≥ 0, y(t) = z(t) − ν(t)x(t), and differentiate, ˙y(t) = ˙z(t) − ν(t) ˙x(t) − ν′(t)x(t). We finally
+rewrite (26) as,
+�
+ε(t) ˙x(t) + β∇f(x(t))
++ν(t)x(t) + y(t) = 0
+˙y(t) + ν′(t)x(t)
++ ν(t)
+β x(t) + 1
+βy(t) = 0 .
+which is (gVM-DIN-AVD). Finally, the initial condition on y is
+y(0) = z(0) − ν(0)x(0) = −ε0 ˙x0 − β∇f(x0) − (α0 − ε′
+0 − 1
+β ε0)x0.
+Remark A.1. Notice that the quantity ν(t) = α(t) − ε′(t) − 1
+βε(t) involved in (gVM-DIN-AVD) also
+plays a key role in our analysis of Section 3, see e.g., (6). In particular the sign of ν(t) changes the
+nature of (VM-DIN-AVD) and is related to Assumption 1.
+22
+
+A.2
+Local Solutions are Global
+Using the formulation (gVM-DIN-AVD), we proved local existence and uniqueness of solutions of
+(VM-DIN-AVD) in Section 2. Using the same notations, we justify that the local solution (x, y) actually
+exists globally. According to Lemma 3.4, the Lyapunov function U(t) = ε(t)
+2 ∥ ˙x(t)∥2 +f(x(t))−f(x⋆)
+is non-negative and decreasing. Thus, it is uniformly bounded on R+ and the same holds for t �→
+f(x(t)) since for all t ≥ 0, U(t) ≥ f(x(t)). Then, f is coercive by assumption, so x is uniformly
+bounded on R+ (otherwise f(x) could not remain bounded). We now prove that y is also uniformly
+bounded. From (gVM-DIN-AVD), for all t > 0, ˙y(t) = − 1
+βy(t) − ( ν(t)
+β + ν′(t))x(t) so we can use the
+following integrating factor,
+y(t) = e− t
+β y0 − e− t
+β
+� t
+0
+1
+β e
+s
+β (ν(s) + βν′(s))x(s) ds.
+Using triangle inequalities, for all t ≥ 0,
+∥y(t)∥ ≤ e− t
+β ∥y0∥ + sup
+s≥0
+∥(ν(s) + βν′(s))x(s)∥e− t
+β
+� t
+0
+1
+β e
+s
+β ds ≤ ∥y0∥ + sup
+s≥0
+∥(ν(s) + βν′(s))x(s)∥.
+(27)
+Using the definition of ε and α from Sections 1.1 and 2, observe that ε, α, ε′ and α′ are all bounded
+on R+, and ε′′ is assumed to be bounded. So ν and ν′ are bounded, and since we also proved that x
+is uniformly bounded on R+, we deduce from (27) that y is uniformly bounded as well. Hence, the
+unique local solution (x, y) is global.
+B
+Proof of Theorem 3.8
+This section is devoted to proving the general result of Section 3. Fix some constants c1, c2 > 0 and let
+ε and α such that Assumption 2 is satisfied with these constants. Let x be the corresponding solution
+of (VM-DIN-AVD), xN, and xLM that of (CN) and (LM), respectively. Following the same arguments
+as in the beginning of the proof of Theorem 3.1, for all t ≥ 0, x(t), xN(t) and xLM(t) belong to the
+bounded set K0 defined in that proof. Since f is µ-strongly convex on K0, the proof relies again on
+bounding difference of gradients, indeed, for all t ≥ 0,
+∥x(t) − xN(t)∥ ≤ 1
+µ∥∇f(x(t)) − ∇f(xN(t))∥ and ∥x(t) − xLM(t)∥ ≤ 1
+µ∥∇f(x(t)) − ∇f(xLM(t))∥.
+(28)
+Recall also that the closed form of ∇f(xN) is given in (4).
+Expressing ∇f(x).
+We follow the exact same steps as in the proof of Theorem 3.1 to obtain the
+expression of ∇f(x) given in (5), which we recall, for all t ≥ 0,
+β∇f(x(t)) = βe− t
+β ∇f(x0) + e− t
+β ε0 ˙x0 − ε(t) ˙x(t) +
+� t
+0
+e
+s−t
+β
+� 1
+β ε(s) + ε′(s) − α(s)
+�
+˙x(s) ds.
+23
+
+Here we do not assume any relation between ε and α, and we thus need to find a more suitable expres-
+sion for ∇f(x(t)). We first expand the terms in the integral, for all t ≥ 0,
+β∇f(x(t)) = βe− t
+β ∇f(x0) + e− t
+β ε0 ˙x0 − ε(t) ˙x(t)
++
+� t
+0
+e
+s−t
+β
+� 1
+β ε(s) + ε′(s)
+�
+˙x(s) ds −
+� t
+0
+e
+s−t
+β α(s) ˙x(s) ds.
+(29)
+Then, for all s ≥ 0, we have the identity,
+e
+s
+β ˙x(s) = e
+s
+β ˙x(s) + 1
+β e
+s
+β x(s) − 1
+β e
+s
+β x(s) = d
+ds(e
+s
+β x(s)) − 1
+β e
+s
+β x(s),
+(30)
+which we use to perform an integration by part on the last integral in (29),
+� t
+0
+e
+s
+β α(s) ˙x(s) ds =
+�
+α(s)e
+s
+β x(s)
+�t
+0 −
+� t
+0
+�
+α′(s) + α(s)
+β
+�
+e
+s
+β x(s) ds.
+Therefore,
+e− t
+β
+� t
+0
+e
+s
+β α(s) ˙x(s) ds = α(t)x(t) − e− t
+β α0x0 −
+� t
+0
+e
+s−t
+β
+�
+α′(s) + α(s)
+β
+�
+x(s) ds,
+(31)
+and we can substitute in (29),
+β∇f(x(t)) = βe− t
+β ∇f(x0) + e− t
+β ε0 ˙x0 − ε(t) ˙x(t) +
+� t
+0
+e
+s−t
+β
+� 1
+β ε(s) + ε′(s)
+�
+˙x(s) ds
+− α(t)x(t) + e− t
+β α0x0 +
+� t
+0
+e
+s−t
+β
+�
+α′(s) + α(s)
+β
+�
+x(s) ds.
+(32)
+Uniform boundedness.
+In view of exploiting (32), we recall that for all (ε, α), x is uniformly
+bounded. So there exists R > 0 such that for all (ε, α), the corresponding solution x of (VM-DIN-AVD)
+is such that
+sup
+t≥0
+∥x(t)∥ ≤ R.
+(33)
+We are now in position to prove (8).
+Distance from x to xN.
+We first gather (4) and (32). For all t ≥ 0,
+β∇f(x(t)) − β∇f(xN(t)) = e− t
+β ε0 ˙x0 − ε(t) ˙x(t) +
+� t
+0
+e
+s−t
+β
+� 1
+β ε(s) + ε′(s)
+�
+˙x(s) ds
++ e− t
+β α0x0 − α(t)x(t) +
+� t
+0
+e
+s−t
+β
+�
+α′(s) + α(s)
+β
+�
+x(s) ds.
+We then use (28) and triangle inequalities,
+βµ∥x(t) − xN(t)∥ ≤ e− t
+β ε0∥ ˙x0∥ + ε(t)∥ ˙x(t)∥ +
+� t
+0
+e
+s−t
+β
+����
+ε(s)
+β
++ ε′(s)
+���� ∥ ˙x(s)∥ ds
++ e− t
+β α0∥x0∥ + α(t)∥x(t)∥ +
+� t
+0
+e
+s−t
+β
+����
+α(s)
+β
++ α′(s)
+���� ∥x(s)∥ ds.
+(34)
+24
+
+By Assumption 2, for all s ≥ 0, | ε(s)
+β + ε′(s)| ≤ ( 1
+β + c1)ε(s) and | α(s)
+β
++ α′(s)| ≤ ( 1
+β + c2)α(s). We
+then use Lemma 3.5 (denoting by C > 0 the constant stated in the lemma) on the first line of (34), and
+we use the boundedness (33) on the second line to obtain,
+βµ∥x(t) − xN(t)∥ ≤ e− t
+β ε0∥ ˙x0∥ + C
+�
+ε(t) + C
+� 1
+β + c1
+� � t
+0
+e
+s−t
+β �
+ε(s) ds
++ e− t
+β α0∥x0∥ + Rα(t) + R
+� 1
+β + c2
+� � t
+0
+e
+s−t
+β α(s) ds.
+This proves (8).
+Expressing ∇f(xLM).
+We now repeat previous arguments but for (LM). First, (LM) is equivalent to
+d
+dt∇f(xLM(t)) + 1
+β ∇f(xLM(t)) = −α(t) ˙xLM(t).
+So using an integrating factor one can check that for all t ≥ 0,
+∇f(xLM(t)) = e− t
+β ∇f(x0) − e− t
+β
+� t
+0
+1
+β e
+s
+β α(s) ˙xLM(s) ds.
+We can then follow exactly steps (30) to (31) so as to obtain,
+e− t
+β
+� t
+0
+e
+s
+β α(s) ˙xLM(s) ds = α(t)xLM(t) − e− t
+β α0x0 − e− t
+β
+� t
+0
+�
+α′(s) + α(s)
+β
+�
+e
+s
+β xLM(s) ds.
+Finally, remark that for all t ≥ 0,
+d
+dtf(xLM(t)) = −α(t)∥ ˙xLM(t)∥2 − β⟨ ˙xLM(t), ∇2f(xLM(t)) ˙xLM(t)⟩ ≤ 0.
+So f(xLM(t)) ≤ f(x0) and using the coercivity of f as before we deduce that for all choices α,
+sup
+t≥0
+∥xLM(t)∥ ≤ R.
+(35)
+Distance from x to xLM.
+We substract gradients,
+β∇f(x(t)) − β∇f(xLM(t))) = e− t
+β ε0 ˙x0 − ε(t) ˙x(t) +
+� t
+0
+e
+s−t
+β
+� 1
+β ε(s) + ε′(s)
+�
+˙x(s) ds
+− α(t)(x(t) − xLM(t)) −
+� t
+0
+e
+s−t
+β
+�
+α′(s) + α(s)
+β
+�
+(x(s) − xLM(s)) ds,
+and we proceed as before using (28), Assumption 2 and Lemma 3.5. It holds that,
+βµ∥x(t) − xLM(t)∥ ≤ e− t
+β ε0∥ ˙x0∥ + C
+�
+ε(t) + C
+� 1
+β + c1
+� � t
+0
+e
+1
+β (s−t)�
+ε(s) ds
++ α(t)∥x(t) − xLM(t)∥ +
+� 1
+β + c2
+� � t
+0
+e
+1
+β (s−t)∥x(s) − xLM(s)∥ ds.
+Finally, using (33) and (35), for all s ≥ 0, ∥x(s) − xLM(s)∥ ≤ 2R, which concludes the proof.
+25
+
+C
+Integrability of ϕ and Additional Asymptotic Computations
+Below we prove Lemma 4.7.
+Proof. We suppose that Assumptions 3 and 4 hold. As stated in Remark 4.3, since ϕ is continuous,
+we only need to check its integrability when t tends to +∞. Let t > 0, we first establish some useful
+identities, we omit the dependence on t for the sake of readability.
+p′ = α′ε − (α + βλ)ε′
+ε2
+,
+p′′ = α′′ε2 − 2α′ε′ε − (α + βλ)ε′′ε + 2(α + βλ)(ε′)2
+ε3
+.
+Then,
+r = p2
+4
+�
+1 + 2p′
+p2 − 4λ
+εp2
+�
+= (α + βλ)2
+4ε2
+�
+1 +
+2p′ε2
+(α + βλ)2 −
+4λε
+(α + βλ)2
+�
+= (α + βλ)2
+4ε2
+�
+1 +
+2α′ε
+(α + βλ)2 −
+2ε′
+(α + βλ) −
+4λε
+(α + βλ)2
+�
+.
+(36)
+An important consequence of Assumption 4 is that |ε′(t)| = o(ε(t)), |ε′′(t)| = o(ε′(t)) (and the same
+holds for α w.r.t. to its derivatives). Therefore, we deduce from (36) that
+r(t) ∼+∞
+(α(t) + βλ)2
+4ε(t)2
+,
+and we note that 1/r decays at the same speed as ε2, which will be useful later. In order to study ϕ, we
+now differentiate r,
+r′ = p′p
+2
+�
+1 + 2p′
+p2 − 4λ
+εp2
+�
++ 1
+4
+�
+2p′′ − 4(p′)2
+p
++ 8λp′
+εp + 4λε′
+ε2
+�
+= 2p′
+p r + 1
+4
+�
+2p′′ − 4(p′)2
+p
++ 8λp′
+εp + 4λε′
+ε2
+�
+,
+and
+r′′ =2p′′p − (p′)2
+p2
+r + 2p′
+p r′
++ 1
+4
+�
+2p′′′ + 4(p′)3 − 2p′′p′p
+p2
++ 8λp′′pε − (p′)2ε − p′pε′
+ε2p2
++ 4λε′′
+ε2
+− 8λ(ε′)2
+ε3
+�
+.
+(37)
+Then, to justify that ϕ is integrable, we prove that
+r′′
+r3/2 and (r′)2
+r5/2 are integrable. Since we know that 1/r
+decays at the same speed as ε2, we can equivalently show that ε3r′′ and ε5(r′)2 are integrable. To this
+aim we fully expand all the terms in (37), and compute (r′)2, which is extremely long and involved.
+26
+
+On the one hand, it holds that,
+r′2ε5 =
+�
+�−
+(α + βλ)2 �
+−
+4λε
+(α+βλ)2 + 1 + (−2(α+βλ)ε′+2α′ε)
+(α+βλ)2
+�
+ε′
+2√ε
++ (α + βλ)2√ε
+4
+�
+−
+4λε′
+(α + βλ)2 +
+8λα′ε
+(α + βλ)3
++
+2
+�
+− 2(α+βλ)ε′
+ε
++ 2α′ε
+�
+ε′
+(α + βλ)2
++
+�
+4(α+βλ)ε′2
+ε
+− 2(α + βλ)ε′′ − 4α′ε′ + 2α′′ε
+�
+(α + βλ)2
+− 2 (−2(α + βλ)ε′ + 2α′ε) α′
+(α + βλ)3
+�
+�
++(α + βλ)
+2
+�
+−
+4λε
+(α + βλ)2 + 1 + (−2(α + βλ)ε′ + 2α′ε)
+(α + βλ)2
+�
+α′√ε
+�2
+.
+On the other hand, we have,
+r′′ε(t)3 =
+− λε′′ε + 4λα′ε′ε
+α + βλ + 2λα′′ε2
+α + βλ −
+6λα′2ε2
+(α + βλ)2 + (α + βλ)2
+2
+�
+−3ε′2
+ε
++ ε′′
+� �
+4λε
+(α + βλ)2 − 1 + 2 ((α + βλ)ε′ − α′ε)
+(α + βλ)2
+�
++ 2(α + βλ)
+�
+4λε
+(α + βλ)2 − 1 + 2 ((α + βλ)ε′ − α′ε)
+(α + βλ)2
+�
+α′ε′ −
+�
+(α + βλ)α′′ + α′2�
+2
+�
+4λε
+(α + βλ)2 − 1 + 2 ((α + βλ)ε′ − α′ε)
+(α + βλ)2
+�
+ε
+−
+�(α + βλ)ε′
+ε
+− α′
+�
+ε′2 −
+�
+(α + βλ)ε′ − α′ε
+�
+ε′′ − 2
+�
+−2(α + βλ)ε′2
+ε
++ (α + βλ)ε′′ + 2α′ε′ − α′′ε
+�
+ε′
++ 2
+�
+2λε′ − 4λα′ε
+α + βλ + 2
+�(α + βλ)ε′
+ε
+− α′
+�
+ε′ +
+�
+−2(α + βλ)ε′2
+ε
++ (α + βλ)ε′′ + 2α′ε′ − α′′ε
+�
+− 2 ((α + βλ)ε′ − α′ε) α′
+α + βλ
+�
+ε′
+− 1
+2
+�6(α + βλ)ε′3
+ε
+− 6(α + βλ)ε′ε′′ + (α + βλ)ε′′′ε − 6α′ε′2 + 3α′ε′′ε + 3α′′ε′ε − α′′′ε2
+�
++
+1
+α + βλ
+�
+4
+�
+(α + βλ)ε′ − α′ε
+�
+α′ε′ +
+�
+(α + βλ)ε′ − α′ε
+�
+α′′ε + 2
+�
+−2(α + βλ)ε′2 + (α + βλ)ε′′ε + 2α′ε′ε − α′′ε2�
+α′�
+−
+2
+α + βλ
+�
+2λε′ − 4λα′ε
+α + βλ + 2
+�(α + βλ)ε′
+ε
+− α′
+�
+ε′ +
+�
+−2(α + βλ)ε′2
+ε
++ (α + βλ)ε′′ + 2α′ε′ − α′′ε
+�
+− 2 ((α + βλ)ε′ − α′ε) α′
+α + βλ
+�
+α′ε
+− 3
+�
+(α + βλ)ε′ε − α′ε2�
+α′2
+(α + βλ)2
+.
+27
+
+We then analyze the integrability of each of the terms above. By Assumption 4, ε′, ε′′ and ε′′′ are
+integrable and the same goes for α′, α′′ and α′′′, which is enough to justify the integrability of almost
+all the terms above. We finally see that we also need (ε′)2
+ε
+and (ε′)3
+ε
+to be integrable, which holds by
+Assumption 4. Overall, ϕ is integrable on R+.
+We now state and prove the following result which was used at the end of the proof of Theorem 4.8.
+Lemma C.1. Under Assumptions 3 and 4, for all s ≥ 0,
+1
+16
+�
+2p′(s)
+p(s)3/2 −
+4λ
+�
+ε(s)
+(α(s) + βλ)3/2
+�2
+=
+λ2ε(s)
+(α(s) + βλ)3 + o(ε(s)).
+Proof. We omit the time dependence on s ≥ 0 for the sake of readability. Using Assumption 3 we can
+define and expand the following quantity,
+1
+16
+� 2p′
+p3/2 −
+4λ√ε
+(α + βλ)3/2
+�2
+=
+λ2ε
+(α + βλ)3 −
+p′λ√ε
+p3/2(α + βλ)3/2 + (p′)2
+4p3
+=
+λ2ε
+(α + βλ)3 − λ(α′ε − (α + βλ)ε′) +
+1
+4(α + βλ)3
+�
+(α′)2ε + (ε′)2
+ε (α + βλ)2 − 2α′ε′(α + βλ)
+�
+.
+Assumption 4, implies in particular that |ε′(t)| = o(ε(t)) and that α′(t) → 0, which we use in the
+equality above to obtain the desired conclusion.
+D
+Additional Experiments and Details
+We first detail the discretization that we used for approximating the solutions of the three ODEs con-
+sidered in Section 5. We use Euler discretization schemes with fixed step-size γ > 0 and approximate
+the solutions at times tk = γk, for all k ∈ N. For a trajectory x, we use the notation x(tk)
+def
+= x(k). The
+approximation of (CN) is obtained by explicit discretization, so that for all k ∈ N, we have,
+x(k+1)
+N
+= x(k)
+N − γ
+�
+β∇2f(x(k)
+N )
+�−1
+∇f(x(k)
+N ).
+(38)
+Then, defining εk = ε(tk) and αk = α(tk), (LM) and (VM-DIN-AVD) are obtained via Euler semi-
+implicit discretization. The solution of (LM) is approximated by,
+x(k+1)
+LM
+= x(k)
+LM − γ
+�
+αkIn + β∇2f(x(k)
+LM)
+�−1
+∇f(x(k)
+LM),
+(39)
+where In is the identity matrix on Rn. The solution of (VM-DIN-AVD) is obtained similarly,
+x(k+1) = x(k) +
+�
+(εk + γαk)In + γβ∇2f(x(k))
+�−1 �
+εk(x(k) − x(k−1)) − γ2∇f(x(k))
+�
+.
+(40)
+As safety check, one can see that for εk = 0, (40) is equivalent to (39), which is itself equivalent to (38)
+when αk = 0.
+28
+
+0
+10
+20
+30
+40
+50
+t
+10
+21
+10
+17
+10
+13
+10
+9
+10
+5
+10
+1
+Distance to CN
+(t) =
+0/log t,
+(t) =
+0/t
+(t) =
+0/ t,
+(t) =
+0/t
+(t) =
+0/t,
+(t) =
+0/t
+(t) =
+0/t2,
+(t) =
+0/t
+(t) =
+0e
+t/ ,
+(t) =
+0/t
+0
+10
+20
+30
+40
+50
+t
+Distance to LM
+0
+10
+20
+30
+40
+50
+t
+Speed of convergence
+Levenberg-Marquardt
+Continuous Newton
+0
+10
+20
+30
+40
+50
+t
+10
+22
+10
+18
+10
+14
+10
+10
+10
+6
+10
+2
+102
+Distance to CN
+(t) =
+0/log t,
+(t) =
+0/t2
+(t) =
+0/ t,
+(t) =
+0/t2
+(t) =
+0/t,
+(t) =
+0/t2
+(t) =
+0/t2,
+(t) =
+0/t2
+(t) =
+0e
+t/ ,
+(t) =
+0/t2
+0
+10
+20
+30
+40
+50
+t
+Distance to LM
+0
+10
+20
+30
+40
+50
+t
+Speed of convergence
+Levenberg-Marquardt
+Continuous Newton
+0
+20
+40
+60
+80
+100
+120
+140
+t
+10
+16
+10
+13
+10
+10
+10
+7
+10
+4
+10
+1
+Distance to CN
+(t) =
+0/log t,
+(t) =
+0/t
+(t) =
+0/ t,
+(t) =
+0/t
+(t) =
+0/t,
+(t) =
+0/t
+(t) =
+0/t2,
+(t) =
+0/t
+(t) =
+0e
+t/ ,
+(t) =
+0/t
+0
+20
+40
+60
+80
+100
+120
+140
+t
+Distance to LM
+0
+20
+40
+60
+80
+100
+120
+140
+t
+Speed of convergence
+Levenberg-Marquardt
+Continuous Newton
+0
+20
+40
+60
+80
+100
+120
+140
+t
+10
+16
+10
+13
+10
+10
+10
+7
+10
+4
+10
+1
+Distance to CN
+(t) =
+0/log t,
+(t) =
+0/t2
+(t) =
+0/ t,
+(t) =
+0/t2
+(t) =
+0/t,
+(t) =
+0/t2
+(t) =
+0/t2,
+(t) =
+0/t2
+(t) =
+0e
+t/ ,
+(t) =
+0/t2
+0
+20
+40
+60
+80
+100
+120
+140
+t
+Distance to LM
+0
+20
+40
+60
+80
+100
+120
+140
+t
+Speed of convergence
+Levenberg-Marquardt
+Continuous Newton
+Figure 6: Similar experiment and figures as those described in Figure 2, but for a poorly conditioned
+quadratic f(x) = 1
+2∥Ax∥2 (first-two rows) and the function f(x) = log (�n
+i=1 exi) + 1
+2∥Ax∥2 (last-two
+rows).
+29
+
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+
diff --git a/aNFAT4oBgHgl3EQf4h4_/content/tmp_files/load_file.txt b/aNFAT4oBgHgl3EQf4h4_/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..f15ec6145fdaac9f88bdca39b268ada028878fce
--- /dev/null
+++ b/aNFAT4oBgHgl3EQf4h4_/content/tmp_files/load_file.txt
@@ -0,0 +1,1096 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf,len=1095
+page_content='Continuous Newton-like Methods featuring Inertia and  Variable Mass  Camille Castera∗  Department of Mathematics  University of T¨ubingen  Germany  Hedy Attouch  IMAG  Universit´e Montpellier  CNRS, France  Jalal Fadili  ENSICAEN  Normandie Universit´e  CNRS, GREYC, France  Peter Ochs  Department of Mathematics  University of T¨ubingen  Germany  ABSTRACT  We introduce a new dynamical system, at the interface between second-order dynamics  with inertia and Newton’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' This system extends the class of inertial Newton-like  dynamics by featuring a time-dependent parameter in front of the acceleration, called  variable mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' For strongly convex optimization, we provide guarantees on how the  Newtonian and inertial behaviors of the system can be non-asymptotically controlled by  means of this variable mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' A connection with the Levenberg–Marquardt (or regular-  ized Newton’s) method is also made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We then show the effect of the variable mass on the  asymptotic rate of convergence of the dynamics, and in particular, how it can turn the lat-  ter into an accelerated Newton method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We provide numerical experiments supporting  our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' This work represents a significant step towards designing new algorithms  that benefit from the best of both first- and second-order optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  1  Introduction  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1  Problem Statement  A major challenge in modern unconstrained convex optimization consists in building fast algorithms  while maintaining low computational cost and memory footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' This plays a central role in many key  applications such as large-scale machine learning problems or data processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The problems we are  aiming to study are of the form  min  x∈Rn f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  ∗Corresponding author: camille.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='castera@protonmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='com  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='08726v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='OC]  20 Jan 2023  Large values of n demand for algorithms at the interface of first- and second-order optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Lim-  ited computational capabilities explain why gradient-based (first-order) algorithms remain prominent  in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Unfortunately, they often require many iterations which is true even for the provably best  algorithms for certain classes of optimization problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' for example that of convex and strongly con-  vex functions with Lipschitz continuous gradient [37, 33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' On the other hand, algorithms using  second-order information (the Hessian of f)—with Newton’s method as prototype—adapt locally to  the geometry of the objective, allowing them to progress much faster towards a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' However,  each iteration comes with high computational and memory costs, which highlights a challenging trade-  off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' It is therefore essential to develop algorithms that take the best of both worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' They are commonly  referred to as (limited-memory) quasi-Newton methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Several quasi-Newton algorithms partly ad-  dress this issue, for example BFGS methods [18, 23, 25, 39, 31], yet, in very large-scale applications,  first-order algorithms often remain the preferred choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  In order to reach a new level of efficiency, deep insights into the mechanism and relations between algo-  rithms are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' To that aim, an insightful approach is to see optimization algorithms as discretiza-  tion of ordinary differential equations (ODEs): for small-enough step-sizes, iterates can be modeled  by a continuous-time trajectory [32, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Obtaining a fast algorithm following this strategy depends  on two ingredients: choosing an ODE for which rapid convergence to a solution can be proved, and  discretizing it with an appropriate scheme that preserves the favorable properties of the ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Both steps are highly challenging, our work focuses on the ODE matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We study the following  second-order dynamical system in a general setting:  ε(t)¨x(t) + α(t) ˙x(t) + β∇2f(x(t)) ˙x(t) + ∇f(x(t)) = 0,  t ≥ 0,  (VM-DIN-AVD)  where f : Rn → R is a smooth convex twice continuously differentiable function, with gradient ∇f and  Hessian ∇2f defined on Rn equipped with scalar product ⟨·, ·⟩, and induced norm ∥·∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Additionally, f is  assumed to be coercive, and strongly convex on bounded subsets of RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The functions ε, α: R+ → R+  (where R+ = [0, +∞[) are differentiable, non-increasing, and ε(t) > 0 for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Together with  β > 0, they are control parameters that define the type of dynamics that drives the trajectory (or  solution) x: R+ → Rn, whose first- and second-order derivatives are denoted ˙x and ¨x respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  We call the above dynamics (VM-DIN-AVD), which stands for “Variable Mass Dynamical Inertial  Newton-like system with Asymptotically Vanishing Damping” since it generalizes a broad class of  ODEs whose original member is DIN [2], where ε and α were constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' DIN was then extended to  the case of non-constant asymptotically vanishing dampings (AVD) α [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' In this work we introduce  the non-constant parameter ε called variable mass (VM) in front of the acceleration ¨x, in the same  way that α is called (viscous) damping by analogy with classical mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' A key feature of these  ODEs, that positions them at the interface of first- and second-order optimization, is that they possess  equivalent forms involving only ∇f but not ∇2f, significantly reducing computational costs, hence  enabling the design of practical algorithms, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=', [20, 10, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The key idea behind this is the  relation ∇2f(x(t)) ˙x(t) = d  dt∇f(x(t)), see Section 2 for an equivalent formulation of (VM-DIN-AVD)  exploiting this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  This paper emphasizes the relation between (VM-DIN-AVD) and well-studied special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Indeed,  taking ε = α = 0, one obtains1 the Continuous Newton (CN) method [24]  β∇2f(xN(t)) ˙xN(t) + ∇f(xN(t)) = 0,  t ≥ 0,  (CN)  1CN is usually considered with β = 1, we put β in the system to ease the discussions below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  2  = 0  (t)  0  > 0 constant  Viscous damping  = 0  (t)  0  > 0 const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Variable mass  0  0  0  (t)  (t)  max(  (t),  (t))  max(  ,  )  (Alvarez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=')  Initialization x0  Continuous Newton  Continuous gradient descent  DIN-AVD (no variable mass)  Minimizer x  of f  Level lines of f  fast decay  slow decay  Decay speed of (t)  Figure 1: Left: phase diagram on distances from (VM-DIN-AVD) to (CN) and (LM) (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  For each patch, the color indicates which of the distances ∥xN(t) − x(t)∥ and ∥xLM(t) − x(t)∥ is  considered, the scaling of a corresponding upper-bound on this distance is written;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' in white for prior  work and in black for our contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The green line separates the cases ε ≥ α (above) and ε ≤  α (below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Right: 2D illustration of the trajectories of (VM-DIN-AVD) for several choices of ε on  a quadratic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Using fast-vanishing ε(t) (dark-blue solid curves), one can bring the solution  of (VM-DIN-AVD) close to that of (CN), making it, for example, more robust to bad conditioning  compared to first-order dynamics (such as gradient descent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  known notably for being invariant to affine transformations and yielding fast vanishing of the gradient  (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' In fact, this observation shows that (VM-DIN-AVD) is a singular perturbation of (CN),  which also justifies the terminology “Newton-like” in DIN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' When α ̸= 0 but ε = 0, we recover the  Levenberg–Marquardt (LM) method,  α(t) ˙xLM(t) + β∇2f(xLM(t)) ˙xLM(t) + ∇f(xLM(t)) = 0,  t ≥ 0,  (LM)  also known as regularized Newton method since it stabilizes (CN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' In the rest of the paper, the solutions  of (CN) and (LM) will always be denoted by xN and xLM respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Alvarez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' [2] showed that  for α = 0, β = 1, and ε constant and small, (VM-DIN-AVD) is a “perturbed” Newton method since  the distance between the solutions of (VM-DIN-AVD) and (CN) is at most proportional to √ε at all  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Yet, despite the benefits of this class of ODEs, such as stabilization properties, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=', [9, 10],  no improvement2 of the rate of convergence (in values) has been shown compared to inertial first-order  dynamics [37, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' This raises the question:  “are these ODEs really of Newton type?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=',  which is crucial in view of designing faster algorithms from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  2DIN-like systems were thought to yield faster vanishing of the gradient compared to inertial first-order dynamics, until  recently [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  3  xn(t) - x(t)II  Both  [XLM(t) 一 x(t)  Frontier ε(t) = α(t)Table 1: Informal summary of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Comparison of (VM-DIN-AVD) with other dynamics  Parameters of (VM-DIN-AVD)  Speed of convergence  Dominant parameter Integrability in +∞  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' (CN)  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' (LM)  variable mass ε  yes  as fast  as fast  no  faster  faster  viscous damping α  yes  as fast  only depends  on ε  no  slower  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2  Main Contributions  We show that the answer to this question is partially positive, and closely related to the choices of ε  and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We provide general results on the role played by these two control parameters and how they  can be chosen to control (VM-DIN-AVD), and make it close to (CN) for all time, as illustrated on the  right-hand side of Figure 1, but also to obtain fast convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' This represents a first step towards  building new fast practical algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Our main contributions are the following:  – We provide a first-order equivalent formulation of (VM-DIN-AVD), and address the questions of  existence and uniqueness of the solutions of (VM-DIN-AVD) under mild assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  – We generalize the perturbed Newtonian property discussed above to non-constant and possibly van-  ishing variable masses ε, and “not too large” positive dampings α, and derive bounds that (formally)  take the form ∥x(t) − xN(t)∥ = O(  �  ε(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We then extend these results to larger dampings α and  make the connection between (VM-DIN-AVD) and (LM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' This contribution is summarized in the phase  diagram of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  – Using quadratic functions as a model for strongly convex functions, we shed light on techniques to  efficiently approximate solutions of (VM-DIN-AVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We then show how ε and α affect the speed of  convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Depending on their setting, the solutions of (VM-DIN-AVD) may either converge as fast  as that of (CN), faster, or rather have a (LM) nature, as summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  – We provide numerical experiments supporting our theoretical findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='3  Related work  The system (VM-DIN-AVD) belongs to the class of inertial systems with viscous and geometric  (“Hessian-driven”) dampings, initially introduced with constant ε = 1 and constant α in [2] and called  DIN (for Dynamical Inertial Newton-like system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Except in a few cases [20, 19], most of the follow-up  work then considered extensions of DIN with non-constant AVD α, with in particular the DIN-AVD  system with α(t) = α0/t as introduced in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The reason for this popular choice for α is its link with  Nesterov’s method [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Non-constant choices for β have been considered [10, 1, 29, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We keep  it constant here, and rather focus on non-constant ε, unlike prior work that used constant ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The  mass ε was only considered in the original work [2], but only for fixed ε, β = 1 and constant α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  VM-DIN-AVD is however closely related to the IGS system considered in [11] as it is actually equiv-  alent to the latter after dividing both sides of (VM-DIN-AVD) by ε(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Our approach—which consists  in studying the connections with other second-order dynamics as ε vanishes asymptotically—is how-  ever different from the one followed in [11], which is of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The literature on DIN  is rich, let us mention further connections with Nesterov’s method [40, 1], extensions with Tikhonov  4  regularization [16] and closed-loop damping [12, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The non-smooth and possibly non-convex cases  have been considered in [5, 6, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Finally, avoidance of strict saddle points in smooth non-convex  optimization has been shown in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  The influence of the damping α on the (LM) dynamics has been studied in [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Interestingly, the  conditions enforced on α in these papers (formally a sub-exponential decay) are very similar to those  we make on ε and α for (VM-DIN-AVD) (see Assumptions 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Regarding the second part of our analysis, which deals with the case where f is quadratic, Attouch  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' [10] provided closed-form solutions to (VM-DIN-AVD) for ε ≡ 1 and special choices of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Our  work rather deals with approximate solutions which allows considering a wide class of functions ε and  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We rely on the Liouville–Green (LG) method [30, 26] presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Generalizations of  LG are also often referred to as WKB methods [44, 28, 17] and seem to be mostly used in physics so  far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' To the best of the authors’ knowledge, the current work seem to be one of the first to use the LG  method in optimization, and the first for DIN-like systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='4  Organization  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We discuss the existence of solutions in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Our main results,  from a non-asymptotic control perspective, are then presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' An analysis of the role  played asymptotically by ε and α is then carried out on quadratic functions in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Finally,  numerical experiments are presented in Section 5, and some conclusions are then drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  2  Existence and Uniqueness of Solutions  In the sequel, we fix x0 ∈ Rn and ˙x0 ∈ Rn, such that, unless stated otherwise, (VM-DIN-AVD) is  always considered with initial condition (x(0), ˙x(0)) = (x0, ˙x0), and (CN) and (LM) with initial con-  dition xN(0) = xLM(0) = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We also fix initial values for the control parameters ε(0) = ε0 > 0,  ε′(0) = ε′  0 ≤ 0 and α(0) = α0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' In addition to the definitions of ε and α in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1, we assume  that ε is twice differentiable, with bounded second derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' In order to use the Cauchy–Lipschitz  Theorem, we reformulate (VM-DIN-AVD) into a first-order (in time) system by introducing an auxil-  iary variable y : R+ → Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Notably, our reformulation does not involve ∇2f, in the same fashion as  [2, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' For all t, defining ν(t) = α(t) − ε′(t) − 1  βε(t), we show in Appendix A that (VM-DIN-AVD) is  equivalent to  �  ε(t) ˙x(t) + β∇f(x(t)) + ν(t)x(t) + y(t)  = 0  ˙y(t) + ν′(t)x(t) + ν(t)  β x(t) + 1  βy(t)  = 0  (gVM-DIN-AVD)  with initial conditions (x(0), y(0)) =  �  x0, −ε0 ˙x0 − β∇f(x0) − (α0 − ε′  0 − 1  βε0)x0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' One can notice  that in the special case where ε is taken constant and equal to 1 (that is when (VM-DIN-AVD) is simply  the DIN-AVD system [9]), we recover the same first-order formulation as that in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  We can then apply the Cauchy–Lipschitz Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' For all t ≥ 0 and (u, v) ∈ Rn × Rn, define the  mapping  G (t, (u, v)) =  � 1  ε(t)(−β∇f(u) − ν(t)u − v)  −ν′(t)u − ν(t)  β u − 1  βv  �  ,  5  so that (gVM-DIN-AVD) rewrites ( ˙x(t), ˙y(t)) = G (t, (x(t), y(t))) for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Since f is twice  continuously differentiable, one can see that G is continuously differentiable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' its second argument  (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Consequently G is locally Lipschitz continuous w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' (u, v) and by the Cauchy–Lipschitz The-  orem, for each initial condition, there exists a unique local solution to (gVM-DIN-AVD) and thus to  (VM-DIN-AVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We then show that this solution is actually global (in Appendix A) by proving the  boundedness of (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We omit the existence and uniqueness of the solutions of (CN) and (LM) since  these are standard results, obtained with similar arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  3  Non-asymptotic Control of (VM-DIN-AVD)  The purpose of this section is to understand how close x might be to xN and xLM, as a function of α  and ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Since f is coercive and strongly convex on bounded sets, it has a unique minimizer x⋆ ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Consequently, any two trajectories that converge to x⋆ will eventually be arbitrarily close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Thus, asymptotic results of the form ∥x(t) − xN(t)∥ −−−−→  t→+∞ 0 are not precise enough to claim, for  example, that x has a “Newtonian behavior”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Instead, we will derive upper bounds on the distance  between trajectories that hold for all time t ≥ 0, and which typically depend on ε and/or α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We first  present the case where α is small relative to ε and then generalize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1  Comparison with (CN) under Moderate Viscous Dampings  When the viscous damping α remains moderate w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' the variable mass ε, one can expect the solutions  of (VM-DIN-AVD) to be close to that of (CN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We make the following assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' There exists c1, c2 ≥ 0 such that for all t ≥ 0, |ε′(t)| ≤ c1ε(t) and α(t) ≤ c2ε(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  The assumption states that α must decrease at least as fast as ε (up to a constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='3 The reason for  assuming |ε′(t)| ≤ c1ε(t) is technical and will appear more clearly in the proofs below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' It formally  means that ε can decrease at most exponentially fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='4 This is a relatively mild assumption that holds,  for example, for any polynomial decay ε0/(t + 1)a, a ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  We start with the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Let xN be the solution of (CN), and let c1, c2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' There exist C0, C1, C2 ≥ 0, de-  pending on c1, c2, such that for all (ε, α) for which Assumption 1 holds with constants c1 and c2, the  corresponding solution x of (VM-DIN-AVD) is such that for all t ≥ 0,  ∥x(t) − xN(t)∥ ≤ C0e− t  β ε0∥ ˙x0∥ + C1  �  ε(t) + C2  � t  s=0  e  1  β (s−t)�  ε(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (1)  This extends a previous result from [2, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1] which states a similar bound for constant ε,  α ≡ 0 and β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1 corresponds to the blue parts in the phase diagram of Figure 1 (see also  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='6 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The strength of the above result comes from the fact that the constants C0, C1, C2 do not  depend on ε and α, and that the result is non-asymptotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' This allows in particular choosing (ε, α) to  control the distance from x to xN, for all time t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  3Assumption 1 can actually hold only after some large-enough t0 ≥ 0, we take t0 = 0 for the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  4This is a consequence of Gronwall’s lemma, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=', [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  6  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Under Assumption 1, the dynamics (VM-DIN-AVD) is dominated by the variable mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  The damping α does not appear in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  The above theorem and remarks emphasize the “Newtonian nature” of (VM-DIN-AVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We present  two lemmas before proving Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1, and then state a simpler bound than (1), see Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Let (ε, α), and let x be the corresponding solution of (VM-DIN-AVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' For all t ≥ 0,  define the function,  U(t) = ε(t)  2 ∥ ˙x(t)∥2 + f(x(t)) − f(x⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Then U is differentiable and for all t > 0,  dU  dt (t) = ε′(t)  2 ∥ ˙x(t)∥2 − α(t)∥ ˙x(t)∥2 − β⟨∇2f(x(t)) ˙x(t), ˙x(t)⟩ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Therefore, in particular, U is non-increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Let t ≥ 0, since x is twice differentiable, U is differentiable and,  dU  dt (t) = ε′(t)  2 ∥ ˙x(t)∥2 + ε(t)⟨ ˙x(t), ¨x(t)⟩ + ⟨ ˙x(t), ∇f(x(t))⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  We use the fact that x is solution of (VM-DIN-AVD), to substitute ε(t)¨x(t) by its expression,  dU  dt (t) = ε′(t)  2 ∥ ˙x(t)∥2 − α(t)∥ ˙x(t)∥2 − β⟨∇2f(x(t)) ˙x(t), ˙x(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  By assumption ε is non-increasing so for all t > 0, ε′(t) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Furthermore f is convex so  ⟨∇2f(x(t)) ˙x(t), ˙x(t)⟩ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Hence U is non-increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  We then state the following bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' There exists C ≥ 0 such that for any (ε, α) and the corresponding solution x of  (VM-DIN-AVD), for all t ≥ 0 it holds that,  ε(t)∥ ˙x(t)∥ ≤ C  �  ε(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Let t ≥ 0, according to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='4, U is non-increasing so U(t) ≤ U(0), or equivalently,  ε(t)  2 ∥ ˙x(t)∥2 + f(x(t)) − f(x⋆) ≤ ε0  2 ∥ ˙x0∥2 + f(x0) − f(x⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  This implies in particular that,  ε(t)∥ ˙x(t)∥2 ≤ ε0∥ ˙x0∥2 + 2(f(x0) − f(x⋆)),  and hence by multiplying both sides by ε(t) and composing with the square-root we obtain that,  ε(t)∥ ˙x(t)∥ ≤ C  �  ε(t),  where C =  �  ε0∥ ˙x0∥2 + 2(f(x0) − f(x⋆)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  7  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Let (ε, α) as defined in Sections 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1 and 2, and let x be the corresponding solu-  tion of (VM-DIN-AVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Then, according to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='4, for all t ≥ 0, U(t) ≤ U(0), so in particular  f(x(t)) ≤ f(x0) + ε0  2 ∥ ˙x0∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Denoting M0 = f(x0) + ε0  2 ∥ ˙x0∥2, the set K0 = {y ∈ Rn | f(y) ≤ M0} is bounded, since f is coercive  (lim∥y∥→+∞ f(y) = +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' So for all t ≥ 0, x(t) ∈ K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Since M0 (and hence K0) depends only on ε0,  x0 and ˙x0, we have proved that for any choice (ε, α), the corresponding solution x of (VM-DIN-AVD)  is inside K0 at all time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Let xN be the solution of (CN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' One can see similarly that for all t ≥ 0,  f(xN(t)) ≤ f(xN(0)) = f(x0) ≤ M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' So we also have xN(t) ∈ K0 for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Now, fix c1, c2 > 0, and let (ε, α) such that Assumption 1 is satisfied with constants c1, c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Let x be the  corresponding solution of (VM-DIN-AVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Since f is strongly convex on bounded sets, it is strongly  convex on K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We denote µ > 0 the strong-convexity parameter of f on K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Equivalently, we have that  ∇f is strongly monotone on K0, that is, ∀y1, y2 ∈ K0,  ⟨∇f(y1) − ∇f(y2), y1 − y2⟩ ≥ µ∥y1 − y2∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (2)  Let t ≥ 0, since x(t) ∈ K0 and xN(t) ∈ K0, by combining (2) with the Cauchy–Schwarz inequality, we  deduce that  ∥x(t) − xN(t)∥ ≤ 1  µ∥∇f(x(t)) − ∇f(xN(t))∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (3)  Therefore, it is sufficient to bound the difference of gradients in order to bound ∥x(t) − xN(t)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' First,  remark that (CN) can be rewritten as follows:  d  dt∇f(xN(t)) + 1  β∇f(xN(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' So we can integrate,  for all t ≥ 0,  ∇f(xN(t)) = e− t  β ∇f(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (4)  We now turn our attention to ∇f(x(t)), for which we cannot find a closed-form solution in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  We rewrite (VM-DIN-AVD) in the following equivalent form  d  dt [ε(t) ˙x(t) + β∇f(x(t))] + 1  β ε(t) ˙x(t) + ∇f(x(t)) =  � 1  β ε(t) + ε′(t) − α(t)  �  ˙x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Introducing the variable ω(t) = ε(t) ˙x(t) + β∇f(x(t)), the latter is thus solution to  �  ˙ω(t) + 1  βω(t) =  �  1  βε(t) + ε′(t) − α(t)  �  ˙x(t),  t ≥ 0,  ω(0) = ε0 ˙x0 + β∇f(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  This is a non-homogeneous first-order ODE in ω, whose solution can be expressed using the integrating  factor  ω(t) = e− t  β (ε0 ˙x0 + β∇f(x0)) + e− t  β  � t  0  e  s  β  � 1  β ε(s) + ε′(s) − α(s)  �  ˙x(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  We thus have the following expression for ∇f(x), for all t ≥ 0,  β∇f(x(t)) = βe− t  β ∇f(x0)+e− t  β ε0 ˙x0−ε(t) ˙x(t)+e− t  β  � t  0  e  s  β  � 1  β ε(s) + ε′(s) − α(s)  �  ˙x(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' (5)  8  We can now use (4) and (5) in (3) to get  ∥x(t) − xN(t)∥ ≤ 1  βµ  ����e− t  β ε0 ˙x0 − ε(t) ˙x(t) + e− t  β  � t  0  e  s  β  � 1  β ε(s) + ε′(s) − α(s)  �  ˙x(s) ds  ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Using the triangle inequality, we obtain,  ∥x(t) − xN(t)∥ ≤ ε0∥ ˙x0∥  βµ  e− t  β + ε(t)∥ ˙x(t)∥  βµ  + 1  βµ  � t  0  e  1  β (s−t)  ����  1  β ε(s) + ε′(s) − α(s)  ���� ∥ ˙x(s)∥ ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' (6)  The first term in (6) corresponds to the first one in (1) with C0 = 1/(βµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' As for the second-one, by  direct application of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='5, there exists C > 0 such that for all t ≥ 0, ε(t)∥ ˙x(t)∥  βµ  ≤ C  �  ε(t), so we  set C1 = C/(βµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Regarding the last term in (6), using Assumption 1 and again Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='5, it holds  that, for all s ≥ 0,  ����  1  β ε(s) + ε′(s) − α(s)  ���� ∥ ˙x(s)∥ ≤  � 1  β + c1 + c2  �  ε(s)∥ ˙x(s)∥ ≤  � 1  β + c1 + c2  �  C  �  ε(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  This proves the theorem with C2 = C  βµ  �  1  β + c1 + c2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Let us analyze the bound in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The first term in (1) decays exponentially fast and can even  be zero if the initial speed is ˙x0 = 0, the second-one decays like  �  ε(t), however, the rate at which the  last term decreases is less obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The following corollary gives a less-tight but easier-to-understand  estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Consider the same assumptions and variables as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' If furthermore c1 < 2  β,  then there exists C3 > 0 such that for all t ≥ 0,  ∥x(t) − xN(t)∥ ≤ C0e− t  β ε0∥ ˙x0∥ + C3  �  ε(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Proof of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' For all t ≥ 0, define J(t) =  � t  0 e  s  β �  ε(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' An integration by parts yields  J(t) =  �  βe  s  β �  ε(s)  �t  s=0 −  � t  s=0  βe  s  β  ε′(s)  2  �  ε(s)  ds = βe  t  β �  ε(t)−βε0 +  � t  s=0  βe  s  β −ε′(s)  2ε(s)  �  ε(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' (7)  By assumption, 0 ≤ c1 < 2  β such that for all s > 0, |ε′(s)| ≤ c1ε(s), which in our setting is equivalent  to −ε′(s)  ε(s) ≤ c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' So we deduce from (7) that  J(t) ≤ βe  t  β �  ε(t) + c1  β  2  � t  s=0  e  s  β �  ε(s) ds = βe  t  β �  ε(t) + c1  β  2 J(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  So,  �  1 − c1  β  2  �  J(t) ≤ βet�  ε(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' By assumption 1 − c1  β  2 > 0, therefore, J(t) ≤  2  2−c1βe  t  β �  ε(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Finally, using this in (1) and setting C3 = C1 + C2  2  2−c1β, we obtain the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The above proofs suggest that an extension to the case where f is non-smooth but strongly  convex is possible using regularization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' This is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  So far our results only cover the case where α is “not too large” w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' ε, and do not study (LM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We  now state a more general result that covers these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2  Generalization to Arbitrary Viscous Dampings with Sub-exponential Decay  This time we do not assume a link between ε and α but only sub-exponential decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' There exists c1, c2 ≥ 0 such that for all t ≥ 0, |ε′(t)| ≤ c1ε(t) and |α′(t)| ≤ c2α(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  We are now in position to state the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Let xN and xLM be the solution of (CN) and (LM) respectively, and let c1, c2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  There exist constants C, ˜C ≥ 0, depending on c1, c2, such that for all ε and α for which Assumption 2  holds with c1 and c2, the corresponding solution x of (VM-DIN-AVD) is such that for all t ≥ 0,  ∥x(t) − xN(t)∥ ≤ C  �  e− t  β +  �  ε(t) + α(t) +  � t  s=0  e  1  β (s−t)(  �  ε(s) + α(s)) ds  �  ,  (8)  and,  ∥x(t) − xLM(t)∥ ≤ ˜C  �  e− t  β +  �  ε(t) + α(t) +  � t  s=0  e  1  β (s−t)(  �  ε(s) + α(s)) ds  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (9)  The proof is postponed to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Although it follows a similar reasoning as that of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1,  more involved estimates are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Let us comment on these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The bound (8) generalizes Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1, although the constant in-  volved will, in general, be larger than those in (1) (see the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='8 in appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Theo-  rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='8 allows for far more flexibility in the choice of ε and α in order to control x and make it possibly  close to xN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The bound in (9) is the same as that in (8) (up to a constant), but this time w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' xLM, thus  connecting (VM-DIN-AVD) to (LM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We make the following two important remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' First (9) involves  α, suggesting that making x close to xLM requires not only ε but also α to vanish asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Ad-  ditionally, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='8 does not state to which of xN and xLM the solution of (VM-DIN-AVD) is the  closest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' It remains an open question to know whether one can make (9) independent of α, and to state  to which trajectory x is the closest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Yet, the numerical experiments in Section 5 suggest that neither are  possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Indeed, we observe that for some functions f, x is sometimes closer to xN than to xLM, even  when ε(t) ≤ α(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Nevertheless, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='8 answers the question asked in the introduction: yes, (VM-DIN-AVD) is  really of second-order nature since it can be brought close to the second-order dynamics (CN) and  (LM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Doing so, it benefits from the good properties of these methods, such as the robustness to bad  conditioning, as previously illustrated on the right of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' This concludes the analysis from a  control perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We will now derive an approximation of the solution x in order to study the impact  that ε and α have on the speed of convergence of x to x⋆ compared to the speeds of convergence of xN  and xLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  4  Approximate Solutions and Asymptotic Analysis on Quadratic Functions  We consider the particular case where f is a strongly-convex quadratic function in order to study the  asymptotic behavior of (VM-DIN-AVD) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' (CN) and (LM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Quadratic functions are the prototypical  example of strongly-convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' In particular, any strongly-convex function can be locally ap-  proximated by a quadratic one around its minimizer, making the latter a good model for understanding  the local behavior of dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' In this section, f is quadratic: f(y) = 1  2∥Ay − b∥2  2 for all y ∈ Rn,  10  where A ∈ Rn×n is symmetric positive definite and b ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Without loss of generality, we take b = 0,  so that the unique minimum is x⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1  Setting: the Special Case of Quadratic Functions  Quadratic functions are particularly interesting in our setting since DIN-like ODEs take a simpler form  (as observed in [10, 40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Indeed, ∀y ∈ Rn, ∇f(y) = ATAy and ∇2f(y) = ATA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Since ∇2f(y) is  independent of y we can rewrite (VM-DIN-AVD) in an eigenspace5 of ATA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' That is, we can study the  ODE coordinate-wise by looking at one-dimensional problems of the form  ε(t)¨x(t) + (α(t) + βλ) ˙x(t) + λx(t) = 0,  t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (Q1-VM-DIN-AVD)  Here (and throughout what follows) λ > 0 denotes any eigenvalue of ATA and x: R+ → R now  denotes the corresponding coordinate (function) of the solution of (VM-DIN-AVD) in an eigenspace  of ATA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The dynamics (Q1-VM-DIN-AVD) is a linear second-order ODE in x with non-constant  coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Similarly, (LM) can be rewritten coordinate-wise as  (α(t) + βλ) ˙xLM(t) + λxLM(t) = 0,  t ≥ 0,  (Q1-LM)  where xLM : R+ → R, and (CN) becomes  β ˙xN(t) + xN(t) = 0,  t ≥ 0,  (Q1-CN)  where again, xN : R+ → R is one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Observe in particular that (CN) and (LM) are now  first-order linear ODEs, whose solutions have the closed forms, for all t ≥ 0,  xN(t) = x0e− t  β  and  xLM(t) = x0 exp  �  −  � t  0  λ  α(s) + βλ ds  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (10)  Since the minimizer is x⋆ = 0, we directly see that xN converges exponentially fast to x⋆, with a rate  independent of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The rate of xLM depends however on λ and how fast α vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Unfortunately, except for some special choices of ε and α (see [10]), one cannot solve the second-order  linear ODE (Q1-VM-DIN-AVD) in closed form in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Additionally, it is hopeless to circumvent  the difficulty by finding a closed form for ∇f(x), accordingly to what we did in Section 3, since here  ∇f(x) = λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' In order to study the speed of convergence of x despite not having access to a closed  form, we will approximate it with a controlled error, via a method that we now present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2  The Liouville–Green Method  In what follows, we rely on the Liouville–Green method [30, 26], a technique for obtaining non-  asymptotic approximations to solutions of linear second-order ODEs with non-constant coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  First, we give the intuition behind the method, following the presentation of [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Consider the differ-  ential equation  ¨z(t) − r(t)z(t) = 0,  t ≥ 0,  (11)  5This can be generalized to the case where AT A is only semi-definite by considering orthogonal projections on an  eigenspace spanned by the positive eigenvalues of AT A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  11  where r is real-valued, positive, and twice continuously differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Any linear second-order ODE  can be reformulated in the form (11), see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='5 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Since for all t ≥ 0, r(t) ̸= 0, we can use  the changes of variables τ =  � t  0  �  r(s) ds and w = r1/4z and show that w is solution to  ¨w(τ) − (1 + ψ(τ))w(τ) = 0,  t ≥ 0,  (12)  where6 ψ(τ) = 4r(t)r′′(t)−5r′(t)2  16r(t)3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The LG method consists in neglecting the term ψ(τ) in (12), which  simply yields two approximate solutions ˆw1(τ) = eτ and ˆw2(τ) = e−τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Expressing this in terms of z  and t, we obtain  ˆz1(t) = r(t)−1/4 exp  �� t  0  �  r(s) ds  �  and  ˆz2(t) = r(t)−1/4 exp  �� t  0  −  �  r(s) ds  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (13)  Those are the LG approximations of the solutions of (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' They are formally valid on any interval  [0, T], T > 0 when ψ is “not too large”, provided that √r is integrable on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' There exists other (but less intuitive) ways to derive the approximations above, which  allow for generalization to higher-order linear ODEs, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=', [14, Chapter 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  The advantage of this approach is the possibility to estimate the error made using (13) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' the true  solutions of (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' This is expressed in the following theorem which gathers results from [15, 36, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2 (Olver [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Let r: R+ → R be a real, positive, twice continuously differentiable func-  tion, and define ϕ(t) = 4r(t)r′′(t)−5r′(t)2  16r(t)5/2  for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Then for any T > 0, the differential equation,  ¨z(t) − r(t)z(t) = 0,  t ∈ [0, T],  (14)  has two real and twice continuously differentiable solutions defined for all t ∈ [0, T] by,  z1(t) = 1 + δ1(t)  r(t)1/4 exp  �� t  0  �  r(s) ds  �  and  z2(t) = 1 + δ2(t)  r(t)1/4 exp  �  −  � t  0  �  r(s) ds  �  ,  where |δ1(t)| ≤ exp  �1  2  � t  0  |ϕ(s)| ds  �  − 1 and |δ2(t)| ≤ exp  �  −1  2  � T  t  |ϕ(s)| ds  �  − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' If in addition  � +∞  0  |ϕ(s)| ds < +∞, then the results above also hold for T = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We make the following remarks regarding the above result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  – Note that z1 and z2 in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2 are exact solutions to (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The LG approximations ˆz1 and ˆz2  are obtained by neglecting the unknown functions δ1 and δ2 in z1 and z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The theorem gives a non-  asymptotic bound for the errors |z1(t) − ˆz1(t)| and |z2(t) − ˆz2(t)|, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  – Since we assumed r to be twice continuously differentiable and positive, ϕ is continuous, so it is  integrable except maybe for t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  – For the sake of simplicity, the formulation of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2 slightly differs from that in [35], the original  formulation can be recovered by a change of variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  6We express ψ(τ) using t for the sake of readability, using the one-to-one correspondence between τ and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  12  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='3  Liouville–Green Approximation of (VM-DIN-AVD)  We now proceed to make use of the LG method for approximating the solutions of (Q1-VM-DIN-AVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  The reader only interested in the result can jump directly to the Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We first make the following  assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The functions α and ε are three times continuously differentiable, and ε0 is such that  ∀t ≥ 0, ε0 <  (βλ)2  2|α′(t)|+4λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The condition on ε0 in Assumption 3 is only technical, so that r defined below is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  It can be easily satisfied since |α′(t)| is uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Indeed, α is non-increasing and non-  negative (see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1), from which one can deduce that  � +∞  0  |α′(s)| ds ≤ α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  We now rewrite (Q1-VM-DIN-AVD) in the form (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Suppose that Assumption 3 holds, and let x be the solution of (Q1-VM-DIN-AVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' For  all t ≥ 0, define  p(t) = α(t) + βλ  ε(t)  and  r(t) = p(t)2  4  + p′(t)  2  −  λ  ε(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (15)  Then, p and r are twice continuously differentiable, r is positive and the function y defined for all t ≥ 0  by y(t) = x(t) exp  �� t  0  p(s)  2 ds  �  is a solution to  ¨y(t) − r(t)y(t) = 0,  t ≥ 0,  (16)  with initial condition (y(0), ˙y(0)) = (x0, ˙x0 + p(0)  2 x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We first check that for all t ≥ 0, r(t) is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Let t > 0,  r(t) > 0 ⇐⇒ (α(t) + βλ)2  4ε(t)2  + α′(t)  2ε(t) − (α(t) + βλ)ε′(t)  ε(t)2  −  λ  ε(t) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (17)  Since ε′(t) ≤ 0 and α′(t) ≤ 0, one can check that a sufficient condition for (17) to hold is,  r(t) > 0 ⇐= (α(t) + βλ)2  4  >  �|α′(t)|  2  + λ  �  ε(t) ⇐=  (βλ)2  2|α′(t)| + 4λ > ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  So under Assumption 3, for all t ≥ 0, r(t) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We then check that y is indeed solution to (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Let  t > 0,  ˙y(t) = p(t)  2 x(t) exp  �� t  0  p(s)  2  ds  �  + ˙x(t) exp  �� t  0  p(s)  2  ds  �  ,  and  ¨y(t) = exp  �� t  0  p(s)  2  ds  � ��p(t)2  4  + p′(t)  2  �  x(t) + p(t) ˙x(t) + ¨x(t)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Since x solves (Q1-VM-DIN-AVD), it holds that ¨x(t) = −p(t) ˙x(t) −  λ  ε(t)x(t), so,  ¨y(t) = exp  �� t  0  p(s)  2  ds  � �p(t)2  4  + p′(t)  2  −  λ  ε(t)  �  x(t)  =  �p(t)2  4  + p′(t)  2  −  λ  ε(t)  �  y(t) = r(t)y(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  13  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='5 gives a reformulation of (Q1-VM-DIN-AVD) suited to apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' To use the  theorem for all t ≥ 0, we need to ensure that ϕ(t) = 4r(t)r′′(t)−5r′(t)2  16r(t)5/2  is integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' To this aim we make  the following assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The functions ε and α have first, second and third-order derivatives that are integrable  on [0, +∞[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' In addition, lim  t→∞ ε(t) = 0 and ε′(t)2/ε(t) is integrable on [0, +∞[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Assumption 4 holds for most decays used in practice, with in particular any polynomial  decay of the form  ε0  (t+1)a and  α0  (t+1)b, a ∈ N \\ {0} and b ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Note that ε and α need not be integrable  and α can even be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  The next lemma states the integrability of ϕ on [0, +∞[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Under Assumption 3 and 4,  � +∞  0  |ϕ(s)| ds < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  The proof of this lemma, relies on relatively simple arguments but involves long computations and is  thus postponed to Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We can now use Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2 to obtain an exact form for the solution  of (Q1-VM-DIN-AVD) based on the LG approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Suppose that Assumptions 3 and 4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' There exists A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' B ∈ R such that x(0) = x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  ˙x(0) = ˙x0 and for all t ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' the solution of (Q1-VM-DIN-AVD) is  x(t) = A1 + δ1(t)  r(t)1/4  �  α(t) + βλ  �  ε(t)  exp  �� t  0  −  λ  α(s) + βλ −  λ2ε(s)  (α(s) + βλ)3 + o(ε(s)) ds  �  +B 1 + δ2(t)  r(t)1/4  �  ε(t)  �  α(t) + βλ  exp  �� t  0  −α(s) + βλ  ε(s)  +  λ  α(s) + βλ +  λ2ε(s)  (α(s) + βλ)3 + o(ε(s)) ds  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (18)  where for all t ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  |δ1(t)| ≤ exp  �1  2  � t  0  |ϕ(s)| ds  �  − 1  and  |δ2(t)| ≤ exp  �  −1  2  � +∞  t  |ϕ(s)| ds  �  − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (19)  Thanks to the bounds (19), we now have an approximation of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We will use it in particular to compare  x asymptotically to the solutions of (Q1-LM) and (Q1-CN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Before this, we prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Let x be the solution of (Q1-VM-DIN-AVD) define p, r as in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Let us also  define y(t)  def  = x(t) exp  �� t  0  p(s)  2 ds  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' According to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='5, r is positive and y is solution to (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Then, from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='7,  � T  t |ϕ(s)| ds < +∞, so we can apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2 to y on [0, +∞[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Therefore,  there exists A, B ∈ R, such that ∀t ≥ 0,  y(t) = A1 + δ1(t)  r(t)1/4 exp  �� t  0  �  r(s) ds  �  + B 1 + δ2(t)  r(t)1/4 exp  �� t  0  −  �  r(s) ds  �  ,  where A and B are determined by the initial conditions, and δ1, δ2 are such that (19) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Going back to x(t) = y(t) exp  �� t  0 − p(s)  2 ds  �  , we obtain that for all t ≥ 0,  x(t) = A1 + δ1(t)  r(t)1/4 exp  �� t  0  −p(s)  2  +  �  r(s) ds  �  + B 1 + δ2(t)  r(t)1/4 exp  �� t  0  −p(s)  2  −  �  r(s) ds  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (20)  14  It now remains to expand the terms in the two exponentials in (20) in order to obtain (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' To this aim,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  we approximate  �  r(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' let s ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='r(s) = p(s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1 + 2p′(s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='p(s)2 −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='4λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='ε(s)p(s)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='= p(s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1 + p′(s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='p(s)2 −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='ε(s)p(s)2 − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�2p′(s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='p(s)2 −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='4λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='ε(s)p(s)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='+ o(ε(s)2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='= p(s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='+ p′(s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2p(s) −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='ε(s)p(s) − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='� 2p′(s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='p(s)3/2 −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='4λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='ε(s)p(s)3/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='+ o(ε(s))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='= p(s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='p′(s)ε(s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2(α(s) + βλ) −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='α(s) + βλ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2p′(s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='p(s)3/2 −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='4λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='ε(s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='(α(s) + βλ)3/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='+ o(ε(s))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='= p(s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='α′(s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2(α(s) + βλ) − ε′(s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2ε(s) −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='α(s) + βλ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2p′(s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='p(s)3/2 −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='4λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='ε(s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='(α(s) + βλ)3/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='+ o(ε(s))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='(21)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='Focusing on the first exponential term in (20),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' we deduce from (21) that for all t ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  exp  �� t  0  −p(s)  2  +  �  r(s) ds  �  = exp  �  �  � t  0  α′(s)/2  α(s) + βλ − ε′(s)  2ε(s) −  λ  α(s) + βλ − 1  16  �  2p′(s)  p(s)3/2 −  4λ  �  ε(s)  (α(s) + βλ)3/2  �2  + o(ε(s)) ds  �  �  =  �  α(t) + βλ)  √α0 + βλ  √ε0  �  ε(t)  exp  �  �  � t  0  −λ  α(s) + βλ − 1  16  �  2p′(s)  p(s)3/2 −  4λ  �  ε(s)  (α(s) + βλ)3/2  �2  + o(ε(s)) ds  �  �  =  �  α(t) + βλ)  √α0 + βλ  √ε0  �  ε(t)  exp  �� t  0  −λ  α(s) + βλ −  λ2ε(s)  (α(s) + βλ)3 + o(ε(s)) ds  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  where the last line relies on further computations postponed to Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1 in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Performing  the exact same type of computations on exp  �� t  0 − p(s)  2 −  �  r(s) ds  �  , and up to redefining A and B so  as to encompass all the constants, we obtain (18) and the result is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='4  Comparison of x with xLM and xN  We now have an expression for x which is almost explicit: we do not know δ1 and δ2 in closed form,  but they are uniformly bounded (by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We will now compare the asymptotic behavior of (18)  with those of the solutions of (Q1-LM) and (Q1-CN) that we denoted xLM and xN respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Our  main result of Section 4 is the following, where ∼+∞ denotes the asymptotic equivalence7 between two  functions as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  7Two real-valued functions g1 and g2 are asymptotically equivalent in +∞ if and only if limt→∞  g1(t)  g2(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  15  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Let x be the solution of (Q1-VM-DIN-AVD), given in (18), and xLM and xN whose  closed forms are stated in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Under Assumptions 3 and 4, there exists C > 0 such that the following  asymptotic equivalences hold:  x(t) ∼+∞ xLM(t)C exp  �� t  0  −  λ2ε(s)  (α(s) + βλ)3 + o(ε(s)) ds  �  ,  and  x(t) ∼+∞ xN(t)C exp  �� t  0  α(s)  β(α(s) + βλ) −  λ2ε(s)  (α(s) + βλ)3 + o(ε(s)) ds  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (22)  As a consequence, the convergence of x to x⋆ is:  (i) Faster than that of xLM if ε is non-integrable and as fast otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (ii) Slower than that of xN if α is non-integrable and as fast if α is integrable, in the case where  ∀t ≥ 0, α(t) > ε(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (iii) Faster than that of xN if ε is non-integrable and as fast if ε is integrable, in the case where  ∀t ≥ 0, α(t) < ε(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  While the results of Section 3 were related to the closeness of (VM-DIN-AVD) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' (CN) and (LM)  from a control perspective, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='9 provides a different type of insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' First, the results are  asymptotic, so they only allow to control (VM-DIN-AVD) for large t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' They provide however a clear  understanding of the nature of the solutions of (VM-DIN-AVD) and their convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The conclusions  (summarized in Table 1) are in accordance with what we would expect: when the viscous damping is  larger than the variable mass, (VM-DIN-AVD) behaves more like the Levenberg–Marquardt method  than the Newton one, but it actually becomes an accelerated Levenberg–Marquardt dynamics when ε  is non-integrable but vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' However, when the variable mass ε is larger than α, the dynamics is  closer to the one of the Newton method, and can actually be an accelerated Newton dynamics, again for  non-integrable ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' This is analogous to the necessary condition that α must be non-integrable in order  to accelerate first-order methods in convex optimization (see [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We conclude this section by proving  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Thanks to Assumptions 3 and 4, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='8 tells us that x has the form (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  We now analyze the two terms in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  First, we know from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='8 that δ1(0) = 0 and limt→+∞ δ2(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' In addition, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='7,  δ1 and δ2 are uniformly bounded by some positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Then r(t)−1/4 decays asymptotically like  �  ε(t) and α is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' So A 1+δ1(t)  r(t)1/4  √  α(t)+βλ  √  ε(t)  is asymptotically equivalent to some constant c1 ∈ R as  t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Similarly, B 1+δ2(t)  r(t)1/4  √  ε(t)  √  α(t)+βλ is equivalent to c2ε(t), with c2 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  We now analyze the “exponential factors” in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' On the one hand,  λ  α(s)+βλ +  λ2ε(s)  (α(s)+βλ)3 + o(ε(s))  converges to 1  β as s → ∞, while α(s)+βλ  ε(s)  diverges to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Therefore, we deduce that,  exp  �� t  0  −α(s) + βλ  ε(s)  +  λ  α(s) + βλ +  λ2ε(s)  (α(s) + βλ)3 + o(ε(s)) ds  �  = o  �  exp  �� t  0  −  λ  α(s) + βλ −  λ2ε(s)  (α(s) + βλ)3 + o(ε(s)) ds  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  16  As a consequence, the second term in (18) will decrease to 0 faster than the first-one (let alone the  additional ε(t) decay that we have just discussed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The asymptotic behavior of x will thus be governed  by the first term in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Let us now focus on the first term in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Observe that exp  �� t  0 −  λ  α(s)+βλ ds  �  is exactly the decay  of xLM in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Thus, we have proved that there exists C > 0, such that the following asymptotic  equivalence holds,  A1 + δ1(t)  r(t)1/4  �  α(t) + βλ  �  ε(t)  exp  �� t  0  −  λ  α(s) + βλ −  λ2ε(s)  (α(s) + βλ)3 + o(ε(s)) ds  �  ∼+∞ xLM(t)C exp  �� t  0  −  λ2ε(s)  (α(s) + βλ)3 + o(ε(s)) ds  �  ,  which proves the first part of (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The second equivalence in (22) is obtained using the following  identity,  � t  0  −  λ  α(s) + βλ ds =  � t  0  − 1  β +  α(s)  β(α(s) + βλ) ds = − t  β +  � t  0  α(s)  β(α(s) + βλ) ds  (23)  and e−t/β is precisely the rate at which xN decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' So (22) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  It finally remains to deduce the conclusions of the theorem from (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  – Regarding the comparison with xLM, the integral  � t  0 −  λ2ε(s)  (α(s)+βλ)3 + o(ε(s)) ds converges if and only  if ε is integrable on [0, +∞[, and diverges to −∞ when ε is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' So x converges to 0 at least as fast as  xLM and faster when ε is not integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  – As for the comparison with xN, if α(s) > ε(s) ≥ 0 for all s ≥ 0, then the integral  � t  0  α(s)  β(α(s)+βλ) −  λ2ε(s)  (α(s)+βλ)3 + o(ε(s)) ds is convergent in +∞ if and only if α is integrable and diverges to +∞ when  α is non-integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' So when α is integrable, the speed of convergence of x is the same as that of  xN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' When α is not integrable, the convergence to 0 is slower but still holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Indeed, for all s ≥ 0  α(s)  β(α(s)+ 1  β ) <  α(s)  βα(s) = 1  β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Thus for all t > 0, − t  β +  � t  0  α(s)  β(α(s)+βλ) ds < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  – Finally, the comparison with xN in the case ε(s) > α(s) is exactly the same as the comparison with  xLM using (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  5  Numerical Experiments  We present two set of experiments that illustrate our main results from Sections 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We first detail  the general methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1  Methodology  We compare the solutions of (CN), (LM) and (VM-DIN-AVD) obtained for strongly-convex functions  in dimension n = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Since closed-form solutions are not available, they are estimated via discretiza-  tion schemes with small step-sizes γ = 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We used Euler semi-explicit schemes, where a linear  system is solved at each iteration, for the sake of stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The resulting algorithms are detailed in  Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  17  0  10  20  30  40  50  t  10  20  10  17  10  14  10  11  10  8  10  5  10  2  101  Distance to CN  (t) =  0/log t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  (t) =  0/ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  (t) =  0/t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  (t) =  0/t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  (t) =  0e  t/ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  0  10  20  30  40  50  t  Distance to LM  0  10  20  30  40  50  t  Speed of convergence  Levenberg-Marquardt  Continuous Newton  0  10  20  30  40  50  t  10  21  10  17  10  13  10  9  10  5  10  1  Distance to CN  (t) =  0/log t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  (t) =  0/ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  (t) =  0/t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  (t) =  0/t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  (t) =  0e  t/ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  0  10  20  30  40  50  t  Distance to LM  0  10  20  30  40  50  t  Speed of convergence  Levenberg-Marquardt  Continuous Newton  Figure 2:  Comparison of the solutions xN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' xLM and x of (CN),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' (LM) and (VM-DIN-AVD) respec-  tively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' for a strongly-convex function of the form f(x) = e−∥x∥2 + 1  2∥Ax∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Left figures: distance  ∥x(t) − xN(t)∥ versus time t, each curve corresponds to a different choice of ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' middle figures: dis-  tance ∥x(t) − xLM(t)∥, again for several ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Right figures: distance to the optimum x⋆ for reference,  xN and xLM are in dotted and dashed lines, other curves correspond to (VM-DIN-AVD) for several  choices of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The brown curve is often hidden behind the purple (and sometimes the pink) curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Top  and bottom rows show results respectively for non-integrable and integrable viscous dampings α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The  theoretical bounds from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='8 are only displayed on Figure 4 below, for the sake of readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2  First Experiment: Distance between Trajectories  We begin with an empirical validation of the results of Section 3 on the distance between x, xLM  and xN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Each of Figures 2, 3 and 4 (as well as Figures 6 in Appendix D) corresponds to a different  strongly-convex function, specified below its corresponding figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' In order to ensure strong convexity,  each function contains a quadratic term of the form ∥Ax∥2, where A is symmetric positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Several observations can be made from the numerical results, but we first note on the right plots of  Figures 2 to 4 that xN always converges asymptotically linearly (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=', exponentially fast).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' This is also  the case for x and xLM in some (but not all) cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' This is important because ∥x(t) − xN(t)∥ ≤  ∥x(t) − x⋆∥ + ∥xN(t) − x⋆∥, so if both x and xN converge linearly, then the bounds of Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1  and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='8 need not be tight asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' That being said, the strength of these results is to be non-  asymptotic and this is highlighted by the experiments as we now explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Looking at the left and middle plots of Figures 2, 3 and 4, we observe that Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='8 seem  empirically validated, since the distances ∥x(t) − xN(t)∥ and ∥x(t) − xLM(t)∥ decrease relatively fast  to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Again,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' when x converges rapidly to x⋆ this is not very insightful,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' however,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' the main interest  18  0  20  40  60  80  100  120  140  t  10  64  10  55  10  46  10  37  10  28  10  19  10  10  10  1  Distance to CN  (t) =  0/log t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  (t) =  0/ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  (t) =  0/t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  (t) =  0/t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  (t) =  0e  t/ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  0  20  40  60  80  100  120  140  t  Distance to LM  0  20  40  60  80  100  120  140  t  Speed of convergence  Levenberg-Marquardt  Continuous Newton  0  20  40  60  80  100  120  140  t  10  64  10  55  10  46  10  37  10  28  10  19  10  10  10  1  Distance to CN  (t) =  0/log t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  (t) =  0/ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  (t) =  0/t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  (t) =  0/t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  (t) =  0e  t/ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  0  20  40  60  80  100  120  140  t  Distance to LM  0  20  40  60  80  100  120  140  t  Speed of convergence  Levenberg-Marquardt  Continuous Newton  Figure 3: Similar experiment and figures as those described in Figure 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' but for the function f(x) =  log (�n  i=1 exi + e−xi) + 1  2∥Ax∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  of our theorems appears on the left of Figures 3 and 4: the blue and green curves, corresponding to  slowly decaying choices of ε, converge more slowly than other trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' However, when taking  faster decays, we recover fast convergence and closeness to xN (this is particularly true for the purple  curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Very similar observations are made w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' xLM on the middle plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Despite not being stated in  the theorems of Section 3, the experiments match the intuition that when ε > α, x may be closer to  xN and when ε ≤ α, x would rather be closer to xLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' This is more noticeable on the top rows of the  figures, where α is not integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Figure 4 suggests that the bounds in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='8 are rather tight for small t, since, for example, the  blue and green curves on the left show a relatively slow vanishing of ∥x(t)−xN(t)∥ for slowly decaying  ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The experiments show that the bounds seem however often too pessimistic for large t, for which the  second part of our study provides better insights (see Section 4 and below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Interestingly, slow decays  of ε may sometimes result in faster convergence for x than fast decays (and also faster convergence  than xLM), notably on Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We also note that ε(t) = ε0/t combined either with α(t) = α0/t or  α(t) = α0/t2 seems to always yield fast convergence on these experiments (and sometimes the fastest  of all dynamics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='3  Second Experiment: Empirical Validation of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='9  We now turn our attention to the solutions x, xN and xLM for a quadratic function of the form f(y) =  1  2∥Ay∥2, y ∈ Rn, and for several choices of ε and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The results in Figure 5 exactly match the expected  behavior summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' looking first at the right-hand side of Figure 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' x is as fast  19  0  20  40  60  80  100  120  140  t  10  46  10  39  10  32  10  25  10  18  10  11  10  4  103  Distance to CN  (t) =  0/log t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  (t) =  0/ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  (t) =  0/t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  (t) =  0/t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  (t) =  0e  t/ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  0  20  40  60  80  100  120  140  t  Distance to LM  0  20  40  60  80  100  120  140  t  Speed of convergence  Levenberg-Marquardt  Continuous Newton  0  20  40  60  80  100  120  140  t  10  59  10  50  10  41  10  32  10  23  10  14  10  5  104  Distance to CN  (t) =  0/log t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  (t) =  0/ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  (t) =  0/t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  (t) =  0/t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  (t) =  0e  t/ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  0  20  40  60  80  100  120  140  t  Distance to LM  0  20  40  60  80  100  120  140  t  Speed of convergence  Levenberg-Marquardt  Continuous Newton  Figure 4: Similar experiment and figures as those described in Figure 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' but for the function f(x) =  ∥x∥50 + 1  2∥Ax∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The thin “dash dotted” curves represent the theoretical bounds from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='8 for  each choice of (ε, α) considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  as the corresponding8 xLM when ε is not integrable and regardless of α, and x is faster when ε is non-  integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Then on the left-hand side, when comparing to xN, x is slower in settings where α is larger  than ε and non-integrable (red curves), or almost as fast when α is integrable (pink curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' However,  acceleration w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' to xN is indeed achieved for non-integrable ε regardless of α (first-two blue curves),  and the rate is the same as that of xN when ε is integrable (third blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  6  Conclusions and Perspectives  We introduced a general ODE (VM-DIN-AVD) featuring variable mass, and provided a deep under-  standing on the behavior of its solutions w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' time dependent control parameters ε and α, both, asymp-  totically and non-asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We can conclude that (VM-DIN-AVD) is indeed of (regularized)  Newton type, since it can be controlled to be close to both (CN) and (LM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Yet we also showed  that (VM-DIN-AVD) fundamentally differs from the other two dynamics in its nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' In particular,  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='9 and the numerical experiments emphasized that ε and α can accelerate (or slow down)  (VM-DIN-AVD) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' (CN) and (LM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We also note that our bounds in Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='8 seem  relatively tight, in particular for functions with large gradients (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Our contribution yields  a complete and satisfying picture on the relation between the three systems, which was only partially  understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We believe that our results build a strong foundation for the development of algorithms  that combine the best properties of first- and second-order optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  8That is, the solution of (LM) for the same α as that considered for (VM-DIN-AVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  20  10  20  30  40  50  t  10  21  10  18  10  15  10  12  10  9  10  6  10  3  100  x(t)  x  Continuous Newton  (t) =  0/ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  (t) =  0/t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  (t) =  0/t3/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  (t) =  0/t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/ t  (t) =  0/t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  (t) =  0/t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t3/2  10  20  30  40  50  t  10  21  10  18  10  15  10  12  10  9  10  6  10  3  100  x(t)  x  LM:  0/t  (t) =  0/ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  (t) =  0/t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  LM:  0/t2  (t) =  0/ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  (t) =  0/t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  Figure 5: Numerical validation of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='9: distance to the optimum x⋆ as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' on  a quadratic function f(x) = 1  2∥Ax∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Left: speed comparison w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' (CN) for several choices of ε and  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Right: Comparison with LM in for α integrable or not and several choices of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Shades of blue  represent cases where ε(t) > α(t) while shades of red represent the opposite setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  As for future work, we showed that (VM-DIN-AVD) is promising from an optimization perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  So far we approximated solutions of (VM-DIN-AVD) via schemes that required solving a linear system  at each iteration (this is also true for (CN) and (LM)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Our new understanding on (ε, α) paves the way  towards designing new Newton-like algorithms with a significantly reduced computational cost, which  is crucial for large-scale optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Another open question is whether it is possible to preserve the  properties evidenced in this work when ε is defined in a closed-loop manner (formally depending on  x rather than on t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Finally, it would be worth investigating how the current work can be extended to  general convex and/or non-smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Acknowledgment  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Castera, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Fadili and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Ochs are supported by the ANR-DFG joint project TRINOM-DS under  number ANR-20-CE92-0037-01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The numerical experiments were made thanks to the development  teams of the following libraries: Python [38], Numpy [43] and Matplotlib [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  21  Appendices  A  Equivalent First-order System and Global Existence of Solutions  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1  First-order Equivalent Formulation  We reformulate (VM-DIN-AVD) as a system of ODE involving only first-order time derivatives and  the gradient of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' For this purpose, notice that for all t > 0 (VM-DIN-AVD) can be rewritten as,  d  dt [ε(t) ˙x(t)] + β d  dt∇f(x(t)) + α(t) ˙x(t) − ε′(t) ˙x(t) + ∇f(x(t)) = 0,  t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (24)  We then integrate (24) for all t ≥ 0,  ε(t) ˙x(t) + β∇f(x(t)) − ε0 ˙x0 − β∇f(x0) +  � t  0  (α(s) − ε′(s)) ˙x(s) + ∇f(x(s)) ds = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (25)  For all t ≥ 0, we define the variable,  z(t) =  � t  0  (α(s) − ε′(s)) ˙x(s) + ∇f(x(s)) ds − ε0 ˙x0 − β∇f(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  We differentiate z, for all t > 0, ˙z(t) = (α(t) − ε′(t)) ˙x(t) + ∇f(x(t)), so that we can rewrite (25) as,  �  ε(t) ˙x(t) + β∇f(x(t)) + z(t) = 0  ˙z(t) − (α(t) − ε′(t)) ˙x(t) − ∇f(x(t)) = 0  ,  t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  We substitute the first line in the second-one,  �  ε(t) ˙x(t) + β∇f(x(t)) + z(t) = 0  β ˙z(t) − β(α(t) − ε′(t) − 1  βε(t)) ˙x(t) + z(t) = 0  ,  t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (26)  To ease the readability, we recall the notation ν(t) = α(t) − ε′(t) − 1  βε(t) from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Then define  for all t ≥ 0, y(t) = z(t) − ν(t)x(t), and differentiate, ˙y(t) = ˙z(t) − ν(t) ˙x(t) − ν′(t)x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We finally  rewrite (26) as,  �  ε(t) ˙x(t) + β∇f(x(t))  +ν(t)x(t) + y(t) = 0  ˙y(t) + ν′(t)x(t)  + ν(t)  β x(t) + 1  βy(t) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  which is (gVM-DIN-AVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Finally, the initial condition on y is  y(0) = z(0) − ν(0)x(0) = −ε0 ˙x0 − β∇f(x0) − (α0 − ε′  0 − 1  β ε0)x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Notice that the quantity ν(t) = α(t) − ε′(t) − 1  βε(t) involved in (gVM-DIN-AVD) also  plays a key role in our analysis of Section 3, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=', (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' In particular the sign of ν(t) changes the  nature of (VM-DIN-AVD) and is related to Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  22  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2  Local Solutions are Global  Using the formulation (gVM-DIN-AVD), we proved local existence and uniqueness of solutions of  (VM-DIN-AVD) in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Using the same notations, we justify that the local solution (x, y) actually  exists globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' According to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='4, the Lyapunov function U(t) = ε(t)  2 ∥ ˙x(t)∥2 +f(x(t))−f(x⋆)  is non-negative and decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Thus, it is uniformly bounded on R+ and the same holds for t �→  f(x(t)) since for all t ≥ 0, U(t) ≥ f(x(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Then, f is coercive by assumption, so x is uniformly  bounded on R+ (otherwise f(x) could not remain bounded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We now prove that y is also uniformly  bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' From (gVM-DIN-AVD), for all t > 0, ˙y(t) = − 1  βy(t) − ( ν(t)  β + ν′(t))x(t) so we can use the  following integrating factor,  y(t) = e− t  β y0 − e− t  β  � t  0  1  β e  s  β (ν(s) + βν′(s))x(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Using triangle inequalities, for all t ≥ 0,  ∥y(t)∥ ≤ e− t  β ∥y0∥ + sup  s≥0  ∥(ν(s) + βν′(s))x(s)∥e− t  β  � t  0  1  β e  s  β ds ≤ ∥y0∥ + sup  s≥0  ∥(ν(s) + βν′(s))x(s)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (27)  Using the definition of ε and α from Sections 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1 and 2, observe that ε, α, ε′ and α′ are all bounded  on R+, and ε′′ is assumed to be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' So ν and ν′ are bounded, and since we also proved that x  is uniformly bounded on R+, we deduce from (27) that y is uniformly bounded as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Hence, the  unique local solution (x, y) is global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  B  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='8  This section is devoted to proving the general result of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Fix some constants c1, c2 > 0 and let  ε and α such that Assumption 2 is satisfied with these constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Let x be the corresponding solution  of (VM-DIN-AVD), xN, and xLM that of (CN) and (LM), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Following the same arguments  as in the beginning of the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1, for all t ≥ 0, x(t), xN(t) and xLM(t) belong to the  bounded set K0 defined in that proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Since f is µ-strongly convex on K0, the proof relies again on  bounding difference of gradients, indeed, for all t ≥ 0,  ∥x(t) − xN(t)∥ ≤ 1  µ∥∇f(x(t)) − ∇f(xN(t))∥ and ∥x(t) − xLM(t)∥ ≤ 1  µ∥∇f(x(t)) − ∇f(xLM(t))∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (28)  Recall also that the closed form of ∇f(xN) is given in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Expressing ∇f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  We follow the exact same steps as in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1 to obtain the  expression of ∇f(x) given in (5), which we recall, for all t ≥ 0,  β∇f(x(t)) = βe− t  β ∇f(x0) + e− t  β ε0 ˙x0 − ε(t) ˙x(t) +  � t  0  e  s−t  β  � 1  β ε(s) + ε′(s) − α(s)  �  ˙x(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  23  Here we do not assume any relation between ε and α, and we thus need to find a more suitable expres-  sion for ∇f(x(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We first expand the terms in the integral, for all t ≥ 0,  β∇f(x(t)) = βe− t  β ∇f(x0) + e− t  β ε0 ˙x0 − ε(t) ˙x(t)  +  � t  0  e  s−t  β  � 1  β ε(s) + ε′(s)  �  ˙x(s) ds −  � t  0  e  s−t  β α(s) ˙x(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (29)  Then, for all s ≥ 0, we have the identity,  e  s  β ˙x(s) = e  s  β ˙x(s) + 1  β e  s  β x(s) − 1  β e  s  β x(s) = d  ds(e  s  β x(s)) − 1  β e  s  β x(s),  (30)  which we use to perform an integration by part on the last integral in (29),  � t  0  e  s  β α(s) ˙x(s) ds =  �  α(s)e  s  β x(s)  �t  0 −  � t  0  �  α′(s) + α(s)  β  �  e  s  β x(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Therefore,  e− t  β  � t  0  e  s  β α(s) ˙x(s) ds = α(t)x(t) − e− t  β α0x0 −  � t  0  e  s−t  β  �  α′(s) + α(s)  β  �  x(s) ds,  (31)  and we can substitute in (29),  β∇f(x(t)) = βe− t  β ∇f(x0) + e− t  β ε0 ˙x0 − ε(t) ˙x(t) +  � t  0  e  s−t  β  � 1  β ε(s) + ε′(s)  �  ˙x(s) ds  − α(t)x(t) + e− t  β α0x0 +  � t  0  e  s−t  β  �  α′(s) + α(s)  β  �  x(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (32)  Uniform boundedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  In view of exploiting (32), we recall that for all (ε, α), x is uniformly  bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' So there exists R > 0 such that for all (ε, α), the corresponding solution x of (VM-DIN-AVD)  is such that  sup  t≥0  ∥x(t)∥ ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (33)  We are now in position to prove (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Distance from x to xN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  We first gather (4) and (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' For all t ≥ 0,  β∇f(x(t)) − β∇f(xN(t)) = e− t  β ε0 ˙x0 − ε(t) ˙x(t) +  � t  0  e  s−t  β  � 1  β ε(s) + ε′(s)  �  ˙x(s) ds  + e− t  β α0x0 − α(t)x(t) +  � t  0  e  s−t  β  �  α′(s) + α(s)  β  �  x(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  We then use (28) and triangle inequalities,  βµ∥x(t) − xN(t)∥ ≤ e− t  β ε0∥ ˙x0∥ + ε(t)∥ ˙x(t)∥ +  � t  0  e  s−t  β  ����  ε(s)  β  + ε′(s)  ���� ∥ ˙x(s)∥ ds  + e− t  β α0∥x0∥ + α(t)∥x(t)∥ +  � t  0  e  s−t  β  ����  α(s)  β  + α′(s)  ���� ∥x(s)∥ ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (34)  24  By Assumption 2, for all s ≥ 0, | ε(s)  β + ε′(s)| ≤ ( 1  β + c1)ε(s) and | α(s)  β  + α′(s)| ≤ ( 1  β + c2)α(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We  then use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='5 (denoting by C > 0 the constant stated in the lemma) on the first line of (34), and  we use the boundedness (33) on the second line to obtain,  βµ∥x(t) − xN(t)∥ ≤ e− t  β ε0∥ ˙x0∥ + C  �  ε(t) + C  � 1  β + c1  � � t  0  e  s−t  β �  ε(s) ds  + e− t  β α0∥x0∥ + Rα(t) + R  � 1  β + c2  � � t  0  e  s−t  β α(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  This proves (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Expressing ∇f(xLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  We now repeat previous arguments but for (LM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' First, (LM) is equivalent to  d  dt∇f(xLM(t)) + 1  β ∇f(xLM(t)) = −α(t) ˙xLM(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  So using an integrating factor one can check that for all t ≥ 0,  ∇f(xLM(t)) = e− t  β ∇f(x0) − e− t  β  � t  0  1  β e  s  β α(s) ˙xLM(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  We can then follow exactly steps (30) to (31) so as to obtain,  e− t  β  � t  0  e  s  β α(s) ˙xLM(s) ds = α(t)xLM(t) − e− t  β α0x0 − e− t  β  � t  0  �  α′(s) + α(s)  β  �  e  s  β xLM(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Finally, remark that for all t ≥ 0,  d  dtf(xLM(t)) = −α(t)∥ ˙xLM(t)∥2 − β⟨ ˙xLM(t), ∇2f(xLM(t)) ˙xLM(t)⟩ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  So f(xLM(t)) ≤ f(x0) and using the coercivity of f as before we deduce that for all choices α,  sup  t≥0  ∥xLM(t)∥ ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (35)  Distance from x to xLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  We substract gradients,  β∇f(x(t)) − β∇f(xLM(t))) = e− t  β ε0 ˙x0 − ε(t) ˙x(t) +  � t  0  e  s−t  β  � 1  β ε(s) + ε′(s)  �  ˙x(s) ds  − α(t)(x(t) − xLM(t)) −  � t  0  e  s−t  β  �  α′(s) + α(s)  β  �  (x(s) − xLM(s)) ds,  and we proceed as before using (28), Assumption 2 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' It holds that,  βµ∥x(t) − xLM(t)∥ ≤ e− t  β ε0∥ ˙x0∥ + C  �  ε(t) + C  � 1  β + c1  � � t  0  e  1  β (s−t)�  ε(s) ds  + α(t)∥x(t) − xLM(t)∥ +  � 1  β + c2  � � t  0  e  1  β (s−t)∥x(s) − xLM(s)∥ ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Finally, using (33) and (35), for all s ≥ 0, ∥x(s) − xLM(s)∥ ≤ 2R, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  25  C  Integrability of ϕ and Additional Asymptotic Computations  Below we prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We suppose that Assumptions 3 and 4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' As stated in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='3, since ϕ is continuous,  we only need to check its integrability when t tends to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Let t > 0, we first establish some useful  identities, we omit the dependence on t for the sake of readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  p′ = α′ε − (α + βλ)ε′  ε2  ,  p′′ = α′′ε2 − 2α′ε′ε − (α + βλ)ε′′ε + 2(α + βλ)(ε′)2  ε3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Then,  r = p2  4  �  1 + 2p′  p2 − 4λ  εp2  �  = (α + βλ)2  4ε2  �  1 +  2p′ε2  (α + βλ)2 −  4λε  (α + βλ)2  �  = (α + βλ)2  4ε2  �  1 +  2α′ε  (α + βλ)2 −  2ε′  (α + βλ) −  4λε  (α + βλ)2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (36)  An important consequence of Assumption 4 is that |ε′(t)| = o(ε(t)), |ε′′(t)| = o(ε′(t)) (and the same  holds for α w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' to its derivatives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Therefore, we deduce from (36) that  r(t) ∼+∞  (α(t) + βλ)2  4ε(t)2  ,  and we note that 1/r decays at the same speed as ε2, which will be useful later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' In order to study ϕ, we  now differentiate r,  r′ = p′p  2  �  1 + 2p′  p2 − 4λ  εp2  �  + 1  4  �  2p′′ − 4(p′)2  p  + 8λp′  εp + 4λε′  ε2  �  = 2p′  p r + 1  4  �  2p′′ − 4(p′)2  p  + 8λp′  εp + 4λε′  ε2  �  ,  and  r′′ =2p′′p − (p′)2  p2  r + 2p′  p r′  + 1  4  �  2p′′′ + 4(p′)3 − 2p′′p′p  p2  + 8λp′′pε − (p′)2ε − p′pε′  ε2p2  + 4λε′′  ε2  − 8λ(ε′)2  ε3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (37)  Then, to justify that ϕ is integrable, we prove that  r′′  r3/2 and (r′)2  r5/2 are integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Since we know that 1/r  decays at the same speed as ε2, we can equivalently show that ε3r′′ and ε5(r′)2 are integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' To this  aim we fully expand all the terms in (37), and compute (r′)2, which is extremely long and involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  26  On the one hand, it holds that,  r′2ε5 =  �  �−  (α + βλ)2 �  −  4λε  (α+βλ)2 + 1 + (−2(α+βλ)ε′+2α′ε)  (α+βλ)2  �  ε′  2√ε  + (α + βλ)2√ε  4  �  −  4λε′  (α + βλ)2 +  8λα′ε  (α + βλ)3  +  2  �  − 2(α+βλ)ε′  ε  + 2α′ε  �  ε′  (α + βλ)2  +  �  4(α+βλ)ε′2  ε  − 2(α + βλ)ε′′ − 4α′ε′ + 2α′′ε  �  (α + βλ)2  − 2 (−2(α + βλ)ε′ + 2α′ε) α′  (α + βλ)3  �  �  +(α + βλ)  2  �  −  4λε  (α + βλ)2 + 1 + (−2(α + βλ)ε′ + 2α′ε)  (α + βλ)2  �  α′√ε  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='r′′ε(t)3 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='− λε′′ε + 4λα′ε′ε  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='α + βλ + 2λα′′ε2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='α + βλ −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='6λα′2ε2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='(α + βλ)2 + (α + βλ)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='−3ε′2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='ε  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='+ ε′′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='� �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='4λε  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='(α + βλ)2 − 1 + 2 ((α + βλ)ε′ − α′ε)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='(α + βλ)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='+ 2(α + βλ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='4λε  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='(α + βλ)2 − 1 + 2 ((α + βλ)ε′ − α′ε)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='(α + βλ)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='α′ε′ −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='(α + βλ)α′′ + α′2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='4λε  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='(α + βλ)2 − 1 + 2 ((α + βλ)ε′ − α′ε)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='(α + βλ)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='ε  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�(α + βλ)ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='ε  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='− α′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='ε′2 −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='(α + βλ)ε′ − α′ε  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='ε′′ − 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='−2(α + βλ)ε′2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='ε  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='+ (α + βλ)ε′′ + 2α′ε′ − α′′ε  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='+ 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2λε′ − 4λα′ε  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='α + βλ + 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�(α + βλ)ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='ε  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='− α′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='ε′ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='−2(α + βλ)ε′2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='ε  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='+ (α + βλ)ε′′ + 2α′ε′ − α′′ε  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='− 2 ((α + βλ)ε′ − α′ε) α′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='α + βλ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='− 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�6(α + βλ)ε′3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='ε  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='− 6(α + βλ)ε′ε′′ + (α + βλ)ε′′′ε − 6α′ε′2 + 3α′ε′′ε + 3α′′ε′ε − α′′′ε2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='α + βλ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='(α + βλ)ε′ − α′ε  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='α′ε′ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='(α + βλ)ε′ − α′ε  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='α′′ε + 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='−2(α + βλ)ε′2 + (α + βλ)ε′′ε + 2α′ε′ε − α′′ε2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='α′�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='α + βλ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='2λε′ − 4λα′ε  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='α + βλ + 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�(α + βλ)ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='ε  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='− α′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='ε′ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='−2(α + βλ)ε′2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='ε  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='+ (α + βλ)ε′′ + 2α′ε′ − α′′ε  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='− 2 ((α + βλ)ε′ − α′ε) α′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='α + βλ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='α′ε  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='− 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='(α + βλ)ε′ε − α′ε2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='α′2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='(α + βλ)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  27  We then analyze the integrability of each of the terms above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' By Assumption 4, ε′, ε′′ and ε′′′ are  integrable and the same goes for α′, α′′ and α′′′, which is enough to justify the integrability of almost  all the terms above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We finally see that we also need (ε′)2  ε  and (ε′)3  ε  to be integrable, which holds by  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Overall, ϕ is integrable on R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  We now state and prove the following result which was used at the end of the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Under Assumptions 3 and 4, for all s ≥ 0,  1  16  �  2p′(s)  p(s)3/2 −  4λ  �  ε(s)  (α(s) + βλ)3/2  �2  =  λ2ε(s)  (α(s) + βλ)3 + o(ε(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We omit the time dependence on s ≥ 0 for the sake of readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Using Assumption 3 we can  define and expand the following quantity,  1  16  � 2p′  p3/2 −  4λ√ε  (α + βλ)3/2  �2  =  λ2ε  (α + βλ)3 −  p′λ√ε  p3/2(α + βλ)3/2 + (p′)2  4p3  =  λ2ε  (α + βλ)3 − λ(α′ε − (α + βλ)ε′) +  1  4(α + βλ)3  �  (α′)2ε + (ε′)2  ε (α + βλ)2 − 2α′ε′(α + βλ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  Assumption 4, implies in particular that |ε′(t)| = o(ε(t)) and that α′(t) → 0, which we use in the  equality above to obtain the desired conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  D  Additional Experiments and Details  We first detail the discretization that we used for approximating the solutions of the three ODEs con-  sidered in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' We use Euler discretization schemes with fixed step-size γ > 0 and approximate  the solutions at times tk = γk, for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' For a trajectory x, we use the notation x(tk)  def  = x(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The  approximation of (CN) is obtained by explicit discretization, so that for all k ∈ N, we have,  x(k+1)  N  = x(k)  N − γ  �  β∇2f(x(k)  N )  �−1  ∇f(x(k)  N ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (38)  Then, defining εk = ε(tk) and αk = α(tk), (LM) and (VM-DIN-AVD) are obtained via Euler semi-  implicit discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The solution of (LM) is approximated by,  x(k+1)  LM  = x(k)  LM − γ  �  αkIn + β∇2f(x(k)  LM)  �−1  ∇f(x(k)  LM),  (39)  where In is the identity matrix on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The solution of (VM-DIN-AVD) is obtained similarly,  x(k+1) = x(k) +  �  (εk + γαk)In + γβ∇2f(x(k))  �−1 �  εk(x(k) − x(k−1)) − γ2∇f(x(k))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (40)  As safety check, one can see that for εk = 0, (40) is equivalent to (39), which is itself equivalent to (38)  when αk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  28  0  10  20  30  40  50  t  10  21  10  17  10  13  10  9  10  5  10  1  Distance to CN  (t) =  0/log t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  (t) =  0/ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  (t) =  0/t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  (t) =  0/t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  (t) =  0e  t/ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  0  10  20  30  40  50  t  Distance to LM  0  10  20  30  40  50  t  Speed of convergence  Levenberg-Marquardt  Continuous Newton  0  10  20  30  40  50  t  10  22  10  18  10  14  10  10  10  6  10  2  102  Distance to CN  (t) =  0/log t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  (t) =  0/ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  (t) =  0/t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  (t) =  0/t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  (t) =  0e  t/ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  0  10  20  30  40  50  t  Distance to LM  0  10  20  30  40  50  t  Speed of convergence  Levenberg-Marquardt  Continuous Newton  0  20  40  60  80  100  120  140  t  10  16  10  13  10  10  10  7  10  4  10  1  Distance to CN  (t) =  0/log t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  (t) =  0/ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  (t) =  0/t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  (t) =  0/t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  (t) =  0e  t/ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t  0  20  40  60  80  100  120  140  t  Distance to LM  0  20  40  60  80  100  120  140  t  Speed of convergence  Levenberg-Marquardt  Continuous Newton  0  20  40  60  80  100  120  140  t  10  16  10  13  10  10  10  7  10  4  10  1  Distance to CN  (t) =  0/log t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  (t) =  0/ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  (t) =  0/t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  (t) =  0/t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  (t) =  0e  t/ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  (t) =  0/t2  0  20  40  60  80  100  120  140  t  Distance to LM  0  20  40  60  80  100  120  140  t  Speed of convergence  Levenberg-Marquardt  Continuous Newton  Figure 6: Similar experiment and figures as those described in Figure 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' but for a poorly conditioned  quadratic f(x) = 1  2∥Ax∥2 (first-two rows) and the function f(x) = log (�n  i=1 exi) + 1  2∥Ax∥2 (last-two  rows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  29  References  [1] Cristian Daniel Alecsa, Szil´ard Csaba L´aszl´o, and Titus Pint¸a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' An extension of the second or-  der dynamical system that models Nesterov’s convex gradient method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Applied Mathematics &  Optimization, 84(2):1687–1716, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  [2] Felipe Alvarez, Hedy Attouch, J´erˆome Bolte, and Patrick Redont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' A second-order gradient-like  dissipative dynamical system with Hessian-driven damping: Application to optimization and me-  chanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Journal de Math´ematiques Pures et Appliqu´ees, 81(8):747–779, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  [3] Hedy Attouch and Alexandre Cabot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Asymptotic stabilization of inertial gradient dynamics with  time-dependent viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
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+page_content=' From the ravine method to the Nesterov method and vice versa: A  dynamical system perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
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+page_content=' Newton-like inertial dynamics and proximal algorithms  governed by maximally monotone operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
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+page_content=' Continuous Newton-like inertial dynamics for monotone  inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
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+page_content=' A continuous dynamical Newton-like approach to solving  monotone inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
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+page_content=' Global convergence of a closed-loop regu-  larized Newton method for solving monotone inclusions in hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
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+page_content=' Fast convex optimization via inertial dy-  namics with Hessian driven damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Journal of Differential Equations, 261(10):5734–5783,  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  [10] Hedy Attouch, Zaki Chbani, Jalal Fadili, and Hassan Riahi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
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+page_content='  Fast convex optimization via  inertial dynamics combining viscous and Hessian-driven damping with time rescaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
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+page_content='  [13] Michel Bena¨ım, Josef Hofbauer, and Sylvain Sorin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Stochastic approximations and differential  inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
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+page_content='  [14] Carl M Bender and Steven A Orszag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Asymptotic methods and perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Springer,  1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  [15] Otto Blumenthal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' ¨Uber asymptotische Integration linearer Differentialgleichungen mit Anwen-  dung auf eine asymptotische Theorie der Kugelfunktionen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Archiv der Mathematik und Physik,  19:136–174, 1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content='  30  [16] Radu Ioan Bot¸, Ern¨o Robert Csetnek, and Szil´ard Csaba L´aszl´o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Tikhonov regularization of a  second order dynamical system with Hessian driven damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
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+page_content='  [17] L´eon Brillouin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' Remarques sur la m´ecanique ondulatoire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
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+page_content='  [18] Charles George Broyden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
+page_content=' The convergence of a class of double-rank minimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFAT4oBgHgl3EQf4h4_/content/2301.08726v1.pdf'}
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diff --git a/adE0T4oBgHgl3EQf4QLf/content/tmp_files/2301.02737v1.pdf.txt b/adE0T4oBgHgl3EQf4QLf/content/tmp_files/2301.02737v1.pdf.txt
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@@ -0,0 +1,1375 @@
+Understanding the (In)Effectiveness of Content Moderation:
+A Case Study of Facebook in the Context of the U.S. Capitol Riot∗
+Ian Goldstein
+New York University
+USA
+Laura Edelson
+New York University
+USA
+Damon McCoy
+New York University
+USA
+Tobias Lauinger
+New York University
+USA
+ABSTRACT
+Social media networks commonly employ content moderation as
+a tool to limit the spread of harmful content. However, the effi-
+cacy of this strategy in limiting the delivery of harmful content to
+users is not well understood. In this paper, we create a framework
+to quantify the efficacy of content moderation and use our met-
+rics to analyze content removal on Facebook within the U.S. news
+ecosystem. In a data set of over 2 𝑀 posts with 1.6 𝐵 user engage-
+ments collected from 2,551 U.S. news sources before and during
+the Capitol Riot on January 6, 2021, we identify 10,811 removed
+posts. We find that the active engagement life cycle of Facebook
+posts is very short, with 90 % of all engagement occurring within
+the first 30 hours after posting. Thus, even relatively quick inter-
+vention allowed significant accrual of engagement before removal,
+and prevented only 21 % of the predicted engagement potential
+during a baseline period before the U.S. Capitol attack. Nearly a
+week after the attack, Facebook began removing older content, but
+these removals occurred so late in these posts’ engagement life
+cycles that they disrupted less than 1 % of predicted future engage-
+ment, highlighting the limited impact of this intervention. Content
+moderation likely has limits in its ability to prevent engagement,
+especially in a crisis, and we recommend that other approaches
+such as slowing down the rate of content diffusion be investigated.
+1
+INTRODUCTION
+Social media companies such as Google (YouTube) and Meta (Face-
+book) have become inextricably linked to contemporary civil soci-
+ety [33] through their communication and content sharing “plat-
+forms.” Billions of users rely on these services to read or comment
+on the news, entertain themselves, and conduct business. Despite
+stated intentions [18] and the commonly used term “platform,” most
+social media companies are not impartial channels for users’ com-
+munications. On the one hand, social media companies set the
+operating parameters of algorithms that promote “interesting” con-
+tent to users. On the other hand, when it is deemed necessary to
+retain users [29] or to survive public scrutiny [66], social media
+companies may intervene against extreme or offensive material.
+Content moderation is currently the primary strategy that so-
+cial media companies employ to counter the spread of harmful
+content [5, 24]. Nearly all services, including venues for adult (On-
+lyFans [48]) or extreme (8kun [1]) content, have established policies
+that dictate what they are willing to host; content deemed incon-
+sistent with those policies may be removed [23]. Public debate has
+∗Version 1 (January 2023): Non-peer-reviewed technical report.
+questioned the adequacy of these policies [67] (i.e., which content
+should be deleted) or surfaced instances of objectionable content
+that was not removed [59]. Prior research studied potential biases
+in moderation of YouTube comments [41, 42], the impact of content
+moderation on user behavior on Reddit [40, 58], and the magnitude
+of content moderation regarding third-party links to conspiracy
+theory stories [49]. It is still an open question how quickly content
+is removed, independent of the specific policy. We argue that to
+judge the effectiveness of content moderation, it is important to
+understand the spread of content until the time of deletion, i.e.,
+whether content is removed before it reaches a large audience.
+In this paper, we present what we believe to be the first mea-
+surement study of the speed and impact of content moderation on
+Facebook, both under normal times and during a crisis. We were
+lucky to obtain from Edelson et al. [17] a data set of public Facebook
+posts from 2,551 U.S. news publishers and “influencers” incidentally
+collected around the time of the U.S. Capitol Riot on January 6,
+2021. Because Facebook currently does not make any post-level
+data transparent about content moderation, we develop a novel
+methodology to infer content removals and their approximate tim-
+ing from daily observations of active posts. For our analysis, we
+develop novel metrics to estimate the impact of content removal
+on user engagement, that is, the combined number of comments,
+shares, and reactions such as “like.” We do so both in terms of
+past engagement that was allowed to occur because of the time
+it took to remove a post, and in terms of future engagement that
+was prevented. We estimate the latter based on predictions of the
+engagement potential of posts, which on aggregate have a net error
+of 4.5 % for viral posts, and 3.2 % for normal posts.
+For a baseline period selected to represent “normal” times, we
+found that the active engagement life cycle of Facebook posts was
+very short, with 90 % of all engagement occurring within the first
+30 hours after posting. An implication of this is that even rela-
+tively quick removal of content (we observed a median of 21 hours)
+allowed significant accrual of engagement before removal (3.8 𝑀 en-
+gagements with 3,843 posts), and prevented only 21.2 % of the pre-
+dicted engagement potential. By the time a removal decision was
+made, the content had already reached the vast majority of its typi-
+cal audience, and the removal had only a minor impact in terms of
+preventing engagement or exposure to users.
+During the crisis surrounding January 6, we initially observed
+similar levels of content removals and prevented engagement rates,
+but with generally higher engagement across both removed and
+non-removed posts. It took 6 days, and two announcements of
+1
+arXiv:2301.02737v1  [cs.SI]  6 Jan 2023
+
+Ian Goldstein, Laura Edelson, Damon McCoy, and Tobias Lauinger
+changes to content moderation by Facebook, before we observed
+a meaningful change in our metrics. Our results show that on
+January 12, Facebook began removing older content, but because
+these 1,416 removals occurred so late in these posts’ engagement
+life cycles, we estimate that they disrupted less than 1 % of future
+engagement, and made hardly any difference in practice. In this
+regard, Facebook did not appear to be prepared to contain the fallout
+of the crisis in a timely manner.
+Our results suggest that Facebook’s content moderation of U.S.
+news publishers and “influencers” could not keep up with the speed
+at which content spread on their social network, neither during
+normal times nor in times of crisis. They highlight that content
+moderation on a social network such as Facebook operates within
+the constraints of content recommendation, and its efficacy cannot
+be assessed in isolation. Merely tallying numbers of removed posts,
+without taking into account enforcement delays or the rates at
+which deleted posts reached their projected audiences, does not
+adequately characterize the outcomes of content moderation. Un-
+fortunately, this is not reflected in the transparency metrics that
+Facebook is currently reporting.
+Our work makes the following contributions:
+(1) We propose a methodology for inferring content moderation
+of public posts from current transparency data.
+(2) We introduce more insightful metrics for quantifying the im-
+pact of delayed content removals on accrued and prevented
+user engagement.
+(3) We show that moderation of public posts by U.S. news sources
+happened late in their engagement life cycle, resulting in
+only 21.2 % prevented engagement.
+2
+BACKGROUND & RELATED WORK
+Facebook and other social media networks reach billions of users
+by promoting content generated by large content producers such
+as news organizations or “influencers,” and also by individual users.
+Unfortunately, social media has been utilized for harmful purposes,
+such as spreading disinformation [8] or planning and publicizing
+violent activities [25]. This has caused these networks to come
+under increasing societal, regulatory and legal pressures to prevent
+the promotion of dangerous and harmful content on their systems.
+Content Moderation Processes. Virtually all social media networks
+have established rules of conduct and acceptable content for their
+services. They have a range of techniques at their disposal to enforce
+these standards, such as deleting [40, 58], downranking [31], quaran-
+tining [11, 56], labeling [69], or demonetizing [15] undesirable con-
+tent, or temporarily or permanently banning the accounts of users
+who repeatedly post violating content (“deplatforming”) [19, 39, 51].
+In this paper, we study content moderation [5, 22, 24, 28] from the
+angle of removal of violating content.
+The technical and human challenges of implementing reliable
+mechanisms for content moderation at scale have been topics of
+repeated study [37, 60]. Social media networks often use automated
+systems to monitor user content and communications, and ide-
+ally interdict violating content before it is published. Such proac-
+tive policy enforcement is typically built as sophisticated pattern
+matching systems, comparing content to “blocklists” of known
+examples [30, 34], and is thereby limited to predefined classes of
+violations (graphic violence, sexual content, child abuse, spam) [35].
+To identify new patterns of violations, and address more ambiguous
+or context-sensitive cases that cannot be handled well by automated
+systems, social media networks also commonly have content exam-
+ined by human reviewers after publication, for example when the
+content reaches a certain popularity threshold, exhibits suspicious
+interaction patterns, or causes complaints from other users [14, 55].
+Human review of the often violent content has been documented to
+have a detrimental effect to the mental well-being of those engaged
+in the process [52, 55].
+Outcomes of Content Moderation. Prior research on content mod-
+eration (specifically, content removals) has often focused on legal
+and socio-political issues (e.g., [12, 35, 44, 53]), or on qualitative
+work aiming to understand users’ experiences and perceptions of
+content moderation (e.g., [36, 38, 60, 68]). Quantitative research
+measuring content removals is more scarce. Srinivasan et al. [58]
+studied the impact of content removals on subsequent user behavior
+within a single subreddit, whereas Jhaver et al. [40] measured the
+impact that explanations accompanying content removals had on
+future user behavior across the entire platform. Jiang et al. [41, 42]
+evaluated partisan bias in content and comment moderation on
+YouTube and found no evidence of left or right bias in removals.
+Papakyriakopoulos et al. [49] quantified the sharing of conspiracy
+theory-related URLs on various social media and modeled the im-
+pact of content moderation; in contrast to our work, they did not
+delve into the timing of engagement accrual or deletion delays. To
+the best of our knowledge, we are the first to study the interplay
+between the timeliness of content removals and the accrual (or
+prevention) of user engagement with the removed content.
+Community Standards on Facebook. The rules for user behavior
+and acceptable content on Facebook are called “Community Stan-
+dards” [23]. They are divided into six broad areas ranging from
+violence and incitement to intellectual property concerns. These
+policies have evolved over time, often in response to crises.
+Our study period around the U.S. Capitol Riot on January 6, 2021
+encompasses several such policy changes. On December 14, 2020,
+Biden was declared the winner of the Electoral College vote [13]
+following the 2020 U.S. presidential election that had taken place
+on November 3. However, President Trump refused to concede and
+announced a rally in Washington, D.C. for January 6. After that
+“Stop the Steal” rally, a group of armed people breached security
+and stormed the U.S. Capitol building [7]. Social media platforms
+were at the center of the spread of false and misleading information
+about the election, including the use of Facebook to promote and
+livestream the Capitol attack [6, 10, 62]. In reaction to the attack,
+Facebook publicly updated their Community Standards to explicitly
+prohibit ‘praise and support’ of the storming of the U.S. Capitol,
+calls to bring weapons to locations in the U.S., video posts and pho-
+tos from Capitol insurrectionists, violations of the D.C. curfew, and
+future calls to violence. In addition, Facebook committed to enforce-
+ment actions targeting militarized social movements, specifically
+naming the Oathkeepers and QAnon [21]. On January 11, Facebook
+announced that they would remove content containing “Stop the
+Steal” [20]. In Section 4.2, we analyze how these announced policy
+changes were reflected in observable content moderation activity.
+2
+
+Understanding the (In)Effectiveness of Content Moderation
+Other Related Work. More loosely related to our work are prior
+studies of deleted content (e.g., [4, 9, 70]), which however were
+focused on characterizing users who voluntarily deleted their own
+content after “regretting” its publication rather than measuring
+content moderation imposed by the social network. Similarly, there
+has been ample prior work on the spread of (and engagement
+with) misinformation on social media (e.g., [3, 17, 57]), but it has
+largely left open the question of how fast such misinformation
+may be removed by the social network. More generally, researchers
+have also studied the diffusion of content on social media and the
+concept of “viral” content, for example from the perspectives of
+user intention [2] and the structure of diffusion [32]. Studies of
+viral content typically take a set number of the ‘most popular’ or
+widely spreading content. Examples of this include Pressgrove et
+al. [50] with Twitter content, and Vallet et al. [64] for YouTube
+content spreading via Twitter. Past research has also tackled the
+problem of predicting the popularity of content (e.g., for Facebook
+and YouTube videos based on visual features [63]). Unfortunately,
+our data set does not contain the multimedia content from removed
+posts, thus we cannot use approaches based on content features to
+estimate the engagement potential of deleted posts.
+3
+DATA SET
+After the events of January 6, 2021, we sought to study the changes
+Facebook made to their content moderation efforts. Unfortunately,
+CrowdTangle, Facebook’s major transparency tool, does not reveal
+anything about posts blocked before publication, and states that
+once a published post is deleted from Facebook, it is also removed
+from CrowdTangle (with a delay).1 Thus, in order to measure post-
+publication content moderation, it is necessary to discover newly
+published posts in near real time, before they may be deleted, but
+we had not set up a measurement process designed to detect content
+moderation in advance of the Capitol Riot. However, we were able
+to obtain a data set of public Facebook posts by U.S. news publishers
+collected by the authors of another study [17], which happened
+to include the time period of interest. Below, we summarize the
+original data collection methodology, providing additional detail
+where it pertains to the timeliness of detecting new posts. We
+then describe our methodology for detecting deleted posts and
+estimating content moderation delays in a data set that was not
+specifically designed for that purpose. We derive three post data
+sets for our study, as shown in Table 1: The Removed Set, which we
+further split into smaller sets for specific analyses; all non-removed
+posts from Facebook pages with at least one deleted post (Impacted
+Publisher Set); and all non-removed posts across all pages.
+3.1
+Monitoring News Publishers on Facebook
+The data set we use in this paper is comprised of the public Facebook
+posts of 2,551 U.S. news publishers during the 2020 U.S. presidential
+election, provided to us by the authors of a study on user engage-
+ment with misinformation [17]. Their selection of Facebook pages
+defines the scope of our study. The authors of that study based their
+selection of Facebook pages on third-party data provided by the
+news rating organizations NewsGuard [47] and Media Bias/Fact
+Check [45]. Based on the assessments of these organizations, the
+1https://help.crowdtangle.com/en/articles/3323105-academics-researchers-faq
+authors derived the political leaning of each news publisher (Far
+Left to Far Right) as well as a binary “misinformation” attribute
+indicating whether a news publisher had a known history of spread-
+ing misinformation [17]. To exclude largely inactive news publish-
+ers, the authors of that study also removed Facebook pages with
+fewer than 100 followers or 100 total engagements during their
+five-month study period; our study inherits this filter. Due to the
+selection of Facebook pages in the original data set, we can study
+post-publication moderation of public posts (not comments) on
+the Facebook pages of U.S. news sources. These news sources are
+a mix of publishers known for reliable reporting as well as less
+reputable sources with a history of spreading misinformation, both
+mainstream and niche, from across the political spectrum.
+3.2
+Original Data Collection Methodology
+The original study authors collected all public Facebook posts and
+corresponding engagement metadata from the 2,551 U.S. news pub-
+lishers using two types of crawls of the CrowdTangle API: (1) A
+daily crawl to discover all new posts since the last crawl, and (2)
+a second, separate daily history crawl to update the engagement
+metadata such as the number of likes, shares, and top-level com-
+ments of existing posts. This latter crawl went back in history as far
+as the remaining daily time permitted; in practice, existing posts
+fell into the observable history window for at least two weeks (or
+longer) after publication. Both crawls were running daily from
+August 10, 2020 until January 18, 2021.
+3.3
+Detecting Deleted Posts & Post Lifetime
+In September 2021, we performed a follow-up crawl of the same
+news publisher pages to determine whether each post from the
+original data set was still available on CrowdTangle or had been
+deleted. This follow-up crawl, however, does not reveal when a
+post was deleted. To detect deletions with daily granularity, we
+leverage the daily history crawls. In the original data set, these
+history crawls ran until January 18, thus we limit our analysis
+to posts published between December 14, 2020 and January 15,
+2021 so that we have sufficient buffer to detect possible deletions.
+We conservatively estimate post lifetimes as the time difference
+between the publication date of the post (provided by CrowdTangle)
+and the last time the post was observed in a daily history crawl. This
+is a lower bound on actual post lifetimes, i.e., we underestimate
+rather than overestimating post lifetimes, as a post could have been
+deleted any time in the 24 h between the last time the post was
+observed, and the first time the post was missing in a history crawl.
+We do not know why any particular post was removed or whether
+it was deleted voluntarily or forcibly moderated by Facebook; we
+only know that a post is not publicly accessible any more when it
+fails to show up in the daily history crawl or our September follow-
+up crawl. We also note that very short-lived posts (published and
+then quickly deleted within less than 24 h) may not be visible to us,
+depending on how these events fall between consecutive crawls.
+3.4
+Data Sets & Time Periods
+Table 1 shows a summary of the data sets used in this paper. Our
+methodology identified 10,811 posts from 878 U.S. news publishers
+3
+
+Ian Goldstein, Laura Edelson, Damon McCoy, and Tobias Lauinger
+Data Set
+Post Count
+Total Engagement
+Delayed Removals
+Delayed Engagement
+Full Data Set (posts published Dec 14–Jan 15)
+2.02 M
+1.61 B
+Removed Set (last observation of removed posts)
+10,811
+10,889,679
+1,416
+2,875,064
+Baseline Period (Dec 14–Jan 3)
+3,843
+3,785,464
+281
+301,741
+January 6 Period (Jan 4–Jan 11)
+1,171
+3,302,379
+60
+374,066
+January 12 Period (Jan 12–Jan 15)
+2,300
+3,257,523
+1,075
+2,496,708
+Removed Pages (Dec 14–Jan 15)
+3,497
+544,313
+NA
+NA
+Impacted Publisher Set (posts pub. Dec 14–Jan 15)
+297,774
+784,842,977
+Non-Removed Set (posts pub. Dec 14–Jan 15)
+2.02 M
+1.61 B
+Table 1: Overview of data sets. Posts created December 14, 2020–January 15, 2021. Engagement is the total accrued by posts at
+the time of final observation. We define delayed removals as those occurring more than 30 hours after post creation.
+that are no longer accessible via CrowdTangle (i.e., no longer pub-
+licly visible on Facebook). We refer to these posts and associated
+metadata as the removed set. We divide the removed set into four
+subsets. First, we set aside the subset of posts from deleted pages.
+When a page is deleted, the deletion cascades to all of its posts.
+Since the deletion times of posts deleted with their page are identi-
+cal (likely independent from their original posting times), we study
+such pages and their posts separately. We subdivide the remaining
+deleted posts into three time periods for more targeted analysis:
+(1) The baseline period contains posts removed prior to Janu-
+ary 4, 2021, and is used to assess “normal” behavior prior to
+the events around January 6.
+(2) The January 6 period contains posts removed between Janu-
+ary 4–11, 2021, and is used to analyze the immediate reaction
+in the days leading up to January 6 and just afterwards.
+(3) The January 12 period contains posts removed on January 12
+or later. We analyze this period separately because we no-
+ticed increased deletion delays after January 12 (deleted posts
+were older than usual); this different deletion behavior may
+correspond to changes on Facebook’s side.
+We create two additional data sets of posts that were not deleted.
+The impacted publisher set comprises the 297,774 remaining
+posts from the Facebook pages that had likely experienced content
+moderation (at least one post deleted). This data set serves to inves-
+tigate the non-moderated content produced by these pages. To give
+an overview of the overall ecosystem, the non-removed set con-
+tains all non-deleted posts collected during the observation period.
+This data set serves as a reference for comparison and consists of
+2.76 M posts from 2,315 news publishers (over 2.5 B engagements).
+3.5
+Post Engagement and Virality
+In the context of Facebook posts, we define engagement as the
+sum of all types of interactions that users can have with a post
+(and that are made transparent through CrowdTangle). Specifically,
+engagement consists of the number of “likes” (and other reaction
+types available on Facebook, such as “angry” or “sad”), the number
+of times a post is shared, and the number of top-level comments
+below the post. Unfortunately, other types of popularity metrics
+are not available in CrowdTangle (e.g., the number of times a post
+has been shown to users, or how many unique users have viewed
+the post), thus we cannot include them in our study.
+For posts that were still available in September 2021, we obtain
+long-term engagement numbers from our follow-up crawl (Sec-
+tion 3.3); for deleted posts, we use the engagement numbers from
+the last time they were observed on CrowdTangle. Similar to our
+estimates of post lifetime, engagement numbers for deleted posts
+are a lower bound. Additional engagement could have accrued be-
+tween the last observation and the time the post was removed, but
+as it is unobserved, it does not factor into our analysis.
+To study the speed of engagement accrual of (non-deleted) posts,
+we leverage time series data extracted from CrowdTangle in Sep-
+tember 2021. These show engagement values in increasing time
+steps relative to the publication time of the post.
+Virality. In the non-removed data set, posts receive a mean total
+engagement of 896 user interactions. However, the uneven distribu-
+tion of engagement with content on social media is well known: a
+small number of posts go viral, and the vast majority do not [61]. We
+are not aware of a commonly accepted engagement-based threshold
+for virality. We define as viral any post that reaches 14.6 K engage-
+ments (three standard deviations above the mean in the set of all
+non-removed posts). This definition of virality is based purely on
+popularity, independent of the speed of engagement accrual.
+3.6
+Estimating Prevented Engagement for
+Removed Posts
+To understand the impact of content removals, it is useful to esti-
+mate how much more user engagement a post might have accrued
+had it not been deleted. We do so by first estimating a post’s en-
+gagement potential, defined as the total engagement a post would
+typically receive in the long run (i.e., if it is not deleted). Because
+accrual of engagement is different for each post, but potentially
+more similar among posts from the same publisher, we estimate a
+post’s engagement potential based on non-deleted posts from the
+same Facebook page. From the engagement time series obtained
+in September 2021 for the surviving posts published during the
+baseline period (which are old enough that we expect them to have
+exhausted their engagement potential), we calculate the mean en-
+gagement at each time step over all non-deleted posts from the same
+page. Since our observations show that there is a distinct engage-
+ment curve for viral posts, we estimate engagement separately for
+viral posts. For pages with more than 10 non-removed viral posts,
+we estimate viral mean engagement per time step individually; for
+4
+
+Understanding the (In)Effectiveness of Content Moderation
+all other pages, we fall back to using global viral engagement es-
+timates. We define prevented engagement for a deleted post as
+the difference between its engagement potential and engagement
+actually accrued before the deletion. It is estimated by summing all
+engagement on the applicable mean time series after the last obser-
+vation of the post. We likely overestimate prevented engagement
+because posts may have remained active for up to 24 h after the
+last observation in the crawl, thus they might have accrued more
+real engagement (and less engagement was prevented) compared
+to the estimate. In content moderation, it is desirable to remove
+policy-violating posts quickly and prevent as much engagement as
+possible, thus our overestimation of prevented engagement paints
+content moderation efforts as more effective than they may be.
+To validate the quality of these engagement predictions, we
+tested our methodology on a sample of 10,000 non-deleted posts
+published during the baseline period. We simulated removals by
+randomly selecting “deletion” timestamps with selection weights
+based on the typical lifetimes of deleted posts in the baseline period
+(Section 4.1). The prediction accuracy for engagement potential was
+89.7 % for non-viral posts, and 84.4 % for viral posts. The net error
+across all 9,929 non-viral sample post predictions was within 3.2 % of
+the total ground truth engagement potential (lifetime engagement)
+for these posts (4.5 % for the 71 viral posts). As our intention is to
+apply our prediction methodology to a large number of posts and
+not to draw inferences about the performance of individual posts,
+these results suggest that our methodology is sufficiently robust.
+4
+RESULTS
+In this work, we aim to characterize how Facebook moderated
+public content published on the Facebook pages of U.S. news pub-
+lishers. When a post is rendered unavailable, we do not know who
+removed the content. However, as we discuss more thoroughly
+in Section 5.2, the patterns we observe across publishers suggest
+that the vast majority of post removals were not voluntary, but
+instances of content moderation, that is, posts deleted by Facebook
+for violation of their platform policies. Both user engagement with
+content and Facebook’s apparent content moderation changed in
+the days during and after the events of January 6, 2021, thus we
+analyze these time periods separately.
+4.1
+Impacts of Content Removals
+To understand Facebook’s content moderation performance during
+“normal” times, we begin with a baseline period from December 14,
+2020 to January 3, 2021. During this time, the 2,551 U.S. news pages
+published 1.35 M posts. Facebook users engaged with these posts
+(i.e., “liked,” commented, or shared) a total of 948 M times. In terms
+of likely content moderation, 3,843 posts (0.28 %) were removed
+from 640 pages during this period, on average 183 removals per
+day. However, at the last time they were observed active in our data
+set, these posts had already accumulated 3.8 M user engagements
+(0.32 % of total engagement during the baseline period). If Facebook
+deemed these posts inappropriate to remain on their platform, they
+still allowed a considerable number of users to see and interact with
+the offending content before reaching the decision to take it down.
+0
+1
+2
+3
+4
+5
+6
+7
+Time since publication (days)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Proportion of removed
+posts (cumulative)
+30 hours
+Time Period
+Baseline
+January 6
+January 12
+Figure 1: Post lifetime: Days between posting and last obser-
+vation. In the baseline and January 6 periods, nearly all last
+observations were at most 30 h after posting, whereas dur-
+ing the January 12 period, much older posts were removed.
+The sharp pivot at 30 h suggests different removal processes.
+We turn to the lifetimes of removed posts to understand how
+these presumably policy-violating posts could accrue so much en-
+gagement before being deleted. As Figure 1 shows, the average time
+between publication of a post to its last observation by the crawler
+is 23.7 hours (median: 21 hours). (Note that the actual time of re-
+moval could be up to 24 hours later because the crawler checked a
+post’s status only once every 24 hours, thus we likely underestimate
+post lifetimes.) The distribution of post lifetimes pivots 30 hours
+after post creation; approximately 90 % of removed posts have had
+their last observation at this point. The remaining 10 % are deleted
+at a noticeably slower pace over a much longer time span, which
+suggests that a different process might be at play for those deletions.
+In summary, most decisions to take down posts appear to be rela-
+tively fast in absolute terms, but they do take time, and significant
+amounts of engagement are allowed to accrue during that time.
+The situation of viral posts is slightly different from the overall
+post distribution. The baseline period saw 9,768 posts accumulate
+sufficient engagement to be considered viral (425 M combined); 61
+of them were removed during this period (more than twice the
+rate of all content removals). Until their last observation, these 61
+removed viral posts had been able to accrue 1.7 M engagements,
+44.8 % of all engagement with posts removed during this period.
+This share illustrates the importance of handling viral posts well;
+quick intervention on a relatively small number of violating viral
+posts could disproportionately reduce the odds of Facebook users
+seeing and engaging with inappropriate content. However, our data
+suggest that viral posts may be more difficult for Facebook to handle,
+as viral posts tended to be active for longer before being removed
+(mean time to last observation: 31 hours, more than 7 hours longer
+than the overall mean; median: 24 hours, or 3 hours longer).
+To put the speed of deletions into context, we compare it to the
+rate of accrual of engagement. Most engagement with posts in our
+data set happens relatively soon after the content is created. This is
+true both for viral and non-viral posts, as shown in Figure 2. Among
+non-removed content created during our baseline period, 78.1 % of
+normal posts reached at least 80 % of their total engagement within
+one day, while 93.3 % of normal posts achieved it within two days. In
+5
+
+Ian Goldstein, Laura Edelson, Damon McCoy, and Tobias Lauinger
+0
+1
+2
+3
+4
+5
+6
+7
+Time since publication (days)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Proportion of posts
+(cumulative)
+Post Type
+viral
+non-viral
+Figure 2: Days until non-removed baseline posts reach 80 %
+of their lifetime engagement. Around 78 % of non-viral posts
+reach 80 % of their lifetime engagement within one day,
+whereas it takes two days for viral posts.
+other words, while the typical content moderation delays observed
+in our data set may seem fast in absolute terms, in fact they do
+come late in a post’s life cycle because most of a post’s engagement
+usually occurs before the typical content moderation delays. As a
+result, these instances of content moderation may not make a big
+difference in practice because most of a post’s audience will already
+have seen and interacted with the post before it is deleted.
+Viral posts have a longer period of active engagement accrual
+compared to non-viral posts, with 55.2 % reaching 80 % of their
+total engagement in one day, and 80.6 % reaching it within two
+days. However, this does not mean that slower content moderation
+would be appropriate due to the slower relative rate of engagement
+accrual. In fact, total engagement with viral posts is so much higher
+that despite the slower relative increase, the absolute increase is
+substantial. In the hour before their respective median removal
+time, viral posts accrued an average of 1,583 additional engage-
+ments, whereas it was only 7.5 for non-viral posts. With the goal
+of reducing total exposure to harmful content, it is important to
+moderate viral posts with the same (or higher) speed than non-viral
+posts, yet our data suggest the opposite is occurring in practice.
+We assess the benefit of typical content moderation delays by
+estimating how much engagement was prevented by removing
+the post, that is, how much of the typical audience did not see
+and interact with the offending post. We estimate a post’s typi-
+cal engagement on a per-page basis and separately for viral and
+non-viral posts, as described in Section 3.6, and calculate prevented
+engagement as the difference between that estimate and the last
+observed engagement count before deletion. During the baseline
+period, we estimate that post removals disrupted 21.2 % of potential
+engagement (314 or fewer prevented engagements each for 90 % of
+removed posts). For the 3,782 removed non-viral posts, this breaks
+down to a 20.5 % prevention rate, or a mean of 151 prevented en-
+gagements per non-viral post. The removal of each of the 61 viral
+posts had a much larger impact in absolute terms, with a mean of
+7,289 prevented engagements. Viral posts accrue engagement for a
+longer time (median 1.4 times longer) than non-viral posts, offering
+the opportunity to disrupt a larger fraction of their engagement
+potential, but because they were also removed more slowly (median
+1.2 times slower), the engagement prevention rate for viral posts
+was only slightly higher at 22.0 %. Timelier enforcement of the
+comparatively few removed viral posts could have made an outsize
+difference in preventing Facebook users from engaging with sub-
+sequently moderated posts. Overall, we estimate that the content
+removals we observed during the baseline period prevented 996 K
+engagements from occurring, which is a consequential number
+in absolute terms. Unfortunately though, these removals came so
+late that the posts had already reached over three quarters of their
+engagement potential, and only a small minority of their potential
+engagement was actually prevented by the removal.
+4.2
+Content Removals After the Capitol Riot
+To understand how Facebook carried out content moderation dur-
+ing and after the Capitol Riot, we repeat the analyses of the baseline
+period above for the time just before and after the event of Janu-
+ary 6, 2021. In the eight days from January 4–11, which we refer to
+as the January 6 period, we observed comparable levels of content
+removals in absolute terms, but higher engagement with content
+by the Facebook users. In detail, the U.S. news pages published
+393 K posts that received a total of 409 M engagements. The pe-
+riod saw 1,171 posts removed from 246 pages, a mean of 146 per
+day, which is roughly comparable to the baseline period (𝑡 = 1.83,
+𝑝 = 0.079). However, by the time they were last observed, these
+posts had received 3.3 M engagements, 0.81 % of all user engage-
+ment, significantly (𝑡 = 4.06, 𝑝 < 0.001) more than during the
+baseline period. The median post lifetime in the January 6 period
+was consistent with the baseline period for non-viral (𝜒2 = 1.24,
+𝑝 = 0.266) and viral posts (𝜒2 = .043, 𝑝 = 0.835), also evidenced by
+the closely overlapping curves seen in Figure 1. Thus, the increased
+engagement accrual did not stem from slower removals, but higher
+user engagement. Although we saw no increase in the proportion
+of posts moderated during the January 6 period, and no significant
+increase of non-removed viral posts (𝑡 = 0.62, 𝑝 = 0.536), there
+was a significant increase in the removal of viral posts as compared
+with the baseline period (𝑡 = 2.38, 𝑝 = 0.024), which explains the in-
+crease in engagement-weighted removals. 80% of engagement with
+removed viral posts in this period was associated with extremely
+partisan sources, and the remainder originated from a single page.
+We estimate that post removals during the January 6 period pre-
+vented a similar share of potential engagement (25.8 %) as in the
+baseline period (a total of 306 K engagements with 1,117 non-viral
+posts, and 521 K engagements with 54 viral posts). In summary,
+while we observed increased engagement from users, any changes
+Facebook may have made to content moderation in the immediate
+aftermath of January 6 do not stand out in our metrics.
+We do observe increased levels of content removals during the
+four days from January 12–15 (the January 12 period), after Face-
+book announced another content moderation policy change [21]
+on January 11th. U.S. news pages published 186 K posts during
+these four days, reaching 161 M total engagements. Facebook re-
+moved 12 entire pages with all their 3,463 posts created between
+December 14, 2020 and January 15, 2021. We discuss these removed
+pages in Section 4.5. In addition, we observed 2,300 removals of
+individual posts from 377 pages, a mean of 575 per day, which was
+a significant increase (𝑡 = 8.63, 𝑝 < 0.001) from the baseline period.
+6
+
+Understanding the (In)Effectiveness of Content Moderation
+These posts had received a total of 3.3 M engagements at the time
+of their final observation. This last observation came considerably
+later in a post’s lifetime than in the baseline and January 6 periods.
+In the January 12 period, the mean time between post creation and
+last observation was 152 hours (median: 24 hours), significantly
+longer than the baseline period for both non-viral (𝜒2 = 126.7,
+𝑝 < 0.001) and viral posts (𝜒2 = 36.8, 𝑝 < 0.001). As shown in
+Figure 1, 48 % of last observations occurred more than 30 hours
+after publication, whereas it was only approximately 6.8 % in the
+baseline and January 6 periods. This sharp increase in delayed con-
+tent removals suggests that Facebook may have applied changes
+in content moderation announced on January 11 retroactively to
+older posts, which were originally published in the time around
+January 6, but deleted only after January 11. (We note that all of the
+posts removed after more than 30 hours were from “repeat offender”
+pages, i.e., we had already observed removed posts from these pages
+in the past.) This further suggests that a policy change, which took
+nearly a week to implement, was necessary for Facebook to deal
+with the fallout of January 6 on their platform.
+The unusually late removals are reflected in our estimates of
+prevented engagement. For all posts removed during the Janu-
+ary 12 period, we estimate that 569 K engagements were disrupted,
+representing only 14.9 % of removed posts’ predicted engagement
+potential. In particular, for the subset of posts removed with a long
+delay (of more than 30 hours), the deletions appear almost incon-
+sequential because they prevented only 0.60 % of the engagement
+we predicted these posts to achieve without being deleted. In other
+words, if these deletions were indeed instances of “retroactive” con-
+tent moderation, Facebook allowed these posts to reach virtually
+their entire potential audience before adjudicating that they were
+in violation of platform policy and had to be taken down.
+4.3
+Impact of Removals by Misinformation and
+Partisanship Reputation
+Over the last several years, there has been intense public scrutiny
+of the partisan impact of content moderation policies on Face-
+book [26, 43, 65]. We now analyze content removals taking into
+account the partisanship and factualness classifications in our data
+set. These labels reflect the reputation of the pages in question
+rather than characterizing individual posts, but they still allow us
+to compare the effects of content removal along partisan lines. We
+calculate the engagement-weighted rate of removal as the ob-
+served engagement from removed posts in a category of pages over
+the total observed engagement from all posts in that category.
+Over the full time range of our data set, posts made by pages
+with a reputation for misinformation had a higher engagement-
+weighted rate of removal than posts from pages not classified as
+misinformation providers. This finding of higher misinformation
+removal rates held true within each of the five partisanship cat-
+egories, as shown in Table 2. At the same time, within the same
+factualness category, engagement-weighted rates of removal varied
+considerably according to the partisanship of pages; in the misin-
+formation category, they ranged from 1.02 % for posts made by Far
+Left pages to 6.97 % for posts from Slightly Right pages. We note
+that misinformation pages of extreme partisanship (Far Left and
+Far Right) had the lowest post removal rates among misinformation
+Partisanship
+Misinformation
+Non-misinformation
+Far Left
+1.02 %
+0.57 %
+Slightly Left
+2.67 %
+0.19 %
+Center
+3.34 %
+0.29 %
+Slightly Right
+6.97 %
+0.07 %
+Far Right
+1.19 %
+0.53 %
+Table 2: Post removal rates by partisanship and factualness
+of the source (weighted by engagement, entire data set).
+Across the political spectrum, news sources known to spread
+misinformation saw a higher fraction of their accumulated
+user engagement affected by content removals.
+pages, whereas non-misinformation pages of extreme partisanship
+had the highest post removal rates of all non-misinformation pages.
+We do not currently have an explanation for this effect.
+So far, our analysis of removal rates covered the entire duration
+of our data set, about a month roughly split in half by the events
+of January 6, 2021. These events were of a deeply partisan nature;
+we now investigate whether any corresponding effects based on
+partisanship are observable among the removed posts. To better
+isolate the impact of these events, we again split the data set into
+the baseline, January 6, and January 12 periods. (To streamline the
+analysis, we no longer differentiate pages by their misinformation
+reputation.) The upper two rows of Figure 3 show the engagement-
+weighted partisan breakdown of posts removed during the baseline
+period (above), and non-removed posts published during the same
+period (below). While not perfectly balanced, there does not appear
+to be any major partisan bias in the posts removed during the
+baseline period compared to the posts that were not removed.
+The picture looks very different in the January 6 period (mid-
+dle two rows of Figure 3). The majority of engagement with posts
+removed during these eight days corresponded to Far Right pages
+(50.8 %), while Slightly Right pages accounted for the next largest
+share (21.9 %). The partisan breakdown of engagement with posts
+that were not removed does not exhibit this effect, and more closely
+resembles the baseline period (albeit not exactly). During the four
+days of the January 12 period, the proportion of engagement attrib-
+utable to removed posts from Slightly Right pages receded to a level
+comparable to the baseline period. However, the corresponding
+proportion from Far Right pages remained more than double the
+baseline share of engagement-weighted post removals. (This does
+not necessarily correspond to “new” engagement accrued during
+this period, but may be partially due to delayed enforcement; 65.7 %
+of these removals were posts originally published in the days around
+January 6, and deleted only on or after January 12.) In conclusion,
+the partisan breakdown of engagement-weighted removal rates
+was somewhat balanced along partisan lines before the events of
+January 6, and remained at comparable levels across all three time
+periods for non-removed posts. Likely content moderation around
+January 6 disproportionately affected posts from Slightly Right and
+Far Right pages. This is in line with the policy changes that Face-
+book announced on January 6 and January 11, which specifically
+targeted so-called “Stop the Steal” [20, 21] misinformation.
+7
+
+Ian Goldstein, Laura Edelson, Damon McCoy, and Tobias Lauinger
+0
+1
+Removed
+Far Left
+S. Left
+Center
+S. Right
+Far Right
+0
+1
+Non-removed
+0
+1
+Removed
+0
+1
+Non-removed
+0
+1
+Removed
+0
+1
+Non-removed
+ Baseline Period
+ January 6 Period
+January 12 Period
+Political Leaning (% Engagement)
+Figure 3: Engagement with removed and non-removed posts
+from U.S. news pages by partisanship for the baseline, Jan-
+uary 6, and January 12 periods, respectively. Around Janu-
+ary 6, the vast majority of engagement with posts that were
+ultimately removed went to posts from sources classified as
+Slightly Right and Far Right. The rate of engagement with
+misinformation sources across all time periods was 66.1 %
+for removed content and 28.2 % for non-removed content.
+4.4
+Most Impacted Publishers
+For illustrative purposes to complement our quantitative findings,
+we identified the U.S. news pages with most user engagement that
+likely experienced content moderation during our entire observa-
+tion period (the Impacted Publisher Set). The two most popular
+pages with significant post removals were ‘Dan Bongino’ and ‘Oc-
+cupy Democrats,’ both classified as sources of misinformation with
+Far Right2 and Far Left3 partisanship, respectively. Taken together,
+both pages generated 835 K engagements with content that was
+later removed, though this represented only a fraction of the tens
+of millions of engagements each of these pages generated in our
+observation window. ‘The Blaze,’ a conservative4 media company
+with a reputation for misinformation, received the greatest total
+engagement (1.37 M) with 310 ultimately removed posts, account-
+ing for 28.3 % of all engagement that the page received between
+December 14, 2020 and January 15, 2021. At 91.5 %, the share of
+engagement attributed to deleted posts was highest for ‘I Love
+My Freedom,’ a page associated with a conservatively oriented
+online apparel retailer.5 Three quarters of engagement-weighted
+removals from this page occurred after Facebook’s January 11 pol-
+icy announcement, with a mean delay of 8.5 days, suggesting that
+the affected content might not have been in clear violation of the
+policies in effect prior to that date. In line with the unusually long
+removal delays, we estimate that only 6.2 % of potential engage-
+ment was prevented. In summary, the pages most heavily impacted
+by removed content tended to be pages categorized as extremely
+partisan, misinformation providers, or both.
+2https://mediabiasfactcheck.com/dan-bongino/
+3https://mediabiasfactcheck.com/occupy-democrats/
+4https://mediabiasfactcheck.com/the-blaze/
+5https://www.facebook.com/ILMFOrg/about
+4.5
+Page Deletions
+So far, we have only considered removals of individual posts from
+pages that themselves remained active. In addition, we observed the
+removal of 14 entire pages (the Removed Pages Set), which included
+their 3,497 posts created between December 14, 2020 and Janu-
+ary 15, 2021. These posts had accumulated an engagement of 544 K.
+Deleted pages fall into three categories: known sources of health
+and vaccine misinformation (‘Erin at Health Nut News,’ ‘LifeSite-
+News.com’), extreme Far Right media with poor records for factual-
+ness and newsworthiness (‘TruNews,’ ‘ConservativeOpinion.com’),
+and pages frequently linked to conspiracy sources (‘The Drudge
+Report,’ ‘Zero Hedge’). Many of these were widely known as spread-
+ers of inaccurate information, and their forced deplatforming by
+Facebook and the related debate were widely reported [16, 27, 46].
+5
+DISCUSSION
+Content moderation aims to limit the exposure of users to harmful
+content by way of deleting user-generated content that is deemed
+unacceptable. In this framework, content moderation needs to com-
+pete with content recommendation systems, in that the speed of
+enforcement needs to keep up with the potentially viral spread of
+toxic content in order to effectively contain undesirable content.
+Our results suggest that Facebook’s content moderation was not
+effective at meaningfully limiting the spread of harmful content
+from U.S. news publishers. In the most common case, content mod-
+eration was relatively quick in absolute terms; half of removed posts
+in the baseline period were last observed 21 hours after publica-
+tion. Yet, the speed of removal pales in comparison to how quickly
+content accrues engagement, and how quickly interest in content
+fades. By the time of removal, the posts in our data set had already
+been shown to millions of Facebook users, accumulating a total of
+10.9 million engagements. We estimate that the removals prevented
+only 21.2 % of engagement from occurring because the vast majority
+of the posts’ potential audience had already seen them. Further-
+more, it appears that Facebook needed to change their policy in
+order to moderate some content after the Capitol riot of January 6,
+2021. This policy change took nearly a week to implement, likely
+delaying the removal of 995 posts by multiple days, with the result
+that they received over 2 M engagements in the meantime, and
+their removal prevented virtually none (< 1 %).
+Whether or not content moderation can be effective depends
+on factors that are under Facebook’s control. For one, Facebook
+could aim to delete content more quickly by decreasing delays in
+human review and implementing a content policy that does not
+require lengthy adjustments to allow reaction to major real-world
+events. It is unclear, however, how much potential there is for such
+improvements. As we have shown, the currently very short ac-
+tive engagement periods of posts, and the speed of viral accrual
+of engagement create a limited window during which content re-
+moval can be effective as a strategy. Therefore, the second avenue
+to making content moderation more effective would be changing
+the parameters of the content recommendation algorithm to en-
+large this time window and give Facebook more time to intervene
+effectively. This could take various forms. To increase the time for
+intervention, new content could be recommended more slowly. To
+limit the impact of harmful content, the relatively small number
+8
+
+Understanding the (In)Effectiveness of Content Moderation
+of posts predicted to “go viral” could be reviewed manually before
+they accrue significant engagement.
+Admittedly, the scenario for content moderation that we stud-
+ied in this paper may be more challenging than the ecosystem of
+Facebook as a whole. Public posts from U.S. news publishers, due
+to their newsworthiness, may require a more thorough review than
+other types of policy violations. Furthermore, due to their larger
+audience, delayed removals have a much larger aggregate impact
+than moderation of posts in smaller groups, or private messages.
+These challenges highlight, however, the limits of what can be
+achieved with the tool of content moderation under the constraints
+of a recommendation system that has been designed and optimized
+to spread engaging content as quickly as possible.
+5.1
+Limitations
+At the time of this writing, Facebook does not provide fine-grained
+transparency for content moderation. Our efforts to study content
+moderation rely on what is observable indirectly through Crowd-
+Tangle. As a result, we can only study posts deleted after publication.
+Other types of content moderation, such as rejecting posts before
+publication, or deliberately downranking posts in the recommen-
+dation algorithm (without deleting them) are outside the scope of
+our study. Facebook claims that up to 90 % of content violations
+(including hate speech, nudity, violence, and bullying) are deleted
+before they are likely to have been seen [22]. Our study can only
+include posts that slip through pre-publication content filters. Yet,
+collectively Facebook users engaged with the deleted posts in our
+data set more than 10.8 million times. While we do not have exact
+numbers about how this relates to impressions and unique users,
+it is reasonable to estimate that millions of Facebook users were
+exposed to these posts that Facebook later deemed inappropriate.
+Because we rely on a repurposed data set that was not originally
+collected to study content moderation, our analysis is subject to
+additional limitations. The authors of the data set did not archive
+multimedia content such as images or videos associated with posts.
+Posts may have been deleted for offending image or video content,
+thus we cannot conduct a meaningful study of the contents of
+deleted posts. Furthermore, the timing of post discovery in the
+original crawl causes a bias towards longer-lived posts, that is,
+when posts are deleted quickly after publication, they are more
+likely to be missing in the data set. Similarly, the daily status checks
+of older posts in the original crawl mean that the inferred post
+lifetimes are only a rough estimate.
+Our analysis is based on public posts published by the Face-
+book pages of 2,551 U.S. news organizations. The content produced
+by these pages is not representative of the Facebook ecosystem
+as a whole. Furthermore, the selection of these pages and their
+categorization with regard to their political leaning and history
+of spreading misinformation is subject to the methodology and
+limitations described by Edelson et al. [17].
+5.2
+Content Moderation vs. Voluntary
+Removals
+Because we cannot observe post deletions directly, we do not know
+for certain why content was removed; it could have been moder-
+ated by Facebook, deleted by the page owner, or the visibility could
+have been changed to private. However, taken together, several
+independent observations from our analysis suggest that a majority
+of removed posts were indeed subject to content moderation by
+Facebook. First, the pattern of deleted post lifetimes with a sharp
+pivot at 30 hours was consistent across pages, which suggests inter-
+vention by a common process as opposed to independent voluntary
+deletions by page owners. We were able to confirm one instance of
+voluntary deletions by contacting the page administrators of the
+National Center for Missing & Exploited Children,6 a non-partisan,
+charitable organization who confirmed to us that they routinely
+take down posts about missing children when those children are
+found. Statistically, their 185 voluntary removals in our data set
+(with a total of 344 K engagements) looked very different from ag-
+gregate removal patterns, suggesting that the bulk of removals are
+not voluntary. In terms of post lifetimes, we last observed their vol-
+untarily deleted posts after a mean of 7.3 days (median: 2.4 days), as
+opposed to 2.7 days (0.9 days) across all removed posts in our data
+set. This supports our hypothesis based on patterns in post lifetimes
+that most observed removals were likely deletions by Facebook.
+Second, the unusually long removal delays only observed in the Jan-
+uary 12 period and the coinciding, prior announcement by Facebook
+of a content moderation policy change suggest that these removals
+were instances of content moderation. Third, the higher prevalence
+of removals among pages with a reputation for spreading misinfor-
+mation makes it less likely these are voluntary deletions, unless we
+hypothesize that these misinformation providers are more prone to
+self-censorship. While only improved transparency from Facebook
+could provide clarity, we are confident that the removal patterns
+we observed are predominantly a reflection of content moderation.
+5.3
+Improved Measurement Design
+If we were to design a measurement of content moderation of public
+posts from scratch, and assuming that a transparency tool such
+as CrowdTangle continues to impose low rate limits and does not
+provide specific transparency around content moderation, what
+would our measurement design look like?
+• Archive multimedia content such as images and video. If space
+is an issue, keep only the content of deleted posts, or of all
+posts from pages that have deleted posts.
+• Detect new posts more frequently (as opposed to daily) to
+observe short-lived posts. Prioritize the time of day when
+most posts are published.
+• Check the status of existing posts more frequently than daily
+to get more fine-grained post lifetimes, at least during the
+first 2-3 days, which see the vast majority of deletions.
+For example, hourly daytime crawls to detect publication or deletion
+of recent posts (0-3 days old) could be combined with nightly history
+crawls to detect deletion of older posts with daily granularity.
+5.4
+Transparency Metrics & Reporting
+Meta’s primary mechanism for making data relating to content
+moderation transparent is their Community Standards Enforce-
+ment Report [22]. In these quarterly reports, the company shares
+aggregate, global data about eleven categories of removals, reported
+6https://www.missingkids.org/
+9
+
+Ian Goldstein, Laura Edelson, Damon McCoy, and Tobias Lauinger
+as the absolute number of posts removed, and the relative share
+of global user impressions attributable to moderated content. The
+discrepancy between absolute and relative numbers makes them
+difficult to compare, and they are not broken down at a country
+level. Furthermore, the raw number of removed posts is not per se
+informative for understanding the efficacy of content moderation.
+Based on the metrics we used in our analysis, we recommend re-
+porting how long content remains active before it is taken down,
+how many people are exposed to content that is later removed, and
+how much of a removed post’s potential audience did not see the
+post as a result of the intervention, that is, whether removal of the
+post made a meaningful difference in safeguarding users.
+6
+CONCLUSION
+In this work, we repurposed a data set of over 2 M Facebook posts
+from 2,551 U.S. news sources In the absence of explicit transparency
+about content moderation, we developed a methodology to identify
+removed posts within an existing data set, and infer when removals
+occurred. We also developed a method for predicting lifetime en-
+gagement of posts based on the engagement of other posts from the
+same Facebook page, and proposed novel metrics to quantify the
+impact of content moderation based on accrued and prevented user
+engagement. We found that content moderation during “normal”
+times was relatively quick in absolute terms (median post lifetime
+of 21 hours), yet posts tended to exhaust their engagement poten-
+tial at an even faster rate, and we estimate that at most 21.3 % of
+predicted future engagement was prevented over the entire data set.
+In summary moderation of public content from U.S. news sources
+and influencers on Facebook was too slow to keep up with the
+spread of content and ensuing engagement; it is unclear whether
+this race to limit user exposure to harmful content can be won
+purely by speeding up content moderation. We recommend that
+other strategies be explored further, such as changes to content rec-
+ommendation algorithms that slow the diffusion of content through
+social networks overall.
+7
+ETHICS STATEMENT
+In conducting this research and creating this paper, we honored
+the guidelines and principles established for the conduct of ethical
+research. We have laid out the limitations of our data and methods
+in Section 5.1.
+In our research, we relied on an existing data set created us-
+ing CrowdTangle, an official tool from Facebook, with API access
+granted to both us and the original authors [17]. CrowdTangle con-
+tains only public posts of Facebook pages and aggregate, purely
+quantitative engagement data, but no personally identifiable infor-
+mation. We did not have access to information about individual
+news consumers, and expose no personally identifiable information.
+The purpose of our paper is to measure the circumstances and
+impact of content moderation on social media, particularly during
+a crisis. We are cognizant that our work could cast a poor light
+on Facebook, or on the authors of moderated posts, who might
+object to a work that emphasizes the classification of some of their
+content as objectionable. We argue that the benefit of better un-
+derstanding content moderation outweighs the potential impact to
+reputation, particularly when Facebook’s conduct has been thor-
+oughly reported [6, 54].
+In identifying the strong connection between removed content
+and sources’ reputations for mis- and disinformation (Section 4.3),
+our intention is not to adjudge the merits of any publisher, merely
+to record engagement based on existing reputations for factualness,
+for which we rely on multiple third-party sources for classification.
+Ultimately, we reason that it is a better outcome for society in
+general that Facebook not become a haven for harmful content than
+that any of these stakeholders maintain a sterling reputation. To this
+end, we contribute recommendations for improving moderation
+transparency and efficacy for the benefit of Facebook and its users.
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf,len=1087
+page_content='Understanding the (In)Effectiveness of Content Moderation:  A Case Study of Facebook in the Context of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Capitol Riot∗  Ian Goldstein  New York University  USA  Laura Edelson  New York University  USA  Damon McCoy  New York University  USA  Tobias Lauinger  New York University  USA  ABSTRACT  Social media networks commonly employ content moderation as  a tool to limit the spread of harmful content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' However, the effi-  cacy of this strategy in limiting the delivery of harmful content to  users is not well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In this paper, we create a framework  to quantify the efficacy of content moderation and use our met-  rics to analyze content removal on Facebook within the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' news  ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In a data set of over 2 𝑀 posts with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='6 𝐵 user engage-  ments collected from 2,551 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' news sources before and during  the Capitol Riot on January 6, 2021, we identify 10,811 removed  posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We find that the active engagement life cycle of Facebook  posts is very short, with 90 % of all engagement occurring within  the first 30 hours after posting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Thus, even relatively quick inter-  vention allowed significant accrual of engagement before removal,  and prevented only 21 % of the predicted engagement potential  during a baseline period before the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Capitol attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Nearly a  week after the attack, Facebook began removing older content, but  these removals occurred so late in these posts’ engagement life  cycles that they disrupted less than 1 % of predicted future engage-  ment, highlighting the limited impact of this intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Content  moderation likely has limits in its ability to prevent engagement,  especially in a crisis, and we recommend that other approaches  such as slowing down the rate of content diffusion be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  1  INTRODUCTION  Social media companies such as Google (YouTube) and Meta (Face-  book) have become inextricably linked to contemporary civil soci-  ety [33] through their communication and content sharing “plat-  forms.” Billions of users rely on these services to read or comment  on the news, entertain themselves, and conduct business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Despite  stated intentions [18] and the commonly used term “platform,” most  social media companies are not impartial channels for users’ com-  munications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' On the one hand, social media companies set the  operating parameters of algorithms that promote “interesting” con-  tent to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' On the other hand, when it is deemed necessary to  retain users [29] or to survive public scrutiny [66], social media  companies may intervene against extreme or offensive material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Content moderation is currently the primary strategy that so-  cial media companies employ to counter the spread of harmful  content [5, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Nearly all services, including venues for adult (On-  lyFans [48]) or extreme (8kun [1]) content, have established policies  that dictate what they are willing to host;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' content deemed incon-  sistent with those policies may be removed [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Public debate has  ∗Version 1 (January 2023): Non-peer-reviewed technical report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  questioned the adequacy of these policies [67] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=', which content  should be deleted) or surfaced instances of objectionable content  that was not removed [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Prior research studied potential biases  in moderation of YouTube comments [41, 42], the impact of content  moderation on user behavior on Reddit [40, 58], and the magnitude  of content moderation regarding third-party links to conspiracy  theory stories [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' It is still an open question how quickly content  is removed, independent of the specific policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We argue that to  judge the effectiveness of content moderation, it is important to  understand the spread of content until the time of deletion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=',  whether content is removed before it reaches a large audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  In this paper, we present what we believe to be the first mea-  surement study of the speed and impact of content moderation on  Facebook, both under normal times and during a crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We were  lucky to obtain from Edelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' [17] a data set of public Facebook  posts from 2,551 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' news publishers and “influencers” incidentally  collected around the time of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Capitol Riot on January 6,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Because Facebook currently does not make any post-level  data transparent about content moderation, we develop a novel  methodology to infer content removals and their approximate tim-  ing from daily observations of active posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' For our analysis, we  develop novel metrics to estimate the impact of content removal  on user engagement, that is, the combined number of comments,  shares, and reactions such as “like.” We do so both in terms of  past engagement that was allowed to occur because of the time  it took to remove a post, and in terms of future engagement that  was prevented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We estimate the latter based on predictions of the  engagement potential of posts, which on aggregate have a net error  of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='5 % for viral posts, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='2 % for normal posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  For a baseline period selected to represent “normal” times, we  found that the active engagement life cycle of Facebook posts was  very short, with 90 % of all engagement occurring within the first  30 hours after posting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' An implication of this is that even rela-  tively quick removal of content (we observed a median of 21 hours)  allowed significant accrual of engagement before removal (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='8 𝑀 en-  gagements with 3,843 posts), and prevented only 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='2 % of the pre-  dicted engagement potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' By the time a removal decision was  made, the content had already reached the vast majority of its typi-  cal audience, and the removal had only a minor impact in terms of  preventing engagement or exposure to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  During the crisis surrounding January 6, we initially observed  similar levels of content removals and prevented engagement rates,  but with generally higher engagement across both removed and  non-removed posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' It took 6 days, and two announcements of  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='02737v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='SI]  6 Jan 2023  Ian Goldstein, Laura Edelson, Damon McCoy, and Tobias Lauinger  changes to content moderation by Facebook, before we observed  a meaningful change in our metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Our results show that on  January 12, Facebook began removing older content, but because  these 1,416 removals occurred so late in these posts’ engagement  life cycles, we estimate that they disrupted less than 1 % of future  engagement, and made hardly any difference in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In this  regard, Facebook did not appear to be prepared to contain the fallout  of the crisis in a timely manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Our results suggest that Facebook’s content moderation of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  news publishers and “influencers” could not keep up with the speed  at which content spread on their social network, neither during  normal times nor in times of crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' They highlight that content  moderation on a social network such as Facebook operates within  the constraints of content recommendation, and its efficacy cannot  be assessed in isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Merely tallying numbers of removed posts,  without taking into account enforcement delays or the rates at  which deleted posts reached their projected audiences, does not  adequately characterize the outcomes of content moderation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Un-  fortunately, this is not reflected in the transparency metrics that  Facebook is currently reporting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Our work makes the following contributions:  (1) We propose a methodology for inferring content moderation  of public posts from current transparency data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  (2) We introduce more insightful metrics for quantifying the im-  pact of delayed content removals on accrued and prevented  user engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  (3) We show that moderation of public posts by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' news sources  happened late in their engagement life cycle, resulting in  only 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='2 % prevented engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  2  BACKGROUND & RELATED WORK  Facebook and other social media networks reach billions of users  by promoting content generated by large content producers such  as news organizations or “influencers,” and also by individual users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Unfortunately, social media has been utilized for harmful purposes,  such as spreading disinformation [8] or planning and publicizing  violent activities [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' This has caused these networks to come  under increasing societal, regulatory and legal pressures to prevent  the promotion of dangerous and harmful content on their systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Content Moderation Processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Virtually all social media networks  have established rules of conduct and acceptable content for their  services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' They have a range of techniques at their disposal to enforce  these standards, such as deleting [40, 58], downranking [31], quaran-  tining [11, 56], labeling [69], or demonetizing [15] undesirable con-  tent, or temporarily or permanently banning the accounts of users  who repeatedly post violating content (“deplatforming”) [19, 39, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  In this paper, we study content moderation [5, 22, 24, 28] from the  angle of removal of violating content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  The technical and human challenges of implementing reliable  mechanisms for content moderation at scale have been topics of  repeated study [37, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Social media networks often use automated  systems to monitor user content and communications, and ide-  ally interdict violating content before it is published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Such proac-  tive policy enforcement is typically built as sophisticated pattern  matching systems, comparing content to “blocklists” of known  examples [30, 34], and is thereby limited to predefined classes of  violations (graphic violence, sexual content, child abuse, spam) [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  To identify new patterns of violations, and address more ambiguous  or context-sensitive cases that cannot be handled well by automated  systems, social media networks also commonly have content exam-  ined by human reviewers after publication, for example when the  content reaches a certain popularity threshold, exhibits suspicious  interaction patterns, or causes complaints from other users [14, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Human review of the often violent content has been documented to  have a detrimental effect to the mental well-being of those engaged  in the process [52, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Outcomes of Content Moderation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Prior research on content mod-  eration (specifically, content removals) has often focused on legal  and socio-political issues (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=', [12, 35, 44, 53]), or on qualitative  work aiming to understand users’ experiences and perceptions of  content moderation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=', [36, 38, 60, 68]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Quantitative research  measuring content removals is more scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Srinivasan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' [58]  studied the impact of content removals on subsequent user behavior  within a single subreddit, whereas Jhaver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' [40] measured the  impact that explanations accompanying content removals had on  future user behavior across the entire platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' [41, 42]  evaluated partisan bias in content and comment moderation on  YouTube and found no evidence of left or right bias in removals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Papakyriakopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' [49] quantified the sharing of conspiracy  theory-related URLs on various social media and modeled the im-  pact of content moderation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' in contrast to our work, they did not  delve into the timing of engagement accrual or deletion delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' To  the best of our knowledge, we are the first to study the interplay  between the timeliness of content removals and the accrual (or  prevention) of user engagement with the removed content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Community Standards on Facebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' The rules for user behavior  and acceptable content on Facebook are called “Community Stan-  dards” [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' They are divided into six broad areas ranging from  violence and incitement to intellectual property concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' These  policies have evolved over time, often in response to crises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Our study period around the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Capitol Riot on January 6, 2021  encompasses several such policy changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' On December 14, 2020,  Biden was declared the winner of the Electoral College vote [13]  following the 2020 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' presidential election that had taken place  on November 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' However, President Trump refused to concede and  announced a rally in Washington, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' for January 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' After that  “Stop the Steal” rally, a group of armed people breached security  and stormed the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Capitol building [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Social media platforms  were at the center of the spread of false and misleading information  about the election, including the use of Facebook to promote and  livestream the Capitol attack [6, 10, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In reaction to the attack,  Facebook publicly updated their Community Standards to explicitly  prohibit ‘praise and support’ of the storming of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Capitol,  calls to bring weapons to locations in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=', video posts and pho-  tos from Capitol insurrectionists, violations of the D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' curfew, and  future calls to violence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In addition, Facebook committed to enforce-  ment actions targeting militarized social movements, specifically  naming the Oathkeepers and QAnon [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' On January 11, Facebook  announced that they would remove content containing “Stop the  Steal” [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='2, we analyze how these announced policy  changes were reflected in observable content moderation activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  2  Understanding the (In)Effectiveness of Content Moderation  Other Related Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' More loosely related to our work are prior  studies of deleted content (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=', [4, 9, 70]), which however were  focused on characterizing users who voluntarily deleted their own  content after “regretting” its publication rather than measuring  content moderation imposed by the social network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Similarly, there  has been ample prior work on the spread of (and engagement  with) misinformation on social media (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=', [3, 17, 57]), but it has  largely left open the question of how fast such misinformation  may be removed by the social network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' More generally, researchers  have also studied the diffusion of content on social media and the  concept of “viral” content, for example from the perspectives of  user intention [2] and the structure of diffusion [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Studies of  viral content typically take a set number of the ‘most popular’ or  widely spreading content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Examples of this include Pressgrove et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' [50] with Twitter content, and Vallet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' [64] for YouTube  content spreading via Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Past research has also tackled the  problem of predicting the popularity of content (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=', for Facebook  and YouTube videos based on visual features [63]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Unfortunately,  our data set does not contain the multimedia content from removed  posts, thus we cannot use approaches based on content features to  estimate the engagement potential of deleted posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  3  DATA SET  After the events of January 6, 2021, we sought to study the changes  Facebook made to their content moderation efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Unfortunately,  CrowdTangle, Facebook’s major transparency tool, does not reveal  anything about posts blocked before publication, and states that  once a published post is deleted from Facebook, it is also removed  from CrowdTangle (with a delay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='1 Thus, in order to measure post-  publication content moderation, it is necessary to discover newly  published posts in near real time, before they may be deleted, but  we had not set up a measurement process designed to detect content  moderation in advance of the Capitol Riot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' However, we were able  to obtain a data set of public Facebook posts by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' news publishers  collected by the authors of another study [17], which happened  to include the time period of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Below, we summarize the  original data collection methodology, providing additional detail  where it pertains to the timeliness of detecting new posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We  then describe our methodology for detecting deleted posts and  estimating content moderation delays in a data set that was not  specifically designed for that purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We derive three post data  sets for our study, as shown in Table 1: The Removed Set, which we  further split into smaller sets for specific analyses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' all non-removed  posts from Facebook pages with at least one deleted post (Impacted  Publisher Set);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' and all non-removed posts across all pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='1  Monitoring News Publishers on Facebook  The data set we use in this paper is comprised of the public Facebook  posts of 2,551 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' news publishers during the 2020 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' presidential  election, provided to us by the authors of a study on user engage-  ment with misinformation [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Their selection of Facebook pages  defines the scope of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' The authors of that study based their  selection of Facebook pages on third-party data provided by the  news rating organizations NewsGuard [47] and Media Bias/Fact  Check [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Based on the assessments of these organizations, the  1https://help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='crowdtangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='com/en/articles/3323105-academics-researchers-faq  authors derived the political leaning of each news publisher (Far  Left to Far Right) as well as a binary “misinformation” attribute  indicating whether a news publisher had a known history of spread-  ing misinformation [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' To exclude largely inactive news publish-  ers, the authors of that study also removed Facebook pages with  fewer than 100 followers or 100 total engagements during their  five-month study period;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' our study inherits this filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Due to the  selection of Facebook pages in the original data set, we can study  post-publication moderation of public posts (not comments) on  the Facebook pages of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' news sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' These news sources are  a mix of publishers known for reliable reporting as well as less  reputable sources with a history of spreading misinformation, both  mainstream and niche, from across the political spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='2  Original Data Collection Methodology  The original study authors collected all public Facebook posts and  corresponding engagement metadata from the 2,551 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' news pub-  lishers using two types of crawls of the CrowdTangle API: (1) A  daily crawl to discover all new posts since the last crawl, and (2)  a second, separate daily history crawl to update the engagement  metadata such as the number of likes, shares, and top-level com-  ments of existing posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' This latter crawl went back in history as far  as the remaining daily time permitted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' in practice, existing posts  fell into the observable history window for at least two weeks (or  longer) after publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Both crawls were running daily from  August 10, 2020 until January 18, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='3  Detecting Deleted Posts & Post Lifetime  In September 2021, we performed a follow-up crawl of the same  news publisher pages to determine whether each post from the  original data set was still available on CrowdTangle or had been  deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' This follow-up crawl, however, does not reveal when a  post was deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' To detect deletions with daily granularity, we  leverage the daily history crawls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In the original data set, these  history crawls ran until January 18, thus we limit our analysis  to posts published between December 14, 2020 and January 15,  2021 so that we have sufficient buffer to detect possible deletions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  We conservatively estimate post lifetimes as the time difference  between the publication date of the post (provided by CrowdTangle)  and the last time the post was observed in a daily history crawl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' This  is a lower bound on actual post lifetimes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=', we underestimate  rather than overestimating post lifetimes, as a post could have been  deleted any time in the 24 h between the last time the post was  observed, and the first time the post was missing in a history crawl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  We do not know why any particular post was removed or whether  it was deleted voluntarily or forcibly moderated by Facebook;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' we  only know that a post is not publicly accessible any more when it  fails to show up in the daily history crawl or our September follow-  up crawl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We also note that very short-lived posts (published and  then quickly deleted within less than 24 h) may not be visible to us,  depending on how these events fall between consecutive crawls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='4  Data Sets & Time Periods  Table 1 shows a summary of the data sets used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Our  methodology identified 10,811 posts from 878 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' news publishers  3  Ian Goldstein, Laura Edelson, Damon McCoy, and Tobias Lauinger  Data Set  Post Count  Total Engagement  Delayed Removals  Delayed Engagement  Full Data Set (posts published Dec 14–Jan 15)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='02 M  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='61 B  Removed Set (last observation of removed posts)  10,811  10,889,679  1,416  2,875,064  Baseline Period (Dec 14–Jan 3)  3,843  3,785,464  281  301,741  January 6 Period (Jan 4–Jan 11)  1,171  3,302,379  60  374,066  January 12 Period (Jan 12–Jan 15)  2,300  3,257,523  1,075  2,496,708  Removed Pages (Dec 14–Jan 15)  3,497  544,313  NA  NA  Impacted Publisher Set (posts pub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Dec 14–Jan 15)  297,774  784,842,977  Non-Removed Set (posts pub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Dec 14–Jan 15)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='02 M  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='61 B  Table 1: Overview of data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Posts created December 14, 2020–January 15, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Engagement is the total accrued by posts at  the time of final observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We define delayed removals as those occurring more than 30 hours after post creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  that are no longer accessible via CrowdTangle (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=', no longer pub-  licly visible on Facebook).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We refer to these posts and associated  metadata as the removed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We divide the removed set into four  subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' First, we set aside the subset of posts from deleted pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  When a page is deleted, the deletion cascades to all of its posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Since the deletion times of posts deleted with their page are identi-  cal (likely independent from their original posting times), we study  such pages and their posts separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We subdivide the remaining  deleted posts into three time periods for more targeted analysis:  (1) The baseline period contains posts removed prior to Janu-  ary 4, 2021, and is used to assess “normal” behavior prior to  the events around January 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  (2) The January 6 period contains posts removed between Janu-  ary 4–11, 2021, and is used to analyze the immediate reaction  in the days leading up to January 6 and just afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  (3) The January 12 period contains posts removed on January 12  or later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We analyze this period separately because we no-  ticed increased deletion delays after January 12 (deleted posts  were older than usual);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' this different deletion behavior may  correspond to changes on Facebook’s side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  We create two additional data sets of posts that were not deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  The impacted publisher set comprises the 297,774 remaining  posts from the Facebook pages that had likely experienced content  moderation (at least one post deleted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' This data set serves to inves-  tigate the non-moderated content produced by these pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' To give  an overview of the overall ecosystem, the non-removed set con-  tains all non-deleted posts collected during the observation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  This data set serves as a reference for comparison and consists of  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='76 M posts from 2,315 news publishers (over 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='5 B engagements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='5  Post Engagement and Virality  In the context of Facebook posts, we define engagement as the  sum of all types of interactions that users can have with a post  (and that are made transparent through CrowdTangle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Specifically,  engagement consists of the number of “likes” (and other reaction  types available on Facebook, such as “angry” or “sad”), the number  of times a post is shared, and the number of top-level comments  below the post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Unfortunately, other types of popularity metrics  are not available in CrowdTangle (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=', the number of times a post  has been shown to users, or how many unique users have viewed  the post), thus we cannot include them in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  For posts that were still available in September 2021, we obtain  long-term engagement numbers from our follow-up crawl (Sec-  tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' for deleted posts, we use the engagement numbers from  the last time they were observed on CrowdTangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Similar to our  estimates of post lifetime, engagement numbers for deleted posts  are a lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Additional engagement could have accrued be-  tween the last observation and the time the post was removed, but  as it is unobserved, it does not factor into our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  To study the speed of engagement accrual of (non-deleted) posts,  we leverage time series data extracted from CrowdTangle in Sep-  tember 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' These show engagement values in increasing time  steps relative to the publication time of the post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Virality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In the non-removed data set, posts receive a mean total  engagement of 896 user interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' However, the uneven distribu-  tion of engagement with content on social media is well known: a  small number of posts go viral, and the vast majority do not [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We  are not aware of a commonly accepted engagement-based threshold  for virality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We define as viral any post that reaches 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='6 K engage-  ments (three standard deviations above the mean in the set of all  non-removed posts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' This definition of virality is based purely on  popularity, independent of the speed of engagement accrual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='6  Estimating Prevented Engagement for  Removed Posts  To understand the impact of content removals, it is useful to esti-  mate how much more user engagement a post might have accrued  had it not been deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We do so by first estimating a post’s en-  gagement potential, defined as the total engagement a post would  typically receive in the long run (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=', if it is not deleted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Because  accrual of engagement is different for each post, but potentially  more similar among posts from the same publisher, we estimate a  post’s engagement potential based on non-deleted posts from the  same Facebook page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' From the engagement time series obtained  in September 2021 for the surviving posts published during the  baseline period (which are old enough that we expect them to have  exhausted their engagement potential), we calculate the mean en-  gagement at each time step over all non-deleted posts from the same  page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Since our observations show that there is a distinct engage-  ment curve for viral posts, we estimate engagement separately for  viral posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' For pages with more than 10 non-removed viral posts,  we estimate viral mean engagement per time step individually;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' for  4  Understanding the (In)Effectiveness of Content Moderation  all other pages, we fall back to using global viral engagement es-  timates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We define prevented engagement for a deleted post as  the difference between its engagement potential and engagement  actually accrued before the deletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' It is estimated by summing all  engagement on the applicable mean time series after the last obser-  vation of the post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We likely overestimate prevented engagement  because posts may have remained active for up to 24 h after the  last observation in the crawl, thus they might have accrued more  real engagement (and less engagement was prevented) compared  to the estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In content moderation, it is desirable to remove  policy-violating posts quickly and prevent as much engagement as  possible, thus our overestimation of prevented engagement paints  content moderation efforts as more effective than they may be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  To validate the quality of these engagement predictions, we  tested our methodology on a sample of 10,000 non-deleted posts  published during the baseline period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We simulated removals by  randomly selecting “deletion” timestamps with selection weights  based on the typical lifetimes of deleted posts in the baseline period  (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' The prediction accuracy for engagement potential was  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='7 % for non-viral posts, and 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='4 % for viral posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' The net error  across all 9,929 non-viral sample post predictions was within 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='2 % of  the total ground truth engagement potential (lifetime engagement)  for these posts (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='5 % for the 71 viral posts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' As our intention is to  apply our prediction methodology to a large number of posts and  not to draw inferences about the performance of individual posts,  these results suggest that our methodology is sufficiently robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  4  RESULTS  In this work, we aim to characterize how Facebook moderated  public content published on the Facebook pages of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' news pub-  lishers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' When a post is rendered unavailable, we do not know who  removed the content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' However, as we discuss more thoroughly  in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='2, the patterns we observe across publishers suggest  that the vast majority of post removals were not voluntary, but  instances of content moderation, that is, posts deleted by Facebook  for violation of their platform policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Both user engagement with  content and Facebook’s apparent content moderation changed in  the days during and after the events of January 6, 2021, thus we  analyze these time periods separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='1  Impacts of Content Removals  To understand Facebook’s content moderation performance during  “normal” times, we begin with a baseline period from December 14,  2020 to January 3, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' During this time, the 2,551 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' news pages  published 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='35 M posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Facebook users engaged with these posts  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=', “liked,” commented, or shared) a total of 948 M times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In terms  of likely content moderation, 3,843 posts (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='28 %) were removed  from 640 pages during this period, on average 183 removals per  day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' However, at the last time they were observed active in our data  set, these posts had already accumulated 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='8 M user engagements  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='32 % of total engagement during the baseline period).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' If Facebook  deemed these posts inappropriate to remain on their platform, they  still allowed a considerable number of users to see and interact with  the offending content before reaching the decision to take it down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  0  1  2  3  4  5  6  7  Time since publication (days)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='0  Proportion of removed  posts (cumulative)  30 hours  Time Period  Baseline  January 6  January 12  Figure 1: Post lifetime: Days between posting and last obser-  vation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In the baseline and January 6 periods, nearly all last  observations were at most 30 h after posting, whereas dur-  ing the January 12 period, much older posts were removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  The sharp pivot at 30 h suggests different removal processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  We turn to the lifetimes of removed posts to understand how  these presumably policy-violating posts could accrue so much en-  gagement before being deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' As Figure 1 shows, the average time  between publication of a post to its last observation by the crawler  is 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='7 hours (median: 21 hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' (Note that the actual time of re-  moval could be up to 24 hours later because the crawler checked a  post’s status only once every 24 hours, thus we likely underestimate  post lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=') The distribution of post lifetimes pivots 30 hours  after post creation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' approximately 90 % of removed posts have had  their last observation at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' The remaining 10 % are deleted  at a noticeably slower pace over a much longer time span, which  suggests that a different process might be at play for those deletions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  In summary, most decisions to take down posts appear to be rela-  tively fast in absolute terms, but they do take time, and significant  amounts of engagement are allowed to accrue during that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  The situation of viral posts is slightly different from the overall  post distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' The baseline period saw 9,768 posts accumulate  sufficient engagement to be considered viral (425 M combined);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' 61  of them were removed during this period (more than twice the  rate of all content removals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Until their last observation, these 61  removed viral posts had been able to accrue 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='7 M engagements,  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='8 % of all engagement with posts removed during this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  This share illustrates the importance of handling viral posts well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  quick intervention on a relatively small number of violating viral  posts could disproportionately reduce the odds of Facebook users  seeing and engaging with inappropriate content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' However, our data  suggest that viral posts may be more difficult for Facebook to handle,  as viral posts tended to be active for longer before being removed  (mean time to last observation: 31 hours, more than 7 hours longer  than the overall mean;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' median: 24 hours, or 3 hours longer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  To put the speed of deletions into context, we compare it to the  rate of accrual of engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Most engagement with posts in our  data set happens relatively soon after the content is created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' This is  true both for viral and non-viral posts, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Among  non-removed content created during our baseline period, 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='1 % of  normal posts reached at least 80 % of their total engagement within  one day, while 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='3 % of normal posts achieved it within two days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In  5  Ian Goldstein, Laura Edelson, Damon McCoy, and Tobias Lauinger  0  1  2  3  4  5  6  7  Time since publication (days)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='0  Proportion of posts  (cumulative)  Post Type  viral  non-viral  Figure 2: Days until non-removed baseline posts reach 80 %  of their lifetime engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Around 78 % of non-viral posts  reach 80 % of their lifetime engagement within one day,  whereas it takes two days for viral posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  other words, while the typical content moderation delays observed  in our data set may seem fast in absolute terms, in fact they do  come late in a post’s life cycle because most of a post’s engagement  usually occurs before the typical content moderation delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' As a  result, these instances of content moderation may not make a big  difference in practice because most of a post’s audience will already  have seen and interacted with the post before it is deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Viral posts have a longer period of active engagement accrual  compared to non-viral posts, with 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='2 % reaching 80 % of their  total engagement in one day, and 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='6 % reaching it within two  days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' However, this does not mean that slower content moderation  would be appropriate due to the slower relative rate of engagement  accrual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In fact, total engagement with viral posts is so much higher  that despite the slower relative increase, the absolute increase is  substantial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In the hour before their respective median removal  time, viral posts accrued an average of 1,583 additional engage-  ments, whereas it was only 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='5 for non-viral posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' With the goal  of reducing total exposure to harmful content, it is important to  moderate viral posts with the same (or higher) speed than non-viral  posts, yet our data suggest the opposite is occurring in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  We assess the benefit of typical content moderation delays by  estimating how much engagement was prevented by removing  the post, that is, how much of the typical audience did not see  and interact with the offending post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We estimate a post’s typi-  cal engagement on a per-page basis and separately for viral and  non-viral posts, as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='6, and calculate prevented  engagement as the difference between that estimate and the last  observed engagement count before deletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' During the baseline  period, we estimate that post removals disrupted 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='2 % of potential  engagement (314 or fewer prevented engagements each for 90 % of  removed posts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' For the 3,782 removed non-viral posts, this breaks  down to a 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='5 % prevention rate, or a mean of 151 prevented en-  gagements per non-viral post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' The removal of each of the 61 viral  posts had a much larger impact in absolute terms, with a mean of  7,289 prevented engagements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Viral posts accrue engagement for a  longer time (median 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='4 times longer) than non-viral posts, offering  the opportunity to disrupt a larger fraction of their engagement  potential, but because they were also removed more slowly (median  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='2 times slower), the engagement prevention rate for viral posts  was only slightly higher at 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='0 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Timelier enforcement of the  comparatively few removed viral posts could have made an outsize  difference in preventing Facebook users from engaging with sub-  sequently moderated posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Overall, we estimate that the content  removals we observed during the baseline period prevented 996 K  engagements from occurring, which is a consequential number  in absolute terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Unfortunately though, these removals came so  late that the posts had already reached over three quarters of their  engagement potential, and only a small minority of their potential  engagement was actually prevented by the removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='2  Content Removals After the Capitol Riot  To understand how Facebook carried out content moderation dur-  ing and after the Capitol Riot, we repeat the analyses of the baseline  period above for the time just before and after the event of Janu-  ary 6, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In the eight days from January 4–11, which we refer to  as the January 6 period, we observed comparable levels of content  removals in absolute terms, but higher engagement with content  by the Facebook users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In detail, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' news pages published  393 K posts that received a total of 409 M engagements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' The pe-  riod saw 1,171 posts removed from 246 pages, a mean of 146 per  day, which is roughly comparable to the baseline period (𝑡 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='83,  𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='079).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' However, by the time they were last observed, these  posts had received 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='3 M engagements, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='81 % of all user engage-  ment, significantly (𝑡 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='06, 𝑝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='001) more than during the  baseline period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' The median post lifetime in the January 6 period  was consistent with the baseline period for non-viral (𝜒2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='24,  𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='266) and viral posts (𝜒2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='043, 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='835), also evidenced by  the closely overlapping curves seen in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Thus, the increased  engagement accrual did not stem from slower removals, but higher  user engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Although we saw no increase in the proportion  of posts moderated during the January 6 period, and no significant  increase of non-removed viral posts (𝑡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='62, 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='536), there  was a significant increase in the removal of viral posts as compared  with the baseline period (𝑡 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='38, 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='024), which explains the in-  crease in engagement-weighted removals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' 80% of engagement with  removed viral posts in this period was associated with extremely  partisan sources, and the remainder originated from a single page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  We estimate that post removals during the January 6 period pre-  vented a similar share of potential engagement (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='8 %) as in the  baseline period (a total of 306 K engagements with 1,117 non-viral  posts, and 521 K engagements with 54 viral posts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In summary,  while we observed increased engagement from users, any changes  Facebook may have made to content moderation in the immediate  aftermath of January 6 do not stand out in our metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  We do observe increased levels of content removals during the  four days from January 12–15 (the January 12 period), after Face-  book announced another content moderation policy change [21]  on January 11th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' news pages published 186 K posts during  these four days, reaching 161 M total engagements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Facebook re-  moved 12 entire pages with all their 3,463 posts created between  December 14, 2020 and January 15, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We discuss these removed  pages in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In addition, we observed 2,300 removals of  individual posts from 377 pages, a mean of 575 per day, which was  a significant increase (𝑡 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='63, 𝑝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='001) from the baseline period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  6  Understanding the (In)Effectiveness of Content Moderation  These posts had received a total of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='3 M engagements at the time  of their final observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' This last observation came considerably  later in a post’s lifetime than in the baseline and January 6 periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  In the January 12 period, the mean time between post creation and  last observation was 152 hours (median: 24 hours), significantly  longer than the baseline period for both non-viral (𝜒2 = 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='7,  𝑝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='001) and viral posts (𝜒2 = 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='8, 𝑝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' As shown in  Figure 1, 48 % of last observations occurred more than 30 hours  after publication, whereas it was only approximately 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='8 % in the  baseline and January 6 periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' This sharp increase in delayed con-  tent removals suggests that Facebook may have applied changes  in content moderation announced on January 11 retroactively to  older posts, which were originally published in the time around  January 6, but deleted only after January 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' (We note that all of the  posts removed after more than 30 hours were from “repeat offender”  pages, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=', we had already observed removed posts from these pages  in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=') This further suggests that a policy change, which took  nearly a week to implement, was necessary for Facebook to deal  with the fallout of January 6 on their platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  The unusually late removals are reflected in our estimates of  prevented engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' For all posts removed during the Janu-  ary 12 period, we estimate that 569 K engagements were disrupted,  representing only 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='9 % of removed posts’ predicted engagement  potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In particular, for the subset of posts removed with a long  delay (of more than 30 hours), the deletions appear almost incon-  sequential because they prevented only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='60 % of the engagement  we predicted these posts to achieve without being deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In other  words, if these deletions were indeed instances of “retroactive” con-  tent moderation, Facebook allowed these posts to reach virtually  their entire potential audience before adjudicating that they were  in violation of platform policy and had to be taken down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='3  Impact of Removals by Misinformation and  Partisanship Reputation  Over the last several years, there has been intense public scrutiny  of the partisan impact of content moderation policies on Face-  book [26, 43, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We now analyze content removals taking into  account the partisanship and factualness classifications in our data  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' These labels reflect the reputation of the pages in question  rather than characterizing individual posts, but they still allow us  to compare the effects of content removal along partisan lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We  calculate the engagement-weighted rate of removal as the ob-  served engagement from removed posts in a category of pages over  the total observed engagement from all posts in that category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Over the full time range of our data set, posts made by pages  with a reputation for misinformation had a higher engagement-  weighted rate of removal than posts from pages not classified as  misinformation providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' This finding of higher misinformation  removal rates held true within each of the five partisanship cat-  egories, as shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' At the same time, within the same  factualness category, engagement-weighted rates of removal varied  considerably according to the partisanship of pages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' in the misin-  formation category, they ranged from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='02 % for posts made by Far  Left pages to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='97 % for posts from Slightly Right pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We note  that misinformation pages of extreme partisanship (Far Left and  Far Right) had the lowest post removal rates among misinformation  Partisanship  Misinformation  Non-misinformation  Far Left  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='02 %  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='57 %  Slightly Left  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='67 %  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='19 %  Center  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='34 %  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='29 %  Slightly Right  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='97 %  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='07 %  Far Right  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='19 %  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='53 %  Table 2: Post removal rates by partisanship and factualness  of the source (weighted by engagement, entire data set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Across the political spectrum, news sources known to spread  misinformation saw a higher fraction of their accumulated  user engagement affected by content removals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  pages, whereas non-misinformation pages of extreme partisanship  had the highest post removal rates of all non-misinformation pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  We do not currently have an explanation for this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  So far, our analysis of removal rates covered the entire duration  of our data set, about a month roughly split in half by the events  of January 6, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' These events were of a deeply partisan nature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  we now investigate whether any corresponding effects based on  partisanship are observable among the removed posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' To better  isolate the impact of these events, we again split the data set into  the baseline, January 6, and January 12 periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' (To streamline the  analysis, we no longer differentiate pages by their misinformation  reputation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=') The upper two rows of Figure 3 show the engagement-  weighted partisan breakdown of posts removed during the baseline  period (above), and non-removed posts published during the same  period (below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' While not perfectly balanced, there does not appear  to be any major partisan bias in the posts removed during the  baseline period compared to the posts that were not removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  The picture looks very different in the January 6 period (mid-  dle two rows of Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' The majority of engagement with posts  removed during these eight days corresponded to Far Right pages  (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='8 %), while Slightly Right pages accounted for the next largest  share (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='9 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' The partisan breakdown of engagement with posts  that were not removed does not exhibit this effect, and more closely  resembles the baseline period (albeit not exactly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' During the four  days of the January 12 period, the proportion of engagement attrib-  utable to removed posts from Slightly Right pages receded to a level  comparable to the baseline period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' However, the corresponding  proportion from Far Right pages remained more than double the  baseline share of engagement-weighted post removals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' (This does  not necessarily correspond to “new” engagement accrued during  this period, but may be partially due to delayed enforcement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='7 %  of these removals were posts originally published in the days around  January 6, and deleted only on or after January 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=') In conclusion,  the partisan breakdown of engagement-weighted removal rates  was somewhat balanced along partisan lines before the events of  January 6, and remained at comparable levels across all three time  periods for non-removed posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Likely content moderation around  January 6 disproportionately affected posts from Slightly Right and  Far Right pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' This is in line with the policy changes that Face-  book announced on January 6 and January 11, which specifically  targeted so-called “Stop the Steal” [20, 21] misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  7  Ian Goldstein, Laura Edelson, Damon McCoy, and Tobias Lauinger  0  1  Removed  Far Left  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Left  Center  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Right  Far Right  0  1  Non-removed  0  1  Removed  0  1  Non-removed  0  1  Removed  0  1  Non-removed  Baseline Period  January 6 Period  January 12 Period  Political Leaning (% Engagement)  Figure 3: Engagement with removed and non-removed posts  from U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' news pages by partisanship for the baseline, Jan-  uary 6, and January 12 periods, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Around Janu-  ary 6, the vast majority of engagement with posts that were  ultimately removed went to posts from sources classified as  Slightly Right and Far Right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' The rate of engagement with  misinformation sources across all time periods was 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='1 %  for removed content and 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='2 % for non-removed content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='4  Most Impacted Publishers  For illustrative purposes to complement our quantitative findings,  we identified the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' news pages with most user engagement that  likely experienced content moderation during our entire observa-  tion period (the Impacted Publisher Set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' The two most popular  pages with significant post removals were ‘Dan Bongino’ and ‘Oc-  cupy Democrats,’ both classified as sources of misinformation with  Far Right2 and Far Left3 partisanship, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Taken together,  both pages generated 835 K engagements with content that was  later removed, though this represented only a fraction of the tens  of millions of engagements each of these pages generated in our  observation window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' ‘The Blaze,’ a conservative4 media company  with a reputation for misinformation, received the greatest total  engagement (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='37 M) with 310 ultimately removed posts, account-  ing for 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='3 % of all engagement that the page received between  December 14, 2020 and January 15, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' At 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='5 %, the share of  engagement attributed to deleted posts was highest for ‘I Love  My Freedom,’ a page associated with a conservatively oriented  online apparel retailer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='5 Three quarters of engagement-weighted  removals from this page occurred after Facebook’s January 11 pol-  icy announcement, with a mean delay of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='5 days, suggesting that  the affected content might not have been in clear violation of the  policies in effect prior to that date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In line with the unusually long  removal delays, we estimate that only 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='2 % of potential engage-  ment was prevented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In summary, the pages most heavily impacted  by removed content tended to be pages categorized as extremely  partisan, misinformation providers, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  2https://mediabiasfactcheck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='com/dan-bongino/  3https://mediabiasfactcheck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='com/occupy-democrats/  4https://mediabiasfactcheck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='com/the-blaze/  5https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='facebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='com/ILMFOrg/about  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='5  Page Deletions  So far, we have only considered removals of individual posts from  pages that themselves remained active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In addition, we observed the  removal of 14 entire pages (the Removed Pages Set), which included  their 3,497 posts created between December 14, 2020 and Janu-  ary 15, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' These posts had accumulated an engagement of 544 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Deleted pages fall into three categories: known sources of health  and vaccine misinformation (‘Erin at Health Nut News,’ ‘LifeSite-  News.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='com’), extreme Far Right media with poor records for factual-  ness and newsworthiness (‘TruNews,’ ‘ConservativeOpinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='com’),  and pages frequently linked to conspiracy sources (‘The Drudge  Report,’ ‘Zero Hedge’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Many of these were widely known as spread-  ers of inaccurate information, and their forced deplatforming by  Facebook and the related debate were widely reported [16, 27, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  5  DISCUSSION  Content moderation aims to limit the exposure of users to harmful  content by way of deleting user-generated content that is deemed  unacceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In this framework, content moderation needs to com-  pete with content recommendation systems, in that the speed of  enforcement needs to keep up with the potentially viral spread of  toxic content in order to effectively contain undesirable content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Our results suggest that Facebook’s content moderation was not  effective at meaningfully limiting the spread of harmful content  from U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' news publishers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In the most common case, content mod-  eration was relatively quick in absolute terms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' half of removed posts  in the baseline period were last observed 21 hours after publica-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Yet, the speed of removal pales in comparison to how quickly  content accrues engagement, and how quickly interest in content  fades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' By the time of removal, the posts in our data set had already  been shown to millions of Facebook users, accumulating a total of  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='9 million engagements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We estimate that the removals prevented  only 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='2 % of engagement from occurring because the vast majority  of the posts’ potential audience had already seen them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Further-  more, it appears that Facebook needed to change their policy in  order to moderate some content after the Capitol riot of January 6,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' This policy change took nearly a week to implement, likely  delaying the removal of 995 posts by multiple days, with the result  that they received over 2 M engagements in the meantime, and  their removal prevented virtually none (< 1 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Whether or not content moderation can be effective depends  on factors that are under Facebook’s control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' For one, Facebook  could aim to delete content more quickly by decreasing delays in  human review and implementing a content policy that does not  require lengthy adjustments to allow reaction to major real-world  events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' It is unclear, however, how much potential there is for such  improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' As we have shown, the currently very short ac-  tive engagement periods of posts, and the speed of viral accrual  of engagement create a limited window during which content re-  moval can be effective as a strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Therefore, the second avenue  to making content moderation more effective would be changing  the parameters of the content recommendation algorithm to en-  large this time window and give Facebook more time to intervene  effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' This could take various forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' To increase the time for  intervention, new content could be recommended more slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' To  limit the impact of harmful content, the relatively small number  8  Understanding the (In)Effectiveness of Content Moderation  of posts predicted to “go viral” could be reviewed manually before  they accrue significant engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Admittedly, the scenario for content moderation that we stud-  ied in this paper may be more challenging than the ecosystem of  Facebook as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Public posts from U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' news publishers, due  to their newsworthiness, may require a more thorough review than  other types of policy violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Furthermore, due to their larger  audience, delayed removals have a much larger aggregate impact  than moderation of posts in smaller groups, or private messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  These challenges highlight, however, the limits of what can be  achieved with the tool of content moderation under the constraints  of a recommendation system that has been designed and optimized  to spread engaging content as quickly as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='1  Limitations  At the time of this writing, Facebook does not provide fine-grained  transparency for content moderation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Our efforts to study content  moderation rely on what is observable indirectly through Crowd-  Tangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' As a result, we can only study posts deleted after publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Other types of content moderation, such as rejecting posts before  publication, or deliberately downranking posts in the recommen-  dation algorithm (without deleting them) are outside the scope of  our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Facebook claims that up to 90 % of content violations  (including hate speech, nudity, violence, and bullying) are deleted  before they are likely to have been seen [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Our study can only  include posts that slip through pre-publication content filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Yet,  collectively Facebook users engaged with the deleted posts in our  data set more than 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='8 million times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' While we do not have exact  numbers about how this relates to impressions and unique users,  it is reasonable to estimate that millions of Facebook users were  exposed to these posts that Facebook later deemed inappropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Because we rely on a repurposed data set that was not originally  collected to study content moderation, our analysis is subject to  additional limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' The authors of the data set did not archive  multimedia content such as images or videos associated with posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Posts may have been deleted for offending image or video content,  thus we cannot conduct a meaningful study of the contents of  deleted posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Furthermore, the timing of post discovery in the  original crawl causes a bias towards longer-lived posts, that is,  when posts are deleted quickly after publication, they are more  likely to be missing in the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Similarly, the daily status checks  of older posts in the original crawl mean that the inferred post  lifetimes are only a rough estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Our analysis is based on public posts published by the Face-  book pages of 2,551 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' news organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' The content produced  by these pages is not representative of the Facebook ecosystem  as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Furthermore, the selection of these pages and their  categorization with regard to their political leaning and history  of spreading misinformation is subject to the methodology and  limitations described by Edelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='2  Content Moderation vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Voluntary  Removals  Because we cannot observe post deletions directly, we do not know  for certain why content was removed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' it could have been moder-  ated by Facebook, deleted by the page owner, or the visibility could  have been changed to private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' However, taken together, several  independent observations from our analysis suggest that a majority  of removed posts were indeed subject to content moderation by  Facebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' First, the pattern of deleted post lifetimes with a sharp  pivot at 30 hours was consistent across pages, which suggests inter-  vention by a common process as opposed to independent voluntary  deletions by page owners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We were able to confirm one instance of  voluntary deletions by contacting the page administrators of the  National Center for Missing & Exploited Children,6 a non-partisan,  charitable organization who confirmed to us that they routinely  take down posts about missing children when those children are  found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Statistically, their 185 voluntary removals in our data set  (with a total of 344 K engagements) looked very different from ag-  gregate removal patterns, suggesting that the bulk of removals are  not voluntary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In terms of post lifetimes, we last observed their vol-  untarily deleted posts after a mean of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='3 days (median: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='4 days), as  opposed to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='7 days (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='9 days) across all removed posts in our data  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' This supports our hypothesis based on patterns in post lifetimes  that most observed removals were likely deletions by Facebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Second, the unusually long removal delays only observed in the Jan-  uary 12 period and the coinciding, prior announcement by Facebook  of a content moderation policy change suggest that these removals  were instances of content moderation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Third, the higher prevalence  of removals among pages with a reputation for spreading misinfor-  mation makes it less likely these are voluntary deletions, unless we  hypothesize that these misinformation providers are more prone to  self-censorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' While only improved transparency from Facebook  could provide clarity, we are confident that the removal patterns  we observed are predominantly a reflection of content moderation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='3  Improved Measurement Design  If we were to design a measurement of content moderation of public  posts from scratch, and assuming that a transparency tool such  as CrowdTangle continues to impose low rate limits and does not  provide specific transparency around content moderation, what  would our measurement design look like?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Archive multimedia content such as images and video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' If space  is an issue, keep only the content of deleted posts, or of all  posts from pages that have deleted posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Detect new posts more frequently (as opposed to daily) to  observe short-lived posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Prioritize the time of day when  most posts are published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Check the status of existing posts more frequently than daily  to get more fine-grained post lifetimes, at least during the  first 2-3 days, which see the vast majority of deletions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  For example, hourly daytime crawls to detect publication or deletion  of recent posts (0-3 days old) could be combined with nightly history  crawls to detect deletion of older posts with daily granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='4  Transparency Metrics & Reporting  Meta’s primary mechanism for making data relating to content  moderation transparent is their Community Standards Enforce-  ment Report [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In these quarterly reports, the company shares  aggregate, global data about eleven categories of removals, reported  6https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='missingkids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='org/  9  Ian Goldstein, Laura Edelson, Damon McCoy, and Tobias Lauinger  as the absolute number of posts removed, and the relative share  of global user impressions attributable to moderated content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' The  discrepancy between absolute and relative numbers makes them  difficult to compare, and they are not broken down at a country  level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Furthermore, the raw number of removed posts is not per se  informative for understanding the efficacy of content moderation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Based on the metrics we used in our analysis, we recommend re-  porting how long content remains active before it is taken down,  how many people are exposed to content that is later removed, and  how much of a removed post’s potential audience did not see the  post as a result of the intervention, that is, whether removal of the  post made a meaningful difference in safeguarding users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  6  CONCLUSION  In this work, we repurposed a data set of over 2 M Facebook posts  from 2,551 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' news sources In the absence of explicit transparency  about content moderation, we developed a methodology to identify  removed posts within an existing data set, and infer when removals  occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We also developed a method for predicting lifetime en-  gagement of posts based on the engagement of other posts from the  same Facebook page, and proposed novel metrics to quantify the  impact of content moderation based on accrued and prevented user  engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We found that content moderation during “normal”  times was relatively quick in absolute terms (median post lifetime  of 21 hours), yet posts tended to exhaust their engagement poten-  tial at an even faster rate, and we estimate that at most 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='3 % of  predicted future engagement was prevented over the entire data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  In summary moderation of public content from U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' news sources  and influencers on Facebook was too slow to keep up with the  spread of content and ensuing engagement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' it is unclear whether  this race to limit user exposure to harmful content can be won  purely by speeding up content moderation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We recommend that  other strategies be explored further, such as changes to content rec-  ommendation algorithms that slow the diffusion of content through  social networks overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  7  ETHICS STATEMENT  In conducting this research and creating this paper, we honored  the guidelines and principles established for the conduct of ethical  research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We have laid out the limitations of our data and methods  in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  In our research, we relied on an existing data set created us-  ing CrowdTangle, an official tool from Facebook, with API access  granted to both us and the original authors [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' CrowdTangle con-  tains only public posts of Facebook pages and aggregate, purely  quantitative engagement data, but no personally identifiable infor-  mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We did not have access to information about individual  news consumers, and expose no personally identifiable information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  The purpose of our paper is to measure the circumstances and  impact of content moderation on social media, particularly during  a crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We are cognizant that our work could cast a poor light  on Facebook, or on the authors of moderated posts, who might  object to a work that emphasizes the classification of some of their  content as objectionable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' We argue that the benefit of better un-  derstanding content moderation outweighs the potential impact to  reputation, particularly when Facebook’s conduct has been thor-  oughly reported [6, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  In identifying the strong connection between removed content  and sources’ reputations for mis- and disinformation (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='3),  our intention is not to adjudge the merits of any publisher, merely  to record engagement based on existing reputations for factualness,  for which we rely on multiple third-party sources for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Ultimately, we reason that it is a better outcome for society in  general that Facebook not become a haven for harmful content than  that any of these stakeholders maintain a sterling reputation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' To this  end, we contribute recommendations for improving moderation  transparency and efficacy for the benefit of Facebook and its users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Tweets are forever: A large-scale quantitative analysis of deleted  tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In Conference on Computer-Supported Cooperative Work (CSCW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' ACM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' Media International Australia 177, 1  (2020), 103–107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' The Big Lie and  Big Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' The Carter Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' ‘Be There.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Will Be Wild!’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=': Trump all but  circled the date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' The New York Times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' Network propaganda: Manip-  ulation, disinformation, and radicalization in American politics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Oxford University  Press, New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' 472 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  [9] Parantapa Bhattacharya and Niloy Ganguly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' In International Conference on Web and Social Media (ICWSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  AAAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content='14803  [10] Center for an Informed Public, Digital Forensic Research Lab, Graphika, and  Stanford Internet Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' The long fuse: Misinformation and the 2020  election.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Stanford Digital Repository: Election Integrity Partnership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  https:  //purl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' ACM Transactions on Computer-Human Interaction 29, 4,  Article 29 (March 2022), 26 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='1145/3490499  [12] MacKenzie F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Fear the Reaper: How content moderation rules are  enforced on social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' International Review of Law, Computers & Technology  34, 2 (2020), 126–152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content='1733762  [13] Nick Corasaniti and Jim Rutenberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' Electoral College vote officially affirms  Biden’s victory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' What is a flag for?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' New Media & Society 18, 3  (2016), 410–428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' Keith, Ethan Zuckerman, Narseo Vallina-Rodriguez,  Brendan O’Connor, and Rishab Nithyanand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Paying attention to the algo-  rithm behind the curtain: Bringing transparency to YouTube’s demonetization  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In Conference on Computer-Supported Cooperative Work (CSCW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' ACM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='1145/3555209  [16] Alison Durkee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Google and Facebook are cracking down on the Far Right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Vanity Fair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' (Mis)information  news sources on Facebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In Internet Measurement Conference (IMC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' ACM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' In response to Oversight Board, Trump suspended for two years;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  will only be reinstated if conditions permit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Terrorism in the digital era: The dark side of  livestreaming, online content and internet subcultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' http://mastersofmedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' Congressional Research Service Report 46662  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Facebook Is finally doing something about the biggest spreaders  of anti-vax lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Vice News.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Custodians of the Internet: Platforms, content moderation,  and the hidden decisions that shape social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Yale University Press, New  Haven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' 288 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Platforms are not intermediaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Georgetown Law Tech-  nology Review 198 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' https://georgetownlawtechreview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content='pdf  [30] Tarleton Gillespie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Content moderation, AI, and the question of scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Big  Data & Society 7 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='1177/2053951720943234  [31] Tarleton Gillespie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Do not recommend?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Reduction as a form of con-  tent moderation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Social Media + Society 8, 3 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content='1177/  20563051221117552  [32] Sharad Goel, Ashton Anderson, Jake Hofman, and Duncan J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Watts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' The  structural virality of online diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Management Science 62, 1 (2015), 180–196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content='2158  [33] Robert Gorwa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' What is platform governance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Information, Communication  & Society 22 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Algorithmic content moderation: Technical and political  challenges in the automation of platform governance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Big Data & Society 7 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content='1177/2053951719897945  [35] James Grimmelmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' The virtues of moderation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Yale Journal of Law &  Technology 42 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Disproportionate removals and differing content moderation experiences for  conservative, transgender, and black social media users: Marginalization and  moderation gray areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In Conference on Computer-Supported Cooperative Work  (CSCW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' ACM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' The political power of platforms: How current attempts  to regulate misinformation amplify opinion power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Digital Journalism 8 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' Computers  in Human Behavior 83 (2018), 278–287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' International Journal of Communication 13  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' Washington  Post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content='2695439  [64] David Vallet, Shlomo Berkovsky, Sebastien Ardon, Anirban Mahanti, and Mo-  hamed Ali Kafaar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' ACM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='  Most Ameri-  cans think social media sites censor political viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Pew Research Cen-  ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content='org/internet/2020/08/19/most-americans-think-  social-media-sites-censor-political-viewpoints/  [66] Jane Wakefield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Christchurch shootings: Social media races to stop attack  footage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' BBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' Facebook can’t be reformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' The New York Times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' https:  //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='nytimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='com/2020/07/01/opinion/facebook-zuckerberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='html  [68] Sarah Myers West.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Censored, suspended, shadowbanned: User interpreta-  tions of content moderation on social media platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' New Media & Society 20,  11 (2018), 4366–4383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content='1177/1461444818773059  [69] Savvas Zannettou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' “I won the election!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=': An empirical analysis of soft  moderation interventions on Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' In International Conference on Web and  11  Ian Goldstein, Laura Edelson, Damon McCoy, and Tobias Lauinger  Social Media (ICWSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' AAAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' Tweet properly: Analyzing deleted  tweets to understand and identify regrettable ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+page_content=' ACM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE0T4oBgHgl3EQf4QLf/content/2301.02737v1.pdf'}
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+DetectGPT: Zero-Shot Machine-Generated Text Detection
+using Probability Curvature
+Eric Mitchell 1 Yoonho Lee 1 Alexander Khazatsky 1 Christopher D. Manning 1 Chelsea Finn 1
+Abstract
+The fluency and factual knowledge of large lan-
+guage models (LLMs) heightens the need for cor-
+responding systems to detect whether a piece of
+text is machine-written. For example, students
+may use LLMs to complete written assignments,
+leaving instructors unable to accurately assess stu-
+dent learning. In this paper, we first demonstrate
+that text sampled from an LLM tends to occupy
+negative curvature regions of the model’s log prob-
+ability function. Leveraging this observation, we
+then define a new curvature-based criterion for
+judging if a passage is generated from a given
+LLM. This approach, which we call DetectGPT,
+does not require training a separate classifier, col-
+lecting a dataset of real or generated passages, or
+explicitly watermarking generated text. It uses
+only log probabilities computed by the model of
+interest and random perturbations of the passage
+from another generic pre-trained language model
+(e.g, T5). We find DetectGPT is more discrimi-
+native than existing zero-shot methods for model
+sample detection, notably improving detection of
+fake news articles generated by 20B parameter
+GPT-NeoX from 0.81 AUROC for the strongest
+zero-shot baseline to 0.95 AUROC for Detect-
+GPT. See ericmitchell.ai/detectgpt
+for code, data, and other project information.
+1. Introduction
+Large language models (LLMs) have proven able to gen-
+erate remarkably fluent responses to a wide variety of user
+queries. Models such as GPT-3 (Brown et al., 2020), PaLM
+(Chowdhery et al., 2022), and ChatGPT (OpenAI, 2022)
+can convincingly answer complex questions about science,
+mathematics, historical and current events, and social trends.
+While recent work has found that cogent-sounding LLM-
+generated responses are often simply wrong (Lin et al.,
+1Stanford University.
+Correspondence to: Eric Mitchell
+<eric.mitchell@cs.stanford.edu>.
+ Candidate passage : 
+“Joe Biden recently made a move to the White House 
+that included bringing along his pet German Shepherd…”
+DetectGPT
+x
+. . .
+GPT-3
+(1) Perturb
+(2) Score
+(3) Compare
+🤖   from GPT-3
+Yes
+(reword 
+with T5)
+“made a move” 
+ “moved”
+→
+“pet” 
+ “dog”
+→
+Delete “bringing along”
+. . .
+🤔   from other source
+No
+Figure 1. We aim to determine whether a piece of text was gener-
+ated by a particular LLM p, such as GPT-3. To classify a candidate
+passage x, DetectGPT first generates minor perturbations of the
+passage ˜xi using a generic pre-trained model such as T5. Then
+DetectGPT compares the log probability under p of the original
+sample x with each perturbed sample ˜xi. If the average log ratio
+is high, the sample is likely from the source model.
+2022), the articulate nature of such generated text may still
+make LLMs attractive for replacing human labor in some
+contexts, notably student essay writing and journalism. At
+least one major news source has released AI-written content
+with limited human review, leading to substantial factual er-
+rors in some articles (Christian, 2023). Such applications of
+LLMs are problematic for a variety of reasons, making fair
+student assessment difficult, impairing student learning, and
+proliferating convincing-but-inaccurate news articles. Un-
+fortunately, humans perform only slightly better than chance
+when classifying machine-generated vs human-written text
+(Gehrmann et al., 2019), leading researchers to consider
+automated detection methods that may identify signals dif-
+ficult for humans to recognize. Such methods might give
+teachers and news-readers more confidence in the human
+origin of the text that they consume.
+As in prior work (Jawahar et al., 2020), we study the
+machine-generated text detection problem as a binary clas-
+sification problem. Specifically, we aim to classify whether
+arXiv:2301.11305v1  [cs.CL]  26 Jan 2023
+
+Zero-Shot Machine-Generated Text Detection using Probability Curvature
+a candidate passage was generated by a particular source
+model. While several works have investigated methods for
+training a second deep network to detect machine-generated
+text, such an approach has several shortcomings, including
+a tendency to overfit to the topics it was trained on as well as
+the need to train a new model for each new source model that
+is released. We therefore consider the zero-shot version of
+machine-generated text detection, where we use the source
+model itself, without fine-tuning or adaptation of any kind,
+to detect its own samples. The most common method for
+zero-shot machine-generated text detection is evaluating the
+average per-token log probability of the generated text and
+thresholding (Solaiman et al., 2019; Gehrmann et al., 2019;
+Ippolito et al., 2020). However, this zeroth-order approach
+to detection ignores the local structure of the learned proba-
+bility function around a candidate passage, which we find
+contains useful information about the source of a passage.
+This paper poses a simple hypothesis: minor rewrites of
+model-generated text tend to have lower log probability un-
+der the model than the original sample, while minor rewrites
+of human-written text may have higher or lower log prob-
+ability than the original sample. In other words, unlike
+human-written text, model-generated text tends to lie in ar-
+eas where the log probability function has negative curvature
+(e.g., local maxima of the log probability). We empirically
+verify this hypothesis, and find that it holds true across a
+diverse body of LLMs, even when the minor rewrites, or
+perturbations, come from alternative language models. We
+leverage this observation to build DetectGPT, a zero-shot
+method for automated machine-generated text detection. To
+test if a passage came from a source model pθ, DetectGPT
+compares the log probability of a candidate passage under
+pθ with the average log probability of several perturbations
+of the passage under pθ (generated with, e.g., T5; Raffel
+et al. (2020)). If the perturbed passages tend to have lower
+average log probability than the original by some margin,
+the candidate passage is likely to have come from pθ. See
+Figure 1 for an overview of the problem and DetectGPT.
+See Figure 2 for an illustration of the underlying hypothesis
+and Figure 3 for empirical evaluation of the hypothesis. Our
+experiments find that DetectGPT is more accurate than exist-
+ing zero-shot methods for detecting machine-generated text,
+improving over the strongest zero-shot baseline by over
+0.1 AUROC for multiple source models when detecting
+machine-generated news articles.
+Contributions. Our main contributions are: (a) the identi-
+fication and empirical validation of the hypothesis that the
+curvature of a model’s log probability function tends to be
+significantly more negative at model samples than for hu-
+man text, and (b) DetectGPT, a practical algorithm inspired
+by this hypothesis that approximates the trace of the log
+probability function’s Hessian to detect a model’s samples.
+log pθ(x)
+xfake ∼ pθ (x)
+˜x fake
+1
+˜x fake
+2
+˜x fake
+3
+˜x fake
+4
+xreal ∼ phuman(x)
+˜xreal
+1
+˜xreal
+2
+˜xreal
+3
+˜xreal
+4
+Fake/real sample
+Perturbed fake/real sample
+Log likelihood
+…
+log pθ(x)
+Figure 2. We identify and exploit the tendency of machine-
+generated passages x ∼ pθ(·) (left) to lie in negative curvature
+regions of log p(x), where nearby samples have lower model
+log probability on average.
+In contrast, human-written text
+x ∼ preal(·) (right) tends not to occupy regions with clear nega-
+tive log probability curvature.
+2. Related Work
+Increasingly large LLMs (Radford et al., 2019; Brown et al.,
+2020; Chowdhery et al., 2022; OpenAI, 2022; Zhang et al.,
+2022) have led to dramatically improved performance on
+many language-related benchmarks and the ability to gen-
+erate convincing and on-topic text. The GROVER model
+(Zellers et al., 2019) was the first LLM trained specifically
+for generating realistic-looking news articles. Human evalu-
+ators found GROVER-generated propaganda at least as trust-
+worthy as human-written propaganda, or even slightly more
+trustworthy, motivating the authors to study GROVER’s abil-
+ity to detect its own generations by fine-tuning a detector on
+top of its features; they found GROVER better able to de-
+tect GROVER-generated text than other pre-trained models.
+However, Bakhtin et al. (2019); Uchendu et al. (2020) note
+that models trained explicitly to detect machine-generated
+text tend to overfit to their training distribution of domains
+or source models.
+Other works have trained supervised models for machine-
+generated text detection on top of neural representations
+(Bakhtin et al., 2019; Solaiman et al., 2019; Uchendu et al.,
+2020; Ippolito et al., 2020; Fagni et al., 2021), bag-of-words
+features (Solaiman et al., 2019; Fagni et al., 2021), and hand-
+crafted statistical features (Gehrmann et al., 2019). Alterna-
+tively, Solaiman et al. (2019) notes the surprising efficacy
+of a simple zero-shot method for machine-generated text
+detection, which thresholds a candidate passage based on its
+average log probability under the generative model, serving
+as a strong baseline for zero-shot machine-generated text
+detection in our work. In our work, we similarly use the
+generating model to detect its own generations in a zero shot
+manner, but through a different approach based on estimat-
+ing local curvature of the log probability around the sample
+rather than the raw log probability of the sample itself. See
+Jawahar et al. (2020) for a complete survey on machine-
+generated text detection. Other work explores watermarks
+for generated text (Kirchenbauer et al., 2023), which modify
+
+Zero-Shot Machine-Generated Text Detection using Probability Curvature
+Algorithm 1 DetectGPT model-generated text detection
+1: Input: passage x, source model pθ, perturbation function q,
+number of perturbations k, decision threshold ϵ
+2: ˜xi ∼ q(· | x), i ∈ [1..k] // mask spans, sample replacements
+3: ˜µ ← 1
+k
+�
+i log pθ(˜xi)
+// approximate expectation in Eq. 1
+4: ˆdx ← log pθ(x) − ˜µ
+// estimate d (x, pθ, q)
+5: ˜σ2
+x ←
+1
+k−1
+�
+i (log pθ(˜xi) − ˜µ)2 // variance for normalization
+6: if
+ˆdx
+√˜σx > ϵ then
+7:
+return true
+// probably model sample
+8: else
+9:
+return false
+// probably not model sample
+a model’s generations to make them easier to detect. Our
+work does not assume text is generated with the goal of easy
+detection; DetectGPT detects text generated from publicly
+available LLMs using standard LLM sampling strategies.
+The problem of machine-generated text detection echoes ear-
+lier work on detecting deepfakes, artificial images or videos
+generated by deep nets, which has spawned substantial ef-
+forts in detection of fake visual content (Dolhansky et al.,
+2020; Zi et al., 2020). While early works in deepfake de-
+tection used relatively general-purpose model architectures
+(G¨uera & Delp, 2018), many deepfake detection methods
+rely on the continuous nature of image data to achieve state-
+of-the-art performance (Zhao et al., 2021; Guarnera et al.,
+2020), making direct application to text difficult.
+3. The Zero-Shot Machine-Generated Text
+Detection Problem
+We study zero-shot machine-generated text detection, the
+problem of detecting whether a piece of text, or candidate
+passage x, is a sample from a source model pθ. The problem
+is zero-shot in the sense that we do not assume access to
+human-written or generated samples to perform detection.
+As in prior work, we study a ‘white box’ setting (Gehrmann
+et al., 2019) in which the detector may evaluate the log prob-
+ability of a sample log pθ(x). The white box setting does
+not assume access to the model architecture or parameters.
+While most public APIs for LLMs (such as GPT-3) enable
+scoring text, some exceptions exist. While most of our ex-
+periments consider the white box setting, see Section 5.2 for
+a suite of experiments in which we score text using models
+other than the source model.
+The detection criterion we propose, DetectGPT, also makes
+use of generic pre-trained mask-filling models in order to
+generate passages that are ‘nearby’ the candidate passage.
+However, these mask-filling models are used off-the-shelf,
+without any fine-tuning or adaptation to the target domain.
+4. DetectGPT: Zero-shot Machine-Generated
+Text Detection with Random Perturbations
+DetectGPT is based on the hypothesis that samples from a
+source model pθ typically lie in areas of negative curvature
+of the log probability function of pθ, unlike human text. In
+other words, if we apply small perturbations to a passage
+x ∼ pθ, producing ˜x, the quantity log pθ(x) − log pθ(˜x)
+should be relatively large on average for machine-generated
+samples compared to human-written text. To leverage this
+hypothesis, first consider a perturbation function q(· | x)
+that gives a distribution over ˜x, slightly modified versions of
+x with similar meaning (we will generally consider roughly
+paragraph-length texts x). As an example, q(· | x) might be
+the result of simply asking a human to rewrite one of the
+sentences of x, while preserving the meaning of x. Using
+the notion of a perturbation function, we can define the
+perturbation discrepancy d (x, pθ, q):
+d (x, pθ, q) ≜ log pθ(x) − E˜x∼q(·|x) log pθ(˜x)
+(1)
+We state our hypothesis more formally in Hypothesis 4.1.
+Hypothesis 4.1. If q produces samples on the data manifold,
+d (x, pθ, q) is positive with high probability for samples
+x ∼ pθ. For human-written text, d (x, pθ, q) tends toward
+zero for all x.
+If we define q(· | x) to be samples from a mask-filling
+model such as T5 (Raffel et al., 2020), rather than human
+rewrites, we can empirically test Hypothesis 4.1 in an au-
+tomated, scalable manner. For real data, we use 500 news
+articles from the XSum dataset (Narayan et al., 2018); for
+model samples, we use the output of four different LLMs
+when prompted with the first 30 tokens of each article in
+XSum. We use T5-3B to apply perturbations, masking out
+randomly-sampled 2-word spans until 15% of the words in
+the article are masked. We approximate the expectation in
+Eq. 1 with 100 samples from T5.1 Figure 3 shows the result
+of this experiment. We find the distribution of perturbation
+discrepancies is significantly different for human-written
+articles and model samples; model samples tend to have
+a larger perturbation discrepancy. Section 5.3 explores a
+relaxation of the assumption that q only produces samples
+on the data manifold, finding that a gap, although reduced,
+still exists in this case.
+Given these results, we can detect if a piece of text was
+generated by a model pθ by simply thresholding the pertur-
+bation discrepancy. In practice, we find that normalizing the
+perturbation discrepancy by the standard deviation of the ob-
+served values used to estimate E˜x∼q(·|x) log pθ(˜x) provides
+a slightly better signal for detection, typically increasing
+AUROC by around 0.020, so we use this normalized version
+of the perturbation discrepancy in our experiments. The
+resulting method, DetectGPT, is summarized in Alg. 1. Hav-
+ing described an application of the perturbation discrepancy
+1We later show in Figure 8 that varying the number of samples
+used to estimate the expectation effectively allows for trading off
+between accuracy and speed.
+
+Zero-Shot Machine-Generated Text Detection using Probability Curvature
+0.0
+0.1
+0.2
+0.3
+0
+20
+40
+60
+gpt2-xl
+Human
+Model
+0.00
+0.05
+0.10
+0.15
+0.20
+EleutherAI/gpt-neo-2.7B
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0
+20
+40
+60
+EleutherAI/gpt-j-6B
+0.0
+0.1
+0.2
+0.3
+EleutherAI/gpt-neox-20b
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Log Likelihood Drop (Perturbation Discrepancy)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Frequency
+Figure 3. The average drop in log probability (perturbation discrep-
+ancy) after rephrasing a passage is consistently higher for model-
+generated passages than for human-written passages. Each plot
+shows the distribution of the perturbation discrepancy d (x, pθ, q)
+for human-written news articles and machine-generated arti-
+cles; of equal word length from models GPT-2 (1.5B), GPT-Neo-
+2.7B (Black et al., 2021), GPT-J (6B; Wang & Komatsuzaki (2021))
+and GPT-NeoX (20B; Black et al. (2022)). Human-written arti-
+cles are a sample of 500 XSum articles; machine-generated text
+is generated by prompting each model with the first 30 tokens of
+each XSum article, sampling from the raw conditional distribution.
+Discrepancies are estimated with 100 T5-3B samples.
+to machine-generated text detection, we next provide an
+interpretation of this quantity.
+Interpretation of the Perturbation Discrepancy as Cur-
+vature
+While Figure 3 suggests that the perturbation dis-
+crepancy may be useful, it is not immediately obvious what
+it measures. In this section, we show that the perturbation
+discrepancy approximates a measure of the local curvature
+of the log probability function near the candidate passage,
+more specifically, that it is proportional to the negative trace
+of the Hessian of the log probability function.2 To han-
+dle the non-differentiability of discrete data, we consider
+candidate passages in a latent semantic space, where small
+displacements correspond to valid edits that retain similar
+meaning to the original. Because our perturbation function
+(T5) models natural text, we expect our perturbations to
+roughly capture such meaningful variations of the original
+passage, rather than arbitrary edits.
+We first invoke Hutchinson’s trace estimator (Hutchinson,
+1990), giving an unbiased estimate of the trace of matrix A:
+tr(A) = Ezz⊤Az
+(2)
+provided that the elements of z ∼ qz are IID with E[zi] = 0
+and Var(zi) = 1. To use Equation 2 to estimate the trace of
+the Hessian, we must therefore compute the expectation of
+2Rather than the Hessian of the log likelihood with respect to
+model parameters (the Fisher Information Matrix), here we refer
+to the Hessian of the log probability with respect to the sample x.
+the directional second derivative z⊤Hf(x)z. We approxi-
+mate this expression with finite differences:
+z⊤Hf(x)z ≈ f(x + hz) + f(x − hz) − 2f(x)
+h2
+(3)
+Combining Equations 2 and 3 and simplifying with h = 1,
+we have an estimate of the negative Hessian trace
+−tr(H)f(x) ≈ 2f(x) − Ez [f(x + z) + f(x − z)] . (4)
+If our noise distribution is symmetric, that is, p(z) = p(−z)
+for all z, then we can simplify Equation 4 to
+−tr(H)f(x)
+2
+≈ f(x) − Ezf(x + z).
+(5)
+We note that the RHS of Equation 5 corresponds to the
+perturbation discrepancy (1) where the perturbation func-
+tion q(˜x | x) is replaced by the distribution qz(z) used
+in Hutchinson’s trace estimator (2). Here, ˜x is a high-
+dimensional sequence of tokens while qz is a vector in a
+compact semantic space. Since the mask-filling model sam-
+ples sentences similar to x with minimal changes to seman-
+tic meaning, we can think of the mask-filling model as first
+sampling a similar semantic embedding (˜z ∼ qz) and then
+mapping this to a token sequence (˜z �→ ˜x). Sampling in
+semantic space ensures that all samples stay near the data
+manifold, which is useful because we would expect the log
+probability to always drop if we randomly perturb tokens.
+We can therefore interpret our objective as approximating
+the curvature restricted to the data manifold.
+5. Experiments
+We conduct experiments to better understand multiple facets
+of machine-generated text detection; we study the effective-
+ness of DetectGPT for zero-shot machine-generated text de-
+tection compared to prior zero-shot approaches, the impact
+of distribution shift on zero-shot and supervised detectors,
+and detection accuracy for the largest publicly-available
+models. To further characterize factors that impact detec-
+tion accuracy, we also study the robustness of zero-shot
+methods to machine-generated text that has been partially
+revised, the impact of alternative decoding strategies on de-
+tection accuracy, and a black-box variant of the detection
+task. Finally, we analyze DetectGPT’s behavior through
+studies of the impact of choice of perturbation function as
+well as the number of samples used to estimate d (x, pθ, q)
+on detection performance.
+Comparisons. We compare DetectGPT with various exist-
+ing zero-shot methods for machine-generated text detection
+that also leverage the predicted token-wise conditional dis-
+tributions of the source model for detection. These methods
+correspond to statistical tests based on token log probabil-
+ities, token ranks, or predictive entropy (Gehrmann et al.,
+
+Zero-Shot Machine-Generated Text Detection using Probability Curvature
+XSum
+SQuAD
+WritingPrompts
+Method
+GPT-2 OPT-2.7 Neo-2.7 GPT-J NeoX
+Avg.
+GPT-2 OPT-2.7 Neo-2.7 GPT-J NeoX
+Avg.
+GPT-2 OPT-2.7 Neo-2.7 GPT-J NeoX
+Avg.
+log p(x)
+0.86
+0.86
+0.86
+0.82
+0.77
+0.83
+0.91
+0.88
+0.84
+0.78
+0.71
+0.82
+0.97
+0.95
+0.95
+0.94
+0.93*
+0.95
+Rank
+0.79
+0.76
+0.77
+0.75
+0.73
+0.76
+0.83
+0.82
+0.80
+0.79
+0.74
+0.80
+0.87
+0.83
+0.82
+0.83
+0.81
+0.83
+LogRank
+0.89*
+0.88*
+0.90*
+0.86*
+0.81*
+0.87*
+0.94*
+0.92*
+0.90*
+0.83*
+0.76*
+0.87*
+0.98*
+0.96*
+0.97*
+0.96*
+0.95
+0.96*
+Entropy
+0.60
+0.50
+0.58
+0.58
+0.61
+0.57
+0.58
+0.53
+0.58
+0.58
+0.59
+0.57
+0.37
+0.42
+0.34
+0.36
+0.39
+0.38
+DetectGPT
+0.99
+0.97
+0.99
+0.97
+0.95
+0.97
+0.99
+0.97
+0.97
+0.90
+0.79
+0.92
+0.99
+0.99
+0.99
+0.97
+0.93*
+0.97
+Diff
+0.10
+0.09
+0.09
+0.11
+0.14
+0.10
+0.05
+0.05
+0.07
+0.07
+0.03
+0.05
+0.01
+0.03
+0.02
+0.01
+-0.02
+0.01
+Table 1. AUROC for detecting samples from the given model on the given dataset for DetectGPT and four previously proposed criteria
+(500 samples used for evaluation). From 1.5B parameter GPT-2 to 20B parameter GPT-NeoX, DetectGPT consistently provides the most
+accurate detections. Bold shows the best AUROC within each column (model-dataset combination); asterisk (*) denotes the second-best
+AUROC. Values in the final row show DetectGPT’s AUROC over the strongest baseline method in that column.
+2019; Solaiman et al., 2019; Ippolito et al., 2020). The
+first method uses the source model’s average token-wise log
+probability to determine if a candidate passage is machine-
+generated or not; passages with high average log probability
+are likely to be generated by the model. The second and
+third methods use the average observed rank or log-rank of
+the tokens in the candidate passage according to the model’s
+conditional distributions. Passages with smaller average
+(log-)rank are likely machine-generated. We also evalu-
+ate an entropy-based approach inspired by the hypothesis
+in Gehrmann et al. (2019) that model-generated texts will
+be more ‘in-distribution’ for the model, leading to more
+over-confident (thus lower entropy) predictive distributions.
+Empirically, we find predictive entropy to be positively cor-
+related with passage fake-ness more often that not; there-
+fore, this baseline uses high average entropy in the model’s
+predictive distribution as a signal that a passage is machine-
+generated. While our main focus is on zero-shot detectors
+as they do not require re-training for new domains or source
+models, for completeness we perform comparisons to su-
+pervised detection models in Section 5.1, using OpenAI’s
+RoBERTa-based (Liu et al., 2019) GPT-2 detector models,3
+which are fine-tuned on millions of samples from various
+GPT-2 model sizes and decoding strategies.
+Datasets & Metrics. Our experiments use six datasets that
+cover a variety of everyday domains and LLM use-cases.
+We use news articles from the XSum dataset (Narayan et al.,
+2018) to represent fake news detection, Wikipedia para-
+graphs from SQuAD contexts (Rajpurkar et al., 2016) to
+represent machine-written academic essays, and prompted
+stories from the Reddit WritingPrompts dataset (Fan et al.,
+2018) to represent detecting machine-generated creative
+writing submissions. To evaluate robustness to distribution
+shift, we also use the English and German splits of WMT16
+(Bojar et al., 2016) as well as long-form answers written by
+human experts in the PubMedQA dataset (Jin et al., 2019).
+Each experiment uses between 150 and 500 examples for
+evaluation, as noted in the text. For each experiment, we
+generate the machine-generated text by prompting with the
+3https://github.com/openai/gpt-2-output-
+dataset/tree/master/detector
+first 30 tokens of the real text (or just the question tokens
+for the PubMedQA experiments). We measure performance
+using the area under the receiver operating characteristic
+curve (AUROC), which can be interpreted as the probability
+that a classifier correctly ranks a randomly-selected posi-
+tive (machine-generated) example higher than a randomly-
+selected negative (human-written) example. All experiments
+use an equal number of positive and negative examples.
+Hyperparameters. The key hyperparameters of Detect-
+GPT are the fraction of words masked for perturbation,
+the length of the masked spans, the model used for mask
+filling, and the sampling hyperparameters for the mask-
+filling model. Using BERT (Devlin et al., 2019) masked
+language modeling as inspiration, we use 15% as the mask
+rate. We performed a small sweep over masked span lengths
+of {2, 5, 10} on a held-out set of XSum data, finding 2 to per-
+form best. We use these settings for all experiments, with-
+out re-tuning. We use T5-3B for almost all experiments,
+except for GPT-NeoX and GPT-3 experiments, where com-
+pute resources allowed for the larger T5-11B model; we also
+use mT5-3B instead of T5-3B for the WMT multilingual
+experiment. We do not tune the hyperparameters for the
+mask filling model, sampling directly with temperature 1.
+5.1. Main Results
+We first present two groups of experiments to evaluate De-
+tectGPT along with existing methods for zero-shot and su-
+pervised detection on models from 1.5B to 175B parameters.
+Zero-Shot Machine-Generated Text Detection.
+We
+present the comparison of different zero-shot detection meth-
+ods in Table 1. In these experiments, model samples are
+generated by sampling from the raw conditional distribu-
+tion with temperature 1. DetectGPT most improves average
+detection accuracy for XSum stories (0.1 AUROC improve-
+ment) and SQuAD Wikipedia contexts (0.05 AUROC im-
+provement). While it also performs accurate detection for
+WritingPrompts, the performance of all methods tends to in-
+crease, and the average margin of improvement is narrow.4
+4The overall ease of detecting machine-generated fake writing
+corroborates anecdotal reporting that machine-generated creative
+
+Zero-Shot Machine-Generated Text Detection using Probability Curvature
+RoB-base
+RoB-lg
+Likelihood
+Rank
+LogRank
+DetectGPT
+(Ours)
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0.981
+0.997
+0.889
+0.800
+0.915
+0.991
+XSum GPT-2 Detection
+RoB-base
+RoB-lg
+Likelihood
+Rank
+LogRank
+DetectGPT
+(Ours)
+0.888
+0.946
+0.838
+0.795
+0.863
+0.957
+WMT16-en mGPT Detection
+RoB-base
+RoB-lg
+Likelihood
+Rank
+LogRank
+DetectGPT
+(Ours)
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0.604
+0.713
+0.768
+0.664
+0.773
+0.836
+PubMedQA PubMedGPT Detection
+Supervised
+Unsupervised
+RoB-base
+RoB-lg
+Likelihood
+Rank
+LogRank
+DetectGPT
+(Ours)
+0.394
+0.537
+0.795
+0.838
+0.861
+0.962
+WMT16-de mGPT Detection
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Detection Method
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Detection AUROC
+Figure 4. Supervised machine-generated text detection models
+trained on large datasets of real and generated texts perform as
+well as or better than DetectGPT on in-distribution (top row)
+text. However, zero-shot methods work out-of-the-box for new do-
+mains (bottom row) such as PubMed medical texts and German
+news data from WMT16. For these domains, supervised detectors
+fail due to excessive distribution shift.
+For 14 of the 15 combinations of dataset and model, Detect-
+GPT provides the most accurate detection performance, with
+a 0.06 AUROC improvement on average. Log-rank thresh-
+olding proves to be a consistently stronger baseline than log
+probability thresholding, although it requires slightly more
+information (full predicted logits), which are not always
+available in public APIs.
+Comparison with Supervised Detectors. While our exper-
+iments generally focus on zero-shot detection, some works
+have evaluated the detection performance of supervised
+methods (typically fine-tuned transformers) for detecting
+machine-generated text. In this section, we explore several
+domains to better understand the relative strengths of super-
+vised and zero-shot detectors. The results are presented in
+Figure 4, using 200 samples from each dataset for evalua-
+tion. We find that supervised detectors can provide similar
+detection performance to DetectGPT on in-distribution data
+like English news, but perform significantly worse than zero-
+shot methods in the case of English scientific writing and
+fail altogether for German writing. This finding echoes past
+work showing that language models trained for machine-
+generated text detection overfit to their training data (source
+model, decoding strategy, topic, language, etc.; Uchendu
+et al. (2020); Ippolito et al. (2020); Jawahar et al. (2020)).
+In contrast, zero-shot methods generalize relatively easily
+to new languages and domains; DetectGPT’s performance
+in particular is mostly unaffected by the change in language
+from English to German.
+writing tends to be noticeably generic, and therefore relatively easy
+to detect (Roose & Newton, 2022).
+PubMedQA
+XSum
+WritingP
+Avg.
+RoBERTa-base
+0.64
+0.92
+0.92
+0.83
+RoBERTa-large
+0.71
+0.92
+0.91
+0.85
+log p(x)
+0.64
+0.76
+0.88
+0.76
+DetectGPT
+0.84
+0.84
+0.87
+0.85
+Table 2. DetectGPT detects GPT-3 generations with average AU-
+ROC on-par with supervised models trained specifically for
+machine-generated text detection. For more ‘typical’ text, such as
+news articles, supervised methods perform strongly.
+While our experiments have shown that DetectGPT is ef-
+fective on a variety of domains and models, it is natural to
+wonder if it is effective for the largest publicly-available
+LMs. Therefore, we also evaluate multiple zero-shot and
+supervised methods on 175B parameter GPT-3 using Ope-
+nAI’s paid API. Because the GPT-3 API does not provide
+access to the complete conditional distribution for each to-
+ken, we cannot compare to the rank, log rank, and entropy-
+based prior methods. We sample 150 examples5 from the
+PubMedQA, XSum, and WritingPrompts datasets and com-
+pare the two pre-trained RoBERTa-based detector models
+with DetectGPT and the probability thresholding baseline.
+We show in Table 2 that DetectGPT can provide detection
+competitive with the stronger supervised model, and it again
+outperforms probability thresholding on average.
+5.2. Variants of Machine-Generated Text Detection
+Detecting Revised Machine-Generated Text.
+In prac-
+tice, humans may manually edit or refine machine-generated
+text rather than blindly use a model’s generations for their
+task of interest. We therefore conduct an experiment to
+simulate the detection problem for model samples that have
+been increasingly heavily revised. We simulate human re-
+vision by replacing 5 word spans of the text with samples
+from T5-3B until r% of the text has been replaced, and
+report performance as r varies. Figure 5 shows that De-
+tectGPT maintains detection AUROC above 0.8 even when
+nearly a quarter of the text in model samples has been re-
+placed. Unsurprisingly, almost all methods show a gradual
+degradation in performance as the sample is more heavily
+revised. The entropy baseline shows surprisingly robust
+performance in this setting (althought it is least accurate
+on average), even slightly improving detection performance
+up to 24% replacement. DetectGPT shows the strongest
+detection performance for all revision levels.
+Impact of Alternative Decoding Strategies on Detection.
+While Table 1 suggests that DetectGPT is effective for
+detecting machine-generated text, prior work notes that
+the decoding strategy (i.e., temperature sampling, top-k,
+nucleus/top-p) can impact the difficulty of detection. We re-
+5We reduce the number of evaluation samples from 500 in our
+main experiments to reduce the API costs of these experiments.
+
+Zero-Shot Machine-Generated Text Detection using Probability Curvature
+0.00
+0.08
+0.16
+0.04
+0.12
+0.24
+0.20
+Fraction of GPT-J-generated news article re-written
+0.6
+0.7
+0.8
+0.9
+1.0
+Detection AUROC
+Rank
+DetectGPT
+LogRank
+Likelihood
+Entropy
+Figure 5. We simulate human edits to machine-generated text by
+replacing varying fractions of model samples with T5-3B gener-
+ated text (masking out random five word spans until r% of text is
+masked to simulate human edits to machine-generated text). The
+four top-performing methods all generally degrade in performance
+with heavier revision, but DetectGPT is consistently most accurate.
+Experiment is conducted on the XSum dataset.
+XSum
+SQuAD
+WritingPrompts
+Method
+top-p
+top-k
+top-p
+top-k
+top-p
+top-k
+log p(x)
+0.92
+0.87
+0.89
+0.85
+0.98
+0.96
+Rank
+0.76
+0.76
+0.81
+0.80
+0.84
+0.83
+LogRank
+0.93*
+0.90*
+0.92*
+0.90*
+0.98
+0.97
+Entropy
+0.53
+0.55
+0.54
+0.56
+0.32
+0.35
+DetectGPT
+0.98
+0.98
+0.94
+0.93
+0.98
+0.97
+Table 3. AUROC for zero-shot methods averaged across the five
+models in Table 1 for both top-k and top-p sampling, with k =
+40 and p = 0.96. Both settings enable slightly more accurate
+detection, and DetectGPT consistently provides the best detection
+performance. See Appendix Tables 4 and 5 for complete results.
+peat the analysis from Section 5.1 using top-k sampling and
+nucleus sampling. Top-k sampling truncates the sampling
+distribution to only the k highest-probability next tokens;
+nucleus sampling samples from only the smallest set of to-
+kens whose combined probability exceeds p. The results
+are summarized in Table 3; Appendix Tables 4 and 5 show
+complete results. We use k = 40, and p = 0.96, in line with
+prior work (Ippolito et al., 2020). We find that both top-k
+and nucleus sampling make detection easier, on average.
+Averaging across domains, DetectGPT provides the clearest
+signal for zero-shot detection.
+Using Likelihoods from Models other than the Source
+Model. While our experiments have focused on the white-
+box setting for machine-generated text detection, in this sec-
+tion, we explore the effect of using a different model to score
+a candidate passage (and perturbed texts) than the model
+that generated the passage. In other words, we aim to clas-
+sify between human-generated text and text from model A,
+GPT-J
+GPT-Neo
+GPT-2
+Scoring Model
+GPT-J
+GPT-Neo
+GPT-2
+Base Model
+0.92
+(0.02)
+0.83
+(0.04)
+0.79
+(0.02)
+0.64
+(0.06)
+0.97
+(0.01)
+0.83
+(0.02)
+0.60
+(0.09)
+0.85
+(0.05)
+0.99
+(0.00)
+0.85
+0.81
+0.81
+0.72
+0.88
+0.87
+Figure 6. DetectGPT performs
+best when scoring samples with
+the same model that generated
+them (diagonal), but the column
+means suggest that some models
+(GPT-Neo, GPT-2) may be bet-
+ter ‘scorers’ than others (GPT-J).
+White values show mean (stan-
+dard error) AUROC over XSum,
+SQuAD, and WritingPrompts;
+black shows row/column mean.
+but
+without
+access
+to
+model A to compute log
+probabilities.
+Instead,
+we use log probabilities
+computed by a surrogate
+model B.
+We consider
+three
+models,
+GPT-J,
+GPT-Neo-2.7, and GPT-2,
+evaluating
+all
+possible
+combinations of source
+model
+and
+surrogate
+model (9 total).
+We
+average the performance
+across 200 samples from
+XSum,
+SQuAD,
+and
+WritingPrompts.
+The
+results are presented in
+Figure 6, showing that
+when the surrogate model
+is
+different
+from
+the
+source model, detection performance is reduced, indicating
+that DetectGPT is most suited to the white-box setting. Yet
+we also observe that if we fix the model used for scoring
+and average across source models whose generations are
+detected (average within column), there is significant
+variation in AUROC; GPT-2 and GPT-Neo-2.7 seem to be
+better ‘scorers’ than GPT-J. These variations in cross-model
+scoring performance suggest ensembling scoring models
+may be a useful direction for future research.
+5.3. Evaluating scaling properties of DetectGPT
+In this section, we evaluate DetectGPT as the size of the
+mask-filling model or the number of perturbations used to
+estimate the expectation in Equation 1 are varied.
+Impact of Source and Mask-Filling Model Scale. Here
+we study the impact of the size of the source model and
+mask-filling model on DetectGPT’s performance; the re-
+sults are shown in Figure 7. In particular, the increased
+discrimination power of DetectGPT for larger mask-filling
+models supports the interpretation that DetectGPT is estimat-
+ing the curvature of the log probability in a latent semantic
+space, rather than in raw token embedding space. Larger T5
+models better represent this latent space, in which random
+directions correspond to meaningful changes in the text on
+the data manifold.
+Impact of Number of Perturbations for DetectGPT. Fi-
+nally, we evaluate the performance of DetectGPT as a func-
+tion of the number of perturbations used to estimate the
+expectation in Equation 1 on three datasets. The results
+are presented in Figure 8. Detection accuracy continues to
+improve until 100 perturbations, where it converges. Evalu-
+ations use 100 examples from each dataset.
+
+Zero-Shot Machine-Generated Text Detection using Probability Curvature
+60M
+220M
+770M
+2.7B
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Detection AUROC
+5 perturbations
+60M
+220M
+770M
+2.7B
+25 perturbations
+Random
+GPT2-sm
+GPT2-md
+GPT2-lg
+GPT2-xl
+Mask filling model size (# parameters)
+Figure 7. There is a clear association between capacity of mask-
+filling model and detection performance, across source model
+scales. Random mask filling (uniform sampling from mask filling
+model vocabulary) performs poorly, reinforcing the idea that the
+perturbation function should produce samples on the data manifold.
+Curves show AUROC scores on 200 SQuAD contexts.
+6. Discussion
+As large language models continue to improve, they will
+become increasingly attractive tools for replacing human
+writers in a variety of contexts, such as education, jour-
+nalism, and art. While legitimate uses of language model
+technologies exist in all of these settings, teachers, readers,
+and consumers are likely to demand tools for verifying the
+human origin of certain content with high educational, so-
+cietal, or artistic significance, particularly when factuality
+(and not just fluency) is crucial.
+In light of these elevated stakes and the regular emergence of
+new large language models, we study the zero-shot machine-
+generated text detection problem, in which we use only the
+raw log probabilities computed by a generative model to
+determine if a candidate passage was sampled from it. We
+identify a property of the log probability function computed
+by a wide variety of large language models, showing that a
+tractable approximation to the trace of the Hessian of the
+model’s log probability function provides a useful signal
+for detecting model samples. Our experiments find that
+this signal is more discriminative than existing zero-shot
+detection methods and is competitive with bespoke detection
+models trained with millions of model samples.
+DetectGPT and Watermarking.
+One interpretation of
+the perturbation function is producing semantically simi-
+lar rephrasings of the original passage. If these rephrasings
+are systematically lower-probability than the original pas-
+sage, the model is exposing its bias toward the specific (and
+roughly arbitrary, by human standards) phrasing used. In
+other words, LLMs that do not perfectly imitate human
+writing essentially watermark themselves implicitly. Under
+this interpretation, efforts to manually add watermarking bi-
+ases to model outputs (Aaronson, 2022; Kirchenbauer et al.,
+2023) may further improve the effectiveness of methods
+such as DetectGPT, even as LLMs continue to improve.
+Limitations. One limitation of probability-based methods
+for zero-shot machine-generated text detection (like Detect-
+1
+10
+100
+1000
+0.6
+0.7
+0.8
+0.9
+1.0
+Detection AUROC
+GPT-2
+XSum
+SQuAD
+WritingPrompts
+1
+10
+100
+1000
+GPT-J
+Number of perturbations
+Figure 8. Impact of varying the number of perturbations (samples
+of mask and mask-fill) used by DetectGPT on auROC (left) and
+auPR (right) to estimate the perturbation discrepancy for clas-
+sification. Averaging up to 100 perturbations greatly increases
+DetectGPT’s discrimination power. Fills sampled from T5-large.
+GPT) is the white-box assumption that we can evaluate log
+probabilities of the model(s) in question. For models be-
+hind APIs that do provide probabilities (such as GPT-3),
+evaluating probabilities nonetheless costs money. Another
+assumption of DetectGPT is access to a reasonable pertur-
+bation function. While in this work, we use off-the-shelf
+mask-filling models such as T5 and mT5 (for non-English
+languages), some domains may see reduced performance
+if existing mask-filling models do not well represent the
+space of meaningful rephrases, reducing the quality of the
+curvature estimate. While DetectGPT provides the best
+available detection performance for PubMedQA, its drop
+in performance compared to other datasets may be a result
+of lower quality perturbations. Finally, DetectGPT is more
+compute-intensive than other methods for detection, as it
+requires sampling and scoring the set of perturbations for
+each candidate passage, rather than just the candidate pas-
+sage; a better tuned perturbation function or more efficient
+curvature approximation may help mitigate these costs.
+Future Work. While the methods in this work make no
+assumptions about the models generating the samples, fu-
+ture work may explore how watermarking algorithms can be
+used in conjunction with detection algorithms like Detect-
+GPT to further improve detection robustness as language
+models continually improve their reproductions of human
+text. Separately, the results in Section 5.2 suggest that ex-
+tending DetectGPT to use ensembles of models for scoring,
+rather than a single model, may improve detection in the
+black box setting. Another topic that remains unexplored
+is the relationship between prompting and detection; that
+is, can a clever prompt successfully prevent a model’s gen-
+erations from being detected by existing methods? Finally,
+future work may explore whether the local log probabil-
+ity curvature property we identify is present for generative
+models in other domains, such as audio, video, or images.
+We hope that the present work serves as inspiration to fu-
+ture work developing effective, general-purpose methods
+for mitigating potential harms of machine-generated media.
+
+Zero-Shot Machine-Generated Text Detection using Probability Curvature
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+Zero-Shot Machine-Generated Text Detection using Probability Curvature
+A. Complete Results for Top-p and Top-k Decoding
+Tables 4 and 5 contain the complete results for XSum, SQuAD, and WritingPrompts for the five models considered in
+Table 1. On average, both top-p and top-k sampling seem to make the detection task easier. This result is perhaps intuitive,
+as both sampling methods strictly increase the average log likelihood of model generations under the model (as they truncate
+low-probability tokens, albeit with different heuristics). Therefore methods based on probability or rank of tokens should
+become more discriminative.
+XSum
+SQuAD
+WritingPrompts
+Method
+GPT-2 OPT-2.7 Neo-2.7 GPT-J
+NeoX
+Avg.
+GPT-2 OPT-2.7 Neo-2.7 GPT-J
+NeoX
+Avg.
+GPT-2 OPT-2.7 Neo-2.7 GPT-J
+NeoX
+Avg.
+log p(x)
+0.93
+0.93
+0.94
+0.91
+0.87
+0.92
+0.96
+0.94
+0.91
+0.87
+0.79
+0.89
+0.99*
+0.98*
+0.98*
+0.97*
+0.97* 0.98
+Rank
+0.80
+0.77
+0.77
+0.75
+0.73
+0.76
+0.84
+0.82
+0.81
+0.80
+0.75
+0.81
+0.87
+0.84
+0.83
+0.83
+0.81
+0.84
+LogRank
+0.95*
+0.94*
+0.96*
+0.93*
+0.89*
+0.93*
+0.98*
+0.96*
+0.94*
+0.90
+0.83
+0.92*
+0.99*
+0.98*
+0.98*
+0.98
+0.98
+0.98
+Entropy
+0.55
+0.46
+0.53
+0.54
+0.58
+0.53
+0.53
+0.50
+0.55
+0.56
+0.57
+0.54
+0.32
+0.37
+0.28
+0.32
+0.32
+0.32
+DetectGPT
+0.99
+0.98
+1.00
+0.98
+0.97
+0.98
+0.99
+0.98
+0.98
+0.90
+0.82*
+0.94
+1.00
+0.99
+0.99
+0.97*
+0.93
+0.98
+Diff
+0.04
+0.04
+0.04
+0.05
+0.08
+0.05
+0.01
+0.02
+0.04
+0.00
+-0.01
+0.02
+0.01
+0.01
+0.01
+-0.01
+-0.05
+0.00
+Table 4. Nucleus (top-p) sampling evaluation with p = 0.96. AUROC for detecting samples from the given model on the given dataset for
+DetectGPT and four previously proposed criteria. Nucleus sampling generally makes detection easier for all methods, but DetectGPT still
+provides the highest average AUROC. For WritingPrompts, however, the LogRank baseline performs as well as DetectGPT.
+XSum
+SQuAD
+WritingPrompts
+Method
+GPT-2 OPT-2.7 Neo-2.7 GPT-J
+NeoX
+Avg.
+GPT-2 OPT-2.7 Neo-2.7 GPT-J
+NeoX
+Avg.
+GPT-2 OPT-2.7 Neo-2.7 GPT-J
+NeoX
+Avg.
+log p(x)
+0.89
+0.89
+0.89
+0.84
+0.81
+0.87
+0.93
+0.90
+0.88
+0.82
+0.74
+0.85
+0.97
+0.95
+0.97
+0.96
+0.95* 0.96
+Rank
+0.79
+0.77
+0.77
+0.75
+0.73
+0.76
+0.84
+0.82
+0.80
+0.80
+0.75
+0.80
+0.87
+0.84
+0.83
+0.82
+0.81
+0.83
+LogRank
+0.92*
+0.91*
+0.93*
+0.89*
+0.85*
+0.90*
+0.96*
+0.94*
+0.92*
+0.87*
+0.79*
+0.90*
+0.98*
+0.97*
+0.98*
+0.97
+0.96
+0.97
+Entropy
+0.58
+0.49
+0.55
+0.56
+0.59
+0.55
+0.55
+0.52
+0.56
+0.56
+0.58
+0.56
+0.35
+0.41
+0.30
+0.34
+0.37
+0.35
+DetectGPT
+0.99
+0.97
+0.99
+0.96
+0.96
+0.98
+0.99
+0.98
+0.98
+0.89
+0.80
+0.93
+0.99
+0.98
+0.99
+0.97
+0.93
+0.97
+Diff
+0.07
+0.06
+0.06
+0.07
+0.11
+0.08
+0.03
+0.04
+0.06
+0.02
+0.01
+0.03
+0.01
+0.01
+0.01
+0.00
+-0.03
+0.00
+Table 5. Top-k sampling evaluation with k = 40. DetectGPT generally provides the most accurate performance (highest AUROC),
+although the gap is narrowed comparing to direct sampling, presumably because top-k generations are more generic.
+
diff --git a/adFIT4oBgHgl3EQflis6/content/tmp_files/load_file.txt b/adFIT4oBgHgl3EQflis6/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..6259a71c4d2af6fd6b41dee758a54802061e1e1c
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf,len=1452
+page_content='DetectGPT: Zero-Shot Machine-Generated Text Detection  using Probability Curvature  Eric Mitchell 1 Yoonho Lee 1 Alexander Khazatsky 1 Christopher D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Manning 1 Chelsea Finn 1  Abstract  The fluency and factual knowledge of large lan-  guage models (LLMs) heightens the need for cor-  responding systems to detect whether a piece of  text is machine-written.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' For example, students  may use LLMs to complete written assignments,  leaving instructors unable to accurately assess stu-  dent learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' In this paper, we first demonstrate  that text sampled from an LLM tends to occupy  negative curvature regions of the model’s log prob-  ability function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Leveraging this observation, we  then define a new curvature-based criterion for  judging if a passage is generated from a given  LLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' This approach, which we call DetectGPT,  does not require training a separate classifier, col-  lecting a dataset of real or generated passages, or  explicitly watermarking generated text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' It uses  only log probabilities computed by the model of  interest and random perturbations of the passage  from another generic pre-trained language model  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='g, T5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We find DetectGPT is more discrimi-  native than existing zero-shot methods for model  sample detection, notably improving detection of  fake news articles generated by 20B parameter  GPT-NeoX from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='81 AUROC for the strongest  zero-shot baseline to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='95 AUROC for Detect-  GPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' See ericmitchell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='ai/detectgpt  for code, data, and other project information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Introduction  Large language models (LLMs) have proven able to gen-  erate remarkably fluent responses to a wide variety of user  queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Models such as GPT-3 (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2020), PaLM  (Chowdhery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2022), and ChatGPT (OpenAI, 2022)  can convincingly answer complex questions about science,  mathematics, historical and current events, and social trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  While recent work has found that cogent-sounding LLM-  generated responses are often simply wrong (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=',  1Stanford University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Correspondence to: Eric Mitchell  <eric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='mitchell@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='edu>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Candidate passage :  “Joe Biden recently made a move to the White House  that included bringing along his pet German Shepherd…”  DetectGPT  x  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  GPT-3  (1) Perturb  (2) Score  (3) Compare  🤖   from GPT-3  Yes  (reword  with T5)  “made a move”  “moved”  →  “pet”  “dog”  →  Delete “bringing along”  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  🤔   from other source  No  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We aim to determine whether a piece of text was gener-  ated by a particular LLM p, such as GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' To classify a candidate  passage x, DetectGPT first generates minor perturbations of the  passage ˜xi using a generic pre-trained model such as T5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Then  DetectGPT compares the log probability under p of the original  sample x with each perturbed sample ˜xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' If the average log ratio  is high, the sample is likely from the source model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  2022), the articulate nature of such generated text may still  make LLMs attractive for replacing human labor in some  contexts, notably student essay writing and journalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' At  least one major news source has released AI-written content  with limited human review, leading to substantial factual er-  rors in some articles (Christian, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Such applications of  LLMs are problematic for a variety of reasons, making fair  student assessment difficult, impairing student learning, and  proliferating convincing-but-inaccurate news articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Un-  fortunately, humans perform only slightly better than chance  when classifying machine-generated vs human-written text  (Gehrmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2019), leading researchers to consider  automated detection methods that may identify signals dif-  ficult for humans to recognize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Such methods might give  teachers and news-readers more confidence in the human  origin of the text that they consume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  As in prior work (Jawahar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2020), we study the  machine-generated text detection problem as a binary clas-  sification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Specifically, we aim to classify whether  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='11305v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='CL]  26 Jan 2023  Zero-Shot Machine-Generated Text Detection using Probability Curvature  a candidate passage was generated by a particular source  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' While several works have investigated methods for  training a second deep network to detect machine-generated  text, such an approach has several shortcomings, including  a tendency to overfit to the topics it was trained on as well as  the need to train a new model for each new source model that  is released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We therefore consider the zero-shot version of  machine-generated text detection, where we use the source  model itself, without fine-tuning or adaptation of any kind,  to detect its own samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' The most common method for  zero-shot machine-generated text detection is evaluating the  average per-token log probability of the generated text and  thresholding (Solaiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Gehrmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Ippolito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' However, this zeroth-order approach  to detection ignores the local structure of the learned proba-  bility function around a candidate passage, which we find  contains useful information about the source of a passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  This paper poses a simple hypothesis: minor rewrites of  model-generated text tend to have lower log probability un-  der the model than the original sample, while minor rewrites  of human-written text may have higher or lower log prob-  ability than the original sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' In other words, unlike  human-written text, model-generated text tends to lie in ar-  eas where the log probability function has negative curvature  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', local maxima of the log probability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We empirically  verify this hypothesis, and find that it holds true across a  diverse body of LLMs, even when the minor rewrites, or  perturbations, come from alternative language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We  leverage this observation to build DetectGPT, a zero-shot  method for automated machine-generated text detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' To  test if a passage came from a source model pθ, DetectGPT  compares the log probability of a candidate passage under  pθ with the average log probability of several perturbations  of the passage under pθ (generated with, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', T5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Raffel  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' If the perturbed passages tend to have lower  average log probability than the original by some margin,  the candidate passage is likely to have come from pθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' See  Figure 1 for an overview of the problem and DetectGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  See Figure 2 for an illustration of the underlying hypothesis  and Figure 3 for empirical evaluation of the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Our  experiments find that DetectGPT is more accurate than exist-  ing zero-shot methods for detecting machine-generated text,  improving over the strongest zero-shot baseline by over  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='1 AUROC for multiple source models when detecting  machine-generated news articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Our main contributions are: (a) the identi-  fication and empirical validation of the hypothesis that the  curvature of a model’s log probability function tends to be  significantly more negative at model samples than for hu-  man text, and (b) DetectGPT, a practical algorithm inspired  by this hypothesis that approximates the trace of the log  probability function’s Hessian to detect a model’s samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  log pθ(x)  xfake ∼ pθ (x)  ˜x fake  1  ˜x fake  2  ˜x fake  3  ˜x fake  4  xreal ∼ phuman(x)  ˜xreal  1  ˜xreal  2  ˜xreal  3  ˜xreal  4  Fake/real sample  Perturbed fake/real sample  Log likelihood  …  log pθ(x)  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We identify and exploit the tendency of machine-  generated passages x ∼ pθ(·) (left) to lie in negative curvature  regions of log p(x), where nearby samples have lower model  log probability on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  In contrast, human-written text  x ∼ preal(·) (right) tends not to occupy regions with clear nega-  tive log probability curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Related Work  Increasingly large LLMs (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=',  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Chowdhery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' OpenAI, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=',  2022) have led to dramatically improved performance on  many language-related benchmarks and the ability to gen-  erate convincing and on-topic text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' The GROVER model  (Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2019) was the first LLM trained specifically  for generating realistic-looking news articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Human evalu-  ators found GROVER-generated propaganda at least as trust-  worthy as human-written propaganda, or even slightly more  trustworthy, motivating the authors to study GROVER’s abil-  ity to detect its own generations by fine-tuning a detector on  top of its features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' they found GROVER better able to de-  tect GROVER-generated text than other pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  However, Bakhtin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Uchendu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' (2020) note  that models trained explicitly to detect machine-generated  text tend to overfit to their training distribution of domains  or source models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Other works have trained supervised models for machine-  generated text detection on top of neural representations  (Bakhtin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Solaiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Uchendu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=',  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Ippolito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Fagni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2021), bag-of-words  features (Solaiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Fagni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2021), and hand-  crafted statistical features (Gehrmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Alterna-  tively, Solaiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' (2019) notes the surprising efficacy  of a simple zero-shot method for machine-generated text  detection, which thresholds a candidate passage based on its  average log probability under the generative model, serving  as a strong baseline for zero-shot machine-generated text  detection in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' In our work, we similarly use the  generating model to detect its own generations in a zero shot  manner, but through a different approach based on estimat-  ing local curvature of the log probability around the sample  rather than the raw log probability of the sample itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' See  Jawahar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' (2020) for a complete survey on machine-  generated text detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Other work explores watermarks  for generated text (Kirchenbauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2023), which modify  Zero-Shot Machine-Generated Text Detection using Probability Curvature  Algorithm 1 DetectGPT model-generated text detection  1: Input: passage x, source model pθ, perturbation function q,  number of perturbations k, decision threshold ϵ  2: ˜xi ∼ q(· | x), i ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='.k] // mask spans, sample replacements  3: ˜µ ← 1  k  �  i log pθ(˜xi)  // approximate expectation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' 1  4: ˆdx ← log pθ(x) − ˜µ  // estimate d (x, pθ, q)  5: ˜σ2  x ←  1  k−1  �  i (log pθ(˜xi) − ˜µ)2 // variance for normalization  6: if  ˆdx  √˜σx > ϵ then  7:  return true  // probably model sample  8: else  9:  return false  // probably not model sample  a model’s generations to make them easier to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Our  work does not assume text is generated with the goal of easy  detection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' DetectGPT detects text generated from publicly  available LLMs using standard LLM sampling strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  The problem of machine-generated text detection echoes ear-  lier work on detecting deepfakes, artificial images or videos  generated by deep nets, which has spawned substantial ef-  forts in detection of fake visual content (Dolhansky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=',  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Zi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' While early works in deepfake de-  tection used relatively general-purpose model architectures  (G¨uera & Delp, 2018), many deepfake detection methods  rely on the continuous nature of image data to achieve state-  of-the-art performance (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Guarnera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=',  2020), making direct application to text difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' The Zero-Shot Machine-Generated Text  Detection Problem  We study zero-shot machine-generated text detection, the  problem of detecting whether a piece of text, or candidate  passage x, is a sample from a source model pθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' The problem  is zero-shot in the sense that we do not assume access to  human-written or generated samples to perform detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  As in prior work, we study a ‘white box’ setting (Gehrmann  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2019) in which the detector may evaluate the log prob-  ability of a sample log pθ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' The white box setting does  not assume access to the model architecture or parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  While most public APIs for LLMs (such as GPT-3) enable  scoring text, some exceptions exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' While most of our ex-  periments consider the white box setting, see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='2 for  a suite of experiments in which we score text using models  other than the source model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  The detection criterion we propose, DetectGPT, also makes  use of generic pre-trained mask-filling models in order to  generate passages that are ‘nearby’ the candidate passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  However, these mask-filling models are used off-the-shelf,  without any fine-tuning or adaptation to the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' DetectGPT: Zero-shot Machine-Generated  Text Detection with Random Perturbations  DetectGPT is based on the hypothesis that samples from a  source model pθ typically lie in areas of negative curvature  of the log probability function of pθ, unlike human text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' In  other words, if we apply small perturbations to a passage  x ∼ pθ, producing ˜x, the quantity log pθ(x) − log pθ(˜x)  should be relatively large on average for machine-generated  samples compared to human-written text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' To leverage this  hypothesis, first consider a perturbation function q(· | x)  that gives a distribution over ˜x, slightly modified versions of  x with similar meaning (we will generally consider roughly  paragraph-length texts x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' As an example, q(· | x) might be  the result of simply asking a human to rewrite one of the  sentences of x, while preserving the meaning of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Using  the notion of a perturbation function, we can define the  perturbation discrepancy d (x, pθ, q):  d (x, pθ, q) ≜ log pθ(x) − E˜x∼q(·|x) log pθ(˜x)  (1)  We state our hypothesis more formally in Hypothesis 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Hypothesis 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' If q produces samples on the data manifold,  d (x, pθ, q) is positive with high probability for samples  x ∼ pθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' For human-written text, d (x, pθ, q) tends toward  zero for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  If we define q(· | x) to be samples from a mask-filling  model such as T5 (Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2020), rather than human  rewrites, we can empirically test Hypothesis 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='1 in an au-  tomated, scalable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' For real data, we use 500 news  articles from the XSum dataset (Narayan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' for  model samples, we use the output of four different LLMs  when prompted with the first 30 tokens of each article in  XSum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We use T5-3B to apply perturbations, masking out  randomly-sampled 2-word spans until 15% of the words in  the article are masked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We approximate the expectation in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' 1 with 100 samples from T5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='1 Figure 3 shows the result  of this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We find the distribution of perturbation  discrepancies is significantly different for human-written  articles and model samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' model samples tend to have  a larger perturbation discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='3 explores a  relaxation of the assumption that q only produces samples  on the data manifold, finding that a gap, although reduced,  still exists in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Given these results, we can detect if a piece of text was  generated by a model pθ by simply thresholding the pertur-  bation discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' In practice, we find that normalizing the  perturbation discrepancy by the standard deviation of the ob-  served values used to estimate E˜x∼q(·|x) log pθ(˜x) provides  a slightly better signal for detection, typically increasing  AUROC by around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='020, so we use this normalized version  of the perturbation discrepancy in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' The  resulting method, DetectGPT, is summarized in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Hav-  ing described an application of the perturbation discrepancy  1We later show in Figure 8 that varying the number of samples  used to estimate the expectation effectively allows for trading off  between accuracy and speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Zero-Shot Machine-Generated Text Detection using Probability Curvature  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='3  0  20  40  60  gpt2-xl  Human  Model  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='20  EleutherAI/gpt-neo-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='7B  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='25  0  20  40  60  EleutherAI/gpt-j-6B  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='3  EleutherAI/gpt-neox-20b  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='0  Log Likelihood Drop (Perturbation Discrepancy)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='0  Frequency  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' The average drop in log probability (perturbation discrep-  ancy) after rephrasing a passage is consistently higher for model-  generated passages than for human-written passages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Each plot  shows the distribution of the perturbation discrepancy d (x, pθ, q)  for human-written news articles and machine-generated arti-  cles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' of equal word length from models GPT-2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='5B), GPT-Neo-  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='7B (Black et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2021), GPT-J (6B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Wang & Komatsuzaki (2021))  and GPT-NeoX (20B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Black et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Human-written arti-  cles are a sample of 500 XSum articles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' machine-generated text  is generated by prompting each model with the first 30 tokens of  each XSum article, sampling from the raw conditional distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Discrepancies are estimated with 100 T5-3B samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  to machine-generated text detection, we next provide an  interpretation of this quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Interpretation of the Perturbation Discrepancy as Cur-  vature  While Figure 3 suggests that the perturbation dis-  crepancy may be useful, it is not immediately obvious what  it measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' In this section, we show that the perturbation  discrepancy approximates a measure of the local curvature  of the log probability function near the candidate passage,  more specifically, that it is proportional to the negative trace  of the Hessian of the log probability function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='2 To han-  dle the non-differentiability of discrete data, we consider  candidate passages in a latent semantic space, where small  displacements correspond to valid edits that retain similar  meaning to the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Because our perturbation function  (T5) models natural text, we expect our perturbations to  roughly capture such meaningful variations of the original  passage, rather than arbitrary edits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  We first invoke Hutchinson’s trace estimator (Hutchinson,  1990), giving an unbiased estimate of the trace of matrix A:  tr(A) = Ezz⊤Az  (2)  provided that the elements of z ∼ qz are IID with E[zi] = 0  and Var(zi) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' To use Equation 2 to estimate the trace of  the Hessian, we must therefore compute the expectation of  2Rather than the Hessian of the log likelihood with respect to  model parameters (the Fisher Information Matrix), here we refer  to the Hessian of the log probability with respect to the sample x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  the directional second derivative z⊤Hf(x)z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We approxi-  mate this expression with finite differences:  z⊤Hf(x)z ≈ f(x + hz) + f(x − hz) − 2f(x)  h2  (3)  Combining Equations 2 and 3 and simplifying with h = 1,  we have an estimate of the negative Hessian trace  −tr(H)f(x) ≈ 2f(x) − Ez [f(x + z) + f(x − z)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' (4)  If our noise distribution is symmetric, that is, p(z) = p(−z)  for all z, then we can simplify Equation 4 to  −tr(H)f(x)  2  ≈ f(x) − Ezf(x + z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  (5)  We note that the RHS of Equation 5 corresponds to the  perturbation discrepancy (1) where the perturbation func-  tion q(˜x | x) is replaced by the distribution qz(z) used  in Hutchinson’s trace estimator (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Here, ˜x is a high-  dimensional sequence of tokens while qz is a vector in a  compact semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Since the mask-filling model sam-  ples sentences similar to x with minimal changes to seman-  tic meaning, we can think of the mask-filling model as first  sampling a similar semantic embedding (˜z ∼ qz) and then  mapping this to a token sequence (˜z �→ ˜x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Sampling in  semantic space ensures that all samples stay near the data  manifold, which is useful because we would expect the log  probability to always drop if we randomly perturb tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  We can therefore interpret our objective as approximating  the curvature restricted to the data manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Experiments  We conduct experiments to better understand multiple facets  of machine-generated text detection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' we study the effective-  ness of DetectGPT for zero-shot machine-generated text de-  tection compared to prior zero-shot approaches, the impact  of distribution shift on zero-shot and supervised detectors,  and detection accuracy for the largest publicly-available  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' To further characterize factors that impact detec-  tion accuracy, we also study the robustness of zero-shot  methods to machine-generated text that has been partially  revised, the impact of alternative decoding strategies on de-  tection accuracy, and a black-box variant of the detection  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Finally, we analyze DetectGPT’s behavior through  studies of the impact of choice of perturbation function as  well as the number of samples used to estimate d (x, pθ, q)  on detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We compare DetectGPT with various exist-  ing zero-shot methods for machine-generated text detection  that also leverage the predicted token-wise conditional dis-  tributions of the source model for detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' These methods  correspond to statistical tests based on token log probabil-  ities, token ranks, or predictive entropy (Gehrmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=',  Zero-Shot Machine-Generated Text Detection using Probability Curvature  XSum  SQuAD  WritingPrompts  Method  GPT-2 OPT-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='7 Neo-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='7 GPT-J NeoX  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  GPT-2 OPT-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='7 Neo-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='7 GPT-J NeoX  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  GPT-2 OPT-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='7 Neo-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='7 GPT-J NeoX  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  log p(x)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content='38  DetectGPT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content='01  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' AUROC for detecting samples from the given model on the given dataset for DetectGPT and four previously proposed criteria  (500 samples used for evaluation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' From 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='5B parameter GPT-2 to 20B parameter GPT-NeoX, DetectGPT consistently provides the most  accurate detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Bold shows the best AUROC within each column (model-dataset combination);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' asterisk (*) denotes the second-best  AUROC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Values in the final row show DetectGPT’s AUROC over the strongest baseline method in that column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Solaiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Ippolito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' The  first method uses the source model’s average token-wise log  probability to determine if a candidate passage is machine-  generated or not;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' passages with high average log probability  are likely to be generated by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' The second and  third methods use the average observed rank or log-rank of  the tokens in the candidate passage according to the model’s  conditional distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Passages with smaller average  (log-)rank are likely machine-generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We also evalu-  ate an entropy-based approach inspired by the hypothesis  in Gehrmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' (2019) that model-generated texts will  be more ‘in-distribution’ for the model, leading to more  over-confident (thus lower entropy) predictive distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Empirically, we find predictive entropy to be positively cor-  related with passage fake-ness more often that not;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' there-  fore, this baseline uses high average entropy in the model’s  predictive distribution as a signal that a passage is machine-  generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' While our main focus is on zero-shot detectors  as they do not require re-training for new domains or source  models, for completeness we perform comparisons to su-  pervised detection models in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='1, using OpenAI’s  RoBERTa-based (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2019) GPT-2 detector models,3  which are fine-tuned on millions of samples from various  GPT-2 model sizes and decoding strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Datasets & Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Our experiments use six datasets that  cover a variety of everyday domains and LLM use-cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  We use news articles from the XSum dataset (Narayan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=',  2018) to represent fake news detection, Wikipedia para-  graphs from SQuAD contexts (Rajpurkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2016) to  represent machine-written academic essays, and prompted  stories from the Reddit WritingPrompts dataset (Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=',  2018) to represent detecting machine-generated creative  writing submissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' To evaluate robustness to distribution  shift, we also use the English and German splits of WMT16  (Bojar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2016) as well as long-form answers written by  human experts in the PubMedQA dataset (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Each experiment uses between 150 and 500 examples for  evaluation, as noted in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' For each experiment, we  generate the machine-generated text by prompting with the  3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='com/openai/gpt-2-output-  dataset/tree/master/detector  first 30 tokens of the real text (or just the question tokens  for the PubMedQA experiments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We measure performance  using the area under the receiver operating characteristic  curve (AUROC), which can be interpreted as the probability  that a classifier correctly ranks a randomly-selected posi-  tive (machine-generated) example higher than a randomly-  selected negative (human-written) example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' All experiments  use an equal number of positive and negative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' The key hyperparameters of Detect-  GPT are the fraction of words masked for perturbation,  the length of the masked spans, the model used for mask  filling, and the sampling hyperparameters for the mask-  filling model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Using BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2019) masked  language modeling as inspiration, we use 15% as the mask  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We performed a small sweep over masked span lengths  of {2, 5, 10} on a held-out set of XSum data, finding 2 to per-  form best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We use these settings for all experiments, with-  out re-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We use T5-3B for almost all experiments,  except for GPT-NeoX and GPT-3 experiments, where com-  pute resources allowed for the larger T5-11B model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' we also  use mT5-3B instead of T5-3B for the WMT multilingual  experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We do not tune the hyperparameters for the  mask filling model, sampling directly with temperature 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Main Results  We first present two groups of experiments to evaluate De-  tectGPT along with existing methods for zero-shot and su-  pervised detection on models from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='5B to 175B parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Zero-Shot Machine-Generated Text Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  We  present the comparison of different zero-shot detection meth-  ods in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' In these experiments, model samples are  generated by sampling from the raw conditional distribu-  tion with temperature 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' DetectGPT most improves average  detection accuracy for XSum stories (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='1 AUROC improve-  ment) and SQuAD Wikipedia contexts (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='05 AUROC im-  provement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' While it also performs accurate detection for  WritingPrompts, the performance of all methods tends to in-  crease, and the average margin of improvement is narrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='4  4The overall ease of detecting machine-generated fake writing  corroborates anecdotal reporting that machine-generated creative  Zero-Shot Machine-Generated Text Detection using Probability Curvature  RoB-base  RoB-lg  Likelihood  Rank  LogRank  DetectGPT  (Ours)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='981  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='997  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='889  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='800  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='915  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='991  XSum GPT-2 Detection  RoB-base  RoB-lg  Likelihood  Rank  LogRank  DetectGPT  (Ours)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='888  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content='838  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='795  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='863  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='957  WMT16-en mGPT Detection  RoB-base  RoB-lg  Likelihood  Rank  LogRank  DetectGPT  (Ours)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content='768  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content='836  PubMedQA PubMedGPT Detection  Supervised  Unsupervised  RoB-base  RoB-lg  Likelihood  Rank  LogRank  DetectGPT  (Ours)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='394  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='537  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content='962  WMT16-de mGPT Detection  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content='0  Detection Method  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='0  Detection AUROC  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Supervised machine-generated text detection models  trained on large datasets of real and generated texts perform as  well as or better than DetectGPT on in-distribution (top row)  text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' However, zero-shot methods work out-of-the-box for new do-  mains (bottom row) such as PubMed medical texts and German  news data from WMT16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' For these domains, supervised detectors  fail due to excessive distribution shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  For 14 of the 15 combinations of dataset and model, Detect-  GPT provides the most accurate detection performance, with  a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='06 AUROC improvement on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Log-rank thresh-  olding proves to be a consistently stronger baseline than log  probability thresholding, although it requires slightly more  information (full predicted logits), which are not always  available in public APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Comparison with Supervised Detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' While our exper-  iments generally focus on zero-shot detection, some works  have evaluated the detection performance of supervised  methods (typically fine-tuned transformers) for detecting  machine-generated text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' In this section, we explore several  domains to better understand the relative strengths of super-  vised and zero-shot detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' The results are presented in  Figure 4, using 200 samples from each dataset for evalua-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We find that supervised detectors can provide similar  detection performance to DetectGPT on in-distribution data  like English news, but perform significantly worse than zero-  shot methods in the case of English scientific writing and  fail altogether for German writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' This finding echoes past  work showing that language models trained for machine-  generated text detection overfit to their training data (source  model, decoding strategy, topic, language, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Uchendu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Ippolito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Jawahar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  In contrast, zero-shot methods generalize relatively easily  to new languages and domains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' DetectGPT’s performance  in particular is mostly unaffected by the change in language  from English to German.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  writing tends to be noticeably generic, and therefore relatively easy  to detect (Roose & Newton, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  PubMedQA  XSum  WritingP  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  RoBERTa-base  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='83  RoBERTa-large  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='71  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='91  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='85  log p(x)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='76  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='76  DetectGPT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='84  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='84  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='87  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='85  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' DetectGPT detects GPT-3 generations with average AU-  ROC on-par with supervised models trained specifically for  machine-generated text detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' For more ‘typical’ text, such as  news articles, supervised methods perform strongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  While our experiments have shown that DetectGPT is ef-  fective on a variety of domains and models, it is natural to  wonder if it is effective for the largest publicly-available  LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Therefore, we also evaluate multiple zero-shot and  supervised methods on 175B parameter GPT-3 using Ope-  nAI’s paid API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Because the GPT-3 API does not provide  access to the complete conditional distribution for each to-  ken, we cannot compare to the rank, log rank, and entropy-  based prior methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We sample 150 examples5 from the  PubMedQA, XSum, and WritingPrompts datasets and com-  pare the two pre-trained RoBERTa-based detector models  with DetectGPT and the probability thresholding baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  We show in Table 2 that DetectGPT can provide detection  competitive with the stronger supervised model, and it again  outperforms probability thresholding on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Variants of Machine-Generated Text Detection  Detecting Revised Machine-Generated Text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  In prac-  tice, humans may manually edit or refine machine-generated  text rather than blindly use a model’s generations for their  task of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We therefore conduct an experiment to  simulate the detection problem for model samples that have  been increasingly heavily revised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We simulate human re-  vision by replacing 5 word spans of the text with samples  from T5-3B until r% of the text has been replaced, and  report performance as r varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Figure 5 shows that De-  tectGPT maintains detection AUROC above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='8 even when  nearly a quarter of the text in model samples has been re-  placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Unsurprisingly, almost all methods show a gradual  degradation in performance as the sample is more heavily  revised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' The entropy baseline shows surprisingly robust  performance in this setting (althought it is least accurate  on average), even slightly improving detection performance  up to 24% replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' DetectGPT shows the strongest  detection performance for all revision levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Impact of Alternative Decoding Strategies on Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  While Table 1 suggests that DetectGPT is effective for  detecting machine-generated text, prior work notes that  the decoding strategy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', temperature sampling, top-k,  nucleus/top-p) can impact the difficulty of detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We re-  5We reduce the number of evaluation samples from 500 in our  main experiments to reduce the API costs of these experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Zero-Shot Machine-Generated Text Detection using Probability Curvature  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='20  Fraction of GPT-J-generated news article re-written  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='0  Detection AUROC  Rank  DetectGPT  LogRank  Likelihood  Entropy  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We simulate human edits to machine-generated text by  replacing varying fractions of model samples with T5-3B gener-  ated text (masking out random five word spans until r% of text is  masked to simulate human edits to machine-generated text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' The  four top-performing methods all generally degrade in performance  with heavier revision, but DetectGPT is consistently most accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Experiment is conducted on the XSum dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  XSum  SQuAD  WritingPrompts  Method  top-p  top-k  top-p  top-k  top-p  top-k  log p(x)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='87  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='89  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='96  Rank  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='76  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='76  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='81  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='84  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='83  LogRank  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='93*  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='90*  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='92*  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='90*  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='97  Entropy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='53  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='54  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='35  DetectGPT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='93  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='97  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' AUROC for zero-shot methods averaged across the five  models in Table 1 for both top-k and top-p sampling, with k =  40 and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Both settings enable slightly more accurate  detection, and DetectGPT consistently provides the best detection  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' See Appendix Tables 4 and 5 for complete results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  peat the analysis from Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='1 using top-k sampling and  nucleus sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Top-k sampling truncates the sampling  distribution to only the k highest-probability next tokens;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  nucleus sampling samples from only the smallest set of to-  kens whose combined probability exceeds p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' The results  are summarized in Table 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Appendix Tables 4 and 5 show  complete results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We use k = 40, and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='96, in line with  prior work (Ippolito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We find that both top-k  and nucleus sampling make detection easier, on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Averaging across domains, DetectGPT provides the clearest  signal for zero-shot detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Using Likelihoods from Models other than the Source  Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' While our experiments have focused on the white-  box setting for machine-generated text detection, in this sec-  tion, we explore the effect of using a different model to score  a candidate passage (and perturbed texts) than the model  that generated the passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' In other words, we aim to clas-  sify between human-generated text and text from model A,  GPT-J  GPT-Neo  GPT-2  Scoring Model  GPT-J  GPT-Neo  GPT-2  Base Model  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content='81  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content='88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='87  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' DetectGPT performs  best when scoring samples with  the same model that generated  them (diagonal), but the column  means suggest that some models  (GPT-Neo, GPT-2) may be bet-  ter ‘scorers’ than others (GPT-J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  White values show mean (stan-  dard error) AUROC over XSum,  SQuAD, and WritingPrompts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  black shows row/column mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  but  without  access  to  model A to compute log  probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Instead,  we use log probabilities  computed by a surrogate  model B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  We consider  three  models,  GPT-J,  GPT-Neo-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='7, and GPT-2,  evaluating  all  possible  combinations of source  model  and  surrogate  model (9 total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  We  average the performance  across 200 samples from  XSum,  SQuAD,  and  WritingPrompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  The  results are presented in  Figure 6, showing that  when the surrogate model  is  different  from  the  source model, detection performance is reduced, indicating  that DetectGPT is most suited to the white-box setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Yet  we also observe that if we fix the model used for scoring  and average across source models whose generations are  detected (average within column), there is significant  variation in AUROC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' GPT-2 and GPT-Neo-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='7 seem to be  better ‘scorers’ than GPT-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' These variations in cross-model  scoring performance suggest ensembling scoring models  may be a useful direction for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Evaluating scaling properties of DetectGPT  In this section, we evaluate DetectGPT as the size of the  mask-filling model or the number of perturbations used to  estimate the expectation in Equation 1 are varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Impact of Source and Mask-Filling Model Scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Here  we study the impact of the size of the source model and  mask-filling model on DetectGPT’s performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' the re-  sults are shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' In particular, the increased  discrimination power of DetectGPT for larger mask-filling  models supports the interpretation that DetectGPT is estimat-  ing the curvature of the log probability in a latent semantic  space, rather than in raw token embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Larger T5  models better represent this latent space, in which random  directions correspond to meaningful changes in the text on  the data manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Impact of Number of Perturbations for DetectGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Fi-  nally, we evaluate the performance of DetectGPT as a func-  tion of the number of perturbations used to estimate the  expectation in Equation 1 on three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' The results  are presented in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Detection accuracy continues to  improve until 100 perturbations, where it converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Evalu-  ations use 100 examples from each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Zero-Shot Machine-Generated Text Detection using Probability Curvature  60M  220M  770M  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='7B  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='0  Detection AUROC  5 perturbations  60M  220M  770M  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='7B  25 perturbations  Random  GPT2-sm  GPT2-md  GPT2-lg  GPT2-xl  Mask filling model size (# parameters)  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' There is a clear association between capacity of mask-  filling model and detection performance, across source model  scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Random mask filling (uniform sampling from mask filling  model vocabulary) performs poorly, reinforcing the idea that the  perturbation function should produce samples on the data manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Curves show AUROC scores on 200 SQuAD contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Discussion  As large language models continue to improve, they will  become increasingly attractive tools for replacing human  writers in a variety of contexts, such as education, jour-  nalism, and art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' While legitimate uses of language model  technologies exist in all of these settings, teachers, readers,  and consumers are likely to demand tools for verifying the  human origin of certain content with high educational, so-  cietal, or artistic significance, particularly when factuality  (and not just fluency) is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  In light of these elevated stakes and the regular emergence of  new large language models, we study the zero-shot machine-  generated text detection problem, in which we use only the  raw log probabilities computed by a generative model to  determine if a candidate passage was sampled from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' We  identify a property of the log probability function computed  by a wide variety of large language models, showing that a  tractable approximation to the trace of the Hessian of the  model’s log probability function provides a useful signal  for detecting model samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Our experiments find that  this signal is more discriminative than existing zero-shot  detection methods and is competitive with bespoke detection  models trained with millions of model samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  DetectGPT and Watermarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  One interpretation of  the perturbation function is producing semantically simi-  lar rephrasings of the original passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' If these rephrasings  are systematically lower-probability than the original pas-  sage, the model is exposing its bias toward the specific (and  roughly arbitrary, by human standards) phrasing used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' In  other words, LLMs that do not perfectly imitate human  writing essentially watermark themselves implicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Under  this interpretation, efforts to manually add watermarking bi-  ases to model outputs (Aaronson, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Kirchenbauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=',  2023) may further improve the effectiveness of methods  such as DetectGPT, even as LLMs continue to improve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' One limitation of probability-based methods  for zero-shot machine-generated text detection (like Detect-  1  10  100  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='0  Detection AUROC  GPT-2  XSum  SQuAD  WritingPrompts  1  10  100  1000  GPT-J  Number of perturbations  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Impact of varying the number of perturbations (samples  of mask and mask-fill) used by DetectGPT on auROC (left) and  auPR (right) to estimate the perturbation discrepancy for clas-  sification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Averaging up to 100 perturbations greatly increases  DetectGPT’s discrimination power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Fills sampled from T5-large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  GPT) is the white-box assumption that we can evaluate log  probabilities of the model(s) in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' For models be-  hind APIs that do provide probabilities (such as GPT-3),  evaluating probabilities nonetheless costs money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Another  assumption of DetectGPT is access to a reasonable pertur-  bation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' While in this work, we use off-the-shelf  mask-filling models such as T5 and mT5 (for non-English  languages), some domains may see reduced performance  if existing mask-filling models do not well represent the  space of meaningful rephrases, reducing the quality of the  curvature estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' While DetectGPT provides the best  available detection performance for PubMedQA, its drop  in performance compared to other datasets may be a result  of lower quality perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Finally, DetectGPT is more  compute-intensive than other methods for detection, as it  requires sampling and scoring the set of perturbations for  each candidate passage, rather than just the candidate pas-  sage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' a better tuned perturbation function or more efficient  curvature approximation may help mitigate these costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Future Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' While the methods in this work make no  assumptions about the models generating the samples, fu-  ture work may explore how watermarking algorithms can be  used in conjunction with detection algorithms like Detect-  GPT to further improve detection robustness as language  models continually improve their reproductions of human  text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Separately, the results in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='2 suggest that ex-  tending DetectGPT to use ensembles of models for scoring,  rather than a single model, may improve detection in the  black box setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Another topic that remains unexplored  is the relationship between prompting and detection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' that  is, can a clever prompt successfully prevent a model’s gen-  erations from being detected by existing methods?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Finally,  future work may explore whether the local log probabil-  ity curvature property we identify is present for generative  models in other domains, such as audio, video, or images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  We hope that the present work serves as inspiration to fu-  ture work developing effective, general-purpose methods  for mitigating potential harms of machine-generated media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Zero-Shot Machine-Generated Text Detection using Probability Curvature  References  Aaronson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' My Projects at OpenAI, Nov 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' URL  https://scottaaronson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='blog/?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='p=6823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Bakhtin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Gross, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Ott, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Deng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Ranzato,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=',  and Szlam,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Real or fake?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Learning  to discriminate machine from human generated text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  arXiv, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' URL http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='org/abs/1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  03351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' cite arxiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='03351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Black, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Gao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Wang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Leahy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', and Biderman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  GPT-Neo: Large Scale Autoregressive Language Model-  ing with Mesh-Tensorflow, March 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' URL https:  //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content='  Black, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Biderman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Hallahan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Anthony, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Gao,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Golding, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', He, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Leahy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', McDonell, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Phang,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Pieler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Prashanth, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Purohit, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Reynolds, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=',  Tow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', and Weinbach, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' GPT-NeoX-20B:  An open-source autoregressive language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' In Pro-  ceedings of the ACL Workshop on Challenges & Perspec-  tives in Creating Large Language Models, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' URL  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content='  Bojar, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content=', Chatterjee, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Federmann, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Graham, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=',  Haddow, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content=',  Logacheva, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content=', Neves,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content=', and Zampieri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Findings  of the 2016 conference on machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' In Pro-  ceedings of the First Conference on Machine Translation,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' 131–198, Berlin, Germany, August 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Associa-  tion for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' URL http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  aclweb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='org/anthology/W/W16/W16-2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='  Brown, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Mann, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Ryder, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Subbiah, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content=', Shyam, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Sastry,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Askell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Agarwal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Herbert-Voss, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Krueger,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Henighan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Child, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Ramesh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Ziegler, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=',  Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Winter, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Hesse, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Sigler, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=',  Litwin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Gray, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Chess, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Clark, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Berner, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=',  McCandlish, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Radford, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', Sutskever, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=', and Amodei,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content=' Nucleus (top-p) sampling evaluation with p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content=' AUROC for detecting samples from the given model on the given dataset for  DetectGPT and four previously proposed criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Nucleus sampling generally makes detection easier for all methods, but DetectGPT still  provides the highest average AUROC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content='  XSum  SQuAD  WritingPrompts  Method  GPT-2 OPT-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content='00  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' Top-k sampling evaluation with k = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
+page_content=' DetectGPT generally provides the most accurate performance (highest AUROC),  although the gap is narrowed comparing to direct sampling, presumably because top-k generations are more generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFIT4oBgHgl3EQflis6/content/2301.11305v1.pdf'}
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+A Quantum-Inspired Binary Optimization Algorithm for Representative Selection
+Anna G. Hughes,1 Jack S. Baker,1 and Santosh Kumar Radha1, ∗
+1Agnostiq Inc., 325 Front St W, Toronto, ON M5V 2Y1
+(Dated: January 6, 2023)
+Advancements in quantum computing are fuelling emerging applications across disciplines, includ-
+ing finance, where quantum and quantum-inspired algorithms can now make market predictions,
+detect fraud, and optimize portfolios. Expanding this toolbox, we propose the selector algorithm:
+a method for selecting the most representative subset of data from a larger dataset. The selected
+subset includes data points that simultaneously meet the two requirements of being maximally close
+to neighboring data points and maximally far from more distant data points where the precise notion
+of distance is given by any kernel or generalized similarity function. The cost function encoding
+the above requirements naturally presents itself as a Quadratic Unconstrained Binary Optimization
+(QUBO) problem, which is well-suited for quantum optimization algorithms - including quantum
+annealing. While the selector algorithm has applications in multiple areas, it is particularly useful
+in finance, where it can be used to build a diversified portfolio from a more extensive selection of
+assets. After experimenting with synthetic datasets, we show two use cases for the selector algorithm
+with real data: (1) approximately reconstructing the NASDAQ 100 index using a subset of stocks,
+and (2) diversifying a portfolio of cryptocurrencies. In our analysis of use case (2), we compare the
+performance of two quantum annealers provided by D-Wave Systems.
+I.
+INTRODUCTION
+The task of choosing representative samples from a
+larger collection of data, often referred to as represen-
+tative selection, has various advantages over studying
+the dataset as a whole (e.g., [1–4]).
+Representative
+selection can reduce the size and complexity of the data,
+simplifying data analysis and processing and reducing
+the memory cost of storing data.
+The computational
+efficiency of data modeling, such as classifier training
+and model application, can be significantly improved.
+Representative selection has been implemented in various
+subjects ranging from computer vision [5] and language
+processing [6] to protein analysis [7]. A variety of repre-
+sentative selection procedures have been offered to reduce
+the volume of training data for some specific supervised
+learning classifiers [8, 9].
+In addition to the methods
+that require additional knowledge for representative se-
+lection, there has been a growing interest in unsupervised
+approaches to finding representative samples [10].
+In this work, we implement an unsupervised represen-
+tative selection algorithm.
+An unlabeled dataset may
+contain some unknown number of classes, or the data in
+the set could be unclustered or only very loosely clustered
+into different categories according to some notion of
+similarity.
+The algorithm aims to find a subset of
+representative data points from a larger dataset.
+The
+selected subset should include only data points dissimilar
+to one another and similar to unselected neighboring
+data points.
+More concretely, our selector algorithm
+finds the k least similar points in a sample of n total
+clustered data points. This is done by both maximizing
+the distance between the chosen data and all other data
+∗ research@agnostiq.ai
+in the dataset while also minimizing the distance between
+selected data and the other similarly clustered data.
+The returned k points are then both representative of
+the data clusters and maximally distant from the other
+groups. This method can be applied across disciplines,
+but in this paper, we explore an application in finance:
+diversification.
+Diversification is a crucial strategy when building a
+robust portfolio. To mitigate the risk of interdependent
+components performing poorly in a market downturn,
+it is critical to invest in assets that are not strongly
+correlated to one another.
+There are many portfolio
+diversification strategies, from a naive 1/N rule [11] to
+more complex methods such as portfolio dimensionality
+or a Bayesian approach [12, 13].
+Many diversification
+methods are designed to weigh assets to minimize the
+portfolio’s overall variance deliberately.
+The recent
+popularity of machine learning in quantitative finance
+has enabled researchers to define new diversification
+methods. In this paper, we use our selector algorithm
+as a method for the diversification of assets.
+In this
+framework, a portfolio is well-diversified when each asset
+is maximally dissimilar to one another and representative
+of their respective sectors or other similarly performing
+assets.
+The core framework of our selector algorithm
+emerges as quadratic unconstrained binary optimization
+(QUBO) problem, which can be tackled with a range
+of metaheuristics [14–17] utilizing classical and quantum
+computation. Because of the large number of variables
+considered in diversification (and other selective repre-
+sentation problems), the resulting QUBO objective has
+many binary variables which, in the quantum setting,
+translates to the requirement of a large number of
+qubits.
+Although gate-model quantum computers are
+continually scaling up qubit numbers, presently, quantum
+inspired hardware like quantum annealers already meet
+this requirement (although the qubits are non-universal).
+arXiv:2301.01836v1  [quant-ph]  4 Jan 2023
+
+2
+Subsequently, we regard our selector algorithm as quan-
+tum inspired.
+II.
+THE SELECTOR ALGORITHM
+The selector algorithm is designed to pick out unique
+and representative data points from a larger dataset by
+finding low-cost solutions to a QUBO objective function.
+This function is constructed such that low-cost solutions
+maximize some notion of distance between selected data
+points and all other data, ensuring that chosen data
+points are unique while simultaneously minimizing the
+distance between selected and similarly clustered data.
+This ensures that each of the selected k data points from
+a dataset containing n > k points represent data that are
+similarly clustered with nearby points whilst each of the
+k data points are not similar to each other.
+The input data are provided as a matrix Y, where each
+row is a vector representative of the ith data point
+Y =
+�
+���������
+⃗y1
+⃗y2
+...
+⃗yi
+...
+⃗yn
+�
+���������
+=
+�
+����������
+y(1)
+1
+y(2)
+1
+y(3)
+1
+. . .
+y(1)
+2
+y(2)
+2
+y(3)
+3
+. . .
+...
+y(1)
+i
+y(2)
+i
+y(3)
+i
+. . .
+...
+y(1)
+n
+y(2)
+n
+y(3)
+n
+. . .
+�
+����������
+.
+(1)
+where n is the size of the dataset.
+Although any
+metric can be used to evaluate the distance between each
+data point, throughout this paper, the choice of distance
+metric is the Euclidean distance unless stated otherwise,
+dij(⃗yi, ⃗yj) =
+��
+m
+�
+⃗y(m)
+i
+− ⃗y(m)
+j
+�2
+.
+(2)
+Alternative distance metrics include standardized Eu-
+clidean, cosine, Minkowski, etc. The distance d is for-
+mally a kernel function, which in principle could also be
+computed and/or learned using quantum/classical neural
+networks (e.g., [18]). It is therefore possible to create a
+cross-paradigm variation of the selector algorithm where
+the d is evaluated on a gate model device as proposed
+in [18], and the QUBO objective is approximately solved
+using a quantum annealer. This however is beyond the
+scope of this work.
+The QUBO objective is defined as
+C(⃗x) = 1
+2k⃗xd⃗xT − 1
+n⃗xd⃗1T + A
+� n
+�
+i=1
+xi − k
+�2
+,
+(3)
+where ⃗1 is the n-length vector of ones, d is the n × n
+distance matrix with elements dij as given in Equation
+FIG. 1. An example of the selector algorithm’s performance
+on clustered data points.
+The clusters are generated by
+randomly choosing data points from a Gaussian distribution
+on a two-dimensional plane centered on two points.
+Each
+blob contains 90 data points, with a standard deviation of
+2.
+The selector algorithm was tasked with choosing two
+representative points from the complete dataset using the
+qbsolv solver. The algorithm-selected points are highlighted
+with blue squares, indicating that the algorithm successfully
+chose visually representative points from each cluster.
+2(or another user-defined distance measure), ⃗x is the
+{0, 1}n vector of binary variables and A is a penalty
+scaling factor used to enforce the equality constraint
+� xi = k.
+The first term on the RHS of Equation
+3 represents the distance between the selected points
+and other points in the cluster, ensuring that selected
+points are representative of their cluster, the second term
+represents the distance between selected points and all
+others in the data set, and the last term enforces the
+equality constraint (i.e., a penalty is applied to the cost
+function if more than k data points are chosen). This
+optimization problem takes on a QUBO form which are,
+in general, NP-hard.
+A more detailed description of
+QUBO models is given in Appendix A and a formal
+extension to weighted selection (i.e, where ⃗x can take
+on a number discrete values not limited to ⃗x ∈ {0, 1}n)
+is given in Appendix C. Now in this form, the problem
+becomes approachable with approximate metaheuristic
+algorithms, including quantum annealing, used in Section
+IV B, which we discuss qualitatively in Appendix B
+III.
+EXPERIMENTS WITH SYNTHETIC DATA
+To demonstrate the basic functionality of the selector
+algorithm, we use it to select points from two synthetic
+
+10
+8
+6
+2
+_12.5
+-10
+-7.5
+-5
+-2.5
+0
+2.5
+V13
+datasets.
+One dataset contains simple and obviously
+clustered data points (in terms of the Euclidean dis-
+tance), while the second dataset contains time series
+data - data points organized in a chronologically ordered
+sequence. We show that in both cases, the selector al-
+gorithm makes reasonable, representative choices. In the
+first application, we use the selector algorithm to choose 2
+data points from a clustered array of points. We generate
+two clusters, or blobs, by randomly choosing data points
+from a Gaussian distribution on a two-dimensional plane
+centered on two points.
+Each blob contains 90 data
+points, with a standard deviation of 2. The coordinates
+of all data points are used as input for the selector
+algorithm, which then chooses k representative points
+by minimizing the cost function in Equation 3 using the
+D-Wave decomposing solver qbsolv [19, 20] which uses
+a modified tabu algorithm [21] to minimize the objective
+function. In this example, we have generated two clusters
+of data points and tasked the selector algorithm with
+choosing k = 2 representative points.
+The blobs and
+selected points are shown in Figure 1; as expected, the
+selector algorithm successfully chose one point from each
+cluster.
+Since the choice of distance is euclidean, the
+chosen point closely resembles the approximate center
+of the cluster.
+This visual representation might not
+always be the case for all metrics; for example, metrics
+like Jensen–Shannon divergence might have no visually
+discernible centers.
+The selector algorithm can similarly choose data points
+of arbitrarily high dimensions. We demonstrate this in
+two examples: choosing points from an array of trigono-
+metric curves and choosing from an array of stochastic
+differential equation (SDE) time series generated ran-
+domly. In each case, the data are generated synthetically
+but clustered into two different classes: sine or cosine in
+the trigonometric case and time series generated from
+coupled Brownian process in the SDE case.
+In both
+cases, the selector algorithm is tasked with choosing two
+representative data points; in the trigonometric case, a
+solution is considered successful if the selector algorithm
+chooses one sine and one cosine curve, and in the SDE
+case, a solution is successful if the selector algorithm
+chooses representatives from the distinct processes.
+The trigonometric time series are generated using,
+ysine
+i
+= sin(ti) + ni
+ycosine
+i
+= cos(ti) + ni,
+(4)
+where ti ranges from 0 to 2π and ni is artificial random
+noise pulled from a Gaussian distribution with mean
+µ = 0 and standard deviation σ ranging from σ = 0.1 to
+σ = 5.5. As σ increases, there is little difference between
+the sine and cosine curves because the introduced noise
+dominates the amplitude of the curves. We generate 50
+sine and 50 cosine time series with 10 different σ values
+and task the selector algorithm with choosing k = 2
+representative data points in each case. We expect the
+selector algorithm to select one sine and one cosine data
+point when σ is low, but as σ is increased, it should be
+harder for the selector algorithm to distinguish the two
+clusters.
+The results are shown in Figure 2.
+In the first two
+panels (a,b), clusters of sine (green) and cosine (orange)
+curves are plotted against time, where t spans from 0
+to 2π. The curves in the top left panel have σ = 0.3,
+while the curves in the top right panel have σ = 3.0.
+In each plot, the 2 representative curves chosen by the
+selector algorithm are plotted in black. As expected, the
+σ = 0.3 time series curves are easily distinguishable, with
+the algorithm choosing one of each. In (b), where the
+standard deviation is increased to σ = 3, the sine and
+cosine curves are no longer visually distinguishable.
+In (c), density plots of the elements of the correla-
+tion distance matrix are shown.
+Each horizontal line
+extending upward represents increasing noise standard
+deviation, from σ = 0.1 to σ = 1.0. As expected, the two
+clusters of curves are clearly separable into two distinct
+peaks at low σ, but start to merge as σ increases. This
+can measure the distinguishability (or lack thereof) of the
+clusters in the dataset as a function of σ.
+In (d), the accuracy of chosen solutions is plotted
+as a function of σ.
+Accuracy here is measured as the
+percentage of algorithm solutions that contained one sine
+and one cosine curve out of 200 total trials. As expected,
+at low σ, the selector algorithm chooses solutions with
+100% accuracy. While that accuracy decreases at higher
+σ, even at σ > 3 where the two sets of density curves
+(figure (c)) are indistinguishable, the selector algorithm
+can pick out curves from each cluster with better-than-
+random accuracy. This demonstrates the robustness of
+the selector algorithm in choosing representative data
+points even as the separation between clusters vanishes.
+Finally, we test the performance of the selector algo-
+rithm at choosing solutions from the solution of a two-
+dimensional SDE given by
+dxi(t) = µixi(t)dt +
+2
+�
+j=1
+σijxi(t)dWj(t),
+(5)
+where µi and σij are the drift vector and volatility matrix
+respectively and Wj(t) is the standard Brownian motion.
+For the case of simulation, we choose µi and σ randomly
+from a uniform distribution [−1, 1] and numerically
+generated multiple two-dimensional time series.
+While the trigonometric time series showed us the
+ability of the selector algorithm to choose representative
+data points from loosely clustered or unclustered data,
+in this example, we show the ability of the selector
+algorithm to evaluate simulated financial data.
+As
+before, the selector algorithm was given an array con-
+taining all-time series and tasked with choosing (apriori
+known) k = 2 representative curves.
+The two sets of
+correlated SDE time series are plotted in Figure 3, with
+the selector algorithm’s choices plotted in black.
+The
+selector algorithm successfully chose one curve from each
+time series cluster, demonstrating its ability to evaluate
+financial time series data such as daily returns.
+
+4
+FIG. 2. Two clusters of 50 sine and cosine curves generated from Equation 4 are used as input for the selector algorithm, with
+varying values of the standard deviation of the Gaussian noise. (a): 50 sine and cosine time series curves with introduced
+Gaussian noise of σ = 0.3 are plotted in green and orange, respectively. The algorithm-selected curves are plotted in black,
+showing that the algorithm chose one curve from each cluster. (b): 50 sine and cosine time series curves are plotted with
+introduced Gaussian noise increased to σ = 3.0. As before, the algorithm-selected curves are plotted in black, but the clusters
+are no longer distinguishable because the amplitude of the noise is too great. (c): the density of curves are shown as a function
+of distance, where each horizontal line extending upward represents increasing noise standard deviation. As expected, the two
+clusters of curves are clearly separable into two distinct peaks at low σ, but start to merge as σ increases. (d): the accuracy
+of chosen solutions is plotted as a function of σ in the bottom right panel. Accuracy here is measured as the percentage of
+algorithm solutions that contained one sine and one cosine curve out of 200 total trials. At low σ, the selector algorithm
+chooses solutions with 100% accuracy. While that accuracy decreases at higher σ, even at σ > 3 where the two sets of curves
+are indistinguishable, the selector algorithm can pick out curves from each cluster with better-than-random accuracy.
+All
+optimizations in this example were performed using the qbsolv solver.
+FIG. 3. Two clusters, each containing 50 synthetic financial
+time series, generated as given in Equation 5, are plotted
+in orange (positive) and teal (negative).
+Each cluster is
+created with a random correlation chosen from a Gaussian
+distribution of µ = 0 and σ = 4.
+For each set of curves,
+the volatility and return rates are randomly chosen from a
+Gaussian distribution with µ = −1 and σ = 1. The selector
+algorithm was tasked with choosing 2 points from the full
+dataset. Plotted in black are the two selected curves, showing
+the selector algorithm’s ability to choose representative time
+series from synthetic financial data reliably.
+IV.
+USE CASE: BUILDING A DIVERSIFIED
+PORTFOLIO
+Diversifying a portfolio by investing in uncorrelated
+assets is an approach to mitigate risks associated with
+downturns in specific markets or unexpected crises.
+Because the selector algorithm is designed to pick out
+data points that are both distinct and representative, it
+is particularly useful for building a diverse portfolio from
+a more extensive list of assets. Quantum annealers are
+particularly well-suited to address QUBO problems at
+this scale. That is, it is possible for the large number
+of variables to be mapped to the large number (∼ 1000)
+of non-universal qubits available on present generation
+quantum annealers. In section IV A, we will start by us-
+ing the selector algorithm as a portfolio diversifier aiming
+to approximate the behaviour of the NASDAQ 100 index
+using a smaller subset of assets.
+For this task we use
+a classical large scale QUBO solver which is presently
+practical to use. We then proceed to section IV B, where
+we perform experiments with the algorithm using D-
+Wave’s quantum annealers. That is, we benchmark and
+compare the performance of two quantum devices for the
+task of diversifying a cryptocurrency portfolio.
+A.
+Reconstructing the NASDAQ 100 with a
+classical QUBO solver
+The selector algorithm can build a diverse portfolio by
+selecting a representative subset of stocks from a market
+index. In this first application on real financial data, we
+use the selector algorithm to approximately reconstruct
+the NASDAQ 100 by choosing a subset (Sk) of k stocks
+from all 102 stocks in the market index. We perform the
+optimization with the D-Wave decomposing solver qbsolv
+[19, 20].
+To choose stocks from the NASDAQ 100 index, we
+treat the daily returns from each stock as individual data
+points in our array. Vectors ⃗yi are composed of the daily
+
+g=.3
+q=3
+2
+(a)
+(b)
+(c)
+(d)
+10
+100 -
+g=1.0
+0=0.9
+Accuracy (%)
+90
+g=0.8
+Count
+0=0.7
+80
+0
+0=0.6
+0=0.5
+70
+0=0.4
+-1 
+g=0.3
+60
+g=0.2
+-10
+g=0.1
+Random probability
+-2 -
+50
+0.98 0.985 0.99 0.995
+1.005
+0
+2
+4
+6
+0
+2
+4
+6
+1
+2
+4
+Time
+Time
+Distance
+Standard deviation (o)5
+Signal (yi)
+0
+-5
+-10
+20
+40
+0
+60
+80
+100
+Time5
+returns from each stock in the index over one trading
+year, or 253 days, starting on 2021-02-01 (YYYY-MM-
+DD) and ending on 2022-02-01 (YYYY-MM-DD). This
+results in a 102 × 253 matrix Y,
+Y =
+�
+�������
+−→
+⃗y1
+−→
+⃗y2
+−→
+⃗y3
+...−→
+⃗yn
+�
+�������
+=
+�
+�������
+−−−−→
+AAPL
+−−−−−→
+ABNB
+−−−−→
+ADBE
+...
+−→
+ZS
+�
+�������
+(6)
+In this familiar form, the data are ready for evaluation
+in the cost function (Equation 3), where n = 102 is the
+total number of vectors in matrix Y, k is the number
+of desired stocks selected by the algorithm, and the
+elements of d are given by the correlation distance,
+dab = 1 −
+�(ai − ¯a) · �(bi − ¯b)
+��(ai − ¯a)2 �(bi − ¯b)2 ,
+(7)
+where ¯a and ¯b are the mean values of vectors ⃗a and ⃗b.
+The chosen stocks can come arbitrarily close to a com-
+plete reproduction of the market index; as the number
+of selected stocks k increases, the combined data from
+the chosen stocks comes closer and closer to an exact
+reproduction of the index itself.
+However, the goal is
+only approximately to reproduce the index with a smaller
+number of stocks, so choosing a large number is not
+necessarily advantageous and is a hyperparameter for the
+end-user.
+In all following analyses, we use only the binary case,
+where the weighted subspace is restricted to only binary
+values, ⃗w → ⃗x ∈ {0, 1}n (see discussion in Appendix
+C for the general discrete weighted case).
+Like in the
+previous examples, this means that specific data points
+are either chosen or not chosen, with no inbetween. This
+is an NP-hard problem [22]. To evaluate the performance
+of the binary selector algorithm in reproducing the
+NASDAQ 100, we create a proxy NASDAQ 100 index by
+linearly combining all stocks and averaging the resulting
+vector. Note that while the NASDAQ 100 index is market
+value-weighted, for simplicity, the stocks in our NASDAQ
+100 index are equally weighted.
+We initially use the selector algorithm to choose
+two stocks from our input matrix Y of returns.
+The
+algorithm-chosen stocks are Fox Corporation (‘FOX’)
+and Synopsys, Inc.
+(‘SNPS’). We use these stocks
+to create Sk=2 := S2, a data point composed of the
+averaged linear combination of our chosen stocks’ daily
+returns.
+A histogram showing the average percentage
+daily returns for both S2 and the proxy NASDAQ 100
+index is shown on the left panel of Figure 4.
+One
+can see that the stock return profile of the selected
+S2 stocks is a good surrogate for the NASDAQ 100
+already at k = 2.
+To further assess the quality of
+the algorithm’s selection, we calculate the value of the
+cost function and the correlation distance between the
+proxy NASDAQ 100 index and some data point S2 for all
+possible combinations of 2 stocks. A histogram showing
+the range of cost function and correlation distance values
+are shown in the middle and right panels of Figure 4,
+respectively. The S2 index had a lower cost function value
+than 74% of all cost function values and a correlation
+distance less than 89% of all values.
+In Figure 5, we show the average (left) and cumulative
+(right) returns from the proxy NASDAQ 100 index
+plotted with the Sk time series, with each row showing
+incremental increases in k.
+As expected, when the
+number of chosen stocks (k) increases, the Sk vector
+comes closer to the proxy NASDAQ 100 index.
+Finally, we compare the algorithm-selected indices Sk
+to the NASDAQ 100 index by computing the mean
+squared error (MSE) between the two. The number of
+stocks selected k increases steadily until all 102 stocks
+in the index are included. The MSE as a function of k
+is plotted in Figure 6, where one can see that NASDAQ
+100 can effectively be reproduced by roughly 40 stocks.
+B.
+Diversifying cryptocurrency portfolios with
+quantum annealers
+Selecting a diverse portfolio from a limited index can
+be challenging, especially when the assets cannot be
+easily divisible into traditional sectors, as in the case
+of cryptocurrencies (crypto).
+Unexpected correlations
+between coins can leave a crypto portfolio vulnerable
+to the volatility of one asset. The selector algorithm is
+thus particularly useful for building a diverse portfolio
+of cryptocurrencies.
+In this use case, we use the
+selector algorithm to build a diverse portfolio from the
+Crescent Crypto Market Index (CCMIX) using quantum
+annealers. We compare the performance of each device,
+investigating how well each device satisfied the problem
+constraints and the quality of solutions.
+The two quantum annealing devices used in the fol-
+lowing experiments are the D-Wave 2000Q and the D-
+Wave Advantage. The D-Wave 2000Q, first introduced
+in 2017, is a 2048-qubit quantum annealing device. The
+2000+ qubits are arranged in a Chimera topology with
+6016 couplers.
+The 2000Q system has been used to
+solve problems ranging from e-commerce listing order to
+cryptography [23, 24]. The D-Wave Advantage quantum
+processing unit (QPU) was released in September 2020
+as an updated, more advanced system. It contains 5000+
+qubits with 35,000+ couplers - an increase from 6 to 15
+couplers per qubit. This increase in qubits and couplers
+enables the D-Wave Advantage QPU not only to solve
+larger problems than the D-Wave 2000Q device but also
+allows for problems with more challenging connectivity
+to be more easily mappable to the device qubits when
+compared to the 2000Q Device. The D-Wave Advantage
+QPU has been used to solve problems such as railway
+dispatching and molecular unfolding [25, 26].
+
+6
+FIG. 4. (a) Histograms and corresponding kernel density estimate (KDE) plots of the daily returns from the proxy NASDAQ
+100 index and the combined index from k = 2 stocks (S2) selected by the algorithm using the qbsolv solver. (b) Histogram and
+corresponding KDE plot showing the value of the cost function for all combinations of 2 stocks in the NASDAQ 100 index,
+where the dashed vertical line shows the value of the cost function for the algorithm-selected k = 2 stocks. (c) Histogram and
+corresponding KDE plot showing the correlation distance for all combinations of 2 stocks in the NASDAQ 100 index, where
+the dashed vertical line shows the distance for the algorithm-selected k = 2 stocks.
+The input data to the selector algorithm consists of the
+daily returns from each coin in the CCMIX over seven
+months, beginning on 2021-04-01 (YYYY-MM-DD) and
+ending on 2021-11-11 (YYYY-MM-DD). Depending on
+the experiment, the selector algorithm chooses some
+desired number of coins in the array, using the specified
+quantum annealer to minimize the cost function.
+1.
+Constraint Satisfaction with Default Parameters
+In the first round of experiments, we investigate the
+ability of each device to choose solutions that satisfy
+the equality constraint imposed by the penalty term in
+Equation 3.
+While satisfying this constraint may be
+trivial for some classical methods, quantum annealers
+are more limited, especially when the constraints are
+imposed as penalties [27]. For experiments in this section,
+we use the default parameters of the annealers, notably
+leading to an annealing time of t = 20 µs. We also use
+a penalty scaling factor of A = 2.
+We run 300 trials
+on each device, where the selector algorithm was tasked
+with choosing k = 3 cryptocurrencies from the 18 coins
+included in the CCMIX. In all experiments, the number
+of shots per run is fixed at 1000.
+The results are shown in Figure 7. There is a significant
+difference in how well each device satisfies the k = 3
+constraint. The D-Wave 2000Q device only chose three
+cryptocurrencies in 49 of the 300 trials – a success rate
+of just 16%. Conversely, the D-Wave Advantage QPU
+chose three coins in 256 of the 300 runs – a success rate
+of over 85%. There is clearly a marked improvement in
+performance for the newer D-Wave Advantage QPU even
+using the default parameters.
+2.
+Consistency of Solutions with Changing Annealing Time
+In subsequent experiments, we investigated the impact
+of the value of the penalty scaling factor A (see Equation
+3) and the annealing time on constraint satisfaction. We
+first investigated tuning the penalty scaling factor in 11
+runs with 50 repeats for each value of A, where the
+value ranged from A = 0 to A = 10 in fixed increments
+while the annealing time was fixed at 20 µs.
+In these
+experiments, we saw very little impact on either the
+quality of the solution (i.e, how low the cost is) or
+satisfaction of the problem constraint.
+For the rest of
+this section, were therefore continue with the default of
+A = 2. Finally, we performed 19 trials on each device
+using annealing times ranging from the default 20 µs to
+990 µs, with 50 repeats at each annealing time.
+As can be seen in Figure 8, the value of the annealing
+time has a negligible effect on the percentage of solutions
+matching the k
+=
+3 constraint for the Advantage
+QPU, which tends towards solutions that satisfy this
+constraint. Conversely, higher annealing times enable the
+2000Q QPU to find more solutions that satisfy the soft
+constraint. Even at high annealing times, most of the
+solutions chosen by the 2000Q device do not satisfy the
+problem constraint, while the Advantage QPU satisfies
+the constraint > 85% of the time at every annealing time.
+The possible impact of the annealing time on the
+overall value of the cost function (k = 3) is more difficult
+to discern. At the intermediate annealing time of 600 µs
+and at the higher annealing time of 900 µs, both the value
+of the cost function and the standard deviation between
+trials hits a low value for the 2000Q QPU. Very little
+such variation is discernible for the Advantage device.
+
+Daily Returms Histogram & KDE
+Cost Function Histogram & KDE
+Correlation Distance Histogram & KDE
+(a)
+100
+(b)
+(c)
+40 
+NASDAQ100
+All k=2 Combinations
+SelectedStocks(S2)
+Selected Stocks (S2)
+Selected Stocks (S2)
+250
+All k=2 Combinations
+35
+80
+30
+200
+Density
+60
+40
+15
+100
+10
+20
+50
++-
+0.050-0.025 0.000 0.025 0.050
+1.95
+2.00
+2.05
+2.10
+2.15
+2.20
+0.000 0.005 0.010 0.015 0.020 0.025
+Daily Returms
+Cost
+Distance7
+FIG. 5. (a)-(e): All daily returns from the proxy NASDAQ 100 index (violet) and the index generated from the selector
+algorithm’s choice of stocks only. The proxy NASDAQ 100 index is created by equally weighting all stocks in the NASDAQ 100
+over 2021-02-01 to 2022-02-01 (YYYY-MM-DD), while the selector algorithm index is created by equally weighting only the
+stocks chosen by the algorithm. The number of selected stocks k varies, starting from k = 2 in the first row and increasing to
+k = 20 in the final row. As k increases, the selector-created index comes closer to the proxy NASDAQ 100 index, as expected.
+(f)-(j): The cumulative returns from both the proxy NASDAQ 100 index and the selector algorithm-generated index, where
+each row represents an increasing number of selected stocks k. As the number of selected stocks is increased, the selector
+algorithm-generated index more closely replicates the features of the proxy NASDAQ 100 index.
+3.
+Comparing solution quality
+We examine the quality of solutions selected by each
+device given the constraint that k = 3. To understand
+how the QPU-chosen solutions compare with all possible
+solutions, we first compute the value of the cost function
+for every combination of coins.
+The CCMIX index
+contains 18 cryptocurrencies, leading to 218 = 262144
+total possible combinations, ranging from choosing 0
+coins to the possibility that all 18 coins are chosen.
+
+(a)
+(f)
+NASDAQ 100
+NASDAQ 100
+Selected Stocks (S2)
+Selected Stocks (S2)
+(b)
+(g)
+NASDAQ 100
+NASDAQ 100
+Selected Stocks (S3)
+Selected Stocks (S3)
+(c)
+(h)
+NASDAQ 100
+NASDAQ 100
+Selected Stocks (S5)
+Selected Stocks (S5)
+(d)
+(1)
+NASDAQ 100
+NASDAQ 100
+Selected Stocks (Sio)
+Selected Stocks (Sio)
+(e)
+()
+NASDAQ 100
+NASDAQ 100
+Selected Stocks (S2o)
+Selected Stocks (S20)8
+FIG. 6. The mean-squared error (MSE) between the proxy
+NASDAQ 100 index and the selector algorithm-generated
+index as a function of the number of selected stocks k. The
+proxy NASDAQ 100 index is created by equally weighting
+all stocks in the NASDAQ 100 over 2021-02-01 to 2022-02-01
+(YYYY-MM-DD). In contrast, the selector algorithm index
+is created by equally weighting only the stocks chosen by the
+algorithm. As k increases from 1 representative stock to all
+102 stocks in the index, the MSE sharply decreases to zero.
+FIG. 7.
+The selector algorithm was tasked with choosing
+three cryptocurrencies from the CCMIX, which contains 18
+total options. Optimization was performed using the D-Wave
+2000Q device (left) and the D-Wave Advantage device (right),
+with 300 trials on each QPU. The solutions chosen by the D-
+Wave 2000Q device satisfied the problem constraint of k = 3
+in only 16% of its trials, while the D-Wave Advantage device
+chose solutions that satisfied the problem constraint in > 85%
+of its trials.
+The cost function values are calculated assuming the
+k = 3 constraint, so all combinations containing more or
+less than three coins are penalized in the cost function.
+A = 2 in these experiments. The mean value of the cost
+function, considering all combinations, is 81.
+To gauge the performance of quantum devices relative
+to a randomized solution, we calculate the average value
+of the cost function for solutions chosen by each device
+FIG. 8. We investigate the ability of each device to choose
+solutions matching problem constraints as a function of
+annealing time, where the annealing time is increased from
+20µs to 990µs. The percentage of solutions matching problem
+constraints (out of 50 trials at each annealing time) is plotted
+for the D-Wave 2000Q (beige) and the D-Wave Advantage
+(violet). The change in annealing time has a negligible effect
+on the percentage of constraint-satisfying solutions for the
+Advantage QPU, while higher annealing times seem to enable
+the 2000Q QPU to satisfy the problem constraint better.
+However, even at high annealing times, most solutions chosen
+by the 2000Q device do not satisfy the problem constraint,
+while > 85% of solutions chosen by the Advantage QPU
+satisfy the constraint at every annealing time.
+for all trials at each device’s optimal annealing time (550
+µs for 2000Q and 900µs for Advantage). For the 2000Q
+device, the average value of the cost function was 4.02,
+while the average cost function value for the Advantage
+QPU was 0.32.
+The results are shown in Figure 9,
+where a histogram showing the density of cost function
+values is plotted, with vertical lines indicating the total
+average cost value and the cost values of the 2000Q and
+Advantage QPUs.
+The average cost function value of both devices was
+significantly lower than the average value considering all
+possible solutions. The average value of the cost function
+for the 2000Q device is in the lowest 4% of possible
+values, while the average value of the cost function for the
+Advantage QPU is in the lowest 0.03% of all solutions.
+Overall, both devices distinctly demonstrate the ability
+to choose solutions with low values of the cost function.
+However, comparing devices, the Advantage QPU shows
+a marked improvement in the ability to choose solutions
+that match the problem constraints and solutions with
+lower cost function values.
+
+0.00012
+0.00010
+Error
+0.00008
+ Squared
+0.00006
+Mean
+0.00004
+0.00002
+0.00000
+0
+20
+40
+60
+80
+100
+k2000Q
+Advantage
+250
+200
+Count
+150
+100
+50
+0
+3
+4
+6
+73
+5
+4
+5
+6
+7
+Number Selected
+Number Selected100
+Constrained Solutions
+90
+80
+70
+60
+50
+Percentage (
+40
+30
+Advantage
+20
+2000Q
+0
+200
+400
+600
+800
+1000
+Annealing Time (μs)9
+FIG. 9. The value of the cost function for all combinations
+of coins is plotted as a histogram, with the black curve fitted
+to the distribution. The average value of the cost function,
+considering all possible combinations, is indicated by the
+vertical dotted red line. The average cost function value of
+solutions chosen by the D-Wave 2000Q and Advantage QPUs
+are indicated by the blue and orange horizontal dotted lines,
+respectively.
+V.
+CONCLUSION
+In this work, we devised a selector algorithm for the
+representative selection of data which relies on solving a
+QUBO problem.
+Our selector algorithm chooses data
+points by maximizing the distance between selected
+points and all other data points in the dataset and
+minimizing the distance between selected points and sim-
+ilarly clustered data points. Given the potentially large
+number of variables in the resulting QUBO problem, the
+optimization stage of selector algorithm is particularly
+suited towards large scale QUBO solvers like quantum
+annealers.
+After experimentation with synthetic datasets, in our
+first practical use case, we used the selector algorithm to
+approximate the NASDAQ 100 with a subset of stocks
+in the index. We created a proxy NASDAQ 100 index
+by linearly combining and averaging all 102 stocks. The
+selector algorithm chose k stocks from the index, whose
+performances were combined, averaged, and compared to
+the proxy 100 indexes. As the number of selected stocks k
+increased, the algorithm-constructed index more closely
+resembled the proxy NASDAQ 100 index. We compared
+the k = 2 point, S2, to all possible combinations of 2
+stocks. We found that the algorithm-selected point S2
+had a cost function value in the bottom 26% of all k = 2
+solutions.
+In our second application, we used the selector algo-
+rithm to build a diversified portfolio of cryptocurrency
+assets from the Crescent Crypto Market Index.
+The
+optimization was performed using 2 quantum annealers:
+the D-Wave 2000Q and the D-Wave Advantage.
+We
+compared the k
+=
+3 selection of both devices by
+evaluating the cost function and comparing that to the
+value of the cost function of all possible coin combinations
+in the index. We found that the average cost function
+value of solutions chosen by the 2000Q device was in
+the lowest 4% of all cost function values, while the D-
+Wave Advantage device’s average cost function value is
+in the lowest 0.03%. The D-Wave Advantage device also
+more consistently chose solutions matching the problem
+constraint - with > 85% of solutions matching the
+constraint, while only 16% of 2000Q solutions matched
+the constraint.
+Overall, we saw clear improvement
+between the newer Advantage QPU and the earlier 2000Q
+QPU, providing meaningful solutions to the combinato-
+rial optimization problem.
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+Appendix A: Quadratic Unconstrained Binary
+Optimization
+Quadratic
+Unconstrained
+Binary
+Optimization
+(QUBO) is the optimization of an objective function
+with binary variables with at-most quadratic order
+coupling between variables. This approach is powerful
+and widespread because many combinatorial problems
+can be written in QUBO form (see Equation A1) and
+solved using general purpose QUBO solvers.
+This is
+in contrast to other more traditional methods which
+are tailored to the exact problem at hand. In general,
+to approximately solve a QUBO problem (the exact
+problem is NP-hard), we must select an approximately
+optimal solution from all possible solutions.
+The basic cost or objective function C(⃗x) of QUBO is,
+C(⃗x) = ⃗xT Q⃗x,
+(A1)
+where ⃗x = {0, 1}n, and Q is an n×n matrix of constants.
+The goal of QUBO is to find the arrangement of binary
+values in ⃗x that minimizes (or maximizes) the value of
+C(⃗x). Other than the requirement that the values in the
+vector ⃗x be binary, there are no formal constraints in this
+simple QUBO model. However, so-called soft constraints
+can be encoded into the QUBO problem using quadratic
+penalty terms (for example, see the last term of the RHS
+of Equation 3). Rather than impose rigid constraints on
+the problem, soft constraints work by discouraging the
+optimizer from solutions that will incur penalties.
+QUBO problems have garnered recent attention due
+to how easily they are adapted mappable to quantum
+annealers and gate-model quantum computers.
+Appendix B: Quantum Annealing
+In metallurgy, annealing strategically applies heat to
+a material to make it more malleable. Annealing works
+by initially raising the temperature of a metal above its
+recrystallization temperature, which causes the atoms in
+the metal to move around rapidly.
+The temperature
+is then slowly lowered, and atoms settle into a new
+crystalline configuration with the desired properties.
+Simulated annealing is a machine-learning technique that
+takes its name from this metallurgic process. Similar to
+how metallurgic annealing causes atoms to explore new
+positions rapidly, simulated annealing rapidly samples
+the solution space of a particular cost function and
+adopts new configurations as desired. The sampling rate
+slowly decreases, allowing the annealer to settle into an
+optimized solution.
+In simulated annealing, the solution starts in some ini-
+tial configuration C and a maximum initial ‘temperature’
+of Tmax, where the temperature is a hyperparameter
+that dictates how randomly the annealer samples the
+cost function.
+The state C is then advanced into a
+new configuration N. The change in the value of the
+cost function can then be calculated. Depending on the
+current temperature and the change in the value of the
+cost function, the new configuration N is either accepted
+as the new state or rejected. A ‘good’ new configuration
+N would have a lower cost function value than the
+previous configuration, indicating that the state has
+advanced closer to the minimum value. However, even
+‘bad’ configurations, where the change in cost function
+value is positive, can be accepted if the temperature is
+very high. Just like with real metallurgic annealing, the
+temperature is then gradually decreased.
+This means
+that initially, fewer new configurations are accepted
+unless the cost function value is significantly lower. This
+method enables the simulated annealer to sample the cost
+landscape very broadly and ideally settle on the global
+minimum rather than any local minimum.
+Quantum annealing uses a similar technique but
+explores quantum states for solution space using the
+strength of quantum fluctuations as a sampling rate
+hyperparameter.
+A quantum annealing cost function
+is a Hamiltonian, expressing the total energy of the
+system for each quantum state.
+Minimizing the cost
+function means finding the quantum state with the lowest
+energy. Quantum annealing has the additional benefit
+of quantum tunneling, where the algorithm can tunnel
+between different low-energy quantum states, avoiding
+the need to sample the space in between these states and
+preventing the algorithm from getting trapped in local
+minima.
+As the sampling rate slowly decreases, so is
+the quantum tunneling radius. Initially, the algorithm is
+
+11
+given a wide radius to explore many low-energy quantum
+states.
+As the system evolves and the sampling rate
+decreases, the tunneling radius slowly decreases as the
+algorithm settles on a final state. Quantum annealing
+provides an approximate solution, not necessarily an
+absolute one.
+Appendix C: The selector algorithm with weights
+In the main article, we explore the case where k out
+of n data points are either “selected" or “not selected".
+This problem is innately QUBO since the weighted
+contribution wi to the objective function is restricted to
+wi = xi ∈ {0, 1}. We now show how this formalism can
+be extended to a more general case where k total points
+are selected, and each selected point is assigned a weight
+with resolution controlled using the integer discretization
+number nD.
+The weight of the ith data point can be expanded with
+nD binary variables,
+wi = wmin + (wmax − wmin)
+nD
+�
+j=1
+2j−1
+2nD − 1Xij,
+(C1)
+for minimum/maximum weights wmin/max, respec-
+tively. In the context of the selector algorithm, a natural
+choice is wmin = 0 and wmax = 1. Xij ∈ {0, 1} are the
+elements of the rectangular n × nD bit matrix
+X =
+�
+��
+X1,1 . . . X1,nD
+...
+...
+Xn,1
+Xn,nD
+�
+�� .
+(C2)
+Increasing the number of terms used in the expansion
+of Equation C1 (i.e, increasing nD) increases the resolu-
+tion of wi which, in the quantum setting, comes at the
+price of increasing the qubit requirement N as given by
+N = n × nD. At nD = 1, the original “selected" or “not
+selected" case is obtained wi = X1,1 := xi.
+We now define a new variable χi as the product of
+the weight and the binary variable xi: χi = wixi. Now,
+within Equation 3, for the first two terms on the RHS,
+we make the change of variable ⃗x → ⃗χ and insert a new
+penalty term on the sum of the elements of ⃗χ to yield the
+objective
+C(⃗χ, ⃗x) = 1
+2k ⃗χd⃗χT − 1
+n ⃗χd⃗1T + A
+� n
+�
+i=1
+xi − k
+�2
++ B
+� n
+�
+i=1
+χi − W
+�2
+(C3)
+where W is the budget constraint on wi: �
+i wi = W
+and B (like A) is a penalty scaling factor. As presented,
+Equation C3 is quartic order in binary variables but, fol-
+lowing the discussion in [28], can always be transformed
+to quadratic and thus reducible to a QUBO problem.
+While such a transformation can be formally derived,
+we use the package PyQUBO [29] to compile the problem
+to a QUBO form intelligently.
+The compiled problem
+can then be solved using metaheuristic optimization
+algorithms, including quantum annealing.
+It should also be noted that Equation C3 is reducible
+to Equation 3 in the main text through some specific
+choices of the variable. Setting wmin = 0, wmax = 1 and
+nD = 1, each χi reduces to χi = xiX1,1 = x2
+i = xi. Now,
+setting W = k, the two quadratic penalty terms can be
+collected into one and the “selected" or “not selected"
+form of Equation 3 is obtained.
+
diff --git a/fNAzT4oBgHgl3EQf3v7_/content/tmp_files/load_file.txt b/fNAzT4oBgHgl3EQf3v7_/content/tmp_files/load_file.txt
new file mode 100644
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--- /dev/null
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@@ -0,0 +1,567 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf,len=566
+page_content='A Quantum-Inspired Binary Optimization Algorithm for Representative Selection  Anna G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Hughes,1 Jack S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Baker,1 and Santosh Kumar Radha1, ∗  1Agnostiq Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=', 325 Front St W, Toronto, ON M5V 2Y1  (Dated: January 6, 2023)  Advancements in quantum computing are fuelling emerging applications across disciplines, includ-  ing finance, where quantum and quantum-inspired algorithms can now make market predictions,  detect fraud, and optimize portfolios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Expanding this toolbox, we propose the selector algorithm:  a method for selecting the most representative subset of data from a larger dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The selected  subset includes data points that simultaneously meet the two requirements of being maximally close  to neighboring data points and maximally far from more distant data points where the precise notion  of distance is given by any kernel or generalized similarity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The cost function encoding  the above requirements naturally presents itself as a Quadratic Unconstrained Binary Optimization  (QUBO) problem, which is well-suited for quantum optimization algorithms - including quantum  annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' While the selector algorithm has applications in multiple areas, it is particularly useful  in finance, where it can be used to build a diversified portfolio from a more extensive selection of  assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' After experimenting with synthetic datasets, we show two use cases for the selector algorithm  with real data: (1) approximately reconstructing the NASDAQ 100 index using a subset of stocks,  and (2) diversifying a portfolio of cryptocurrencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' In our analysis of use case (2), we compare the  performance of two quantum annealers provided by D-Wave Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  INTRODUCTION  The task of choosing representative samples from a  larger collection of data, often referred to as represen-  tative selection, has various advantages over studying  the dataset as a whole (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=', [1–4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Representative  selection can reduce the size and complexity of the data,  simplifying data analysis and processing and reducing  the memory cost of storing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The computational  efficiency of data modeling, such as classifier training  and model application, can be significantly improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Representative selection has been implemented in various  subjects ranging from computer vision [5] and language  processing [6] to protein analysis [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' A variety of repre-  sentative selection procedures have been offered to reduce  the volume of training data for some specific supervised  learning classifiers [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  In addition to the methods  that require additional knowledge for representative se-  lection, there has been a growing interest in unsupervised  approaches to finding representative samples [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  In this work, we implement an unsupervised represen-  tative selection algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  An unlabeled dataset may  contain some unknown number of classes, or the data in  the set could be unclustered or only very loosely clustered  into different categories according to some notion of  similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The algorithm aims to find a subset of  representative data points from a larger dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The  selected subset should include only data points dissimilar  to one another and similar to unselected neighboring  data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  More concretely, our selector algorithm  finds the k least similar points in a sample of n total  clustered data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' This is done by both maximizing  the distance between the chosen data and all other data  ∗ research@agnostiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='ai  in the dataset while also minimizing the distance between  selected data and the other similarly clustered data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The returned k points are then both representative of  the data clusters and maximally distant from the other  groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' This method can be applied across disciplines,  but in this paper, we explore an application in finance:  diversification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Diversification is a crucial strategy when building a  robust portfolio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' To mitigate the risk of interdependent  components performing poorly in a market downturn,  it is critical to invest in assets that are not strongly  correlated to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  There are many portfolio  diversification strategies, from a naive 1/N rule [11] to  more complex methods such as portfolio dimensionality  or a Bayesian approach [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Many diversification  methods are designed to weigh assets to minimize the  portfolio’s overall variance deliberately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The recent  popularity of machine learning in quantitative finance  has enabled researchers to define new diversification  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' In this paper, we use our selector algorithm  as a method for the diversification of assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  In this  framework, a portfolio is well-diversified when each asset  is maximally dissimilar to one another and representative  of their respective sectors or other similarly performing  assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The core framework of our selector algorithm  emerges as quadratic unconstrained binary optimization  (QUBO) problem, which can be tackled with a range  of metaheuristics [14–17] utilizing classical and quantum  computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Because of the large number of variables  considered in diversification (and other selective repre-  sentation problems), the resulting QUBO objective has  many binary variables which, in the quantum setting,  translates to the requirement of a large number of  qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Although gate-model quantum computers are  continually scaling up qubit numbers, presently, quantum  inspired hardware like quantum annealers already meet  this requirement (although the qubits are non-universal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='01836v1  [quant-ph]  4 Jan 2023  2  Subsequently, we regard our selector algorithm as quan-  tum inspired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  THE SELECTOR ALGORITHM  The selector algorithm is designed to pick out unique  and representative data points from a larger dataset by  finding low-cost solutions to a QUBO objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  This function is constructed such that low-cost solutions  maximize some notion of distance between selected data  points and all other data, ensuring that chosen data  points are unique while simultaneously minimizing the  distance between selected and similarly clustered data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  This ensures that each of the selected k data points from  a dataset containing n > k points represent data that are  similarly clustered with nearby points whilst each of the  k data points are not similar to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The input data are provided as a matrix Y, where each  row is a vector representative of the ith data point  Y =  �  ���������  ⃗y1  ⃗y2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  ⃗yi  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  ⃗yn  �  ���������  =  �  ����������  y(1)  1  y(2)  1  y(3)  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  y(1)  2  y(2)  2  y(3)  3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  y(1)  i  y(2)  i  y(3)  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  y(1)  n  y(2)  n  y(3)  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  �  ����������  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  (1)  where n is the size of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Although any  metric can be used to evaluate the distance between each  data point, throughout this paper, the choice of distance  metric is the Euclidean distance unless stated otherwise,  dij(⃗yi, ⃗yj) =  ��  m  �  ⃗y(m)  i  − ⃗y(m)  j  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  (2)  Alternative distance metrics include standardized Eu-  clidean, cosine, Minkowski, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The distance d is for-  mally a kernel function, which in principle could also be  computed and/or learned using quantum/classical neural  networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=', [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' It is therefore possible to create a  cross-paradigm variation of the selector algorithm where  the d is evaluated on a gate model device as proposed  in [18], and the QUBO objective is approximately solved  using a quantum annealer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' This however is beyond the  scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The QUBO objective is defined as  C(⃗x) = 1  2k⃗xd⃗xT − 1  n⃗xd⃗1T + A  � n  �  i=1  xi − k  �2  ,  (3)  where ⃗1 is the n-length vector of ones, d is the n × n  distance matrix with elements dij as given in Equation  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' An example of the selector algorithm’s performance  on clustered data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The clusters are generated by  randomly choosing data points from a Gaussian distribution  on a two-dimensional plane centered on two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Each  blob contains 90 data points, with a standard deviation of  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The selector algorithm was tasked with choosing two  representative points from the complete dataset using the  qbsolv solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The algorithm-selected points are highlighted  with blue squares, indicating that the algorithm successfully  chose visually representative points from each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  2(or another user-defined distance measure), ⃗x is the  {0, 1}n vector of binary variables and A is a penalty  scaling factor used to enforce the equality constraint  � xi = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The first term on the RHS of Equation  3 represents the distance between the selected points  and other points in the cluster, ensuring that selected  points are representative of their cluster, the second term  represents the distance between selected points and all  others in the data set, and the last term enforces the  equality constraint (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=', a penalty is applied to the cost  function if more than k data points are chosen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' This  optimization problem takes on a QUBO form which are,  in general, NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  A more detailed description of  QUBO models is given in Appendix A and a formal  extension to weighted selection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='e, where ⃗x can take  on a number discrete values not limited to ⃗x ∈ {0, 1}n)  is given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Now in this form, the problem  becomes approachable with approximate metaheuristic  algorithms, including quantum annealing, used in Section  IV B, which we discuss qualitatively in Appendix B  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  EXPERIMENTS WITH SYNTHETIC DATA  To demonstrate the basic functionality of the selector  algorithm, we use it to select points from two synthetic  10  8  6  2  _12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='5  10  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='5  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='5  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='5  V13  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  One dataset contains simple and obviously  clustered data points (in terms of the Euclidean dis-  tance), while the second dataset contains time series  data - data points organized in a chronologically ordered  sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' We show that in both cases, the selector al-  gorithm makes reasonable, representative choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' In the  first application, we use the selector algorithm to choose 2  data points from a clustered array of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' We generate  two clusters, or blobs, by randomly choosing data points  from a Gaussian distribution on a two-dimensional plane  centered on two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Each blob contains 90 data  points, with a standard deviation of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The coordinates  of all data points are used as input for the selector  algorithm, which then chooses k representative points  by minimizing the cost function in Equation 3 using the  D-Wave decomposing solver qbsolv [19, 20] which uses  a modified tabu algorithm [21] to minimize the objective  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' In this example, we have generated two clusters  of data points and tasked the selector algorithm with  choosing k = 2 representative points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The blobs and  selected points are shown in Figure 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' as expected, the  selector algorithm successfully chose one point from each  cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Since the choice of distance is euclidean, the  chosen point closely resembles the approximate center  of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  This visual representation might not  always be the case for all metrics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' for example, metrics  like Jensen–Shannon divergence might have no visually  discernible centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The selector algorithm can similarly choose data points  of arbitrarily high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' We demonstrate this in  two examples: choosing points from an array of trigono-  metric curves and choosing from an array of stochastic  differential equation (SDE) time series generated ran-  domly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' In each case, the data are generated synthetically  but clustered into two different classes: sine or cosine in  the trigonometric case and time series generated from  coupled Brownian process in the SDE case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  In both  cases, the selector algorithm is tasked with choosing two  representative data points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' in the trigonometric case, a  solution is considered successful if the selector algorithm  chooses one sine and one cosine curve, and in the SDE  case, a solution is successful if the selector algorithm  chooses representatives from the distinct processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The trigonometric time series are generated using,  ysine  i  = sin(ti) + ni  ycosine  i  = cos(ti) + ni,  (4)  where ti ranges from 0 to 2π and ni is artificial random  noise pulled from a Gaussian distribution with mean  µ = 0 and standard deviation σ ranging from σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='1 to  σ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' As σ increases, there is little difference between  the sine and cosine curves because the introduced noise  dominates the amplitude of the curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' We generate 50  sine and 50 cosine time series with 10 different σ values  and task the selector algorithm with choosing k = 2  representative data points in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' We expect the  selector algorithm to select one sine and one cosine data  point when σ is low, but as σ is increased, it should be  harder for the selector algorithm to distinguish the two  clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The results are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  In the first two  panels (a,b), clusters of sine (green) and cosine (orange)  curves are plotted against time, where t spans from 0  to 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The curves in the top left panel have σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='3,  while the curves in the top right panel have σ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  In each plot, the 2 representative curves chosen by the  selector algorithm are plotted in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' As expected, the  σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='3 time series curves are easily distinguishable, with  the algorithm choosing one of each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' In (b), where the  standard deviation is increased to σ = 3, the sine and  cosine curves are no longer visually distinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  In (c), density plots of the elements of the correla-  tion distance matrix are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Each horizontal line  extending upward represents increasing noise standard  deviation, from σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='1 to σ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' As expected, the two  clusters of curves are clearly separable into two distinct  peaks at low σ, but start to merge as σ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' This  can measure the distinguishability (or lack thereof) of the  clusters in the dataset as a function of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  In (d), the accuracy of chosen solutions is plotted  as a function of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Accuracy here is measured as the  percentage of algorithm solutions that contained one sine  and one cosine curve out of 200 total trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' As expected,  at low σ, the selector algorithm chooses solutions with  100% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' While that accuracy decreases at higher  σ, even at σ > 3 where the two sets of density curves  (figure (c)) are indistinguishable, the selector algorithm  can pick out curves from each cluster with better-than-  random accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' This demonstrates the robustness of  the selector algorithm in choosing representative data  points even as the separation between clusters vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Finally, we test the performance of the selector algo-  rithm at choosing solutions from the solution of a two-  dimensional SDE given by  dxi(t) = µixi(t)dt +  2  �  j=1  σijxi(t)dWj(t),  (5)  where µi and σij are the drift vector and volatility matrix  respectively and Wj(t) is the standard Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  For the case of simulation, we choose µi and σ randomly  from a uniform distribution [−1, 1] and numerically  generated multiple two-dimensional time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  While the trigonometric time series showed us the  ability of the selector algorithm to choose representative  data points from loosely clustered or unclustered data,  in this example, we show the ability of the selector  algorithm to evaluate simulated financial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  As  before, the selector algorithm was given an array con-  taining all-time series and tasked with choosing (apriori  known) k = 2 representative curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The two sets of  correlated SDE time series are plotted in Figure 3, with  the selector algorithm’s choices plotted in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The  selector algorithm successfully chose one curve from each  time series cluster, demonstrating its ability to evaluate  financial time series data such as daily returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Two clusters of 50 sine and cosine curves generated from Equation 4 are used as input for the selector algorithm, with  varying values of the standard deviation of the Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' (a): 50 sine and cosine time series curves with introduced  Gaussian noise of σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='3 are plotted in green and orange, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The algorithm-selected curves are plotted in black,  showing that the algorithm chose one curve from each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' (b): 50 sine and cosine time series curves are plotted with  introduced Gaussian noise increased to σ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' As before, the algorithm-selected curves are plotted in black, but the clusters  are no longer distinguishable because the amplitude of the noise is too great.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' (c): the density of curves are shown as a function  of distance, where each horizontal line extending upward represents increasing noise standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' As expected, the two  clusters of curves are clearly separable into two distinct peaks at low σ, but start to merge as σ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' (d): the accuracy  of chosen solutions is plotted as a function of σ in the bottom right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Accuracy here is measured as the percentage of  algorithm solutions that contained one sine and one cosine curve out of 200 total trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' At low σ, the selector algorithm  chooses solutions with 100% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' While that accuracy decreases at higher σ, even at σ > 3 where the two sets of curves  are indistinguishable, the selector algorithm can pick out curves from each cluster with better-than-random accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  All  optimizations in this example were performed using the qbsolv solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Two clusters, each containing 50 synthetic financial  time series, generated as given in Equation 5, are plotted  in orange (positive) and teal (negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Each cluster is  created with a random correlation chosen from a Gaussian  distribution of µ = 0 and σ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  For each set of curves,  the volatility and return rates are randomly chosen from a  Gaussian distribution with µ = −1 and σ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The selector  algorithm was tasked with choosing 2 points from the full  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Plotted in black are the two selected curves, showing  the selector algorithm’s ability to choose representative time  series from synthetic financial data reliably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  USE CASE: BUILDING A DIVERSIFIED  PORTFOLIO  Diversifying a portfolio by investing in uncorrelated  assets is an approach to mitigate risks associated with  downturns in specific markets or unexpected crises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Because the selector algorithm is designed to pick out  data points that are both distinct and representative, it  is particularly useful for building a diverse portfolio from  a more extensive list of assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Quantum annealers are  particularly well-suited to address QUBO problems at  this scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' That is, it is possible for the large number  of variables to be mapped to the large number (∼ 1000)  of non-universal qubits available on present generation  quantum annealers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' In section IV A, we will start by us-  ing the selector algorithm as a portfolio diversifier aiming  to approximate the behaviour of the NASDAQ 100 index  using a smaller subset of assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  For this task we use  a classical large scale QUBO solver which is presently  practical to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' We then proceed to section IV B, where  we perform experiments with the algorithm using D-  Wave’s quantum annealers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' That is, we benchmark and  compare the performance of two quantum devices for the  task of diversifying a cryptocurrency portfolio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Reconstructing the NASDAQ 100 with a  classical QUBO solver  The selector algorithm can build a diverse portfolio by  selecting a representative subset of stocks from a market  index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' In this first application on real financial data, we  use the selector algorithm to approximately reconstruct  the NASDAQ 100 by choosing a subset (Sk) of k stocks  from all 102 stocks in the market index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' We perform the  optimization with the D-Wave decomposing solver qbsolv  [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  To choose stocks from the NASDAQ 100 index, we  treat the daily returns from each stock as individual data  points in our array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Vectors ⃗yi are composed of the daily  g=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='3  q=3  2  (a)  (b)  (c)  (d)  10  100 -  g=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='0  0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='9  Accuracy (%)  90  g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='8  Count  0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='7  80  0  0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='6  0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='5  70  0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='4  1  g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='3  60  g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='2  10  g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='1  Random probability  2 -  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='995  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='005  0  2  4  6  0  2  4  6  1  2  4  Time  Time  Distance  Standard deviation (o)5  Signal (yi)  0  5  10  20  40  0  60  80  100  Time5  returns from each stock in the index over one trading  year, or 253 days, starting on 2021-02-01 (YYYY-MM-  DD) and ending on 2022-02-01 (YYYY-MM-DD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' This  results in a 102 × 253 matrix Y,  Y =  �  �������  −→  ⃗y1  −→  ⃗y2  −→  ⃗y3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='−→  ⃗yn  �  �������  =  �  �������  −−−−→  AAPL  −−−−−→  ABNB  −−−−→  ADBE  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  −→  ZS  �  �������  (6)  In this familiar form, the data are ready for evaluation  in the cost function (Equation 3), where n = 102 is the  total number of vectors in matrix Y, k is the number  of desired stocks selected by the algorithm, and the  elements of d are given by the correlation distance,  dab = 1 −  �(ai − ¯a) · �(bi − ¯b)  ��(ai − ¯a)2 �(bi − ¯b)2 ,  (7)  where ¯a and ¯b are the mean values of vectors ⃗a and ⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The chosen stocks can come arbitrarily close to a com-  plete reproduction of the market index;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' as the number  of selected stocks k increases, the combined data from  the chosen stocks comes closer and closer to an exact  reproduction of the index itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  However, the goal is  only approximately to reproduce the index with a smaller  number of stocks, so choosing a large number is not  necessarily advantageous and is a hyperparameter for the  end-user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  In all following analyses, we use only the binary case,  where the weighted subspace is restricted to only binary  values, ⃗w → ⃗x ∈ {0, 1}n (see discussion in Appendix  C for the general discrete weighted case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Like in the  previous examples, this means that specific data points  are either chosen or not chosen, with no inbetween.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' This  is an NP-hard problem [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' To evaluate the performance  of the binary selector algorithm in reproducing the  NASDAQ 100, we create a proxy NASDAQ 100 index by  linearly combining all stocks and averaging the resulting  vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Note that while the NASDAQ 100 index is market  value-weighted, for simplicity, the stocks in our NASDAQ  100 index are equally weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  We initially use the selector algorithm to choose  two stocks from our input matrix Y of returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The  algorithm-chosen stocks are Fox Corporation (‘FOX’)  and Synopsys, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  (‘SNPS’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' We use these stocks  to create Sk=2 := S2, a data point composed of the  averaged linear combination of our chosen stocks’ daily  returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  A histogram showing the average percentage  daily returns for both S2 and the proxy NASDAQ 100  index is shown on the left panel of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  One  can see that the stock return profile of the selected  S2 stocks is a good surrogate for the NASDAQ 100  already at k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  To further assess the quality of  the algorithm’s selection, we calculate the value of the  cost function and the correlation distance between the  proxy NASDAQ 100 index and some data point S2 for all  possible combinations of 2 stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' A histogram showing  the range of cost function and correlation distance values  are shown in the middle and right panels of Figure 4,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The S2 index had a lower cost function value  than 74% of all cost function values and a correlation  distance less than 89% of all values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  In Figure 5, we show the average (left) and cumulative  (right) returns from the proxy NASDAQ 100 index  plotted with the Sk time series, with each row showing  incremental increases in k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  As expected, when the  number of chosen stocks (k) increases, the Sk vector  comes closer to the proxy NASDAQ 100 index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Finally, we compare the algorithm-selected indices Sk  to the NASDAQ 100 index by computing the mean  squared error (MSE) between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The number of  stocks selected k increases steadily until all 102 stocks  in the index are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The MSE as a function of k  is plotted in Figure 6, where one can see that NASDAQ  100 can effectively be reproduced by roughly 40 stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Diversifying cryptocurrency portfolios with  quantum annealers  Selecting a diverse portfolio from a limited index can  be challenging, especially when the assets cannot be  easily divisible into traditional sectors, as in the case  of cryptocurrencies (crypto).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Unexpected correlations  between coins can leave a crypto portfolio vulnerable  to the volatility of one asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The selector algorithm is  thus particularly useful for building a diverse portfolio  of cryptocurrencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  In this use case, we use the  selector algorithm to build a diverse portfolio from the  Crescent Crypto Market Index (CCMIX) using quantum  annealers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' We compare the performance of each device,  investigating how well each device satisfied the problem  constraints and the quality of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The two quantum annealing devices used in the fol-  lowing experiments are the D-Wave 2000Q and the D-  Wave Advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The D-Wave 2000Q, first introduced  in 2017, is a 2048-qubit quantum annealing device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The  2000+ qubits are arranged in a Chimera topology with  6016 couplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The 2000Q system has been used to  solve problems ranging from e-commerce listing order to  cryptography [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The D-Wave Advantage quantum  processing unit (QPU) was released in September 2020  as an updated, more advanced system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' It contains 5000+  qubits with 35,000+ couplers - an increase from 6 to 15  couplers per qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' This increase in qubits and couplers  enables the D-Wave Advantage QPU not only to solve  larger problems than the D-Wave 2000Q device but also  allows for problems with more challenging connectivity  to be more easily mappable to the device qubits when  compared to the 2000Q Device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The D-Wave Advantage  QPU has been used to solve problems such as railway  dispatching and molecular unfolding [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  6  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' (a) Histograms and corresponding kernel density estimate (KDE) plots of the daily returns from the proxy NASDAQ  100 index and the combined index from k = 2 stocks (S2) selected by the algorithm using the qbsolv solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' (b) Histogram and  corresponding KDE plot showing the value of the cost function for all combinations of 2 stocks in the NASDAQ 100 index,  where the dashed vertical line shows the value of the cost function for the algorithm-selected k = 2 stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' (c) Histogram and  corresponding KDE plot showing the correlation distance for all combinations of 2 stocks in the NASDAQ 100 index, where  the dashed vertical line shows the distance for the algorithm-selected k = 2 stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The input data to the selector algorithm consists of the  daily returns from each coin in the CCMIX over seven  months, beginning on 2021-04-01 (YYYY-MM-DD) and  ending on 2021-11-11 (YYYY-MM-DD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Depending on  the experiment, the selector algorithm chooses some  desired number of coins in the array, using the specified  quantum annealer to minimize the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Constraint Satisfaction with Default Parameters  In the first round of experiments, we investigate the  ability of each device to choose solutions that satisfy  the equality constraint imposed by the penalty term in  Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  While satisfying this constraint may be  trivial for some classical methods, quantum annealers  are more limited, especially when the constraints are  imposed as penalties [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' For experiments in this section,  we use the default parameters of the annealers, notably  leading to an annealing time of t = 20 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' We also use  a penalty scaling factor of A = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  We run 300 trials  on each device, where the selector algorithm was tasked  with choosing k = 3 cryptocurrencies from the 18 coins  included in the CCMIX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' In all experiments, the number  of shots per run is fixed at 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The results are shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' There is a significant  difference in how well each device satisfies the k = 3  constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The D-Wave 2000Q device only chose three  cryptocurrencies in 49 of the 300 trials – a success rate  of just 16%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Conversely, the D-Wave Advantage QPU  chose three coins in 256 of the 300 runs – a success rate  of over 85%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' There is clearly a marked improvement in  performance for the newer D-Wave Advantage QPU even  using the default parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Consistency of Solutions with Changing Annealing Time  In subsequent experiments, we investigated the impact  of the value of the penalty scaling factor A (see Equation  3) and the annealing time on constraint satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' We  first investigated tuning the penalty scaling factor in 11  runs with 50 repeats for each value of A, where the  value ranged from A = 0 to A = 10 in fixed increments  while the annealing time was fixed at 20 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  In these  experiments, we saw very little impact on either the  quality of the solution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='e, how low the cost is) or  satisfaction of the problem constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  For the rest of  this section, were therefore continue with the default of  A = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Finally, we performed 19 trials on each device  using annealing times ranging from the default 20 µs to  990 µs, with 50 repeats at each annealing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  As can be seen in Figure 8, the value of the annealing  time has a negligible effect on the percentage of solutions  matching the k  =  3 constraint for the Advantage  QPU, which tends towards solutions that satisfy this  constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Conversely, higher annealing times enable the  2000Q QPU to find more solutions that satisfy the soft  constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Even at high annealing times, most of the  solutions chosen by the 2000Q device do not satisfy the  problem constraint, while the Advantage QPU satisfies  the constraint > 85% of the time at every annealing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The possible impact of the annealing time on the  overall value of the cost function (k = 3) is more difficult  to discern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' At the intermediate annealing time of 600 µs  and at the higher annealing time of 900 µs, both the value  of the cost function and the standard deviation between  trials hits a low value for the 2000Q QPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Very little  such variation is discernible for the Advantage device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Daily Returms Histogram & KDE  Cost Function Histogram & KDE  Correlation Distance Histogram & KDE  (a)  100  (b)  (c)  40  NASDAQ100  All k=2 Combinations  SelectedStocks(S2)  Selected Stocks (S2)  Selected Stocks (S2)  250  All k=2 Combinations  35  80  30  200  Density  60  40  15  100  10  20  50  +-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='050-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='050  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='95  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='05  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='15  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='025  Daily Returms  Cost  Distance7  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' (a)-(e): All daily returns from the proxy NASDAQ 100 index (violet) and the index generated from the selector  algorithm’s choice of stocks only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The proxy NASDAQ 100 index is created by equally weighting all stocks in the NASDAQ 100  over 2021-02-01 to 2022-02-01 (YYYY-MM-DD), while the selector algorithm index is created by equally weighting only the  stocks chosen by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The number of selected stocks k varies, starting from k = 2 in the first row and increasing to  k = 20 in the final row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' As k increases, the selector-created index comes closer to the proxy NASDAQ 100 index, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  (f)-(j): The cumulative returns from both the proxy NASDAQ 100 index and the selector algorithm-generated index, where  each row represents an increasing number of selected stocks k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' As the number of selected stocks is increased, the selector  algorithm-generated index more closely replicates the features of the proxy NASDAQ 100 index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Comparing solution quality  We examine the quality of solutions selected by each  device given the constraint that k = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' To understand  how the QPU-chosen solutions compare with all possible  solutions, we first compute the value of the cost function  for every combination of coins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The CCMIX index  contains 18 cryptocurrencies, leading to 218 = 262144  total possible combinations, ranging from choosing 0  coins to the possibility that all 18 coins are chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  (a)  (f)  NASDAQ 100  NASDAQ 100  Selected Stocks (S2)  Selected Stocks (S2)  (b)  (g)  NASDAQ 100  NASDAQ 100  Selected Stocks (S3)  Selected Stocks (S3)  (c)  (h)  NASDAQ 100  NASDAQ 100  Selected Stocks (S5)  Selected Stocks (S5)  (d)  (1)  NASDAQ 100  NASDAQ 100  Selected Stocks (Sio)  Selected Stocks (Sio)  (e)  ()  NASDAQ 100  NASDAQ 100  Selected Stocks (S2o)  Selected Stocks (S20)8  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The mean-squared error (MSE) between the proxy  NASDAQ 100 index and the selector algorithm-generated  index as a function of the number of selected stocks k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The  proxy NASDAQ 100 index is created by equally weighting  all stocks in the NASDAQ 100 over 2021-02-01 to 2022-02-01  (YYYY-MM-DD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' In contrast, the selector algorithm index  is created by equally weighting only the stocks chosen by the  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' As k increases from 1 representative stock to all  102 stocks in the index, the MSE sharply decreases to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The selector algorithm was tasked with choosing  three cryptocurrencies from the CCMIX, which contains 18  total options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Optimization was performed using the D-Wave  2000Q device (left) and the D-Wave Advantage device (right),  with 300 trials on each QPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The solutions chosen by the D-  Wave 2000Q device satisfied the problem constraint of k = 3  in only 16% of its trials, while the D-Wave Advantage device  chose solutions that satisfied the problem constraint in > 85%  of its trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The cost function values are calculated assuming the  k = 3 constraint, so all combinations containing more or  less than three coins are penalized in the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  A = 2 in these experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The mean value of the cost  function, considering all combinations, is 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  To gauge the performance of quantum devices relative  to a randomized solution, we calculate the average value  of the cost function for solutions chosen by each device  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' We investigate the ability of each device to choose  solutions matching problem constraints as a function of  annealing time, where the annealing time is increased from  20µs to 990µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The percentage of solutions matching problem  constraints (out of 50 trials at each annealing time) is plotted  for the D-Wave 2000Q (beige) and the D-Wave Advantage  (violet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The change in annealing time has a negligible effect  on the percentage of constraint-satisfying solutions for the  Advantage QPU, while higher annealing times seem to enable  the 2000Q QPU to satisfy the problem constraint better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  However, even at high annealing times, most solutions chosen  by the 2000Q device do not satisfy the problem constraint,  while > 85% of solutions chosen by the Advantage QPU  satisfy the constraint at every annealing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  for all trials at each device’s optimal annealing time (550  µs for 2000Q and 900µs for Advantage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' For the 2000Q  device, the average value of the cost function was 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='02,  while the average cost function value for the Advantage  QPU was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The results are shown in Figure 9,  where a histogram showing the density of cost function  values is plotted, with vertical lines indicating the total  average cost value and the cost values of the 2000Q and  Advantage QPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The average cost function value of both devices was  significantly lower than the average value considering all  possible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The average value of the cost function  for the 2000Q device is in the lowest 4% of possible  values, while the average value of the cost function for the  Advantage QPU is in the lowest 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='03% of all solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Overall, both devices distinctly demonstrate the ability  to choose solutions with low values of the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  However, comparing devices, the Advantage QPU shows  a marked improvement in the ability to choose solutions  that match the problem constraints and solutions with  lower cost function values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='00012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='00010  Error  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='00008  Squared  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='00006  Mean  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='00004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='00002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='00000  0  20  40  60  80  100  k2000Q  Advantage  250  200  Count  150  100  50  0  3  4  6  73  5  4  5  6  7  Number Selected  Number Selected100  Constrained Solutions  90  80  70  60  50  Percentage (  40  30  Advantage  20  2000Q  0  200  400  600  800  1000  Annealing Time (μs)9  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The value of the cost function for all combinations  of coins is plotted as a histogram, with the black curve fitted  to the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The average value of the cost function,  considering all possible combinations, is indicated by the  vertical dotted red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The average cost function value of  solutions chosen by the D-Wave 2000Q and Advantage QPUs  are indicated by the blue and orange horizontal dotted lines,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  CONCLUSION  In this work, we devised a selector algorithm for the  representative selection of data which relies on solving a  QUBO problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Our selector algorithm chooses data  points by maximizing the distance between selected  points and all other data points in the dataset and  minimizing the distance between selected points and sim-  ilarly clustered data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Given the potentially large  number of variables in the resulting QUBO problem, the  optimization stage of selector algorithm is particularly  suited towards large scale QUBO solvers like quantum  annealers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  After experimentation with synthetic datasets, in our  first practical use case, we used the selector algorithm to  approximate the NASDAQ 100 with a subset of stocks  in the index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' We created a proxy NASDAQ 100 index  by linearly combining and averaging all 102 stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The  selector algorithm chose k stocks from the index, whose  performances were combined, averaged, and compared to  the proxy 100 indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' As the number of selected stocks k  increased, the algorithm-constructed index more closely  resembled the proxy NASDAQ 100 index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' We compared  the k = 2 point, S2, to all possible combinations of 2  stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' We found that the algorithm-selected point S2  had a cost function value in the bottom 26% of all k = 2  solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  In our second application, we used the selector algo-  rithm to build a diversified portfolio of cryptocurrency  assets from the Crescent Crypto Market Index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The  optimization was performed using 2 quantum annealers:  the D-Wave 2000Q and the D-Wave Advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  We  compared the k  =  3 selection of both devices by  evaluating the cost function and comparing that to the  value of the cost function of all possible coin combinations  in the index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' We found that the average cost function  value of solutions chosen by the 2000Q device was in  the lowest 4% of all cost function values, while the D-  Wave Advantage device’s average cost function value is  in the lowest 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='03%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The D-Wave Advantage device also  more consistently chose solutions matching the problem  constraint - with > 85% of solutions matching the  constraint, while only 16% of 2000Q solutions matched  the constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Overall, we saw clear improvement  between the newer Advantage QPU and the earlier 2000Q  QPU, providing meaningful solutions to the combinato-  rial optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
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+page_content=' Zaman, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Tanahashi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Tanaka, Pyqubo:  Python library for mapping combinatorial optimization  problems to qubo form (2021), arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='01708 [quant-  ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Appendix A: Quadratic Unconstrained Binary  Optimization  Quadratic  Unconstrained  Binary  Optimization  (QUBO) is the optimization of an objective function  with binary variables with at-most quadratic order  coupling between variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' This approach is powerful  and widespread because many combinatorial problems  can be written in QUBO form (see Equation A1) and  solved using general purpose QUBO solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  This is  in contrast to other more traditional methods which  are tailored to the exact problem at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' In general,  to approximately solve a QUBO problem (the exact  problem is NP-hard), we must select an approximately  optimal solution from all possible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The basic cost or objective function C(⃗x) of QUBO is,  C(⃗x) = ⃗xT Q⃗x,  (A1)  where ⃗x = {0, 1}n, and Q is an n×n matrix of constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The goal of QUBO is to find the arrangement of binary  values in ⃗x that minimizes (or maximizes) the value of  C(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Other than the requirement that the values in the  vector ⃗x be binary, there are no formal constraints in this  simple QUBO model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' However, so-called soft constraints  can be encoded into the QUBO problem using quadratic  penalty terms (for example, see the last term of the RHS  of Equation 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Rather than impose rigid constraints on  the problem, soft constraints work by discouraging the  optimizer from solutions that will incur penalties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  QUBO problems have garnered recent attention due  to how easily they are adapted mappable to quantum  annealers and gate-model quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Appendix B: Quantum Annealing  In metallurgy, annealing strategically applies heat to  a material to make it more malleable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Annealing works  by initially raising the temperature of a metal above its  recrystallization temperature, which causes the atoms in  the metal to move around rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The temperature  is then slowly lowered, and atoms settle into a new  crystalline configuration with the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Simulated annealing is a machine-learning technique that  takes its name from this metallurgic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Similar to  how metallurgic annealing causes atoms to explore new  positions rapidly, simulated annealing rapidly samples  the solution space of a particular cost function and  adopts new configurations as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The sampling rate  slowly decreases, allowing the annealer to settle into an  optimized solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  In simulated annealing, the solution starts in some ini-  tial configuration C and a maximum initial ‘temperature’  of Tmax, where the temperature is a hyperparameter  that dictates how randomly the annealer samples the  cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The state C is then advanced into a  new configuration N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' The change in the value of the  cost function can then be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Depending on the  current temperature and the change in the value of the  cost function, the new configuration N is either accepted  as the new state or rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' A ‘good’ new configuration  N would have a lower cost function value than the  previous configuration, indicating that the state has  advanced closer to the minimum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' However, even  ‘bad’ configurations, where the change in cost function  value is positive, can be accepted if the temperature is  very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Just like with real metallurgic annealing, the  temperature is then gradually decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  This means  that initially, fewer new configurations are accepted  unless the cost function value is significantly lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' This  method enables the simulated annealer to sample the cost  landscape very broadly and ideally settle on the global  minimum rather than any local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Quantum annealing uses a similar technique but  explores quantum states for solution space using the  strength of quantum fluctuations as a sampling rate  hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  A quantum annealing cost function  is a Hamiltonian, expressing the total energy of the  system for each quantum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Minimizing the cost  function means finding the quantum state with the lowest  energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Quantum annealing has the additional benefit  of quantum tunneling, where the algorithm can tunnel  between different low-energy quantum states, avoiding  the need to sample the space in between these states and  preventing the algorithm from getting trapped in local  minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  As the sampling rate slowly decreases, so is  the quantum tunneling radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Initially, the algorithm is  11  given a wide radius to explore many low-energy quantum  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  As the system evolves and the sampling rate  decreases, the tunneling radius slowly decreases as the  algorithm settles on a final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Quantum annealing  provides an approximate solution, not necessarily an  absolute one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Appendix C: The selector algorithm with weights  In the main article, we explore the case where k out  of n data points are either “selected" or “not selected".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  This problem is innately QUBO since the weighted  contribution wi to the objective function is restricted to  wi = xi ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' We now show how this formalism can  be extended to a more general case where k total points  are selected, and each selected point is assigned a weight  with resolution controlled using the integer discretization  number nD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The weight of the ith data point can be expanded with  nD binary variables,  wi = wmin + (wmax − wmin)  nD  �  j=1  2j−1  2nD − 1Xij,  (C1)  for minimum/maximum weights wmin/max, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' In the context of the selector algorithm, a natural  choice is wmin = 0 and wmax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Xij ∈ {0, 1} are the  elements of the rectangular n × nD bit matrix  X =  �  ��  X1,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' X1,nD  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  Xn,1  Xn,nD  �  �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  (C2)  Increasing the number of terms used in the expansion  of Equation C1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='e, increasing nD) increases the resolu-  tion of wi which, in the quantum setting, comes at the  price of increasing the qubit requirement N as given by  N = n × nD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' At nD = 1, the original “selected" or “not  selected" case is obtained wi = X1,1 := xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  We now define a new variable χi as the product of  the weight and the binary variable xi: χi = wixi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Now,  within Equation 3, for the first two terms on the RHS,  we make the change of variable ⃗x → ⃗χ and insert a new  penalty term on the sum of the elements of ⃗χ to yield the  objective  C(⃗χ, ⃗x) = 1  2k ⃗χd⃗χT − 1  n ⃗χd⃗1T + A  � n  �  i=1  xi − k  �2  + B  � n  �  i=1  χi − W  �2  (C3)  where W is the budget constraint on wi: �  i wi = W  and B (like A) is a penalty scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' As presented,  Equation C3 is quartic order in binary variables but, fol-  lowing the discussion in [28], can always be transformed  to quadratic and thus reducible to a QUBO problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  While such a transformation can be formally derived,  we use the package PyQUBO [29] to compile the problem  to a QUBO form intelligently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  The compiled problem  can then be solved using metaheuristic optimization  algorithms, including quantum annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content='  It should also be noted that Equation C3 is reducible  to Equation 3 in the main text through some specific  choices of the variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Setting wmin = 0, wmax = 1 and  nD = 1, each χi reduces to χi = xiX1,1 = x2  i = xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
+page_content=' Now,  setting W = k, the two quadratic penalty terms can be  collected into one and the “selected" or “not selected"  form of Equation 3 is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQf3v7_/content/2301.01836v1.pdf'}
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+Simultaneous Fermion and Exciton Condensations from a Model Hamiltonian
+LeeAnn M. Sager and David A. Mazziotti∗
+Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, IL 60637
+(Dated: Submitted October 26, 2021)
+Fermion-exciton condensation in which both fermion-pair (i.e. superconductivity) and exciton
+condensations occur simultaneously in a single coherent quantum state has recently been conjec-
+tured to exist. Here, we capture the fermion-exciton condensation through a model Hamiltonian
+that can recreate the physics of this new class of highly-correlated condensation phenomena. We
+demonstrate that the Hamiltonian generates the large-eigenvalue signatures of fermion-pair and ex-
+citon condensations for a series of states with increasing particle numbers. The results confirm that
+the dual-condensate wave function arises from the entanglement of fermion-pair and exciton wave
+functions, which we previously predicted in the thermodynamic limit. This model Hamiltonian—
+generalizing well-known model Hamiltonians for either superconductivity or exciton condensation—
+can explore a wide variety of condensation behavior. It provides significant insights into the required
+forces for generating a fermion-exciton condensate, which will likely be invaluable for realizing such
+condensations in realistic materials with applications from superconductors to excitonic materials.
+PACS numbers: 31.10.+z
+I.
+INTRODUCTION
+Model Hamiltonians are theoretical tools that are of-
+ten useful in simulating the key physics associated with
+large-scale, highly-correlated systems. They are capable
+of modeling an array of quantum phases and many-body
+phenomena such as phase transitions [1–5], superconduc-
+tivity [6–10], quantum magnetism [11–14], exciton con-
+densation [15–21], lattice-like systems [22, 23], etc. Addi-
+tionally, model Hamiltonians which encompass nontriv-
+ial physics are often useful as benchmarks for theoretical
+tools such as many-body approximations [6, 24–26].
+Condensation
+phenomena—which
+are
+inherently
+highly-correlated—have a long history of being computa-
+tionally studied through the lens of model Hamiltonians
+as traditional band theory is inaccurate for such highly-
+entangled materials [6, 7, 10, 15, 27–29].
+Specifically,
+superconductors—materials in which fermion-fermion
+(Cooper/electron-electron) pairs aggregate into a single
+quantum state, resulting in the superfluidity of the
+fermion-fermion
+pairs—are
+often
+explored
+through
+use
+of
+the
+Pairing-Force
+(PF)
+Hamiltonian
+[6–9],
+which is additionally referred to as the Standard Re-
+duced Bardeen-Cooper-Schrieffer (BCS) Hamiltonian
+[10, 30, 31]. This Hamiltonian is a simple representation
+of superconductivity as it describes a system with
+bound Cooper (or Cooper-like particle-particle) pairs
+interacting in an attractive manner with the holes they
+leave behind in a Fermi sea with the high-correlation
+limit
+of
+this
+Hamiltonian
+resulting
+in
+well-known,
+number-projected BCS wave functions [7, 32]. Similarly,
+exciton condensation—in which particle-hole (exciton)
+pairs condense into a single quantum state resulting in
+the superfluidity of the composite excitons [33]—can
+be modeled according to the Lipkin-Meshkov-Glick
+∗ damazz@uchicago.edu
+(LMG) Hamiltonian, which is often simply referred to
+as the Lipkin model [15–21, 29, 34].
+This Hamilto-
+nian is a highly-degenerate system in which partnered
+orbitals are inherently particle-hole paired and whose
+strongly-correlated form results in ground states that
+demonstrate character of exciton condensation.
+Here, we introduce a model Hamiltonian that is ca-
+pable of capturing fermion-exciton condensation, a new
+class of highly-correlated condensation phenomena in
+which both fermion-pair and exciton condensations coex-
+ist in a single quantum state. We demonstrate such coex-
+istent condensate character by calculating the quantum
+signatures of fermion-pair [35, 36] and exciton [37, 38]
+condensations (see Sec.
+II and Appendix A) for sys-
+tems of even particle numbers ranging from N = 4
+to N
+= 10 particles in r
+= 2N orbitals.
+These
+fermion-exciton condensates are shown to be described
+by wavefunctions which are entanglements of wavefunc-
+tions from BCS-like superconductivity and Lipkin-like
+exciton condensation—consistent with our prior predic-
+tions for the large-N thermodynamic limit [39] as well as
+those we observed experimentally on a quantum device
+[40].
+Our determination of a model Hamiltonian that sup-
+ports fermion-exciton condensation provides information
+regarding the nature of the forces necessary to generate
+such systems—an invaluable first step in the realization
+of real-world systems that support such dual condensa-
+tion of excitons and fermion-fermion pairs, which may
+demonstrate some sort of hybrid of the properties of su-
+perconductors and exciton condensates and hence have
+applications in energy transport and electronics.
+The
+extent of these different phases and the transitions be-
+tween these phases can also be studied. Moreover, our
+Hamiltonian provides an important reference in order to
+determine whether a given many-body approximation is
+capable of measuring dual condensate character.
+arXiv:2301.00670v1  [quant-ph]  29 Dec 2022
+
+2
+FIG. 1: A figure of the condensate phase diagram in
+the phase space of the signatures of particle-particle
+condensation, λD, and exciton condensation, λG, is
+shown.
+II.
+THEORY
+A.
+Fermion-Pair Condensation
+Superconductivity results from the condensation of
+bosonic fermion-fermion pairs [10, 41–43] into a single
+geminal—a two-fermion function directly analogous to
+the one-fermion orbital [35, 36, 44–47]—at temperatures
+below a certain critical temperature. This condensation
+of fermion-pairs results in the superfluidity (i.e., fric-
+tionless flow) of the constituent particle-particle pairs
+[10, 43, 48, 49]; if the fermionic pairs are composed
+of electrons (i.e., Cooper pairs), then these superfluid
+electron-electron pairs demonstrate superconductivity.
+As was first demonstrated by Yang [35] and Sasaki
+[36], a computational signature of such superconducting
+states is a large eigenvalue in the particle-particle reduced
+density matrix (2-RDM), whose elements are given by
+2Di,j
+k,l = ⟨Ψ|ˆa†
+iˆa†
+jˆalˆak|Ψ⟩
+(1)
+where |Ψ⟩ is an N-fermion wavefunction and where ˆa†
+i
+and ˆai are fermionic creation and annihilation opera-
+tors for orbital i, respectively. As eigenvalues of the 2-
+RDM can be interpreted as the occupations of the two-
+fermion geminals [50], when the largest eigenvalue of the
+2-RDM—the signature of particle-particle condensation,
+represented by λD—exceeds the Pauli-like limit of one
+(λD > 1), multiple fermion-fermion pairs occupy a sin-
+gle geminal and hence superconducting character is ob-
+served.
+This signature is known to directly probe the
+presence and extent of non-classical (off-diagonal) long-
+range order [45]. (See the Appendix for more details on
+how the signature of superconductivity, λD, was com-
+puted.)
+The
+Pairing-Force
+(PF)
+model
+[6–9]—also
+called
+the Standard Reduced Bardeen-Cooper-Schrieffer (BCS)
+model [10, 30, 31]—is known to exhibit superconduct-
+ing character in the strong correlation limit and hence
+achieve a large λD. The Hamiltonian for the PF model
+is given in second quantization by
+HP F = 1
+2
+�
+σ=↑,↓
+N
+�
+p=1
+ϵpˆa†
+p,σˆap,σ − G
+N
+�
+p=1
+N
+�
+q=1
+ˆa†
+p,↑ˆa†
+p,↓ˆaq,↓ˆaq,↑
+(2)
+where p is a quantum number that represents a pair
+of orbitals denoted as p, ↑ and p, ↓ with the same en-
+ergy, where the energies (ϵp) are considered to be known,
+and where the parameter G is a constant that tunes the
+strength of the pairwise interactions. Note that in the
+limit of strong correlation (G >> ϵp), maximal super-
+conducting character—λD =
+N
+2
+�
+1 − N−2
+r
+�
+[44, 50]—is
+observed.
+B.
+Exciton Condensation
+Directly analogous to superconductivity resulting from
+bosonic particle-particle pairs condensing into a single
+particle-particle function, exciton condensation results
+from the condensation of particle-hole pairs (i.e., exci-
+tons) into a single particle-hole function below a certain
+critical temperature, which results in the superfluidity of
+the excitons [33, 51]. Exciton condensates, while diffi-
+cult to realize experimentally, have been observed in sys-
+tems composed of polaritons (excitons coupled to pho-
+tons) [52–54] and in two-dimensional structures such as
+semiconductors [55], graphene bilayers [56–58], and van
+der Waals heterostructures [59–62].
+The signature of exciton condensation—denoted as
+λG—is similarly analogous to that for fermion-pair con-
+densation; the presence and extent of exciton condensate
+character can be measured from the largest eigenvalue of
+a modified particle-hole reduced density matrix given by
+[37, 38, 63]
+2 ˜Gi,j
+k,l = 2Gi,j
+k,l − 1Di
+k
+1Dj
+l
+= ⟨Ψ|ˆa†
+iˆajˆa†
+l ˆak|Ψ⟩ − ⟨Ψ|ˆa†
+iˆak|Ψ⟩⟨Ψ|ˆa†
+jˆal|Ψ⟩
+(3)
+where 1D is the one-fermion reduced density matrix (1-
+RDM). Note that this modification removes the extrane-
+ous large eigenvalue from a ground-state-to-ground-state
+transition such that a signature above one (λG > 1) is in-
+dicative of exciton condensation. (See the Appendix for
+more details on how the signature of exciton condensa-
+tion, λG was computed.) This computational signature
+has been utilized to study exciton condensation is possi-
+ble in quantum and molecular systems [29, 38–40, 64].
+One model known to achieve a large λG value and
+hence exhibit exciton condensate character in the limit of
+a large correlation is the Lipkin quasispin model [15–21].
+The N-fermion Lipkin quasispin model consists of two
+energy levels
+�
+− ϵ
+2, ϵ
+2
+�
+, each containing N energetically-
+degenerate states.
+The second-quantized Hamiltonian
+
+Superconducting
+Coexistence
+Condensate
+1
+入D
+Exciton
+No Condensation
+Condensation
+Phenomenon
+1 入G3
+can be expressed as [18]
+HL = ϵ
+2
+�
+σ=±1
+σ
+N
+�
+p=1
+ˆa†
+σ,pˆaσ,p
++γ
+2
+�
+σ=±1
+N
+�
+p,q=1
+ˆa†
++σ,pˆa−σ,pˆa†
+−σ,qˆa+σ,q
++λ
+2
+�
+σ=±1
+N
+�
+p,q=1
+ˆa†
++σ,pˆa†
++σ,qˆa−σ,qˆa−σ,p
+(4)
+where σ = ±1 and p = 1, 2, . . . , N are quantum numbers
+that completely characterize the system in which p de-
+scribes the site number labelling the N states in a given
+level and σ represents the upper (+1) or lower (−1) en-
+ergy levels, respectively.
+Note that in this model, the
+λ term allows for double excitations and de-excitations,
+and the γ term allows for a single particle to be scattered
+up while another is simultaneously scattered down; as a
+result, in the Lipkin model, only even excitations are al-
+lowed, and only one particle may occupy a given site (i.e.,
+have a specific quantum number p) such that each site
+in the lower level is particle-hole paired with the corre-
+sponding site in the upper level. By having the terms cor-
+relating orbitals in the Hamiltonian (λ, γ) be sufficiently
+larger than the energy term (i.e., in the limit of high cor-
+relation), maximal exciton condensation—λG = N
+2 [37]—
+can be obtained for λ = γ.
+C.
+Fermion-Exciton Condensation
+A fermion-exciton condensate is a single quantum state
+that simultaneously demonstrates character of supercon-
+ductivity and exciton condensation, i.e., both signatures
+of condensation—the largest eigenvalue of the particle-
+particle RDM (Eq. (1)) and the largest eigenvalue of the
+modified particle-hole RDM (Eq.
+(3))—are simultane-
+ously large (λD, λG > 1). [39].
+FIG. 2: A pictorial representation of the model
+Hamiltonian we introduce in which there are two N-
+degenerate energy levels—with energies − ϵ
+2 and ϵ
+2—
+with double excitations and de-excitations, scattering
+in which one particle is de-excited while another is si-
+multaneously excited, and a pair-wise interaction term
+between sites 2j − 1 and 2j for j ∈ {1, 2, . . . , N} (yellow
+circles) is shown. Note that the Lipkin-like excitations
+must occur within a site (p ↔ p + N, blue arrow).
+To gain insight into such fermion-exciton condensates,
+here we propose a model system that is capable of demon-
+strating simultaneous fermion-pair and exciton conden-
+sate character. In this model, we introduce the pairwise
+interaction from the Pairing-Force model into the scaf-
+folding of the Lipkin model; thus, the model keeps the
+structure of the Lipkin model in which N particles oc-
+cupy two N-degenerate energy levels (− ϵ
+2 and
+ϵ
+2) with
+allowed double excitations on two sites (λ) and simulta-
+neous scattering of a particle up on one site and down on
+another (γ)—where Lipkin-like sites are now given as or-
+bitals p and p+N; however, we additionally pair adjacent
+orbitals—orbitals 2j−1 and 2j for j ∈ {1, 2, . . . , N}—via
+the PF parameter, G. (See Fig. 2.) The Hamiltonian for
+this model is thus given by
+H = − ϵ
+2
+N
+�
+i=1
+ˆa†
+iˆai + ϵ
+2
+N
+�
+i=1
+ˆa†
+i+Nˆai+N
++λ
+2
+N
+�
+p=1
+N
+�
+q=1
+ˆa†
+pˆa†
+qˆaq+Nˆap+N + λ
+2
+N
+�
+p=1
+N
+�
+q=1
+ˆa†
+p+Nˆa†
+q+Nˆaqˆap
++γ
+2
+N
+�
+p=1
+N
+�
+q=1
+ˆa†
+p+Nˆa†
+qˆaq+Nˆap + γ
+2
+N
+�
+p=1
+N
+�
+q=1
+ˆa†
+pˆa†
+q+Nˆaqˆap+N
+−G
+N
+�
+j=1
+N
+�
+k=1
+ˆa†
+2j−1ˆa†
+2jˆa2kˆa2k−1
+(5)
+in second quantization, with a given set of parameters
+(ϵ, λ, γ, G) directly determining the extent of fermion-
+pair and exciton condensation (λD and λG, respectively)
+of the ground state corresponding to this model Hamil-
+tonian.
+While this model Hamiltonian is not the first to com-
+bine the pairwise interaction from the Pairing-Force
+model with the Lipkin model, the model Hamiltonian
+introduced by Plastino and coworkers causes direct com-
+petition between particle-particle and particle-hole cor-
+relations and hence proves incapable of demonstrating a
+fermion-exciton condensate phase (see Appendix B) [65–
+67]. Conversely, due to our introduction of the Pairing-
+Force interactions between adjacent orbitals instead of
+orbitals in the same Lipkin-like site, particle-particle and
+particle hole pairing can coexist and hence fermion-pair-
+exciton (FEC) states can be achieved as is shown in the
+results that follow.
+III.
+RESULTS
+A.
+N = 4, The Minimal FEC
+As the authors have previously demonstrated [39], a
+system with as few as N = 4 particles in r = 8 or-
+bitals can support formation of a fermion-exciton con-
+densate. As such, we first fully explore such a minimal-
+istic FEC system. The ground state of the FEC Hamil-
+tonian that we have introduced—Equation (5)—for four
+
+2
+N+1
+N+2
+2N-1
+2N
+2
+2
+N-1
+N
+14
+particles has contributions from only ten of the seventy
+(rCN) possible configurations. Of these ten basis states,
+there are only five distinct classes composed of degener-
+ate orientations—see Fig. 3—that allow for the direct
+computation of a matrix-form of the Hamiltonian in a
+minimal basis state. The five basis states are defined by
+three quantum numbers, x, y, bool, where the first indi-
+cates the number of particles excited to the upper energy
+level (x), the second indicates the number of BCS-like
+pairs (number of times both 2j − 1 and 2j are occupied,
+y), and the third is a boolean that indicates whether the
+configuration is “Lipkin”-like in the regard that no two
+orbitals representing a “Lipkin” site (denoted as p and
+p + N, see the blue arrow in Fig. 3) are dually occupied
+or dually unoccupied.
+FIG. 3: Configurations representing each of the five
+classes of non-zero basis states for the FEC Hamil-
+tonian for N, r = 4, 8 are shown where each label
+x, y, bool represents the number of particles excited to
+the upper N-degenerate energy level (x), the number
+of BCS-like pairs (y), and whether the configuration is
+consistent with the Lipkin model (bool), where the de-
+generacy of each class of states is given in parenthesis,
+and where green, yellow, and blue configurations repre-
+sent that the corresponding states are consistent with
+only the Lipkin Hamiltonian, only the Pairing-Force
+Hamiltonian, or both Lipkin and PF Hamiltonians, re-
+spectively.
+Utilizing
+the
+basis
+shown
+in
+Fig.
+3—
+|0, 2, T⟩, |2, 2, F⟩, |2, 2, T⟩, |2, 0, T⟩, and |4, 2, T⟩—the
+Hamiltonian from Eq. (5) can be represented by
+H4 =
+�
+�
+�
+�
+�
+�
+�
+−2ϵ − 2G
+−G
+√
+2
+2λ−2G
+√
+2
+2λ
+0
+−G
+√
+2
+−2G + 2γ
+−2G
+0
+−G
+√
+2
+2λ−2G
+√
+2
+−2G
+−2G
+2γ
+√
+2
+2λ−2G
+√
+2
+2λ
+0
+2γ
+√
+2
+2γ
+2λ
+0
+−G
+√
+2
+2λ−2G
+√
+2
+2λ
+2ϵ − 2G
+�
+�
+�
+�
+�
+�
+�
+(6)
+where each term—corresponding to the interaction be-
+tween two classes of basis states, |i⟩ and |j⟩—is obtained
+from programmatically generating all sets of second-
+quantization creation and annihilation operators in Eq.
+(5), taking the expectation value for each combination of
+pairs of configurations in classes |i⟩ and |j⟩, summing the
+results, and normalizing by dividing by the square root
+of the number of configurations for both |i⟩ and |j⟩. For
+example, if |i⟩ = |2, 2, F⟩ = (|1, 2, 5, 6⟩ + |3, 4, 7, 8⟩) /
+√
+2
+and |j⟩ = |2, 2, T⟩ = (|1, 2, 7, 8⟩ + |3, 4, 5, 6⟩) /
+√
+2, the
+Hamiltonian term would be given by
+(⟨1, 2, 5, 6| + ⟨3, 4, 7, 8|)
+√
+2
+H4
+(|1, 2, 7, 8⟩ + |3, 4, 5, 6⟩)
+√
+2
+= 1
+2[⟨1, 2, 5, 6|H4|1, 2, 7, 8⟩ + ⟨1, 2, 5, 6|H4|3, 4, 5, 6⟩
++⟨3, 4, 7, 8|H4|1, 2, 7, 8⟩ + ⟨3, 4, 7, 8|H4|3, 4, 5, 6⟩]
+(7)
+Fig.
+4a scans over the signatures of condensation—
+λD and λG—for the ground state of the Hamiltonian in
+Eq. (6) by systematically varying the parameters ϵ, λ,
+γ, and G where the yellow BCS x’s represent ground
+states in which the PF Hamiltonian is implemented (i.e.,
+λ = γ = 0), the blue Lipkin x’s represent states in which
+the Lipkin Hamiltonian is implemented (i.e., G = 0),
+and where the green FEC x’s represent states with char-
+acter of both PF and Lipkin Hamiltonians. As this fig-
+ure demonstrates, the largest degree of superconducting
+character (the largest λD) is indeed observed in the BCS
+limit of the FEC Hamiltonian (when G >> ϵ, λ = γ ≈
+0), and the largest degree of exciton condensate character
+(the largest λG) is observed in the Lipkin limit of the FEC
+Hamiltonian (λ ≈ γ >> ϵ, G ≈ 0). However, neither the
+BCS nor Lipkin limits of the Hamiltonian is capable of
+demonstrating a dual fermion-exciton condensate as λD
+and λG only simultaneously exceed the Pauli-like limit of
+one when the full FEC Hamiltonian from Eq. (5) is im-
+plemented including both BCS-like (G) and Lipkin-like
+(λ, γ) terms.
+Our model FEC Hamiltonian, however, is capable of
+demonstrating a wide variety of dual condensate charac-
+ter as a variety of input parameters lead to ground state
+configurations in which both λG and λD simultaneously
+exceed one. Additionally, the λD and λG values obtained
+by scanning over the Hamiltonian parameters (in Fig. 4a)
+demonstrate an elliptic nature consistent with the con-
+vex nature of 2-RDMs projected onto a two-dimensional
+space [68–70] that matches predictions for a FEC that
+these authors first presented in Ref.
+39.
+This elliptic
+
+0, 2, T (1)
+2, 2, F (2)
+2, 2, T (2)
+2, 0, T (4)
+4, 2, T (1)5
+boundary as well as the density of points in the zone cor-
+responding to fermion-exciton condensate character indi-
+cate that the FEC model Hamiltonian introduced here is
+capable of spanning the entirety of the FEC region of λD
+versus λG space (i.e., λD, λG > 1).
+In Ref.
+39, these authors theoretically establish
+that in the thermodynamic limit, a possible wavefunc-
+tion demonstrating fermion-exciton condensation can be
+obtained by entangling wavefunctions that separately
+demonstrate superconducting character (|ΨD⟩ with large
+λD) and exciton condensate character (|ΨG⟩ with large
+λG) according to
+|ΨF EC⟩ =
+1
+�
+2 − |∆|
+(|ΨD⟩ − sgn(∆)|ΨG⟩) ,
+(8)
+where ∆ = 2⟨ΨD|ΨG⟩. In Fig. 5 occupation probabili-
+ties for each of the five classes of basis states consistent
+with the N, r = 4, 8 FEC Hamiltonian that contribute
+to a BCS wavefunction (yellow, ϵ, λ, γ, G = 0, 0, 0, 0.7,
+λD = 1.50, λG = 0.67), a Lipkin wavefunction (blue,
+ϵ, λ, γ, G = 0, −0.5, −0.5, 0, λD = 0.50, λG = 2.00), and a
+FEC wavefunction (green, ϵ, λ, γ, G = 0, −0.5, −0.5, 0.7,
+λD = 1.31, λG = 1.32) are given.
+From this data, it
+can be observed that the FEC wavefunction does indeed
+appear to be an entanglement of the individual BCS and
+Lipkin wavefunctions for the case of N = 4; this is con-
+sistent with the theoretical result in the thermodynamic
+limit.
+B.
+Higher-Particle FECs
+In order to observe trends related to system size, we
+employ the methodologies used to explore the N, r = 4, 8
+model system and extrapolate to systems composed of
+N = 6, 8, 10 particles in r = 12, 16, 20 orbitals. Figures
+summarizing the signatures of superconducting charac-
+ter (λD) and exciton condensate character (λG) obtained
+for the ground state wavefunctions of these larger model
+Hamiltonians can be seen in Figs.
+4b-4d.
+Similar to
+the results from the N = 4 data, elliptic fits spanning
+the maximal signature of superconducting character ob-
+served for the BCS wavefunction to the maximal signa-
+ture of exciton condensate character for the Lipkin wave-
+function with a large variety of parameters supporting
+dual fermion-exciton condensation. Note that as the size
+of the system increases from N = 6 to N = 8 to N = 10,
+the number of classes of degenerate, non-zero basis states
+as well as the number of basis states composing each class
+increase from 8 classes with a total of 44 non-zero basis
+states to 14 classes with a total of 230 non-zero basis
+states to 20 classes with a total of 1212 non-zero basis
+states.
+As such, the relative sparsity of the computa-
+tions in λD versus λG as system size is increased is due
+to fewer computations being run with larger increments
+between each of the parameters as they are varied.
+To demonstrate how the classes of non-zero basis states
+vary as system size is increased, Fig. 6—which shows the
+occupation probabilities for each of the fourteen classes
+of basis states consistent with the N, r = 8, 16 FEC
+Hamiltonian that contribute to a BCS wavefunction (yel-
+low, ϵ, λ, γ, G = 0, 0, 0, 0.9, λD = 2.50, λG = 0.57), a
+Lipkin wavefunction (blue, ϵ, λ, γ, G = 0, −0.5, −0.5, 0,
+λD = 0.50, λG = 4.00), and a FEC wavefunction (green,
+ϵ, λ, γ, G = 0, −0.5, −0.5, 0.9, λD = 2.06, λG = 1.87)—
+is included. Note that due to an increase in the possible
+complexity, two more quantum numbers are added to de-
+scribe a few of the classes of basis states; specifically, ζ
+and τ are added to x, y, bool, ζ, τ where ζ corresponds to
+the number of times BCS-like pairs are “stacked” into
+the same site such that orbitals 2j − 1, 2j, 2j − 1 + N,
+and 2j + N are all occupied and where τ corresponds
+to the number of diagonal configurations in which ei-
+ther 2j − 1/2j + N or 2j − 1 + N/2j are both occupied
+where 2j −1 and 2j are adjacent, BCS-paired orbitals. A
+few configurations with the necessary quantum numbers
+specified for N = 8 are included in Fig. 7.
+As can be seen from Fig.
+6,
+the groundstate
+wavefunction for the N
+=
+8 FEC Hamiltonian no
+longer simply contains elements of the BCS wavefunc-
+tion and the Lipkin wavefunction naively entangled to-
+gether. Specifically, while the |4, 4, F, 1, 2⟩ class of ba-
+sis states does include BCS-paired particles (see Fig.
+7), it does not include the maximal number of BCS-
+paired particles, which appears to be a necessary con-
+dition for non-zero occupation of the ground state for
+the BCS Hamiltonian.
+However, this class of basis
+states can interact with other BCS-like and Lipkin-
+like classes of basis states.
+Explicitly, |4, 4, F, 1, 2⟩ in-
+teracts with |2, 4, F⟩ via
+λ
+2 ˆa†
+pˆa†
+qˆaq+Nˆap+N; |4, 4, F, 1⟩
+via λ
+2 ˆa†
+pˆa†
+q+Nˆaqˆap+N; |6, 4, F⟩ via λ
+2 ˆa†
+p+Nˆa†
+q+Nˆaqˆap; and
+|2, 2, T⟩ via −Gˆa†
+2j−1ˆa†
+2jˆa2kˆa2k−1, which does further en-
+tangle the Lipkin-like configurations and BCS-like con-
+figurations in a non-trivial manner. As such, while the
+interaction between the BCS-like classes of basis states
+and Lipkin-like classes of basis states in the formation of
+the FEC ground state wavefunction is not as clear-cut or
+simple as in the N = 4 case, the N = 8 FEC wavefunc-
+tion is still an entanglement of BCS-like and Lipkin-like
+terms.
+A representative configuration as well as the relevant
+quantum numbers for all classes of basis states for the
+N = 6, N = 8, and N = 12 FEC Hamiltonians is given
+in the Supplemental Information.
+IV.
+DISCUSSION AND CONCLUSIONS
+In this study, we introduce a model Hamiltonian that
+successfully demonstrates the physics associated with
+both fermion-pair condensation and exciton condensa-
+tion, as well as encompassing the phase space consisting
+of systems in which fermion-pair condensation and ex-
+citon condensation are simultaneously realized—a phe-
+nomenon which we term fermion-exciton condensation
+
+6
+(a) N, r = 4, 8
+(b) N, r = 6, 12
+(c) N, r = 8, 16
+(d) N, r = 10, 20
+FIG. 4: Plots of λG versus λD where parameters in the FEC Hamiltonian are systematically varied are shown for
+systems involving (a) N = 4, (b) N = 6, (c) N = 8, and (d) N = 10 particles in r = 2N orbitals.
+(FEC). Applying this model to systems composed of
+N = 4, 6, 8, 10 particles in r = 2N orbitals, we confirm
+this fermion-exciton condensate character for a wide va-
+riety of ground state wavefunctions corresponding to a
+diverse range of input parameters in the model Hamilto-
+nian, additionally verifying the prediction made in prior
+investigation [39] that the wavefunction of a fermion-
+exciton condensate is an entanglement of wavefunctions
+of exciton condensates and fermion-pair condensates.
+The introduction of our model Hamiltonian that sup-
+ports fermion-exciton condensation advances our under-
+standing of the forces and orbital correlations necessary
+for the experimental construction of FEC states in real-
+world materials—important insights in the search for
+real-world materials exhibiting fermion-exciton conden-
+sate character. Depending on the interpretation of the
+Hamiltonian elements, this could have ramifications for
+fields such as traditional and molecularly-scaled electron-
+ics, spin systems, and nuclear physics.
+Specifically, if the orbitals in the Hamiltonian are in-
+terpreted as spin orbitals, fermion-exciton condensates
+simultaneously demonstrate the condensation of Cooper
+into a single particle-particle quantum state and the con-
+densation of electron-hole pairs into a single particle-hole
+quantum state; thus, superfluid Cooper pairs—resulting
+in superconductivity—and superfluid excitons—which
+are associated with the dissipationless flow of energy
+[33, 51]—should both be present to a certain extent in
+FEC systems, maybe demonstrating some hybridization
+of the properties of superconductors and exciton conden-
+sates, which may be relevant to the fields of energy trans-
+port and electronics in both macroscopic materials and
+molecular-scaled systems.
+Alternatively, the two Lipkin-like N-degenerate lev-
+els can be interpreted as being representative of specific
+spin states such that the upper level is spin up and the
+lower level is spin down or vice versa. This interpretation
+is most-consistent with ϵ = 0—which does demonstrate
+FEC states for a wide variety of input parameters—, al-
+though in a magnetic field the different spin states could
+be separated by some non-zero energy.
+In this frame-
+work, the Lipkin-like terms could represent simultaneous
+double spin flips that are either aligned (λ) or misaligned
+(γ), and the pairwise Pairing-Force term could be seen as
+
+FEC
+X
+1.4
+BCS
+X
+Lipkin
+X
+1.2
+0.8
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+22
+FEC
+X
+1.8
+BCS
+X
+Lipkin
+X
+1.6
+X
+X
+1.4
+X
+D
+X
+X
+< 1.2
+X
+X
+X
+X
+1
+×X
+XX
+X
+0.8
+X
+XX
+0.6
+1.5
+2
+2.5
+32.5
+FEC
+X
+BCS
+X
+Lipkin
+X
+XXXX
+2
+D 1.5
+X
+X
+X
+X
+X
+1
+X
+XX X
+X
+X
+X
+XX
+0.5
+X
+X
+1.5
+2
+2.5
+3
+3.5
+1
+43
+FEC
+X
+BCS
+X
+XX
+2.5
+Lipkin
+X
+2
+1.5
+1
+X
+0.5
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+TG7
+FIG. 5: The probabilities corresponding to each of the
+five classes of basis states (see Fig. 3) consistent with
+the FEC Hamiltonian for N, r = 4, 8 are shown where
+green, yellow, and blue bars correspond to the lowest
+eigenstate of the Lipkin Hamiltonian, the Pairing-Force
+Hamiltonian, and FEC Hamiltonian, respectively.
+FIG. 6: The probabilities corresponding to each of
+the fourteen classes of basis states consistent with the
+FEC Hamiltonian for N, r = 8, 16 are shown where
+green, yellow, and blue bars correspond to the lowest
+eigenstate of the Lipkin Hamiltonian, the Pairing-Force
+Hamiltonian, and FEC Hamiltonian, respectively. Each
+label x, y, bool, ζ, τ represents the number of particles
+excited to the upper N-degenerate energy level (x), the
+number of BCS-like pairs (y), whether the configuration
+is consistent with the Lipkin model (bool), the number
+of times BCS-like pairs are “stacked” into the same site
+(ζ), and the number of times a diagonal configuration
+occur in which either 2j −1/2j +N or 2j −1+N/2j are
+simultaneously occupied where 2j − 1 and 2j are adja-
+cent, paired orbitals (τ). These values act as quantum
+numbers that define the degenerate classes of non-zero
+basis functions composing the ground state to the FEC
+Hamiltonian.
+a favorable interaction between adjacent particles demon-
+FIG. 7: Configurations representing how the Lipkin-
+like double excitation term (λ) and scattering term (γ)
+in the FEC Hamiltonian relate the |4, 4, F, 1, 2⟩ basis
+state for N, r = 8, 16 to BCS-like basis states.
+strating the same spin.
+Moreover, as both particle-particle (consistent with the
+Pairing-Force Hamiltonian) and particle-hole (consistent
+with the Lipkin Hamiltonian) are utilized in the field of
+nuclear physics to display the essential properties of the
+nuclear interaction [71–73], we can interpret our FEC
+Hamiltonian in this framework. In this interpretation,
+the particles being created and annihilated are nucleons
+such that the Lipkin terms are associated with the inter-
+action of nucleons within a valence shell (γ), the mixing
+of particle-hole excitations with the valence configura-
+tions, and excitations of a nucleon from one valence shell
+to another having an energetic penalty (ϵ) [71, 73]. Ad-
+ditionally, in this interpretation, the PF pairwise inter-
+action is associated with the short-range portion of the
+nuclear interaction [71, 72].
+Overall, this model Hamiltonian is capable of demon-
+strating a wider array of collective behavior than either
+the Lipkin or the Pairing-Force models. Such a Hamil-
+tonian will have a vast degree of applications and will
+be beneficial for the exploration—and for benchmarking
+computational methodologies for the treatment of—the
+nontrivial physics of real-world material and chemical
+systems.
+Author contributions.
+L. M. and D. M. conceived
+of the project, developed the theoretical framework, de-
+signed the computations, wrote the code, performed the
+computations, analyzed the results, and wrote the paper.
+Acknowledgments:
+D.A.M. gratefully acknowledges
+the U.S. National Science Foundation Grants No. CHE-
+1565638, No. CHE-2035876, and No. DMR-2037783 and
+the Department of Energy, Office of Basic Energy Sci-
+ences, Grant DE-SC0019215.
+Data availability. Data will be made available upon
+reasonable request.
+Code availability. Code will be made available on a
+public Github repository upon publication.
+
+4.4.F.1
+4, 4,F, 1, 2
+2,4,FFEC
+0.5
+BCS
+Lipkin
+0.4
+Probability
+0.3
+0.2
+0.1
+0
+0,2,T
+2,2,F
+2,2,T
+2,0,T
+4,2,T
+Basis Classification0.6
+FEC
+BCS
+0.5
+Lipkin
+Probability
+0.4
+0.3
+0.2
+0.1
+0
+0
+乙
+乙
+4
+4
+4
+4
+4
+4
+9
+9
+9
+8
+4
+4
+乙
+0
+4
+4
+乙
+4
+乙
+4
+Z
+F
+F
+F
+L
+F
+Z
+Basis
+s Classification8
+Appendix A: Determination of Signatures of
+Condensation
+To determine the largest eigenvalue of the particle-
+particle RDM (2D, see Eq. (1)—i.e., λD, the signature of
+superconducting character—, only the following N × N
+subblock of the full 2-RDM containing the large eigen-
+value must be computed and diagonalized [50, 74, 75]
+ˆa0ˆa1
+ˆa2ˆa3
+· · ·
+ˆar−2ˆar−1
+ˆa†
+0ˆa†
+1
+ˆa†
+0ˆa†
+1ˆa0ˆa1
+ˆa†
+0ˆa†
+1ˆa2ˆa3
+· · ·
+ˆa†
+0ˆa†
+1ˆar−2ˆar−1
+ˆa†
+2ˆa†
+3
+ˆa†
+2ˆa†
+3ˆa0ˆa1
+ˆa†
+2ˆa†
+3ˆa2ˆa3
+· · ·
+ˆa†
+2ˆa†
+3ˆar−2ˆar−1
+...
+...
+...
+...
+...
+ˆa†
+r−2ˆa†
+r−1 ˆa†
+r−2ˆa†
+r−1ˆa0ˆa1 ˆa†
+r−2ˆa†
+r−1ˆa2ˆa3 · · · ˆa†
+r−2ˆa†
+r−1ˆar−2ˆar−1
+(A1)
+where, again, ˆa†
+i and ˆai are to creation and annihi-
+lation operators corresponding to the orbital with in-
+dex i.
+Each element of this subblock of the 2-RDM
+is the expectation value ⟨Ψ|ˆa†
+2j−1ˆa†
+2jˆa2kˆa2k−1|Ψ⟩ ob-
+tained by programmatically applying the appropriate
+creation and annihilation operators to each pair of
+non-zero basis states composing the previously-obtained
+ground state wavefunction of the Hamiltonian.
+As
+an example, for the N, r
+=
+4 computations, there
+are ten non-zero basis elements composing five distinct
+classes (|0, 2, T⟩, |2, 2, F⟩, |2, 2, T⟩, |2, 0, T⟩, |4, 2, T⟩) that
+are used to construct the Hamiltonian (see the Result
+section). The ground-state wavefunction is obtained in
+terms of these classes with a structure given by
+|Ψ⟩ = v0,2,T |0, 2, T⟩ + v2,2,F |2, 2, F⟩ + v2,2,T |2, 2, T⟩
++v2,0,T |2, 0, T⟩ + v4,2,T |4, 2, T⟩
+(A2)
+where each of the classes is a weighted linear combination
+of the basis states composing it, i.e,
+|2, 0, T⟩ = |1, 3, 6, 8⟩ + |1, 4, 6, 7⟩ + |2, 3, 5, 8⟩ + |2, 4, 5, 7⟩
+√
+4
+(A3)
+Thus, ⟨Ψ|ˆa†
+2j−1ˆa†
+2jˆa2kˆa2k−1|Ψ⟩ is a sum of all expectation
+values of the form
+vc1vc2⟨c1|ˆa†
+2j−1ˆa†
+2jˆa2kˆa2k−1|c2⟩
+(A4)
+where c1 and c2 refer to each of the distinct classes of
+non-zero basis states and where these expectation values
+are sums over
+vb1vb2
+N(cb1)N(cb2)⟨b1|ˆa†
+2j−1ˆa†
+2jˆa2kˆa2k−1|b2⟩
+(A5)
+where b1 and b2 are the basis states composing each class,
+where N(cb1) refers to the size of the class to which basis
+b1 belongs, and where all possible combinations of basis
+states are analyzed.
+Note that only ϵ = 0 calculations were run for the
+N, r = 10, 20 scan such that site symmetry allowed
+the entire matrix to be constructed from three distinct
+types of elements, which lowered computational expense;
+these element types are as follows:
+ˆa†
+2j−1ˆa†
+2jˆa2jˆa2j−1,
+ˆa†
+2j−1ˆa†
+2jˆa2kˆa2k−1, and ˆa†
+2j−1ˆa†
+2jˆa2j±Nˆa2j−1±N.
+The signature of superconductivity (λD) is then com-
+puted from the N ×N subblock of the 2-RDM according
+to the eigenvalue equation
+2Dvi
+D = ϵi
+Dvi
+D
+(A6)
+with the signature corresponding the largest eigenvalue
+(the maximum ϵi
+D).
+The portion of the particle-hole RDM (2G) associated
+with a large eigenvalue is composed of sub-matrices of
+the form
+ˆa†
+qˆaq
+ˆa†
+q+Nˆaq
+ˆa†
+qˆaq+N
+ˆa†
+q+Nˆaq+N
+ˆa†
+pˆap
+ˆa†
+pˆapˆa†
+qˆaq
+ˆa†
+pˆapˆa†
+q+Nˆaq
+ˆa†
+pˆapˆa†
+qˆaq+N
+ˆa†
+pˆapˆa†
+q+Nˆaq+N
+ˆa†
+pˆap+N
+ˆa†
+pˆap+Nˆa†
+qˆaq
+ˆa†
+pˆap+Nˆa†
+q+Nˆaq
+ˆa†
+pˆap+Nˆa†
+qˆaq+N
+ˆa†
+pˆap+Nˆa†
+q+Nˆaq+N
+ˆa†
+p+Nˆap
+ˆa†
+p+Nˆapˆa†
+qˆaq
+ˆa†
+p+Nˆapˆa†
+q+Nˆaq
+ˆa†
+p+Nˆapˆa†
+qˆaq+N
+ˆa†
+p+Nˆapˆa†
+q+Nˆaq+N
+ˆa†
+p+Nˆap+N ˆa†
+p+Nˆap+Nˆa†
+qˆaq ˆa†
+p+Nˆap+Nˆa†
+q+Nˆaq ˆa†
+p+Nˆap+Nˆa†
+qˆaq+N ˆa†
+p+Nˆap+Nˆa†
+q+Nˆaq+N.
+(A7)
+
+9
+tiled in the following manner:
+p=0,q=0
+p=0,q=1
+· · ·
+p=0,q= N
+2 −1
+p=1,q=0
+p=1,q=1
+· · ·
+p=1,q= N
+2 −1
+...
+...
+...
+...
+p= N
+2 −1,q=0
+p= N
+2 −1,q=1 · · ·
+p= N
+2 −1,q= N
+2 −1
+(A8)
+In order to remove the ground-state-to-ground-state
+transition (to form the modified particle-hole RDM, 2 ˜G,
+see Eq. (3)),
+ˆa†
+qˆaq
+ˆa†
+q+Nˆaq
+ˆa†
+qˆaq+N
+ˆa†
+p+Nˆap+N
+ˆa†
+pˆap
+1Dp[0, 0]1Dq[0, 0]
+1Dp[0, 0]1Dq[0, 1]
+1Dp[0, 0]1Dq[1, 0]
+1Dp[0, 0]1Dq[1, 1]
+ˆa†
+pˆap+N
+1Dp[0, 1]1Dq[0, 0]
+1Dp[0, 1]1Dq[0, 1]
+1Dp[0, 1]1Dq[1, 0]
+1Dp[0, 1]1Dq[1, 1]
+ˆa†
+p+Nˆap
+1Dp[1, 0]1Dq[0, 0]
+1Dp[1, 0]1Dq[0, 1]
+1Dp[1, 0]1Dq[1, 0]
+1Dp[1, 0]1Dq[1, 1]
+ˆa†
+p+Nˆap+N
+1Dp[1, 1]1Dq[0, 0]
+1Dp[1, 1]1Dq[0, 1]
+1Dp[1, 1]1Dq[1, 0]
+1Dp[1, 1]1Dq[1, 1]
+is subtracted off from each segment defined by p and q
+where the one-particle density matrix (1D) is given by
+ˆap
+ˆap+N
+ˆa†
+p
+ˆa†
+pˆap
+ˆa†
+pˆap+N
+ˆa†
+p+N ˆa†
+p+Nˆap ˆa†
+p+Nˆap+N
+(A9)
+The signature of exciton condensation (λG) is then ob-
+tained from the eigenvalue equation
+2 ˜Gvi
+G = ϵi
+Gvi
+G
+(A10)
+with the signature corresponding the largest eigenvalue
+(the maximum ϵi
+G).
+Again, for the N, r = 10, 20, ϵ = 0 calculations,
+site symmetry was utilized to decrease computational
+expense.
+Only sub-matrices corresponding to diagonal
+sub-matrices p = q, sub-matrices for BCS-paired or-
+bitals p = 2j − 1, q = 2j, and for unpaired orbitals
+p = 2j − 1, q ̸= p ̸= 2j needed to be computed.
+Appendix B: Plastino’s Model
+In literature that dates back to the 1960s and contin-
+ues to this day, Plastino and coworkers [65–67] explore a
+model Hamiltonian that adds a pairing-force term to the
+Lipkin model in the context of nuclear physics. Introduc-
+ing the Plastino pairing-force term to the Lipkin Hamil-
+tonian from Eq. (4)—which allows for slightly more flex-
+ibility than the formulation given in the Plastino litera-
+ture as that literature is concerned only with the double
+excitation/de-excitation (λ) term and omits the scatter-
+ing term (γ)—yields the following model Hamiltonian:
+HP = − ϵ
+2
+N
+�
+i=1
+ˆa†
+iˆai + ϵ
+2
+N
+�
+i=1
+ˆa†
+i+Nˆai+N
++λ
+2
+N
+�
+p=1
+N
+�
+q=1
+ˆa†
+pˆa†
+qˆaq+Nˆap+N + λ
+2
+N
+�
+p=1
+N
+�
+q=1
+ˆa†
+p+Nˆa†
+q+Nˆaqˆap
++γ
+2
+N
+�
+p=1
+N
+�
+q=1
+ˆa†
+p+Nˆa†
+qˆaq+Nˆap + γ
+2
+N
+�
+p=1
+N
+�
+q=1
+ˆa†
+pˆa†
+q+Nˆaqˆap+N
+−G
+N
+�
+p=1
+N
+�
+q=1
+ˆa†
+p+Nˆa†
+pˆaqˆaq+N
+(B1)
+While the form of this Hamiltonian is similar to the one
+we introduce in Eq.
+(5), the difference is the orbitals
+which the pairing-force term (G) causes to be correlated
+in Cooper-like pairs. Specifically, while our model Hamil-
+tonian pairs adjacent qubits (see Fig. 2), the Plastino
+Hamiltonian pairs orbitals with on the same Lipkin-like
+cite in different layers (i.e., stacked orbitals p and p+N).
+In order to determine whether the Plastino Hamil-
+tonian is capable of probing fermion-exciton conden-
+sate character—where λD and λG simultaneously exceed
+the Pauli-like limit of one and hence character of both
+fermion-pair condensation and exciton condensation are
+observed in a single quantum state—, a systematic scan
+over the input parameters of the Hamiltonian (ϵ, λ, γ, G)
+is conducted. As can in seen by Fig. 8 where the blue
+pluses represent the Lipkin model Hamiltonian, the yel-
+low pluses represent the PF BCS-like Hamiltonian, and
+the green x’s represent the Plastino Hamiltonian, while
+Plastino’s Hamiltonian is capable of reproducing all Lip-
+kin states accessible by the Lipkin model and states that
+
+10
+demonstrate fermion-pair condensation, no dual conden-
+sate character is observed from the Plastino model as the
+region in which both λD and λG exceed one is not probed
+within this model.
+In fact, as noted in Ref.
+, there is direct competi-
+tion between the particle-hole and particle-particle pair-
+ing between Lipkin-like sites which results in each type
+of pairing “driving” the system toward radically differ-
+ent states with the magnitudes of the coupling constants
+causing a transition between the Lipkin-like and BCS-like
+states favored by the different interactions. Conversely,
+because the particle-particle and particle-hole pairing in
+the model we introduce do not occur between the same
+orbitals, they can coexist, allowing for a much larger pos-
+sible range of λD versus λG including the region demon-
+strating a fermion-exciton condensate.
+FIG. 8: A plot of λG versus λD where parameters in
+the Plastino Hamiltonian are systematically varied for
+N = 4 particles in r = 8 orbitals is shown.
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diff --git a/htAyT4oBgHgl3EQfxvkc/content/tmp_files/load_file.txt b/htAyT4oBgHgl3EQfxvkc/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf,len=1108
+page_content='Simultaneous Fermion and Exciton Condensations from a Model Hamiltonian  LeeAnn M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Sager and David A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Mazziotti∗  Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, IL 60637  (Dated: Submitted October 26, 2021)  Fermion-exciton condensation in which both fermion-pair (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' superconductivity) and exciton  condensations occur simultaneously in a single coherent quantum state has recently been conjec-  tured to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Here, we capture the fermion-exciton condensation through a model Hamiltonian  that can recreate the physics of this new class of highly-correlated condensation phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' We  demonstrate that the Hamiltonian generates the large-eigenvalue signatures of fermion-pair and ex-  citon condensations for a series of states with increasing particle numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' The results confirm that  the dual-condensate wave function arises from the entanglement of fermion-pair and exciton wave  functions, which we previously predicted in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' This model Hamiltonian—  generalizing well-known model Hamiltonians for either superconductivity or exciton condensation—  can explore a wide variety of condensation behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' It provides significant insights into the required  forces for generating a fermion-exciton condensate, which will likely be invaluable for realizing such  condensations in realistic materials with applications from superconductors to excitonic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  PACS numbers: 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='+z  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  INTRODUCTION  Model Hamiltonians are theoretical tools that are of-  ten useful in simulating the key physics associated with  large-scale, highly-correlated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' They are capable  of modeling an array of quantum phases and many-body  phenomena such as phase transitions [1–5], superconduc-  tivity [6–10], quantum magnetism [11–14], exciton con-  densation [15–21], lattice-like systems [22, 23], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Addi-  tionally, model Hamiltonians which encompass nontriv-  ial physics are often useful as benchmarks for theoretical  tools such as many-body approximations [6, 24–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Condensation  phenomena—which  are  inherently  highly-correlated—have a long history of being computa-  tionally studied through the lens of model Hamiltonians  as traditional band theory is inaccurate for such highly-  entangled materials [6, 7, 10, 15, 27–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Specifically,  superconductors—materials in which fermion-fermion  (Cooper/electron-electron) pairs aggregate into a single  quantum state, resulting in the superfluidity of the  fermion-fermion  pairs—are  often  explored  through  use  of  the  Pairing-Force  (PF)  Hamiltonian  [6–9],  which is additionally referred to as the Standard Re-  duced Bardeen-Cooper-Schrieffer (BCS) Hamiltonian  [10, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' This Hamiltonian is a simple representation  of superconductivity as it describes a system with  bound Cooper (or Cooper-like particle-particle) pairs  interacting in an attractive manner with the holes they  leave behind in a Fermi sea with the high-correlation  limit  of  this  Hamiltonian  resulting  in  well-known,  number-projected BCS wave functions [7, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Similarly,  exciton condensation—in which particle-hole (exciton)  pairs condense into a single quantum state resulting in  the superfluidity of the composite excitons [33]—can  be modeled according to the Lipkin-Meshkov-Glick  ∗ damazz@uchicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='edu  (LMG) Hamiltonian, which is often simply referred to  as the Lipkin model [15–21, 29, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  This Hamilto-  nian is a highly-degenerate system in which partnered  orbitals are inherently particle-hole paired and whose  strongly-correlated form results in ground states that  demonstrate character of exciton condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Here, we introduce a model Hamiltonian that is ca-  pable of capturing fermion-exciton condensation, a new  class of highly-correlated condensation phenomena in  which both fermion-pair and exciton condensations coex-  ist in a single quantum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' We demonstrate such coex-  istent condensate character by calculating the quantum  signatures of fermion-pair [35, 36] and exciton [37, 38]  condensations (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  II and Appendix A) for sys-  tems of even particle numbers ranging from N = 4  to N  = 10 particles in r  = 2N orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  These  fermion-exciton condensates are shown to be described  by wavefunctions which are entanglements of wavefunc-  tions from BCS-like superconductivity and Lipkin-like  exciton condensation—consistent with our prior predic-  tions for the large-N thermodynamic limit [39] as well as  those we observed experimentally on a quantum device  [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Our determination of a model Hamiltonian that sup-  ports fermion-exciton condensation provides information  regarding the nature of the forces necessary to generate  such systems—an invaluable first step in the realization  of real-world systems that support such dual condensa-  tion of excitons and fermion-fermion pairs, which may  demonstrate some sort of hybrid of the properties of su-  perconductors and exciton condensates and hence have  applications in energy transport and electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  The  extent of these different phases and the transitions be-  tween these phases can also be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Moreover, our  Hamiltonian provides an important reference in order to  determine whether a given many-body approximation is  capable of measuring dual condensate character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='00670v1  [quant-ph]  29 Dec 2022  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 1: A figure of the condensate phase diagram in  the phase space of the signatures of particle-particle  condensation, λD, and exciton condensation, λG, is  shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  THEORY  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Fermion-Pair Condensation  Superconductivity results from the condensation of  bosonic fermion-fermion pairs [10, 41–43] into a single  geminal—a two-fermion function directly analogous to  the one-fermion orbital [35, 36, 44–47]—at temperatures  below a certain critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' This condensation  of fermion-pairs results in the superfluidity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=', fric-  tionless flow) of the constituent particle-particle pairs  [10, 43, 48, 49];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' if the fermionic pairs are composed  of electrons (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=', Cooper pairs), then these superfluid  electron-electron pairs demonstrate superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  As was first demonstrated by Yang [35] and Sasaki  [36], a computational signature of such superconducting  states is a large eigenvalue in the particle-particle reduced  density matrix (2-RDM), whose elements are given by  2Di,j  k,l = ⟨Ψ|ˆa†  iˆa†  jˆalˆak|Ψ⟩  (1)  where |Ψ⟩ is an N-fermion wavefunction and where ˆa†  i  and ˆai are fermionic creation and annihilation opera-  tors for orbital i, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' As eigenvalues of the 2-  RDM can be interpreted as the occupations of the two-  fermion geminals [50], when the largest eigenvalue of the  2-RDM—the signature of particle-particle condensation,  represented by λD—exceeds the Pauli-like limit of one  (λD > 1), multiple fermion-fermion pairs occupy a sin-  gle geminal and hence superconducting character is ob-  served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  This signature is known to directly probe the  presence and extent of non-classical (off-diagonal) long-  range order [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' (See the Appendix for more details on  how the signature of superconductivity, λD, was com-  puted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=')  The  Pairing-Force  (PF)  model  [6–9]—also  called  the Standard Reduced Bardeen-Cooper-Schrieffer (BCS)  model [10, 30, 31]—is known to exhibit superconduct-  ing character in the strong correlation limit and hence  achieve a large λD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' The Hamiltonian for the PF model  is given in second quantization by  HP F = 1  2  �  σ=↑,↓  N  �  p=1  ϵpˆa†  p,σˆap,σ − G  N  �  p=1  N  �  q=1  ˆa†  p,↑ˆa†  p,↓ˆaq,↓ˆaq,↑  (2)  where p is a quantum number that represents a pair  of orbitals denoted as p, ↑ and p, ↓ with the same en-  ergy, where the energies (ϵp) are considered to be known,  and where the parameter G is a constant that tunes the  strength of the pairwise interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Note that in the  limit of strong correlation (G >> ϵp), maximal super-  conducting character—λD =  N  2  �  1 − N−2  r  �  [44, 50]—is  observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Exciton Condensation  Directly analogous to superconductivity resulting from  bosonic particle-particle pairs condensing into a single  particle-particle function, exciton condensation results  from the condensation of particle-hole pairs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=', exci-  tons) into a single particle-hole function below a certain  critical temperature, which results in the superfluidity of  the excitons [33, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Exciton condensates, while diffi-  cult to realize experimentally, have been observed in sys-  tems composed of polaritons (excitons coupled to pho-  tons) [52–54] and in two-dimensional structures such as  semiconductors [55], graphene bilayers [56–58], and van  der Waals heterostructures [59–62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  The signature of exciton condensation—denoted as  λG—is similarly analogous to that for fermion-pair con-  densation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' the presence and extent of exciton condensate  character can be measured from the largest eigenvalue of  a modified particle-hole reduced density matrix given by  [37, 38, 63]  2 ˜Gi,j  k,l = 2Gi,j  k,l − 1Di  k  1Dj  l  = ⟨Ψ|ˆa†  iˆajˆa†  l ˆak|Ψ⟩ − ⟨Ψ|ˆa†  iˆak|Ψ⟩⟨Ψ|ˆa†  jˆal|Ψ⟩  (3)  where 1D is the one-fermion reduced density matrix (1-  RDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Note that this modification removes the extrane-  ous large eigenvalue from a ground-state-to-ground-state  transition such that a signature above one (λG > 1) is in-  dicative of exciton condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' (See the Appendix for  more details on how the signature of exciton condensa-  tion, λG was computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=') This computational signature  has been utilized to study exciton condensation is possi-  ble in quantum and molecular systems [29, 38–40, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  One model known to achieve a large λG value and  hence exhibit exciton condensate character in the limit of  a large correlation is the Lipkin quasispin model [15–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  The N-fermion Lipkin quasispin model consists of two  energy levels  �  − ϵ  2, ϵ  2  �  , each containing N energetically-  degenerate states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  The second-quantized Hamiltonian  Superconducting  Coexistence  Condensate  1  入D  Exciton  No Condensation  Condensation  Phenomenon  1 入G3  can be expressed as [18]  HL = ϵ  2  �  σ=±1  σ  N  �  p=1  ˆa†  σ,pˆaσ,p  +γ  2  �  σ=±1  N  �  p,q=1  ˆa†  +σ,pˆa−σ,pˆa†  −σ,qˆa+σ,q  +λ  2  �  σ=±1  N  �  p,q=1  ˆa†  +σ,pˆa†  +σ,qˆa−σ,qˆa−σ,p  (4)  where σ = ±1 and p = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' , N are quantum numbers  that completely characterize the system in which p de-  scribes the site number labelling the N states in a given  level and σ represents the upper (+1) or lower (−1) en-  ergy levels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Note that in this model, the  λ term allows for double excitations and de-excitations,  and the γ term allows for a single particle to be scattered  up while another is simultaneously scattered down;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' as a  result, in the Lipkin model, only even excitations are al-  lowed, and only one particle may occupy a given site (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=',  have a specific quantum number p) such that each site  in the lower level is particle-hole paired with the corre-  sponding site in the upper level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' By having the terms cor-  relating orbitals in the Hamiltonian (λ, γ) be sufficiently  larger than the energy term (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=', in the limit of high cor-  relation), maximal exciton condensation—λG = N  2 [37]—  can be obtained for λ = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Fermion-Exciton Condensation  A fermion-exciton condensate is a single quantum state  that simultaneously demonstrates character of supercon-  ductivity and exciton condensation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=', both signatures  of condensation—the largest eigenvalue of the particle-  particle RDM (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' (1)) and the largest eigenvalue of the  modified particle-hole RDM (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  (3))—are simultane-  ously large (λD, λG > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 2: A pictorial representation of the model  Hamiltonian we introduce in which there are two N-  degenerate energy levels—with energies − ϵ  2 and ϵ  2—  with double excitations and de-excitations, scattering  in which one particle is de-excited while another is si-  multaneously excited, and a pair-wise interaction term  between sites 2j − 1 and 2j for j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' , N} (yellow  circles) is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Note that the Lipkin-like excitations  must occur within a site (p ↔ p + N, blue arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  To gain insight into such fermion-exciton condensates,  here we propose a model system that is capable of demon-  strating simultaneous fermion-pair and exciton conden-  sate character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' In this model, we introduce the pairwise  interaction from the Pairing-Force model into the scaf-  folding of the Lipkin model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' thus, the model keeps the  structure of the Lipkin model in which N particles oc-  cupy two N-degenerate energy levels (− ϵ  2 and  ϵ  2) with  allowed double excitations on two sites (λ) and simulta-  neous scattering of a particle up on one site and down on  another (γ)—where Lipkin-like sites are now given as or-  bitals p and p+N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' however, we additionally pair adjacent  orbitals—orbitals 2j−1 and 2j for j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' , N}—via  the PF parameter, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=') The Hamiltonian for  this model is thus given by  H = − ϵ  2  N  �  i=1  ˆa†  iˆai + ϵ  2  N  �  i=1  ˆa†  i+Nˆai+N  +λ  2  N  �  p=1  N  �  q=1  ˆa†  pˆa†  qˆaq+Nˆap+N + λ  2  N  �  p=1  N  �  q=1  ˆa†  p+Nˆa†  q+Nˆaqˆap  +γ  2  N  �  p=1  N  �  q=1  ˆa†  p+Nˆa†  qˆaq+Nˆap + γ  2  N  �  p=1  N  �  q=1  ˆa†  pˆa†  q+Nˆaqˆap+N  −G  N  �  j=1  N  �  k=1  ˆa†  2j−1ˆa†  2jˆa2kˆa2k−1  (5)  in second quantization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' with a given set of parameters  (ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' G) directly determining the extent of fermion-  pair and exciton condensation (λD and λG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' respectively)  of the ground state corresponding to this model Hamil-  tonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  While this model Hamiltonian is not the first to com-  bine the pairwise interaction from the Pairing-Force  model with the Lipkin model, the model Hamiltonian  introduced by Plastino and coworkers causes direct com-  petition between particle-particle and particle-hole cor-  relations and hence proves incapable of demonstrating a  fermion-exciton condensate phase (see Appendix B) [65–  67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Conversely, due to our introduction of the Pairing-  Force interactions between adjacent orbitals instead of  orbitals in the same Lipkin-like site, particle-particle and  particle hole pairing can coexist and hence fermion-pair-  exciton (FEC) states can be achieved as is shown in the  results that follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  RESULTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  N = 4, The Minimal FEC  As the authors have previously demonstrated [39], a  system with as few as N = 4 particles in r = 8 or-  bitals can support formation of a fermion-exciton con-  densate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' As such, we first fully explore such a minimal-  istic FEC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' The ground state of the FEC Hamil-  tonian that we have introduced—Equation (5)—for four  2  N+1  N+2  2N-1  2N  2  2  N-1  N  14  particles has contributions from only ten of the seventy  (rCN) possible configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Of these ten basis states,  there are only five distinct classes composed of degener-  ate orientations—see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 3—that allow for the direct  computation of a matrix-form of the Hamiltonian in a  minimal basis state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' The five basis states are defined by  three quantum numbers, x, y, bool, where the first indi-  cates the number of particles excited to the upper energy  level (x), the second indicates the number of BCS-like  pairs (number of times both 2j − 1 and 2j are occupied,  y), and the third is a boolean that indicates whether the  configuration is “Lipkin”-like in the regard that no two  orbitals representing a “Lipkin” site (denoted as p and  p + N, see the blue arrow in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 3) are dually occupied  or dually unoccupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 3: Configurations representing each of the five  classes of non-zero basis states for the FEC Hamil-  tonian for N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' r = 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 8 are shown where each label  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' bool represents the number of particles excited to  the upper N-degenerate energy level (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' the number  of BCS-like pairs (y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' and whether the configuration is  consistent with the Lipkin model (bool),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' where the de-  generacy of each class of states is given in parenthesis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  and where green,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' yellow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' and blue configurations repre-  sent that the corresponding states are consistent with  only the Lipkin Hamiltonian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' only the Pairing-Force  Hamiltonian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' or both Lipkin and PF Hamiltonians,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Utilizing  the  basis  shown  in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  3—  |0, 2, T⟩, |2, 2, F⟩, |2, 2, T⟩, |2, 0, T⟩, and |4, 2, T⟩—the  Hamiltonian from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' (5) can be represented by  H4 =  �  �  �  �  �  �  �  −2ϵ − 2G  −G  √  2  2λ−2G  √  2  2λ  0  −G  √  2  −2G + 2γ  −2G  0  −G  √  2  2λ−2G  √  2  −2G  −2G  2γ  √  2  2λ−2G  √  2  2λ  0  2γ  √  2  2γ  2λ  0  −G  √  2  2λ−2G  √  2  2λ  2ϵ − 2G  �  �  �  �  �  �  �  (6)  where each term—corresponding to the interaction be-  tween two classes of basis states, |i⟩ and |j⟩—is obtained  from programmatically generating all sets of second-  quantization creation and annihilation operators in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  (5), taking the expectation value for each combination of  pairs of configurations in classes |i⟩ and |j⟩, summing the  results, and normalizing by dividing by the square root  of the number of configurations for both |i⟩ and |j⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' For  example, if |i⟩ = |2, 2, F⟩ = (|1, 2, 5, 6⟩ + |3, 4, 7, 8⟩) /  √  2  and |j⟩ = |2, 2, T⟩ = (|1, 2, 7, 8⟩ + |3, 4, 5, 6⟩) /  √  2, the  Hamiltonian term would be given by  (⟨1, 2, 5, 6| + ⟨3, 4, 7, 8|)  √  2  H4  (|1, 2, 7, 8⟩ + |3, 4, 5, 6⟩)  √  2  = 1  2[⟨1, 2, 5, 6|H4|1, 2, 7, 8⟩ + ⟨1, 2, 5, 6|H4|3, 4, 5, 6⟩  +⟨3, 4, 7, 8|H4|1, 2, 7, 8⟩ + ⟨3, 4, 7, 8|H4|3, 4, 5, 6⟩]  (7)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  4a scans over the signatures of condensation—  λD and λG—for the ground state of the Hamiltonian in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' (6) by systematically varying the parameters ϵ, λ,  γ, and G where the yellow BCS x’s represent ground  states in which the PF Hamiltonian is implemented (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=',  λ = γ = 0), the blue Lipkin x’s represent states in which  the Lipkin Hamiltonian is implemented (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=', G = 0),  and where the green FEC x’s represent states with char-  acter of both PF and Lipkin Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' As this fig-  ure demonstrates, the largest degree of superconducting  character (the largest λD) is indeed observed in the BCS  limit of the FEC Hamiltonian (when G >> ϵ, λ = γ ≈  0), and the largest degree of exciton condensate character  (the largest λG) is observed in the Lipkin limit of the FEC  Hamiltonian (λ ≈ γ >> ϵ, G ≈ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' However, neither the  BCS nor Lipkin limits of the Hamiltonian is capable of  demonstrating a dual fermion-exciton condensate as λD  and λG only simultaneously exceed the Pauli-like limit of  one when the full FEC Hamiltonian from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' (5) is im-  plemented including both BCS-like (G) and Lipkin-like  (λ, γ) terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Our model FEC Hamiltonian, however, is capable of  demonstrating a wide variety of dual condensate charac-  ter as a variety of input parameters lead to ground state  configurations in which both λG and λD simultaneously  exceed one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Additionally, the λD and λG values obtained  by scanning over the Hamiltonian parameters (in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 4a)  demonstrate an elliptic nature consistent with the con-  vex nature of 2-RDMs projected onto a two-dimensional  space [68–70] that matches predictions for a FEC that  these authors first presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  This elliptic  0, 2, T (1)  2, 2, F (2)  2, 2, T (2)  2, 0, T (4)  4, 2, T (1)5  boundary as well as the density of points in the zone cor-  responding to fermion-exciton condensate character indi-  cate that the FEC model Hamiltonian introduced here is  capable of spanning the entirety of the FEC region of λD  versus λG space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=', λD, λG > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  39, these authors theoretically establish  that in the thermodynamic limit, a possible wavefunc-  tion demonstrating fermion-exciton condensation can be  obtained by entangling wavefunctions that separately  demonstrate superconducting character (|ΨD⟩ with large  λD) and exciton condensate character (|ΨG⟩ with large  λG) according to  |ΨF EC⟩ =  1  �  2 − |∆|  (|ΨD⟩ − sgn(∆)|ΨG⟩) ,  (8)  where ∆ = 2⟨ΨD|ΨG⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 5 occupation probabili-  ties for each of the five classes of basis states consistent  with the N, r = 4, 8 FEC Hamiltonian that contribute  to a BCS wavefunction (yellow, ϵ, λ, γ, G = 0, 0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='7,  λD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='50, λG = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='67), a Lipkin wavefunction (blue,  ϵ, λ, γ, G = 0, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5, 0, λD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='50, λG = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='00), and a  FEC wavefunction (green, ϵ, λ, γ, G = 0, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='7,  λD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='31, λG = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='32) are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  From this data, it  can be observed that the FEC wavefunction does indeed  appear to be an entanglement of the individual BCS and  Lipkin wavefunctions for the case of N = 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' this is con-  sistent with the theoretical result in the thermodynamic  limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Higher-Particle FECs  In order to observe trends related to system size, we  employ the methodologies used to explore the N, r = 4, 8  model system and extrapolate to systems composed of  N = 6, 8, 10 particles in r = 12, 16, 20 orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Figures  summarizing the signatures of superconducting charac-  ter (λD) and exciton condensate character (λG) obtained  for the ground state wavefunctions of these larger model  Hamiltonians can be seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  4b-4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Similar to  the results from the N = 4 data, elliptic fits spanning  the maximal signature of superconducting character ob-  served for the BCS wavefunction to the maximal signa-  ture of exciton condensate character for the Lipkin wave-  function with a large variety of parameters supporting  dual fermion-exciton condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Note that as the size  of the system increases from N = 6 to N = 8 to N = 10,  the number of classes of degenerate, non-zero basis states  as well as the number of basis states composing each class  increase from 8 classes with a total of 44 non-zero basis  states to 14 classes with a total of 230 non-zero basis  states to 20 classes with a total of 1212 non-zero basis  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  As such, the relative sparsity of the computa-  tions in λD versus λG as system size is increased is due  to fewer computations being run with larger increments  between each of the parameters as they are varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  To demonstrate how the classes of non-zero basis states  vary as system size is increased, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 6—which shows the  occupation probabilities for each of the fourteen classes  of basis states consistent with the N, r = 8, 16 FEC  Hamiltonian that contribute to a BCS wavefunction (yel-  low, ϵ, λ, γ, G = 0, 0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='9, λD = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='50, λG = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='57), a  Lipkin wavefunction (blue, ϵ, λ, γ, G = 0, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5, 0,  λD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='50, λG = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='00), and a FEC wavefunction (green,  ϵ, λ, γ, G = 0, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='9, λD = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='06, λG = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='87)—  is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Note that due to an increase in the possible  complexity, two more quantum numbers are added to de-  scribe a few of the classes of basis states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' specifically, ζ  and τ are added to x, y, bool, ζ, τ where ζ corresponds to  the number of times BCS-like pairs are “stacked” into  the same site such that orbitals 2j − 1, 2j, 2j − 1 + N,  and 2j + N are all occupied and where τ corresponds  to the number of diagonal configurations in which ei-  ther 2j − 1/2j + N or 2j − 1 + N/2j are both occupied  where 2j −1 and 2j are adjacent, BCS-paired orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' A  few configurations with the necessary quantum numbers  specified for N = 8 are included in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  As can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  6,  the groundstate  wavefunction for the N  =  8 FEC Hamiltonian no  longer simply contains elements of the BCS wavefunc-  tion and the Lipkin wavefunction naively entangled to-  gether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Specifically, while the |4, 4, F, 1, 2⟩ class of ba-  sis states does include BCS-paired particles (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  7), it does not include the maximal number of BCS-  paired particles, which appears to be a necessary con-  dition for non-zero occupation of the ground state for  the BCS Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  However, this class of basis  states can interact with other BCS-like and Lipkin-  like classes of basis states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Explicitly, |4, 4, F, 1, 2⟩ in-  teracts with |2, 4, F⟩ via  λ  2 ˆa†  pˆa†  qˆaq+Nˆap+N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' |4, 4, F, 1⟩  via λ  2 ˆa†  pˆa†  q+Nˆaqˆap+N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' |6, 4, F⟩ via λ  2 ˆa†  p+Nˆa†  q+Nˆaqˆap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' and  |2, 2, T⟩ via −Gˆa†  2j−1ˆa†  2jˆa2kˆa2k−1, which does further en-  tangle the Lipkin-like configurations and BCS-like con-  figurations in a non-trivial manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' As such, while the  interaction between the BCS-like classes of basis states  and Lipkin-like classes of basis states in the formation of  the FEC ground state wavefunction is not as clear-cut or  simple as in the N = 4 case, the N = 8 FEC wavefunc-  tion is still an entanglement of BCS-like and Lipkin-like  terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  A representative configuration as well as the relevant  quantum numbers for all classes of basis states for the  N = 6, N = 8, and N = 12 FEC Hamiltonians is given  in the Supplemental Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  DISCUSSION AND CONCLUSIONS  In this study, we introduce a model Hamiltonian that  successfully demonstrates the physics associated with  both fermion-pair condensation and exciton condensa-  tion, as well as encompassing the phase space consisting  of systems in which fermion-pair condensation and ex-  citon condensation are simultaneously realized—a phe-  nomenon which we term fermion-exciton condensation  6  (a) N, r = 4, 8  (b) N, r = 6, 12  (c) N, r = 8, 16  (d) N, r = 10, 20  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 4: Plots of λG versus λD where parameters in the FEC Hamiltonian are systematically varied are shown for  systems involving (a) N = 4, (b) N = 6, (c) N = 8, and (d) N = 10 particles in r = 2N orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  (FEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Applying this model to systems composed of  N = 4, 6, 8, 10 particles in r = 2N orbitals, we confirm  this fermion-exciton condensate character for a wide va-  riety of ground state wavefunctions corresponding to a  diverse range of input parameters in the model Hamilto-  nian, additionally verifying the prediction made in prior  investigation [39] that the wavefunction of a fermion-  exciton condensate is an entanglement of wavefunctions  of exciton condensates and fermion-pair condensates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  The introduction of our model Hamiltonian that sup-  ports fermion-exciton condensation advances our under-  standing of the forces and orbital correlations necessary  for the experimental construction of FEC states in real-  world materials—important insights in the search for  real-world materials exhibiting fermion-exciton conden-  sate character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Depending on the interpretation of the  Hamiltonian elements, this could have ramifications for  fields such as traditional and molecularly-scaled electron-  ics, spin systems, and nuclear physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Specifically, if the orbitals in the Hamiltonian are in-  terpreted as spin orbitals, fermion-exciton condensates  simultaneously demonstrate the condensation of Cooper  into a single particle-particle quantum state and the con-  densation of electron-hole pairs into a single particle-hole  quantum state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' thus, superfluid Cooper pairs—resulting  in superconductivity—and superfluid excitons—which  are associated with the dissipationless flow of energy  [33, 51]—should both be present to a certain extent in  FEC systems, maybe demonstrating some hybridization  of the properties of superconductors and exciton conden-  sates, which may be relevant to the fields of energy trans-  port and electronics in both macroscopic materials and  molecular-scaled systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Alternatively, the two Lipkin-like N-degenerate lev-  els can be interpreted as being representative of specific  spin states such that the upper level is spin up and the  lower level is spin down or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' This interpretation  is most-consistent with ϵ = 0—which does demonstrate  FEC states for a wide variety of input parameters—, al-  though in a magnetic field the different spin states could  be separated by some non-zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  In this frame-  work, the Lipkin-like terms could represent simultaneous  double spin flips that are either aligned (λ) or misaligned  (γ), and the pairwise Pairing-Force term could be seen as  FEC  X  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='4  BCS  X  Lipkin  X  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='8  22  FEC  X  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='8  BCS  X  Lipkin  X  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='6  X  X  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='4  X  D  X  X  < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='2  X  X  X  X  1  ×X  XX  X  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='8  X  XX  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5  FEC  X  BCS  X  Lipkin  X  XXXX  2  D 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5  X  X  X  X  X  1  X  XX X  X  X  X  XX  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5  X  X  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5  1  43  FEC  X  BCS  X  XX  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5  Lipkin  X  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5  1  X  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5  5  TG7  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 5: The probabilities corresponding to each of the  five classes of basis states (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 3) consistent with  the FEC Hamiltonian for N, r = 4, 8 are shown where  green, yellow, and blue bars correspond to the lowest  eigenstate of the Lipkin Hamiltonian, the Pairing-Force  Hamiltonian, and FEC Hamiltonian, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 6: The probabilities corresponding to each of  the fourteen classes of basis states consistent with the  FEC Hamiltonian for N, r = 8, 16 are shown where  green, yellow, and blue bars correspond to the lowest  eigenstate of the Lipkin Hamiltonian, the Pairing-Force  Hamiltonian, and FEC Hamiltonian, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Each  label x, y, bool, ζ, τ represents the number of particles  excited to the upper N-degenerate energy level (x), the  number of BCS-like pairs (y), whether the configuration  is consistent with the Lipkin model (bool), the number  of times BCS-like pairs are “stacked” into the same site  (ζ), and the number of times a diagonal configuration  occur in which either 2j −1/2j +N or 2j −1+N/2j are  simultaneously occupied where 2j − 1 and 2j are adja-  cent, paired orbitals (τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' These values act as quantum  numbers that define the degenerate classes of non-zero  basis functions composing the ground state to the FEC  Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  a favorable interaction between adjacent particles demon-  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 7: Configurations representing how the Lipkin-  like double excitation term (λ) and scattering term (γ)  in the FEC Hamiltonian relate the |4, 4, F, 1, 2⟩ basis  state for N, r = 8, 16 to BCS-like basis states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  strating the same spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Moreover, as both particle-particle (consistent with the  Pairing-Force Hamiltonian) and particle-hole (consistent  with the Lipkin Hamiltonian) are utilized in the field of  nuclear physics to display the essential properties of the  nuclear interaction [71–73], we can interpret our FEC  Hamiltonian in this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' In this interpretation,  the particles being created and annihilated are nucleons  such that the Lipkin terms are associated with the inter-  action of nucleons within a valence shell (γ), the mixing  of particle-hole excitations with the valence configura-  tions, and excitations of a nucleon from one valence shell  to another having an energetic penalty (ϵ) [71, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Ad-  ditionally, in this interpretation, the PF pairwise inter-  action is associated with the short-range portion of the  nuclear interaction [71, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Overall, this model Hamiltonian is capable of demon-  strating a wider array of collective behavior than either  the Lipkin or the Pairing-Force models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Such a Hamil-  tonian will have a vast degree of applications and will  be beneficial for the exploration—and for benchmarking  computational methodologies for the treatment of—the  nontrivial physics of real-world material and chemical  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Author contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' conceived  of the project, developed the theoretical framework, de-  signed the computations, wrote the code, performed the  computations, analyzed the results, and wrote the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Acknowledgments:  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' gratefully acknowledges  the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' National Science Foundation Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' CHE-  1565638, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' CHE-2035876, and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' DMR-2037783 and  the Department of Energy, Office of Basic Energy Sci-  ences, Grant DE-SC0019215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Data availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Data will be made available upon  reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Code availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Code will be made available on a  public Github repository upon publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='1  4, 4,F, 1, 2  2,4,FFEC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5  BCS  Lipkin  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='4  Probability  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='1  0  0,2,T  2,2,F  2,2,T  2,0,T  4,2,T  Basis Classification0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='6  FEC  BCS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='5  Lipkin  Probability  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='1  0  0  乙  乙  4  4  4  4  4  4  9  9  9  8  4  4  乙  0  4  4  乙  4  乙  4  Z  F  F  F  L  F  Z  Basis  s Classification8  Appendix A: Determination of Signatures of  Condensation  To determine the largest eigenvalue of the particle-  particle RDM (2D, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' (1)—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=', λD, the signature of  superconducting character—, only the following N × N  subblock of the full 2-RDM containing the large eigen-  value must be computed and diagonalized [50, 74, 75]  ˆa0ˆa1  ˆa2ˆa3  · ·  ˆar−2ˆar−1  ˆa†  0ˆa†  1  ˆa†  0ˆa†  1ˆa0ˆa1  ˆa†  0ˆa†  1ˆa2ˆa3  · ·  ˆa†  0ˆa†  1ˆar−2ˆar−1  ˆa†  2ˆa†  3  ˆa†  2ˆa†  3ˆa0ˆa1  ˆa†  2ˆa†  3ˆa2ˆa3  · ·  ˆa†  2ˆa†  3ˆar−2ˆar−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  ˆa†  r−2ˆa†  r−1 ˆa†  r−2ˆa†  r−1ˆa0ˆa1 ˆa†  r−2ˆa†  r−1ˆa2ˆa3 · · · ˆa†  r−2ˆa†  r−1ˆar−2ˆar−1  (A1)  where, again, ˆa†  i and ˆai are to creation and annihi-  lation operators corresponding to the orbital with in-  dex i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Each element of this subblock of the 2-RDM  is the expectation value ⟨Ψ|ˆa†  2j−1ˆa†  2jˆa2kˆa2k−1|Ψ⟩ ob-  tained by programmatically applying the appropriate  creation and annihilation operators to each pair of  non-zero basis states composing the previously-obtained  ground state wavefunction of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  As  an example, for the N, r  =  4 computations, there  are ten non-zero basis elements composing five distinct  classes (|0, 2, T⟩, |2, 2, F⟩, |2, 2, T⟩, |2, 0, T⟩, |4, 2, T⟩) that  are used to construct the Hamiltonian (see the Result  section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' The ground-state wavefunction is obtained in  terms of these classes with a structure given by  |Ψ⟩ = v0,2,T |0, 2, T⟩ + v2,2,F |2, 2, F⟩ + v2,2,T |2, 2, T⟩  +v2,0,T |2, 0, T⟩ + v4,2,T |4, 2, T⟩  (A2)  where each of the classes is a weighted linear combination  of the basis states composing it, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  |2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' T⟩ = |1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 8⟩ + |1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 7⟩ + |2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 8⟩ + |2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 7⟩  √  4  (A3)  Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' ⟨Ψ|ˆa†  2j−1ˆa†  2jˆa2kˆa2k−1|Ψ⟩ is a sum of all expectation  values of the form  vc1vc2⟨c1|ˆa†  2j−1ˆa†  2jˆa2kˆa2k−1|c2⟩  (A4)  where c1 and c2 refer to each of the distinct classes of  non-zero basis states and where these expectation values  are sums over  vb1vb2  N(cb1)N(cb2)⟨b1|ˆa†  2j−1ˆa†  2jˆa2kˆa2k−1|b2⟩  (A5)  where b1 and b2 are the basis states composing each class,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  where N(cb1) refers to the size of the class to which basis  b1 belongs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' and where all possible combinations of basis  states are analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Note that only ϵ = 0 calculations were run for the  N, r = 10, 20 scan such that site symmetry allowed  the entire matrix to be constructed from three distinct  types of elements, which lowered computational expense;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  these element types are as follows:  ˆa†  2j−1ˆa†  2jˆa2jˆa2j−1,  ˆa†  2j−1ˆa†  2jˆa2kˆa2k−1, and ˆa†  2j−1ˆa†  2jˆa2j±Nˆa2j−1±N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  The signature of superconductivity (λD) is then com-  puted from the N ×N subblock of the 2-RDM according  to the eigenvalue equation  2Dvi  D = ϵi  Dvi  D  (A6)  with the signature corresponding the largest eigenvalue  (the maximum ϵi  D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
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+page_content=' 1]  is subtracted off from each segment defined by p and q  where the one-particle density matrix (1D) is given by  ˆap  ˆap+N  ˆa†  p  ˆa†  pˆap  ˆa†  pˆap+N  ˆa†  p+N ˆa†  p+Nˆap ˆa†  p+Nˆap+N  (A9)  The signature of exciton condensation (λG) is then ob-  tained from the eigenvalue equation  2 ˜Gvi  G = ϵi  Gvi  G  (A10)  with the signature corresponding the largest eigenvalue  (the maximum ϵi  G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Again, for the N, r = 10, 20, ϵ = 0 calculations,  site symmetry was utilized to decrease computational  expense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Only sub-matrices corresponding to diagonal  sub-matrices p = q, sub-matrices for BCS-paired or-  bitals p = 2j − 1, q = 2j, and for unpaired orbitals  p = 2j − 1, q ̸= p ̸= 2j needed to be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Appendix B: Plastino’s Model  In literature that dates back to the 1960s and contin-  ues to this day, Plastino and coworkers [65–67] explore a  model Hamiltonian that adds a pairing-force term to the  Lipkin model in the context of nuclear physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Introduc-  ing the Plastino pairing-force term to the Lipkin Hamil-  tonian from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' (4)—which allows for slightly more flex-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
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+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='p=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='q=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='ˆa†  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='pˆa†  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='q+Nˆaqˆap+N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='−G  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='p=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='q=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='ˆa†  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='p+Nˆa†  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='pˆaqˆaq+N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='(B1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='While the form of this Hamiltonian is similar to the one  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='we introduce in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  (5), the difference is the orbitals  which the pairing-force term (G) causes to be correlated  in Cooper-like pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Specifically, while our model Hamil-  tonian pairs adjacent qubits (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 2), the Plastino  Hamiltonian pairs orbitals with on the same Lipkin-like  cite in different layers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=', stacked orbitals p and p+N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  In order to determine whether the Plastino Hamil-  tonian is capable of probing fermion-exciton conden-  sate character—where λD and λG simultaneously exceed  the Pauli-like limit of one and hence character of both  fermion-pair condensation and exciton condensation are  observed in a single quantum state—, a systematic scan  over the input parameters of the Hamiltonian (ϵ, λ, γ, G)  is conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' As can in seen by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 8 where the blue  pluses represent the Lipkin model Hamiltonian, the yel-  low pluses represent the PF BCS-like Hamiltonian, and  the green x’s represent the Plastino Hamiltonian, while  Plastino’s Hamiltonian is capable of reproducing all Lip-  kin states accessible by the Lipkin model and states that  10  demonstrate fermion-pair condensation, no dual conden-  sate character is observed from the Plastino model as the  region in which both λD and λG exceed one is not probed  within this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  In fact, as noted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  , there is direct competi-  tion between the particle-hole and particle-particle pair-  ing between Lipkin-like sites which results in each type  of pairing “driving” the system toward radically differ-  ent states with the magnitudes of the coupling constants  causing a transition between the Lipkin-like and BCS-like  states favored by the different interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Conversely,  because the particle-particle and particle-hole pairing in  the model we introduce do not occur between the same  orbitals, they can coexist, allowing for a much larger pos-  sible range of λD versus λG including the region demon-  strating a fermion-exciton condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' 8: A plot of λG versus λD where parameters in  the Plastino Hamiltonian are systematically varied for  N = 4 particles in r = 8 orbitals is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
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+page_content=' Waghmare, First-principles model hamil-  tonians for ferroelectric phase transitions, Ferroelectrics  151, 59 (1994), cited By 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
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+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Mei,  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
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+page_content=' Zhou, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
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+page_content='1038/s41467-021-21425-8 (2021), cited By 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Far´ıas and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Davis, Multiple metastable states  in  an  off-lattice  potts  model,  Physica  A  581,  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='physa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='126215 (2021), cited By 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
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+page_content=' Kafri, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Campbell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
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+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
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+page_content='  Poelmans,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Van  Raemdonck,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content='  Verstichel,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' De Baerdemacker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Torre, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
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+page_content=' Bultinck, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Van Neck, Vari-  ational optimization of the second-order density matrix  corresponding to a seniority-zero configuration interac-  tion wave function, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAyT4oBgHgl3EQfxvkc/content/2301.00670v1.pdf'}
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+Draft version January 12, 2023
+Typeset using LATEX twocolumn style in AASTeX63
+Simulations of Precessing Jets and the Formation of X-shaped Radio Galaxies
+Chris Nolting,1, 2 Jay Ball,2 and Tri Nguyen2
+1Los Alamos National Laboratory, Los Alamos, NM, USA
+2Department of Physics and Astronomy, College of Charleston, Charleston, SC, USA
+ABSTRACT
+Jet precession is sometimes invoked to explain asymmetries in radio galaxy (RG) jets and “X/S/Z-
+shape” radio galaxies, caused by the presence of a binary black hole companion to the source active
+galactic nucleus (AGN) or by accretion instabilities. We present a series of simulations of radio galaxy
+jet precession to examine how these sources would evolve over time, including a passive distribution
+of cosmic ray electrons (CRe) so we can model radio synchrotron emissions and create synthetic radio
+maps of the sources. We find that a single source viewed from different angles can result in differing
+RG morphological classifications, confusing physical implications of these classifications. Additionally,
+the jet trajectories can become unstable due to their own self-interactions and lead to “reorientation
+events” that may look like the effects of external dynamics such as shocks, winds, or cold fronts in the
+medium. Finally, something akin to an “Odd Radio Circle” may be observed in the case of viewing
+the radio remnant of such a precessing source from a line of sight near the precession axis.
+1. INTRODUCTION
+Radio galaxies consist of an active galactic nucleus
+(AGN) and a pair of anti-parallel jets of high temper-
+ature, low density, supersonic plasma that can propa-
+gate to scales larger than the host galaxy. In reality,
+and especially in denser environments like galaxy clus-
+ters, RGs are often distorted, with broken symmetries
+caused by their interactions with their immediate sur-
+roundings. These interactions can be related to motion
+of the host galaxy relative to a cluster or group cen-
+ter due to an orbit or due to bulk motions within the
+local medium itself (Begelman et al. 1979; Jones et al.
+2017). Another way for these symmetries to be broken
+is through nonuniform activity in the host AGN, either
+through variability in the jet properties or changes in its
+direction.
+One class of RG asymmetries that is common and be-
+ing studied more frequently is the so called “X-shaped”
+RG (Leahy & Parma 1992; Bhukta et al. 2022). These
+RGs have two pairs of radio lobes that are misaligned
+from each other. Sometimes one pair is designated as
+the “main lobes” and the other pair as “wings” or “side-
+lobes,” with the designation based on the detection of
+active radio jets or radio hot spots in the main lobes, or
+from surface brightness or spectral aging considerations.
+There have been multiple suggested theories for the
+formation of such structures.
+Some suggest that RG
+wings can form through the backflow of radio plasma
+that rebounds laterally off the hot gaseous halo of the
+host galaxy (Leahy & Williams 1984; Cotton et al. 2020;
+Ignesti et al. 2020). Others invoke a “spin-flip” in the
+AGN due to a merger event between a binary super-
+massive black hole (BSBH) system and a sudden reori-
+entation of the spin axis of the active nucleus (Zier &
+Biermann 2001; Gopal-Krishna et al. 2012). It may also
+be possible for both nuclei in a BSBH system to be ac-
+creting, in which case two sets of jets may exist. These
+may both be active during the same period or undergo-
+ing some cadence of activity depending on their respec-
+tive accretion histories, creating multiple sets of lobes
+(Gopal-Krishna et al. 2012).
+Lastly, and most relevant to what we will discuss here,
+is the idea that the jet axis of the RG may change or
+precess over time. This precession may lead to the for-
+mation of X-shaped RGs if the precession angle is large
+so that the current axis is well separated from the side
+lobes.
+Smaller angles may lead to “S-shaped” or “Z-
+shaped” RGs, which are designations sometimes given
+to RGs that show curvature or sharp turns in their jets
+or lobes (Riley 1972).
+There are many examples of S-shaped radio sources
+that could be fit by a projected helical precessing
+jet such as 4C35.06 in Abell 407 (Biju et al. 2014),
+J1328+2752 (Nandi et al. 2021), or Hydra A (Taylor
+et al. 1990). On smaller scales, SS433 is a striking ex-
+ample of a precessing X-ray binary jet system (Abell &
+Margon 1979; Monceau-Baroux et al. 2014). The pre-
+cession of AGN jets on large scales has been studied ex-
+tensively as a method of creating X-shaped RGs (Ekers
+et al. 1978; Rubinur et al. 2017). In particular, (mag-
+neto)hydrodynamical simulations have proven useful in
+arXiv:2301.04343v1  [astro-ph.HE]  11 Jan 2023
+
+2
+characterizing the evolution and morphology of such jets
+(Smith & Donohoe 2019; Horton et al. 2020; Giri et al.
+2022).
+Precession of an RG jet may occur for a number of
+reasons. Again invoking a BSBH system, if either nu-
+cleus is substantially accreting and hosts an active radio
+jet, then the jet may precess around the orbital axis of
+the BSBH system, assuming their is a misalignment be-
+tween the orientation of the binary orbit and the black
+hole spin axis (Begelman et al. 1980).
+Alternatively,
+precession may be related to accretion instabilities such
+as Lense-Thirring or the related Bardeen Peterson effect
+(Bardeen & Petterson 1975; Nandi et al. 2021). These
+instabilities may cause the precession of the accretion
+disk if its angular momentum is misaligned with the an-
+gular momentum of the central spinning black hole. If
+the disk orientation controls the jet axis, then this would
+induce a precession in the jet axis. This precession mech-
+anism is seen in some general relativistic accretion disk
+simulations (e.g., Liska et al. 2018).
+Regardless of the mechanism for the precession of the
+jet, we present the results of simulations in which we
+assume jets do precess, and discuss the consequences
+of that precession for the radio observables from these
+simulated RGs. One of our main goals is to use these
+simulations to invert this process and be better able to
+understand the dynamical state of observed radio jets
+that appear to be precessing.
+The remainder of this paper is organized as follows:
+Section 2 outlines the geometry of the interaction we
+are examining and the underlying physics. Section 3 de-
+scribes the simulations, including methods and specific
+parameters for each simulation. We describe the results
+of the simulations in section 4 and summarize the major
+findings in section 5.
+2. JET PHYSICS
+2.1. Jet Precession
+The basic geometry of the type of interactions we ex-
+plore in this paper are illustrated in Figure 1. The jet is
+launched from a cylindrical region in which the cylinder
+axis is tilted by ψ degrees from the z-axis of the sim-
+ulation, which we will refer to as the precession angle.
+The other angles in Figure 1 refer to the viewing angles
+used in the synthetic radio observations we present in
+Section 4. θ is the spherical coordinate polar angle, also
+measured from the z-axis.
+Observations are taken at
+four θ angles: 90◦, 45◦, 30◦, and 0◦. φ is the azimulthal
+angle, and observations are taken at four φ angles from
+0◦ ≤ φ ≤ 135◦ in ∆φ = 45◦ intervals.
+ψ
+Δϕ = 45∘
+θ = 90∘
+θ = 45∘
+θ = 30∘
+Figure 1. Diagram of the jet precession. The precession
+angle ψ is the angle between the jet axis and the precession
+axis, and the diagram uses a fiducial value of ψ = 30◦. The
+polar viewing angle, θ, is the angle between the precession
+axis and the viewing angle, with angles used in Figure 3
+shown. Not shown are the azimuthal viewing angles which
+are rotated around the precession axis in increments of ∆φ =
+45◦. A cyan arrow indicates the direction that the jet axis
+precesses.
+The jet will precess through a full 2π radians within
+one precession period, τ. This can also be expressed as
+an angular velocity of precession, ωprec = 2π/τ.
+2.2. Jet Propagation & Bending
+As the jet precesses, it will bend due to its interaction
+with the medium. The character of this bending will be
+determined by a combination of the momentum in the
+jet and the properties of the precession of its launching
+angle.
+The forward propagation of a jet can be described by
+a momentum balance in the frame of the head of the
+jet. As derived in Jones et al. (2017), we find that in
+the absence of relative motion between the surrounding
+medium and the jet source, the propagation of the head
+
+3
+of the jet will follow:
+Mh ≈ Mj
+�
+Aj
+Aa
+�
+Pj
+Pa
+,
+(1)
+where Mh = vh/cs,a is the Mach number of the head of
+the jet propagating at velocity vh relative to the ambi-
+ent medium that has a sound speed of cs,a =
+�
+γPa/ρa.
+Mj
+= vj/cs,j is the Mach number of the material
+within the jet that has velocity vj and has an internal
+sound speed of cs,j =
+�
+γPj/ρj. P(j/a) and ρ(j/a) are
+the thermal pressure and density of the (jet/ambient)
+plasma, respectively, and we use an adiabatic index
+of γ = 5/3.
+�
+Aj/Ah is a ratio of “effective areas”
+over which the momentum balance is taken for the
+jet and ambient plasma.
+In previous simulations of
+steady, non-precessing jets, we have empirically found
+�
+Aj/Ah ∼ 1/2 (Jones et al. 2017). This ratio departs
+from unity mainly due to the existence of backflow of
+jet plasma away from the region of the jet head. In the
+case where the jet is in pressure equilibrium with its sur-
+roundings, i.e. Pj = Pa, Equation 1 can be expressed
+as:
+vh ≈ vj
+�
+Aj
+Ah
+� ρj
+ρa
+,
+(2)
+which is a useful metric when the jet head crosses a
+density discontinuity.
+When there is some relative motion between the jet
+and its environment, the propagation will be affected.
+If that motion has a component transverse to the jet
+propagation direction, then the jet may be deflected or
+bent by the ram pressure it experiences.
+The classic
+“cartoon” formula for the distance, ℓb, over which the
+jet is bent backwards by a “wind” is (Begelman et al.
+1979; Jones et al. 2017):
+ℓb ∼ 2rj
+ρjv2
+j
+ρav2a
+,
+(3)
+where rj is the cross-sectional radius of the jet and
+va is the ambient velocity that provides the ram pres-
+sure to bend the jet. Equation 3 assumes a constant
+va, but if we assume this “wind” comes from the pre-
+cession of the jet through a uniform medium at rest,
+then the relevant ambient velocity is defined by the
+angular velocity of the precession of the jet source.
+va(y) = ωprec · y sin ψ = 2πy sin ψ/τ, where y is the
+distance along the jet direction from the jet source to
+the point of interest. Since the ambient velocity varies
+along the jet, we need to revisit the arguments that lead
+to equation 3.
+Usually, this begins by assuming that
+the ram pressure acts to redirect the jet, without com-
+pressing or accelerating the jet plasma. This leads to an
+incompressible Euler equation:
+Dvj
+Dt =
+
+7
+0
+∂vj
+∂t + (vj · ∇)vj = − 1
+ρj
+∂P
+∂x ≈ − 1
+ρj
+ρav2
+a
+2rj
+,
+(4)
+where we assume that the jet bending leads to a steady
+state (∂vj/∂t = 0) and that the ram pressure, P =
+ρav2
+a, acts across the diameter of the jet, 2rj. If we look
+at the component of the jet velocity perpendicular to
+the jet launching axis (vj,⊥) and integrate along the jet
+length until the jet has been bent into the direction of
+the “wind,” i.e. vj,⊥ = vj:
+� vj
+0
+dv =
+1
+2rj
+ρa
+ρjvj
+�2π sin ψ
+τ
+�2 � ℓb
+0
+y2dy,
+(5)
+ℓb ∼
+�
+rj
+3ρjv2
+j τ 2
+2π2ρa sin2 ψ
+�1/3
+,
+(6)
+showing that the bending length of the jet will be a
+function of the jet properties, the precession angle and
+period, and the density of the medium.
+Now, using Equation 2, we can compare this length to
+the distance that the head of the jet would travel before
+the precession has significantly changed its launching
+direction (ignoring for now how the bending affects this
+propagation). For this duration, we will take the frac-
+tion of the total period that it takes the core of the jet
+to transverse one jet diameter. This will depend on the
+shape and size of the jet launching cylinder:
+∆t = 2rj/(2π∆y sin ψ),
+(7)
+where ∆y is the half length of the cylinder (from the
+center to one end). The ratio ℓb/(vh∆t) is then a good
+measure of whether or not the jet will create a well de-
+fined curved jet, or if the precession will be too extreme
+and cause the jet to break up or be unable to propagate
+to large scales.
+3. NUMERICAL METHODS
+The simulations reported here used the Eulerian
+WOMBAT ideal 3D MHD code (see, e.g.
+Mendygral
+et al. (2012); Nolting et al. (2019)) on a uniform, Carte-
+sian grid employing an adiabatic equation of state with
+γ = 5/3. The simulations utilized the 2nd order TVD al-
+gorithm with constrained transport (CT) magnetic field
+evolution as in Ryu et al. (1998). Specific simulation
+setups are introduced in §3.1 and listed in Table 1.
+Along with the fluid, we track a population of pas-
+sive cosmic ray electrons (CRe) to allow for the cal-
+culation of a synchrotron emissivity anywhere within
+
+4
+the simulation volume1. The CRe momentum distribu-
+tion, f(p), was tracked using the conservative, Eulerian
+“coarse grained momentum volume transport” CGMV
+algorithm in Jones & Kang (2005). f(p) spanned the
+range 10 ≲ p/(mec) ≈ Γe ≲ 1.7 × 105 (so, energies 5
+MeV ≲ ECRe ≈ Γemec2 ≲ 90 GeV) with uniform log-
+arithmic momentum bins, 1 ≤ k ≤ 8. Inside a given
+momentum bin, k, f(p) ∝ p−qk, with qk being bin de-
+pendent and evolving in time and space. Γe here repre-
+sents CRe Lorentz factors.
+At the jet cylinder, the CRe momentum distribution
+had a power law form with index q = q0 = 4.2 across the
+full momentum range simulated. This was chosen be-
+cause this translates to a radio synchrotron spectral in-
+dex of α = α0 = 0.6 (Iν ∝ ν−α), which is a good match
+for many RGs near their sources. The synchrotron emis-
+sion and spectra reported here are calculated numeri-
+cally using f(p) over the full momentum range speci-
+fied above using the standard synchrotron emission ker-
+nel for isotropic electrons in a local magnetic field (e.g.,
+Blumenthal & Gould 1970). In our analysis we calcu-
+lated synthetic synchrotron emission at frequencies 300
+MHz ≤ ν ≤ 600MHz. This emission, as it turns out,
+comes predominantly from regions with magnetic field
+strengths ∼ a few µG, so mostly reflect CRe energies ≳
+1 GeV (Γe ∼ 103–104) (well inside our distribution).
+We included adiabatic, as well as radiative (syn-
+chrotron and inverse Compton) CRe energy changes out-
+side of shocks, along with test-particle diffusive shock
+(re)acceleration (DSA) at any shocks encountered. We
+did not include 2nd order turbulent CRe reacceleration
+or CRe energy losses from Coulomb collisions with am-
+bient plasma.
+The former depends on uncertain ki-
+netic scale turbulence behaviors beyond the scope of this
+study, while the latter is most relevant for CRe with
+energies well below those responsible for the radio syn-
+chrotron emission computed in this work (Sarazin 1999).
+CRe radiative losses combine synchrotron with inverse
+Compton (iC) scattered CMB radiation. The simula-
+tions reported here assumed a low redshift, z = 0.02.
+The resulting radiative lifetime can be written
+τrad ≈ 215
+1
+Γe4 [1 + B2
+3.4] Myr,
+(8)
+where Γe4 = Γe/104 and B3.4 = B/(3.4µG). The first
+term in the denominator on the RHS reflects inverse
+Compton (iC) losses at z = 0.02, while the second rep-
+1 Except for a negligible ICM population included to avoid numer-
+ical singularities in the CRe transport algorithm, all CRe were
+injected onto the computational domain via the jet launch cylin-
+der.
+resents synchrotron losses. Thus, we can see that for
+Γe ∼ 104, of primary interest for the radio emission in
+this work, τrad ∼ 200 Myr, and that iC losses are pre-
+dominant.
+DSA of the CRe was implemented at shock passage by
+setting qk,out = min(qk,in, 3σ/(σ−1)) immediately post-
+shock, where σ is the code-evaluated compression ratio
+of the shock. This simple treatment is appropriate in the
+CRe energy range covered, since likely DSA acceleration
+times to those energies are much less than a typical time
+step in the simulations (∆t ≳ 104 yr). Since our CRe
+have no dynamical impact, we treat the total CRe num-
+ber density, nCRe, as arbitrary. Consequently, while we
+compute meaningful synchrotron brightness and spec-
+tral distributions from our simulations, synchrotron in-
+tensity normalizations are arbitrary.
+3.1. Simulation Setups
+In this work we present a suite of simulations that ex-
+plore the parameter space of jet precession angles and
+precession periods.
+Each simulation consists of a jet
+with constant density, pressure, and velocity being in-
+jected into a medium that is initially homogeneous and
+unmagnetized. This medium is an oversimplification of
+the intracluster medium (ICM), but it is useful for easier
+interpretation of the observable signatures of the preces-
+sion that we wish to study. The dynamics of the sim-
+ulations we present here are scale free, however a scale
+is set when introducing CRe radiative timescales (e.g.,
+Equation 8). Because of this, we will quote the scale of
+the system that is consistent with the chosen radiation
+timescales.
+Inside the simulation volume, a cylindrical region was
+updated at each time step with the jet properties de-
+scribed in Section 3.1.
+This cylinder had a radius of
+rj = 3kpc and length lj = 4kpc. The jet cylinder was
+surrounded by a 2 zone coaxial collar, within which the
+state transitioned from the jet properties to the local
+ambient conditions.
+Inside, the jet density and pres-
+sure were kept constant, and the velocity was ramped
+up along the cylinder’s length from its midpoint (where
+the velocity reverses).
+A toroidal magnetic field was
+also maintained withing the jet cylinder, with Bφ =
+Bj(r/rj)ˆφ. In these jets, βp = 8πPj/B2
+j = 75, giving
+the magnitude of the field at launch to be Bj ≈ 0.54µG.
+Due to this relatively high βp, the fields are initially
+dynamically sub-dominant to the gas pressure, which
+remains true throughout the simulation duration in all
+cases.
+To create the precession of the jet axis, the jet is ini-
+tialized ψ degrees away from the precession axis, defin-
+ing the precession angle. Every time step, the jet cylin-
+
+5
+Simulation
+Name
+Nx
+Ny
+Nz
+Precession
+Period
+Precession
+Angle
+P3A10
+288
+288
+624
+3.2Myr
+10◦
+P3A20
+288
+288
+624
+3.2Myr
+20◦
+P3A30
+288
+288
+624
+3.2Myr
+30◦
+P3A45
+360
+600
+600
+3.2Myr
+45◦
+P32A10
+288
+288
+624
+31.7Myr
+10◦
+P32A20
+288
+288
+624
+31.7Myr
+20◦
+P32A30
+288
+288
+448
+31.7Myr
+30◦
+P32A45
+360
+600
+600
+31.7Myr
+45◦
+P95A10
+288
+288
+1040
+95.1Myr
+10◦
+P95A20
+288
+288
+624
+95.1Myr
+20◦
+P95A30
+288
+288
+448
+95.1Myr
+30◦
+P95A45
+360
+600
+600
+95.1Myr
+45◦
+Table 1. Names, domain size, and precession parameters
+for each simulation.
+der is rotated azimuthally around the precession axis by
+∆φ = 2π∆t/τ, where ∆t is the length of the time step
+and τ is the jet precession period.
+Each simulation had a uniform initial density of
+ρICM = 2.17 × 10−28 g cm−3, and pressure of PICM =
+8.8 × 10−13 dynes cm−2 and was initially unmagnitized
+and at rest with respect to the jet source. This gave the
+medium an initial temperature of 2.5keV.
+Table 1 gives a list of parameters for the 12 simu-
+lations, including domain size, precession period, and
+precession angle. Three precession periods and four pre-
+cession angles were included in this study. Simulation
+names include information about the precession period
+(number following “P” in the name) and the precession
+angle (number following “A” in the name) for ease of
+understanding when referring to individual simulations.
+3.2. Synthetic Observations
+The radio images we report here are calculated from
+the CRe momentum distribution f(p) multiplied by the
+appropriate synchrotron emissivity functions integrated
+across the full momentum range, as specified above.
+This emissivity takes into account the strength and ori-
+entation of the local magnetic field, as well as the his-
+tory of the CRe energy changes (radiative, adiabatic,
+& DSA). The emissivity is calculated as (Ginzburg &
+Syrovatskii 1965; Longair 2011):
+j⊥(ν) =
+�
+ννB⊥
+8
+e2
+c
+� x2
+x1
+n(x)[F(x) + G(x)]
+x3/2
+dx ,
+(9a)
+j∥(ν) =
+�
+ννB⊥
+8
+e2
+c
+� x2
+x1
+n(x)[F(x) − G(x)]
+x3/2
+dx ,
+(9b)
+j(ν) = j⊥(ν) + j∥(ν) =
+�
+ννB⊥
+2
+e2
+c
+� x2
+x1
+n(x)F(x)
+x3/2
+dx ,
+(9c)
+F(x) = x
+� ∞
+x
+K5/3(z)dz ≈ 1.78
+×
+�
+x
+1 − 0.4 exp (−5x)
+�1/3
+exp (−x) ,
+(9d)
+G(x) = xK2/3(x) ≈ 1.56613 ×
+x1/3
+exp (x) + 0.427687 ,
+(9e)
+where F(x) and G(x) are functions that describe the
+synchrotron spectrum of a single CRe, K5/3 and K2/3
+are modified Bessel functions of the second kind, j⊥(ν)
+and j∥(ν) are the two orthogonal polarizations of the
+emissivity, ν is the frequency of the radiation, νB⊥ =
+eB⊥/2πmec is the electron gyrofrequency, and e is the
+elementary charge. In this case, B⊥ is the local mag-
+netic field projected onto the plane of the sky. The in-
+tegration variable x = (2ν/3νB⊥γ2), with γ being the
+electron Lorentz factor.
+The integration limit x1[x2]
+corresponds to the high[low] energy (high[low] γ) end
+of the integration range, covering our available range
+10 < p/(mec) < 1.7 × 105. A change of variables from
+integrating over γ to integrating over x introduces the
+factor of x−3/2 in the integrand and a coefficient of
+(−1/2)
+�
+2ν/3νB⊥. We also present these equations in
+Gaussian units, rather than the SI units used by Lon-
+gair (2011). Equations (9d) and (9e) include our own
+approximations to the synchrotron functions used in our
+calculations. These approximations are based on those
+given in Rybicki & Lightman (1986), but combine the
+small x and large x approximations to produce a single
+fit that is accurate to a few percent (Nolting 2020).
+Using equations (9a) - (9e) we calculate the total
+synchrotron emissivity as well as polarized emissivities.
+Then, we perform radiative transfer integration along a
+defined line of sight to create images of Stokes I, Q, or
+U.
+A radio spectral index is calculated from any two ra-
+dio maps at different frequencies with the radio spectral
+index, α (e.g., f ∝ να), at each pixel being
+α = log10(I(ν2) − log10(I(ν1)))
+log10(ν2) − log10(ν1)
+.
+(10)
+In this work, the two frequencies used to generate spec-
+tral index maps were ν2 = 600 MHz and ν1 = 300 MHz.
+
+6
+Lastly, all radio images presented here are convolved
+with a gaussian convolution kernel.
+The resolution
+for most images is 2.355 arcsec Full Width Half Max
+(FWHM), or a gaussian standard deviation of 1 arcsec-
+ond, while one figure is convolved to a lower resolution
+of 11.775 arcsec FWHM, or 5 arcseconds gaussian stan-
+dard deviation.
+4. DISCUSSION
+4.1. Effects of Precession Period and Precession Angle
+To study the effects of precession period and preces-
+sion angle, we ran a total of 12 simulations varying
+these parameters. The resulting morphological and ra-
+dio spectral properties from these simulations are pre-
+sented in figure 2.
+We show each simulation at time
+equal to one precession period as well as at t = 96
+Myr for comparison. The labels at the bottom of each
+panel describe which simulation is shown and the simu-
+lation time represented. The simulations in the left two
+columns have a precession period of 3.2Myr, the 3rd and
+4th column have a precession period of 31.7 Myr, and the
+5th column has a precession period of 95.1 Myr. When
+comparing the jets all at the same dynamical stage (after
+1 precession period) the reader should examine columns
+1, 3, and 5 together. When comparing at the same time,
+the reader should instead compare columns 2, 4, and 5.
+As Figure 2 demonstrates, these systems evolve quite
+differently in terms of their dynamics.
+The jets with
+large precession angles do not extend as far in any one
+direction and are instead spread over a wider area, with
+cocoons of larger lateral extent.
+Similarly, jets with
+shorter precession periods are also more centrally con-
+densed as they are unable to deposit momentum in a
+sustained direction long enough to substantially push
+back the denser ICM plasma.
+We can quantify the changes in dynamics using the ra-
+tio ℓb/(vh∆t), as described in section 2.2. With our cho-
+sen jet injection parameters and equation 2, the propa-
+gation rate of the head of the jet through the ambient
+medium was 3.45 kpc/Myr. Given, this, we calculated
+the ratio ℓb/(vh∆t) in Table 2. By comparing the value
+of this ratio to the corresponding images in Figure 2, we
+can see that precessing jets with higher values of this ra-
+tio will be more tightly wound and generally have prop-
+agated less far from the source. Conversely, precessing
+jets with lower values of this ratio will be less tightly
+wound and propagate farther before curving and bend-
+ing backward. While this ratio does not have a clear
+critical value that determines when a jet will break up
+from precession or display a clear s-shape morphology, it
+is still useful for comparing how different precession pa-
+rameters lead to differing morphology. In particular, the
+dependencies on τ and ψ are ℓb/(vh∆t) ∝ (sin ψ/τ)1/3.
+Taking this ratio, as well as the general trends with
+precession period from Figure 2, it is clear that as the
+precession period decreases, the jet cocoons are more
+condensed and the jets more dramatically wound up. In
+simulations not presented here with shorter periods, the
+jet began to break up in our simulations.
+These jets
+may even have trouble breaking out of the interstellar
+medium of the host galaxy, which we do not include in
+these simulations.
+Material surrounding the jet source is a combination
+of jet backflow during the initial stages of jet launch-
+ing as well as material that has bled off from the jet
+as it precesses around. This material at late times ra-
+diatively steepens to α ≈ −1.5, while fresh material in-
+jected by the precessing jet remains spectrally flat. In
+the cases with large precession angle, there is more sig-
+nificant mixing of the older and newer jet material, as
+the cocoons are smaller and the jet injection covers a
+larger solid angle during their precession.
+Examining the 5th column of Figure 2, with preces-
+sion period P ∼ 95 Myr, we can see that as the pre-
+cession period begins to approach the cooling timescale
+for the CRe, the surface brightness of the material away
+from the current jet direction drops off. If the period
+becomes too long, the CRe will age enough that they
+may no longer be detectable. This would leave the ob-
+server with little or no evidence of precession, and we
+would undercount precessing sources on the long period
+end of the distribution. However, only the longest lived
+AGN will reach such sustained jet ages, with most hav-
+ing lifetimes > 100 Myr (e.g., Turner & Shabala 2015).
+If the jet precession has such a long period but isn’t
+active for a large enough fraction of the precession pe-
+riod, the source will likely be indistinguishable from a
+non-precessing source. Because of this, radiative cooling
+may not be a major constraint on observing precessing
+radio jets.
+4.2. Effects of Viewing Angle
+Within a single snapshot of a precessing jet system, its
+radio morphology may appear significantly different de-
+pending on the viewing angle. Figure 3 explicitly shows
+this for the P32A30 simulation. Each panel within this
+figure shows the P32A30 simulation at the same time (82
+Myr), but from 12 perspectives. In the online version,
+this Figure is animated to show the evolution of this
+simulation in each of these 12 viewing angles simultane-
+ously. At any given frame in the animation, one can see
+how the morphology changes with viewing angle. The
+differences are so stark that there are frames in which
+
+7
+200
+150
+100
+50
+0
+50
+100
+150
+kpc
+   P3A10 - 3 Myr    P3A10 - 96 Myr    P32A10 - 32 Myr    P32A10 - 96 Myr    P95A10 - 96 Myr 
+150
+100
+50
+0
+50
+100
+150
+kpc
+   P3A20 - 3 Myr    P3A20 - 96 Myr    P32A20 - 32 Myr    P32A20 - 96 Myr    P95A20 - 96 Myr 
+100
+50
+0
+50
+100
+kpc
+   P3A30 - 3 Myr    P3A30 - 96 Myr    P32A30 - 32 Myr    P32A30 - 96 Myr    P95A30 - 96 Myr 
+50
+0
+50
+kpc
+100
+50
+0
+50
+100
+kpc
+   P3A45 - 3 Myr   
+50
+0
+50
+kpc
+ P3A45 - 96 Myr   
+50
+0
+50
+kpc
+ P32A45 - 32 Myr   
+50
+0
+50
+kpc
+ P32A45 - 96 Myr   
+50
+0
+50
+kpc
+ P95A45 - 96 Myr 
+1.0
+0.9
+0.8
+0.7
+0.6
+0.5
+Spectral Index (600-300 MHz)
+Figure 2. Radio spectral index (600-300MHz, cut at 1µJy beam−1 at 300MHz) with 300 MHz radio brightness contours (levels
+= [1, 10, 100]µJy beam−1) for each simulation presented here. Radio images are convolved to 1” resolution. Each row shows
+simulations with different precession angles (row 1: 10◦, 2: 20◦, 3: 30◦, 4: 45◦). Columns show simulations with differing
+precession periods, or the same simulations at different times (period for column 1 & 2: 3.2 Myr, 3 & 4: 31.7 Myr, 5: 95.1 Myr)
+(column 1, 3, & 5 are shown at approximately 1 precession period; columns 2, 4, & 5 are shown at 96 Myr simulation time).
+
+8
+−50
+0
+50
+kpc
+θ = 90◦
+φ = 0◦
+−50
+0
+50
+kpc
+θ = 90◦
+φ = 45◦
+−50
+0
+50
+kpc
+θ = 90◦
+φ = 90◦
+−50
+0
+50
+kpc
+−50
+0
+50
+kpc
+θ = 90◦
+φ = 135◦
+θ = 45◦
+φ = 0◦
+θ = 45◦
+φ = 45◦
+θ = 45◦
+φ = 90◦
+−50
+0
+50
+kpc
+θ = 45◦
+φ = 135◦
+t = 82 Myr
+θ = 30◦
+φ = 0◦
+θ = 30◦
+φ = 45◦
+θ = 30◦
+φ = 90◦
+−50
+0
+50
+kpc
+θ = 30◦
+φ = 135◦
+−1.0
+−0.9
+−0.8
+−0.7
+−0.6
+−0.5
+Spectral Index (600-300 MHz)
+Figure 3.
+Radio spectral index (600-300MHz, cut at 1µJy beam−1 at 300MHz) with 300 MHz radio brightness contours
+(levels = [1, 10, 100]µJy beam−1) for the P32A30 simulation at t = 82 Myr from a variety of viewing angles. Radio images are
+convolved to 2.355” FWHM resolution. Viewing angles are listed in the inset for each panel. For ease of visualization, the polar
+viewing angles are included in the diagram Figure 1. The polar angle is varied by column while the azimuthal angle is varied
+by row. In the online version, an animated figure is available (8 seconds, spanning 100 Myr simulation time) that follows the
+evolution of the source from each viewing angle simultaneously.
+
+9
+Simulation
+Name
+∆t
+vh∆t
+ℓb
+ℓb
+vh∆t
+P3A10
+2.93Myr
+10.1kpc
+19.5kpc
+1.9
+P3A20
+1.5Myr
+5.2kpc
+12.4kpc
+2.38
+P3A30
+1.02Myr
+3.52kpc
+9.6kpc
+2.73
+P3A45
+0.72Myr
+2.5kpc
+7.6kpc
+3.04
+P32A10
+29.1Myr
+100.4kpc
+89.7kpc
+0.89
+P32A20
+14.8Myr
+51.1kpc
+57.1kpc
+1.12
+P32A30
+10.1Myr
+34.9kpc
+44.3kpc
+1.27
+P32A45
+7.1Myr
+24.5kpc
+35.2kpc
+1.44
+P95A10
+87.2Myr
+300.8kpc
+186.6kpc
+0.62
+P95A20
+44.3Myr
+152.8kpc
+118.8kpc
+0.78
+P95A30
+30.3Myr
+104.5kpc
+92.2kpc
+0.88
+P95A45
+21.4Myr
+73.8kpc
+73.2kpc
+0.99
+Table 2. Values of jet bending analysis variables from sec-
+tion 2. ∆t comes from Equation 7, vh from Equation 2, and
+ℓb refers to the definition in Equation 6.
+the radio source would very likely be classified in differ-
+ent ways, and lead to a misunderstanding of the physics
+of the source’s evolution. For example, in the frame at
+t = 82 Myr, the first six frames (counting from left to
+right, top to bottom) and the ninth frame show signs of
+precession with curved jets. In the 7th and 10th frames,
+the jet appears mostly straight, with some deflection at
+the farthest points. In the 8th frame 11th frame, the jets
+appear to have been disrupted or sharply bent by some
+(perhaps external) dynamics. And the radio source in
+the 12th would likely be unclassifiable. By providing the
+animation of the evolution of this source from many an-
+gles, we hope that radio observers can compare these
+images to detected radio galaxies to help identify possi-
+ble precessing sources. We also hope to display the full
+range of complex morphologies that precessing jets may
+take on.
+4.3. Jet Reorientation Events
+In each simulation, the precessing jets would some-
+times undergo a dramatic and sudden change in their
+propagation direction.
+This “reorientation event” oc-
+curred roughly after one precession period had elapsed
+in the simulation.
+As the jet precesses initially, it
+bends strongly due to ram pressure from its interaction
+with the ICM. However, after one precession period has
+elapsed, the jet encounters a region of space in which it
+has previously deposited low density plasma. As the jet
+encounters this density discontinuity, the velocity of the
+head of the jet suddenly increases, as can be inferred
+from Equation 2. Additionally, the ram pressure that
+leads to the bending of the jet is greatly reduced, leading
+to an increase in the bending length (Equation 6). Both
+50
+0
+50
+0
+50
+100
+150
+t = 34.0 Myr
+1.0
+0.8
+0.6
+Spectral Index (600-300 MHz)
+Figure 4. Radio spectral index (600-300MHz, cut at 1µJy
+beam−1 at 300MHz) with 300 MHz radio brightness contours
+(levels = [1, 10, 100]µJy beam−1) for the P32A30 simulation
+at t = 34 Myr. Radio images are convolved to 2.355” FWHM
+resolution. At this time, the jet is undergoing a “jet reori-
+entation event” in which the jet trajectory changes by more
+than 90◦ over about 2.5 Myr. This can be seen here as the
+very sharp bend at the tip of the jet head at coordinates (x,y)
+≈ (-5, 80) kpc. In the online version, an animated figure (5
+seconds) shows the full duration of the reorientation event
+from t = 30 Myr until t = 38 Myr.
+of these effects lead to the jet becoming straighter and
+propagating to further distances before it then begins
+to interact with the higher density ICM when it reaches
+the outskirts of the low density cavity. This process is
+illustrated in Figure 4, which shows the beginning of
+the reorientation event, in which the jet trajectory sud-
+denly changes by > 90◦. This type of sudden change
+in jet propagation direction is often attributed to either
+changes in the jet properties or duty cycle, or to external
+cluster dynamics such as shocks or other waves travel-
+ing through the medium. However, none of these effects
+are included in our simulations. These jet reorientation
+events can come about from jet precession alone, and the
+self interaction of the jet with its own previous activity.
+4.4. Odd Radio Circles (ORCs)
+Odd Radio Circles (ORCs) are recent and mysterious
+radio sources. They are circles of steep spectrum, dif-
+fuse radio emission. Some (but not all) ORCs have a
+detected galaxy near their center. There have been a
+variety of possible physical explanations as to their ori-
+gin, including a spherical shock from a starburst wind,
+a supermassive black hole merger, or radio galaxy lobes
+
+10
+50
+0
+50
+50
+25
+0
+25
+50
+t = 98 Myr
+50
+0
+50
+t = 140 Myr
+1.4
+1.2
+1.0
+0.8
+0.6
+Spectral Index (600-300 MHz)
+Figure 5. Radio spectral index (600-300MHz, cut at 100µJy beam−1 at 300MHz) with 300 MHz radio brightness contours
+(levels = [100, 400, 1600]µJy beam−1) for the P32A30 simulation at 98 Myr (left) and 140 Myr (right) both viewed from above
+the jet precession axis (θ = 0◦). Radio images are convolved to 11.775” FWHM resolution. The jet turns off at 100 Myr and
+the plasma is allowed to radiatively cool. In the online version, a single-panel animated figure (10 seconds) follows the evolution
+of the source from the beginning of the simulation until t = 160 Myr.
+seen end-on. Our simulations may lend some support to
+this last suggestion. Figure 5 shows the P3A30 simula-
+tion along a line of sight parallel to the precession axis
+at two times. The jets turn off just after the time in
+the left panel. The right panel shows the radio emission
+about 40 Myr later. The radio morphology is similar to
+that of ORCs. In the online version of this article, an
+animated version of this figure is available, which shows
+the evolution up from 0-160 Myr, with the jet power-
+ing off around t=100 Myr. Just after the jet turns off,
+there is a full ring of emission visible. However, the ra-
+dio spectrum is still relatively flat with α ≈ −0.8. The
+radio ring ages in place, without expanding further, as
+it is in pressure balance with its surroundings and the
+jets are no longer injecting momentum into the region.
+As the plasma radiatively ages and spectrally steepens,
+the ring becomes broken. There is a period of time in
+which the ring is still mostly intact and visible and it
+approaches the spectral index observed for some ORCs
+α ≈ −1.2 (Norris et al. 2022).
+If this mechanism for explaining some ORCs holds,
+then it may be interesting to reexamine existing ob-
+servations of S-shaped RGs and burn out the images
+looking for regions of low level emission surrounding the
+visible structures as a possible analog to the existing
+ORCs. This low level emission may not appear as circu-
+lar from viewing angles misaligned from the precession
+axis, but it may represent the same material. Addition-
+ally, searches for AGN counterparts in galaxies near the
+centers of ORCs or VLBI observations looking for any
+current small scale jets could help determine if such a
+formation mechanism is possible.
+5. SUMMARY
+We have presented a series of simulations that look
+into the properties of precessing radio jets. These jets
+take on a wide variety of morphologies, depending on the
+properties of the precession itself as well as on the angle
+at which the source is viewed. In particular, the effects
+of viewing angle can greatly affect the classification of
+these sources. In many viewing angles, precession is not
+an obvious mechanism, as the jet may appear straight in
+projection or with very complex and messy morphology.
+Additionally, we observed some interesting and un-
+expected dynamics in our simulations, including self in-
+duced “reorientation events,” in which the jet trajectory
+quickly changed as the jet encounters a region filled with
+previous jet material of lower density than the ambient
+medium. This led to sharp turns in the jet which could
+be misinterpreted as due to external dynamics affecting
+the jet.
+Lastly, we point out the similarities between ORCs
+and the remnants of a precessing jet seen end–on. This
+could be a possible explanation for some ORCs. Fol-
+lowup observations of existing ORCs looking AGN coun-
+terparts could help determine if this is likely.
+
+11
+ACKNOWLEDGMENTS
+CN acknowledges funding through the NSF grant
+AST-1907850 as well as from Los Alamos National Lab-
+oratory through the LDRD program and NASA pro-
+grams through the Astrophysical Theory Program. JB
+and TN were supported by NSF grant AST-1907850.
+We thank L. Rudnick, T. Jones, and P. C. Fragile for
+useful discussions. The simulations presented here were
+run and analyzed at the College of Charleston on their
+high-performance Linux cluster (https://hpc.cofc.edu)
+as well as utilizing the Los Alamos National Laboratory
+Institutional Computing Program.
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diff --git a/j9E3T4oBgHgl3EQfJQn8/content/tmp_files/load_file.txt b/j9E3T4oBgHgl3EQfJQn8/content/tmp_files/load_file.txt
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@@ -0,0 +1,742 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf,len=741
+page_content='Draft version January 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' 2023  Typeset using LATEX twocolumn style in AASTeX63  Simulations of Precessing Jets and the Formation of X-shaped Radio Galaxies  Chris Nolting,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' 2 Jay Ball,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='2 and Tri Nguyen2  1Los Alamos National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Los Alamos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' NM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' USA  2Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' College of Charleston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Charleston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' SC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' USA  ABSTRACT  Jet precession is sometimes invoked to explain asymmetries in radio galaxy (RG) jets and “X/S/Z-  shape” radio galaxies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' caused by the presence of a binary black hole companion to the source active  galactic nucleus (AGN) or by accretion instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' We present a series of simulations of radio galaxy  jet precession to examine how these sources would evolve over time, including a passive distribution  of cosmic ray electrons (CRe) so we can model radio synchrotron emissions and create synthetic radio  maps of the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' We find that a single source viewed from different angles can result in differing  RG morphological classifications, confusing physical implications of these classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Additionally,  the jet trajectories can become unstable due to their own self-interactions and lead to “reorientation  events” that may look like the effects of external dynamics such as shocks, winds, or cold fronts in the  medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Finally, something akin to an “Odd Radio Circle” may be observed in the case of viewing  the radio remnant of such a precessing source from a line of sight near the precession axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' INTRODUCTION  Radio galaxies consist of an active galactic nucleus  (AGN) and a pair of anti-parallel jets of high temper-  ature, low density, supersonic plasma that can propa-  gate to scales larger than the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' In reality,  and especially in denser environments like galaxy clus-  ters, RGs are often distorted, with broken symmetries  caused by their interactions with their immediate sur-  roundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' These interactions can be related to motion  of the host galaxy relative to a cluster or group cen-  ter due to an orbit or due to bulk motions within the  local medium itself (Begelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Another way for these symmetries to be broken  is through nonuniform activity in the host AGN, either  through variability in the jet properties or changes in its  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  One class of RG asymmetries that is common and be-  ing studied more frequently is the so called “X-shaped”  RG (Leahy & Parma 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Bhukta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' These  RGs have two pairs of radio lobes that are misaligned  from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Sometimes one pair is designated as  the “main lobes” and the other pair as “wings” or “side-  lobes,” with the designation based on the detection of  active radio jets or radio hot spots in the main lobes, or  from surface brightness or spectral aging considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  There have been multiple suggested theories for the  formation of such structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Some suggest that RG  wings can form through the backflow of radio plasma  that rebounds laterally off the hot gaseous halo of the  host galaxy (Leahy & Williams 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Cotton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Ignesti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Others invoke a “spin-flip” in the  AGN due to a merger event between a binary super-  massive black hole (BSBH) system and a sudden reori-  entation of the spin axis of the active nucleus (Zier &  Biermann 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Gopal-Krishna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' It may also  be possible for both nuclei in a BSBH system to be ac-  creting, in which case two sets of jets may exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' These  may both be active during the same period or undergo-  ing some cadence of activity depending on their respec-  tive accretion histories, creating multiple sets of lobes  (Gopal-Krishna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Lastly, and most relevant to what we will discuss here,  is the idea that the jet axis of the RG may change or  precess over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' This precession may lead to the for-  mation of X-shaped RGs if the precession angle is large  so that the current axis is well separated from the side  lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Smaller angles may lead to “S-shaped” or “Z-  shaped” RGs, which are designations sometimes given  to RGs that show curvature or sharp turns in their jets  or lobes (Riley 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  There are many examples of S-shaped radio sources  that could be fit by a projected helical precessing  jet such as 4C35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='06 in Abell 407 (Biju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' 2014),  J1328+2752 (Nandi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' 2021), or Hydra A (Taylor  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' On smaller scales, SS433 is a striking ex-  ample of a precessing X-ray binary jet system (Abell &  Margon 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Monceau-Baroux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The pre-  cession of AGN jets on large scales has been studied ex-  tensively as a method of creating X-shaped RGs (Ekers  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Rubinur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' In particular, (mag-  neto)hydrodynamical simulations have proven useful in  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='04343v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='HE]  11 Jan 2023  2  characterizing the evolution and morphology of such jets  (Smith & Donohoe 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Horton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Giri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Precession of an RG jet may occur for a number of  reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Again invoking a BSBH system, if either nu-  cleus is substantially accreting and hosts an active radio  jet, then the jet may precess around the orbital axis of  the BSBH system, assuming their is a misalignment be-  tween the orientation of the binary orbit and the black  hole spin axis (Begelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Alternatively,  precession may be related to accretion instabilities such  as Lense-Thirring or the related Bardeen Peterson effect  (Bardeen & Petterson 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Nandi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' These  instabilities may cause the precession of the accretion  disk if its angular momentum is misaligned with the an-  gular momentum of the central spinning black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' If  the disk orientation controls the jet axis, then this would  induce a precession in the jet axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' This precession mech-  anism is seen in some general relativistic accretion disk  simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=', Liska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Regardless of the mechanism for the precession of the  jet, we present the results of simulations in which we  assume jets do precess, and discuss the consequences  of that precession for the radio observables from these  simulated RGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' One of our main goals is to use these  simulations to invert this process and be better able to  understand the dynamical state of observed radio jets  that appear to be precessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  The remainder of this paper is organized as follows:  Section 2 outlines the geometry of the interaction we  are examining and the underlying physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Section 3 de-  scribes the simulations, including methods and specific  parameters for each simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' We describe the results  of the simulations in section 4 and summarize the major  findings in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' JET PHYSICS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Jet Precession  The basic geometry of the type of interactions we ex-  plore in this paper are illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The jet is  launched from a cylindrical region in which the cylinder  axis is tilted by ψ degrees from the z-axis of the sim-  ulation, which we will refer to as the precession angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  The other angles in Figure 1 refer to the viewing angles  used in the synthetic radio observations we present in  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' θ is the spherical coordinate polar angle, also  measured from the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Observations are taken at  four θ angles: 90◦, 45◦, 30◦, and 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' φ is the azimulthal  angle, and observations are taken at four φ angles from  0◦ ≤ φ ≤ 135◦ in ∆φ = 45◦ intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  ψ  Δϕ = 45∘  θ = 90∘  θ = 45∘  θ = 30∘  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Diagram of the jet precession.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The precession  angle ψ is the angle between the jet axis and the precession  axis, and the diagram uses a fiducial value of ψ = 30◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The  polar viewing angle, θ, is the angle between the precession  axis and the viewing angle, with angles used in Figure 3  shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Not shown are the azimuthal viewing angles which  are rotated around the precession axis in increments of ∆φ =  45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' A cyan arrow indicates the direction that the jet axis  precesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  The jet will precess through a full 2π radians within  one precession period, τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' This can also be expressed as  an angular velocity of precession, ωprec = 2π/τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Jet Propagation & Bending  As the jet precesses, it will bend due to its interaction  with the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The character of this bending will be  determined by a combination of the momentum in the  jet and the properties of the precession of its launching  angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  The forward propagation of a jet can be described by  a momentum balance in the frame of the head of the  jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' As derived in Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' (2017), we find that in  the absence of relative motion between the surrounding  medium and the jet source, the propagation of the head  3  of the jet will follow:  Mh ≈ Mj  �  Aj  Aa  �  Pj  Pa  ,  (1)  where Mh = vh/cs,a is the Mach number of the head of  the jet propagating at velocity vh relative to the ambi-  ent medium that has a sound speed of cs,a =  �  γPa/ρa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Mj  = vj/cs,j is the Mach number of the material  within the jet that has velocity vj and has an internal  sound speed of cs,j =  �  γPj/ρj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' P(j/a) and ρ(j/a) are  the thermal pressure and density of the (jet/ambient)  plasma, respectively, and we use an adiabatic index  of γ = 5/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  �  Aj/Ah is a ratio of “effective areas”  over which the momentum balance is taken for the  jet and ambient plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  In previous simulations of  steady, non-precessing jets, we have empirically found  �  Aj/Ah ∼ 1/2 (Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' This ratio departs  from unity mainly due to the existence of backflow of  jet plasma away from the region of the jet head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' In the  case where the jet is in pressure equilibrium with its sur-  roundings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Pj = Pa, Equation 1 can be expressed  as:  vh ≈ vj  �  Aj  Ah  � ρj  ρa  ,  (2)  which is a useful metric when the jet head crosses a  density discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  When there is some relative motion between the jet  and its environment, the propagation will be affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  If that motion has a component transverse to the jet  propagation direction, then the jet may be deflected or  bent by the ram pressure it experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  The classic  “cartoon” formula for the distance, ℓb, over which the  jet is bent backwards by a “wind” is (Begelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' 2017):  ℓb ∼ 2rj  ρjv2  j  ρav2a  ,  (3)  where rj is the cross-sectional radius of the jet and  va is the ambient velocity that provides the ram pres-  sure to bend the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Equation 3 assumes a constant  va, but if we assume this “wind” comes from the pre-  cession of the jet through a uniform medium at rest,  then the relevant ambient velocity is defined by the  angular velocity of the precession of the jet source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  va(y) = ωprec · y sin ψ = 2πy sin ψ/τ, where y is the  distance along the jet direction from the jet source to  the point of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Since the ambient velocity varies  along the jet, we need to revisit the arguments that lead  to equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Usually, this begins by assuming that  the ram pressure acts to redirect the jet, without com-  pressing or accelerating the jet plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' This leads to an  incompressible Euler equation:  Dvj  Dt =  \x13  \x13\x137  0  ∂vj  ∂t + (vj · ∇)vj = − 1  ρj  ∂P  ∂x ≈ − 1  ρj  ρav2  a  2rj  ,  (4)  where we assume that the jet bending leads to a steady  state (∂vj/∂t = 0) and that the ram pressure, P =  ρav2  a, acts across the diameter of the jet, 2rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' If we look  at the component of the jet velocity perpendicular to  the jet launching axis (vj,⊥) and integrate along the jet  length until the jet has been bent into the direction of  the “wind,” i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' vj,⊥ = vj:  � vj  0  dv =  1  2rj  ρa  ρjvj  �2π sin ψ  τ  �2 � ℓb  0  y2dy,  (5)  ℓb ∼  �  rj  3ρjv2  j τ 2  2π2ρa sin2 ψ  �1/3  ,  (6)  showing that the bending length of the jet will be a  function of the jet properties, the precession angle and  period, and the density of the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Now, using Equation 2, we can compare this length to  the distance that the head of the jet would travel before  the precession has significantly changed its launching  direction (ignoring for now how the bending affects this  propagation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' For this duration, we will take the frac-  tion of the total period that it takes the core of the jet  to transverse one jet diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' This will depend on the  shape and size of the jet launching cylinder:  ∆t = 2rj/(2π∆y sin ψ),  (7)  where ∆y is the half length of the cylinder (from the  center to one end).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The ratio ℓb/(vh∆t) is then a good  measure of whether or not the jet will create a well de-  fined curved jet, or if the precession will be too extreme  and cause the jet to break up or be unable to propagate  to large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' NUMERICAL METHODS  The simulations reported here used the Eulerian  WOMBAT ideal 3D MHD code (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Mendygral  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Nolting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' (2019)) on a uniform, Carte-  sian grid employing an adiabatic equation of state with  γ = 5/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The simulations utilized the 2nd order TVD al-  gorithm with constrained transport (CT) magnetic field  evolution as in Ryu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Specific simulation  setups are introduced in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='1 and listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Along with the fluid, we track a population of pas-  sive cosmic ray electrons (CRe) to allow for the cal-  culation of a synchrotron emissivity anywhere within  4  the simulation volume1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The CRe momentum distribu-  tion, f(p), was tracked using the conservative, Eulerian  “coarse grained momentum volume transport” CGMV  algorithm in Jones & Kang (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' f(p) spanned the  range 10 ≲ p/(mec) ≈ Γe ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='7 × 105 (so, energies 5  MeV ≲ ECRe ≈ Γemec2 ≲ 90 GeV) with uniform log-  arithmic momentum bins, 1 ≤ k ≤ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Inside a given  momentum bin, k, f(p) ∝ p−qk, with qk being bin de-  pendent and evolving in time and space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Γe here repre-  sents CRe Lorentz factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  At the jet cylinder, the CRe momentum distribution  had a power law form with index q = q0 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='2 across the  full momentum range simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' This was chosen be-  cause this translates to a radio synchrotron spectral in-  dex of α = α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='6 (Iν ∝ ν−α), which is a good match  for many RGs near their sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The synchrotron emis-  sion and spectra reported here are calculated numeri-  cally using f(p) over the full momentum range speci-  fied above using the standard synchrotron emission ker-  nel for isotropic electrons in a local magnetic field (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=',  Blumenthal & Gould 1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' In our analysis we calcu-  lated synthetic synchrotron emission at frequencies 300  MHz ≤ ν ≤ 600MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' This emission, as it turns out,  comes predominantly from regions with magnetic field  strengths ∼ a few µG, so mostly reflect CRe energies ≳  1 GeV (Γe ∼ 103–104) (well inside our distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  We included adiabatic, as well as radiative (syn-  chrotron and inverse Compton) CRe energy changes out-  side of shocks, along with test-particle diffusive shock  (re)acceleration (DSA) at any shocks encountered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' We  did not include 2nd order turbulent CRe reacceleration  or CRe energy losses from Coulomb collisions with am-  bient plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  The former depends on uncertain ki-  netic scale turbulence behaviors beyond the scope of this  study, while the latter is most relevant for CRe with  energies well below those responsible for the radio syn-  chrotron emission computed in this work (Sarazin 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  CRe radiative losses combine synchrotron with inverse  Compton (iC) scattered CMB radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The simula-  tions reported here assumed a low redshift, z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  The resulting radiative lifetime can be written  τrad ≈ 215  1  Γe4 [1 + B2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='4] Myr,  (8)  where Γe4 = Γe/104 and B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='4 = B/(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='4µG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The first  term in the denominator on the RHS reflects inverse  Compton (iC) losses at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='02, while the second rep-  1 Except for a negligible ICM population included to avoid numer-  ical singularities in the CRe transport algorithm, all CRe were  injected onto the computational domain via the jet launch cylin-  der.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  resents synchrotron losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Thus, we can see that for  Γe ∼ 104, of primary interest for the radio emission in  this work, τrad ∼ 200 Myr, and that iC losses are pre-  dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  DSA of the CRe was implemented at shock passage by  setting qk,out = min(qk,in, 3σ/(σ−1)) immediately post-  shock, where σ is the code-evaluated compression ratio  of the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' This simple treatment is appropriate in the  CRe energy range covered, since likely DSA acceleration  times to those energies are much less than a typical time  step in the simulations (∆t ≳ 104 yr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Since our CRe  have no dynamical impact, we treat the total CRe num-  ber density, nCRe, as arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Consequently, while we  compute meaningful synchrotron brightness and spec-  tral distributions from our simulations, synchrotron in-  tensity normalizations are arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Simulation Setups  In this work we present a suite of simulations that ex-  plore the parameter space of jet precession angles and  precession periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Each simulation consists of a jet  with constant density, pressure, and velocity being in-  jected into a medium that is initially homogeneous and  unmagnetized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' This medium is an oversimplification of  the intracluster medium (ICM), but it is useful for easier  interpretation of the observable signatures of the preces-  sion that we wish to study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The dynamics of the sim-  ulations we present here are scale free, however a scale  is set when introducing CRe radiative timescales (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=',  Equation 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Because of this, we will quote the scale of  the system that is consistent with the chosen radiation  timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Inside the simulation volume, a cylindrical region was  updated at each time step with the jet properties de-  scribed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  This cylinder had a radius of  rj = 3kpc and length lj = 4kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The jet cylinder was  surrounded by a 2 zone coaxial collar, within which the  state transitioned from the jet properties to the local  ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Inside, the jet density and pres-  sure were kept constant, and the velocity was ramped  up along the cylinder’s length from its midpoint (where  the velocity reverses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  A toroidal magnetic field was  also maintained withing the jet cylinder, with Bφ =  Bj(r/rj)ˆφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' In these jets, βp = 8πPj/B2  j = 75, giving  the magnitude of the field at launch to be Bj ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='54µG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Due to this relatively high βp, the fields are initially  dynamically sub-dominant to the gas pressure, which  remains true throughout the simulation duration in all  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  To create the precession of the jet axis, the jet is ini-  tialized ψ degrees away from the precession axis, defin-  ing the precession angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Every time step, the jet cylin-  5  Simulation  Name  Nx  Ny  Nz  Precession  Period  Precession  Angle  P3A10  288  288  624  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='2Myr  10◦  P3A20  288  288  624  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='2Myr  20◦  P3A30  288  288  624  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='2Myr  30◦  P3A45  360  600  600  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='2Myr  45◦  P32A10  288  288  624  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='7Myr  10◦  P32A20  288  288  624  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='7Myr  20◦  P32A30  288  288  448  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='7Myr  30◦  P32A45  360  600  600  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='7Myr  45◦  P95A10  288  288  1040  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='1Myr  10◦  P95A20  288  288  624  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='1Myr  20◦  P95A30  288  288  448  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='1Myr  30◦  P95A45  360  600  600  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='1Myr  45◦  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Names, domain size, and precession parameters  for each simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  der is rotated azimuthally around the precession axis by  ∆φ = 2π∆t/τ, where ∆t is the length of the time step  and τ is the jet precession period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Each simulation had a uniform initial density of  ρICM = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='17 × 10−28 g cm−3, and pressure of PICM =  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='8 × 10−13 dynes cm−2 and was initially unmagnitized  and at rest with respect to the jet source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' This gave the  medium an initial temperature of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='5keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Table 1 gives a list of parameters for the 12 simu-  lations, including domain size, precession period, and  precession angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Three precession periods and four pre-  cession angles were included in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Simulation  names include information about the precession period  (number following “P” in the name) and the precession  angle (number following “A” in the name) for ease of  understanding when referring to individual simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Synthetic Observations  The radio images we report here are calculated from  the CRe momentum distribution f(p) multiplied by the  appropriate synchrotron emissivity functions integrated  across the full momentum range, as specified above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  This emissivity takes into account the strength and ori-  entation of the local magnetic field, as well as the his-  tory of the CRe energy changes (radiative, adiabatic,  & DSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The emissivity is calculated as (Ginzburg &  Syrovatskii 1965;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Longair 2011):  j⊥(ν) =  �  ννB⊥  8  e2  c  � x2  x1  n(x)[F(x) + G(x)]  x3/2  dx ,  (9a)  j∥(ν) =  �  ννB⊥  8  e2  c  � x2  x1  n(x)[F(x) − G(x)]  x3/2  dx ,  (9b)  j(ν) = j⊥(ν) + j∥(ν) =  �  ννB⊥  2  e2  c  � x2  x1  n(x)F(x)  x3/2  dx ,  (9c)  F(x) = x  � ∞  x  K5/3(z)dz ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='78  ×  �  x  1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='4 exp (−5x)  �1/3  exp (−x) ,  (9d)  G(x) = xK2/3(x) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='56613 ×  x1/3  exp (x) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='427687 ,  (9e)  where F(x) and G(x) are functions that describe the  synchrotron spectrum of a single CRe, K5/3 and K2/3  are modified Bessel functions of the second kind, j⊥(ν)  and j∥(ν) are the two orthogonal polarizations of the  emissivity, ν is the frequency of the radiation, νB⊥ =  eB⊥/2πmec is the electron gyrofrequency, and e is the  elementary charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' In this case, B⊥ is the local mag-  netic field projected onto the plane of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The in-  tegration variable x = (2ν/3νB⊥γ2), with γ being the  electron Lorentz factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  The integration limit x1[x2]  corresponds to the high[low] energy (high[low] γ) end  of the integration range, covering our available range  10 < p/(mec) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='7 × 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' A change of variables from  integrating over γ to integrating over x introduces the  factor of x−3/2 in the integrand and a coefficient of  (−1/2)  �  2ν/3νB⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' We also present these equations in  Gaussian units, rather than the SI units used by Lon-  gair (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Equations (9d) and (9e) include our own  approximations to the synchrotron functions used in our  calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' These approximations are based on those  given in Rybicki & Lightman (1986), but combine the  small x and large x approximations to produce a single  fit that is accurate to a few percent (Nolting 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Using equations (9a) - (9e) we calculate the total  synchrotron emissivity as well as polarized emissivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Then, we perform radiative transfer integration along a  defined line of sight to create images of Stokes I, Q, or  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  A radio spectral index is calculated from any two ra-  dio maps at different frequencies with the radio spectral  index, α (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=', f ∝ να), at each pixel being  α = log10(I(ν2) − log10(I(ν1)))  log10(ν2) − log10(ν1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  (10)  In this work, the two frequencies used to generate spec-  tral index maps were ν2 = 600 MHz and ν1 = 300 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  6  Lastly, all radio images presented here are convolved  with a gaussian convolution kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  The resolution  for most images is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='355 arcsec Full Width Half Max  (FWHM), or a gaussian standard deviation of 1 arcsec-  ond, while one figure is convolved to a lower resolution  of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='775 arcsec FWHM, or 5 arcseconds gaussian stan-  dard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' DISCUSSION  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Effects of Precession Period and Precession Angle  To study the effects of precession period and preces-  sion angle, we ran a total of 12 simulations varying  these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The resulting morphological and ra-  dio spectral properties from these simulations are pre-  sented in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  We show each simulation at time  equal to one precession period as well as at t = 96  Myr for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The labels at the bottom of each  panel describe which simulation is shown and the simu-  lation time represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The simulations in the left two  columns have a precession period of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='2Myr, the 3rd and  4th column have a precession period of 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='7 Myr, and the  5th column has a precession period of 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='1 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' When  comparing the jets all at the same dynamical stage (after  1 precession period) the reader should examine columns  1, 3, and 5 together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' When comparing at the same time,  the reader should instead compare columns 2, 4, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  As Figure 2 demonstrates, these systems evolve quite  differently in terms of their dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  The jets with  large precession angles do not extend as far in any one  direction and are instead spread over a wider area, with  cocoons of larger lateral extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Similarly, jets with  shorter precession periods are also more centrally con-  densed as they are unable to deposit momentum in a  sustained direction long enough to substantially push  back the denser ICM plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  We can quantify the changes in dynamics using the ra-  tio ℓb/(vh∆t), as described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' With our cho-  sen jet injection parameters and equation 2, the propa-  gation rate of the head of the jet through the ambient  medium was 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='45 kpc/Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Given, this, we calculated  the ratio ℓb/(vh∆t) in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' By comparing the value  of this ratio to the corresponding images in Figure 2, we  can see that precessing jets with higher values of this ra-  tio will be more tightly wound and generally have prop-  agated less far from the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Conversely, precessing  jets with lower values of this ratio will be less tightly  wound and propagate farther before curving and bend-  ing backward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' While this ratio does not have a clear  critical value that determines when a jet will break up  from precession or display a clear s-shape morphology, it  is still useful for comparing how different precession pa-  rameters lead to differing morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' In particular, the  dependencies on τ and ψ are ℓb/(vh∆t) ∝ (sin ψ/τ)1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Taking this ratio, as well as the general trends with  precession period from Figure 2, it is clear that as the  precession period decreases, the jet cocoons are more  condensed and the jets more dramatically wound up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' In  simulations not presented here with shorter periods, the  jet began to break up in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  These jets  may even have trouble breaking out of the interstellar  medium of the host galaxy, which we do not include in  these simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Material surrounding the jet source is a combination  of jet backflow during the initial stages of jet launch-  ing as well as material that has bled off from the jet  as it precesses around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' This material at late times ra-  diatively steepens to α ≈ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='5, while fresh material in-  jected by the precessing jet remains spectrally flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' In  the cases with large precession angle, there is more sig-  nificant mixing of the older and newer jet material, as  the cocoons are smaller and the jet injection covers a  larger solid angle during their precession.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Examining the 5th column of Figure 2, with preces-  sion period P ∼ 95 Myr, we can see that as the pre-  cession period begins to approach the cooling timescale  for the CRe, the surface brightness of the material away  from the current jet direction drops off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' If the period  becomes too long, the CRe will age enough that they  may no longer be detectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' This would leave the ob-  server with little or no evidence of precession, and we  would undercount precessing sources on the long period  end of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' However, only the longest lived  AGN will reach such sustained jet ages, with most hav-  ing lifetimes > 100 Myr (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=', Turner & Shabala 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  If the jet precession has such a long period but isn’t  active for a large enough fraction of the precession pe-  riod, the source will likely be indistinguishable from a  non-precessing source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Because of this, radiative cooling  may not be a major constraint on observing precessing  radio jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Effects of Viewing Angle  Within a single snapshot of a precessing jet system, its  radio morphology may appear significantly different de-  pending on the viewing angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Figure 3 explicitly shows  this for the P32A30 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Each panel within this  figure shows the P32A30 simulation at the same time (82  Myr), but from 12 perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' In the online version,  this Figure is animated to show the evolution of this  simulation in each of these 12 viewing angles simultane-  ously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' At any given frame in the animation, one can see  how the morphology changes with viewing angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='differences are so stark that there are frames in which  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='150  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='150  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='kpc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='P3A10 - 3 Myr    ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='P3A10 - 96 Myr    ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='P32A10 - 32 Myr    ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='P32A10 - 96 Myr    ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='P95A10 - 96 Myr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='150  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='150  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='kpc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='P3A20 - 3 Myr    ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='P3A20 - 96 Myr    ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='P32A20 - 32 Myr    ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='P32A20 - 96 Myr    ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='P95A20 - 96 Myr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
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+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='kpc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='P3A30 - 3 Myr    ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='P3A30 - 96 Myr    ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='P32A30 - 32 Myr    ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='P32A30 - 96 Myr    ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='P95A30 - 96 Myr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
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+page_content='kpc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
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+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='kpc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='P3A45 - 3 Myr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
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+page_content='kpc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='P3A45 - 96 Myr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
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+page_content='kpc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='P32A45 - 32 Myr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
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+page_content='kpc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='P32A45 - 96 Myr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
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+page_content='P95A45 - 96 Myr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='5  Spectral Index (600-300 MHz)  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Radio spectral index (600-300MHz, cut at 1µJy beam−1 at 300MHz) with 300 MHz radio brightness contours (levels  = [1, 10, 100]µJy beam−1) for each simulation presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Radio images are convolved to 1” resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Each row shows  simulations with different precession angles (row 1: 10◦, 2: 20◦, 3: 30◦, 4: 45◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Columns show simulations with differing  precession periods, or the same simulations at different times (period for column 1 & 2: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='2 Myr, 3 & 4: 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='7 Myr, 5: 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='1 Myr)  (column 1, 3, & 5 are shown at approximately 1 precession period;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' columns 2, 4, & 5 are shown at 96 Myr simulation time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  8  −50  0  50  kpc  θ = 90◦  φ = 0◦  −50  0  50  kpc  θ = 90◦  φ = 45◦  −50  0  50  kpc  θ = 90◦  φ = 90◦  −50  0  50  kpc  −50  0  50  kpc  θ = 90◦  φ = 135◦  θ = 45◦  φ = 0◦  θ = 45◦  φ = 45◦  θ = 45◦  φ = 90◦  −50  0  50  kpc  θ = 45◦  φ = 135◦  t = 82 Myr  θ = 30◦  φ = 0◦  θ = 30◦  φ = 45◦  θ = 30◦  φ = 90◦  −50  0  50  kpc  θ = 30◦  φ = 135◦  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='9  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='8  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='7  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='6  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='5  Spectral Index (600-300 MHz)  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Radio spectral index (600-300MHz, cut at 1µJy beam−1 at 300MHz) with 300 MHz radio brightness contours  (levels = [1, 10, 100]µJy beam−1) for the P32A30 simulation at t = 82 Myr from a variety of viewing angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Radio images are  convolved to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='355” FWHM resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Viewing angles are listed in the inset for each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' For ease of visualization, the polar  viewing angles are included in the diagram Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The polar angle is varied by column while the azimuthal angle is varied  by row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' In the online version, an animated figure is available (8 seconds, spanning 100 Myr simulation time) that follows the  evolution of the source from each viewing angle simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  9  Simulation  Name  ∆t  vh∆t  ℓb  ℓb  vh∆t  P3A10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='93Myr  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='1kpc  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='5kpc  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='9  P3A20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='5Myr  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='2kpc  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='4kpc  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='38  P3A30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='02Myr  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='52kpc  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='6kpc  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='73  P3A45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='72Myr  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='5kpc  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='6kpc  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='04  P32A10  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='1Myr  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='4kpc  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='7kpc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='89  P32A20  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='8Myr  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='1kpc  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='1kpc  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='12  P32A30  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='1Myr  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='9kpc  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='3kpc  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='27  P32A45  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='1Myr  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='5kpc  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='2kpc  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='44  P95A10  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='2Myr  300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='8kpc  186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='6kpc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='62  P95A20  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='3Myr  152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='8kpc  118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='8kpc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='78  P95A30  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='3Myr  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='5kpc  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='2kpc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='88  P95A45  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='4Myr  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='8kpc  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='2kpc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='99  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Values of jet bending analysis variables from sec-  tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' ∆t comes from Equation 7, vh from Equation 2, and  ℓb refers to the definition in Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  the radio source would very likely be classified in differ-  ent ways, and lead to a misunderstanding of the physics  of the source’s evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' For example, in the frame at  t = 82 Myr, the first six frames (counting from left to  right, top to bottom) and the ninth frame show signs of  precession with curved jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' In the 7th and 10th frames,  the jet appears mostly straight, with some deflection at  the farthest points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' In the 8th frame 11th frame, the jets  appear to have been disrupted or sharply bent by some  (perhaps external) dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' And the radio source in  the 12th would likely be unclassifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' By providing the  animation of the evolution of this source from many an-  gles, we hope that radio observers can compare these  images to detected radio galaxies to help identify possi-  ble precessing sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' We also hope to display the full  range of complex morphologies that precessing jets may  take on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Jet Reorientation Events  In each simulation, the precessing jets would some-  times undergo a dramatic and sudden change in their  propagation direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  This “reorientation event” oc-  curred roughly after one precession period had elapsed  in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  As the jet precesses initially, it  bends strongly due to ram pressure from its interaction  with the ICM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' However, after one precession period has  elapsed, the jet encounters a region of space in which it  has previously deposited low density plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' As the jet  encounters this density discontinuity, the velocity of the  head of the jet suddenly increases, as can be inferred  from Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Additionally, the ram pressure that  leads to the bending of the jet is greatly reduced, leading  to an increase in the bending length (Equation 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Both  50  0  50  0  50  100  150  t = 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='0 Myr  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='6  Spectral Index (600-300 MHz)  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Radio spectral index (600-300MHz, cut at 1µJy  beam−1 at 300MHz) with 300 MHz radio brightness contours  (levels = [1, 10, 100]µJy beam−1) for the P32A30 simulation  at t = 34 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Radio images are convolved to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='355” FWHM  resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' At this time, the jet is undergoing a “jet reori-  entation event” in which the jet trajectory changes by more  than 90◦ over about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='5 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' This can be seen here as the  very sharp bend at the tip of the jet head at coordinates (x,y)  ≈ (-5, 80) kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' In the online version, an animated figure (5  seconds) shows the full duration of the reorientation event  from t = 30 Myr until t = 38 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  of these effects lead to the jet becoming straighter and  propagating to further distances before it then begins  to interact with the higher density ICM when it reaches  the outskirts of the low density cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' This process is  illustrated in Figure 4, which shows the beginning of  the reorientation event, in which the jet trajectory sud-  denly changes by > 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' This type of sudden change  in jet propagation direction is often attributed to either  changes in the jet properties or duty cycle, or to external  cluster dynamics such as shocks or other waves travel-  ing through the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' However, none of these effects  are included in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' These jet reorientation  events can come about from jet precession alone, and the  self interaction of the jet with its own previous activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Odd Radio Circles (ORCs)  Odd Radio Circles (ORCs) are recent and mysterious  radio sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' They are circles of steep spectrum, dif-  fuse radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Some (but not all) ORCs have a  detected galaxy near their center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' There have been a  variety of possible physical explanations as to their ori-  gin, including a spherical shock from a starburst wind,  a supermassive black hole merger, or radio galaxy lobes  10  50  0  50  50  25  0  25  50  t = 98 Myr  50  0  50  t = 140 Myr  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='6  Spectral Index (600-300 MHz)  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Radio spectral index (600-300MHz, cut at 100µJy beam−1 at 300MHz) with 300 MHz radio brightness contours  (levels = [100, 400, 1600]µJy beam−1) for the P32A30 simulation at 98 Myr (left) and 140 Myr (right) both viewed from above  the jet precession axis (θ = 0◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Radio images are convolved to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='775” FWHM resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The jet turns off at 100 Myr and  the plasma is allowed to radiatively cool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' In the online version, a single-panel animated figure (10 seconds) follows the evolution  of the source from the beginning of the simulation until t = 160 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  seen end-on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Our simulations may lend some support to  this last suggestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Figure 5 shows the P3A30 simula-  tion along a line of sight parallel to the precession axis  at two times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The jets turn off just after the time in  the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The right panel shows the radio emission  about 40 Myr later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The radio morphology is similar to  that of ORCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' In the online version of this article, an  animated version of this figure is available, which shows  the evolution up from 0-160 Myr, with the jet power-  ing off around t=100 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Just after the jet turns off,  there is a full ring of emission visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' However, the ra-  dio spectrum is still relatively flat with α ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The  radio ring ages in place, without expanding further, as  it is in pressure balance with its surroundings and the  jets are no longer injecting momentum into the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  As the plasma radiatively ages and spectrally steepens,  the ring becomes broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' There is a period of time in  which the ring is still mostly intact and visible and it  approaches the spectral index observed for some ORCs  α ≈ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='2 (Norris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  If this mechanism for explaining some ORCs holds,  then it may be interesting to reexamine existing ob-  servations of S-shaped RGs and burn out the images  looking for regions of low level emission surrounding the  visible structures as a possible analog to the existing  ORCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' This low level emission may not appear as circu-  lar from viewing angles misaligned from the precession  axis, but it may represent the same material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Addition-  ally, searches for AGN counterparts in galaxies near the  centers of ORCs or VLBI observations looking for any  current small scale jets could help determine if such a  formation mechanism is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' SUMMARY  We have presented a series of simulations that look  into the properties of precessing radio jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' These jets  take on a wide variety of morphologies, depending on the  properties of the precession itself as well as on the angle  at which the source is viewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' In particular, the effects  of viewing angle can greatly affect the classification of  these sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' In many viewing angles, precession is not  an obvious mechanism, as the jet may appear straight in  projection or with very complex and messy morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Additionally, we observed some interesting and un-  expected dynamics in our simulations, including self in-  duced “reorientation events,” in which the jet trajectory  quickly changed as the jet encounters a region filled with  previous jet material of lower density than the ambient  medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' This led to sharp turns in the jet which could  be misinterpreted as due to external dynamics affecting  the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  Lastly, we point out the similarities between ORCs  and the remnants of a precessing jet seen end–on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' This  could be a possible explanation for some ORCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Fol-  lowup observations of existing ORCs looking AGN coun-  terparts could help determine if this is likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  11  ACKNOWLEDGMENTS  CN acknowledges funding through the NSF grant  AST-1907850 as well as from Los Alamos National Lab-  oratory through the LDRD program and NASA pro-  grams through the Astrophysical Theory Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' JB  and TN were supported by NSF grant AST-1907850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content='  We thank L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Rudnick, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Jones, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' Fragile for  useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
+page_content=' The simulations presented here were  run and analyzed at the College of Charleston on their  high-performance Linux cluster (https://hpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfJQn8/content/2301.04343v1.pdf'}
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+arXiv:2301.11901v1  [math.NT]  27 Jan 2023
+SUMS OF CUSP FORM COEFFICIENTS ALONG
+QUADRATIC SEQUENCES
+CHAN IEONG KUAN, DAVID LOWRY-DUDA, AND ALEXANDER WALKER
+Abstract. Let f(z) = � A(n)n(k−1)/2e(nz) be a cusp form of weight
+k ≥ 2 on Γ0(N) with character χ. By studying a certain shifted convo-
+lution sum, we prove that �
+n≤X A(n2 +h) = cf,hX +Of,h,ǫ(X
+6k−8
+8k−11 +ǫ)
+for ǫ > 0, which improves a result of Blomer [Blo08] with error X6/7+ǫ.
+1. Introduction
+In [Hoo63], Hooley considers the average behavior of the divisor function
+d(n) within a quadratic sequence and proves that
+S(X) :=
+�
+n≤X
+d(n2 + h) = chX log X + c′
+hX + Oh,ǫ
+�
+X
+8
+9 log3 X
+�
+for constants ch, c′
+h (when −h is non-square) using the theory of exponen-
+tial sums. Hooley’s error term was improved by Bykovskii [Byk87], who
+uses the spectral theory of automorphic forms to study the generalized sum
+�
+n≤X σν(n2 + h), in which σν(n) = �
+d|n dν. In the case ν = 0, Bykovskii
+obtains S(X) = chX log X + c′
+hX + O(X2/3+ǫ) for any ǫ > 0.
+An analogous question for the normalized coefficients of a GL(2) cusp
+form was introduced by Blomer in [Blo08]. Let f(z) = � a(n)e(nz) be a
+cusp form on Sk(Γ0(N), χ) with weight k ≥ 4 and set A(n) = a(n)/n(k−1)/2.
+For any monic quadratic polynomial q(x) ∈ Z[x], Blomer proves
+�
+n≤X
+A(q(n)) = cf,qX + Of,q,ǫ
+�
+X
+6
+7+ǫ�
+(1.1)
+for some constant cf,q which equals 0 in most but not all cases.
+Both Hooley and Bykovskii rely on the convolution identity σ0 = 1 ∗ 1.
+As this has no analogue for cusp forms, Blomer instead proceeds by writing
+f(z) as a sum of Poincar´e series, whose q(n)-th Fourier coefficients involve
+sums of the form
+�
+c≥1
+1
+NcSχ(m, q(n); Nc)Jk−1
+�4π
+�
+q(n)m
+Nc
+�
+,
+in which Jk−1 is the J-Bessel function and Sχ is a twisted Kloosterman
+sum. Blomer evaluates a smooth version of the sum over n ≤ X using Pois-
+son summation, which converts the sums Sχ(m, q(n); Nc) into half-integral
+1
+
+2
+CHAN IEONG KUAN, DAVID LOWRY-DUDA, AND ALEXANDER WALKER
+weight Kloosterman sums. Blomer’s result then follows from cancellation in
+the latter, as proved using a half-integral weight Kuznetsov formula.
+Two alternative methods for treating the cusp form analogy are given by
+Templier and Tsimerman in [TT13]. The first is inspired by earlier work by
+Sarnak on d(n2 + h) in [Sar84], who relates the shifted convolution sum
+�
+n≥1
+d(n2 + h)
+(n2 + h)s
+to the Petersson inner product ⟨Im(z)
+1
+4 θ(z)E(z, 1
+2), Ph(z, s)⟩, where θ(z) is
+a theta function, E(z, s) is a weight 0 real analytic Eisenstein series, and
+Ph(z, s) is a half-integral weight Poincar´e series. (Sarnak notes that this
+connects S(X) to the spectrum of the half-integral weight Laplacian, going
+no further.) In [TT13, §4], similar methods are applied to study the shifted
+convolution sum
+Dh(s) :=
+�
+n≥0
+r1(n)a(n + h)
+(n + h)s+ k
+2 − 3
+4
+,
+(1.2)
+in which rℓ(n) is the number of representations of n as a sum of ℓ squares.
+Templier and Tsimerman give a meromorphic continuation of Dh(s) and
+prove that Dh(s) grows polynomially in |Im s| in vertical strips.
+In par-
+ticular, for any smooth function g(x) on R+ with Mellin transform �g(s)
+satisfying �g(s) ≪ Γ(s) in vertical strips, [TT13, §4.8] gives a constant c′
+f,h
+depending only on f(z) and h for which
+�
+n≥0
+A(n2 + h)g
+�n2 + h
+X2
+�
+= c′
+f,h �g(1
+2)X + Oǫ
+�X
+1
+2+Θ+ǫ
+h
+1
+2Θ−δ
+�
+,
+in which Θ ≤
+7
+64 (due to [KS03]) denotes progress towards the Selberg
+eigenvalue conjecture and δ ≤ 1
+6 (due to [PY19]) denotes progress towards
+the Ramanujan–Petersson conjecture for half-integral weight cusp forms.
+Templier and Tsimerman give a second proof of their result for A(n2 +h)
+using representation theory.
+This alternative framework allows for equal
+treatment of holomorphic cusp forms and Maass cusp forms. Here as before,
+Templier–Tsimerman restrict to smoothed sums, so their results cannot be
+directly compared to the sharp cutoff (1.1) from [Blo08].
+In this paper, we refine the shifted convolution sum technique described
+in [TT13, §4] to produce the following sharp cutoff result.
+Theorem 1.1. Let f(z) = � a(n)e(nz) be a cusp form in Sk(Γ0(N), χ) and
+define A(n) = a(n)/n(k−1)/2. For k ≥ 2, h > 0, and any ǫ > 0, we have
+�
+n2+h≤X2
+A(n2 + h) = cf,hX + Of,h,ǫ
+�
+X
+1−
+1
+k+3−√
+k(k−2) +ǫ�
+.
+The constant cf,h is given in (4.6).
+
+SUMS OF CUSP FORM COEFFICIENTS ALONG QUADRATIC SEQUENCES
+3
+The error term in Theorem 1.1 may be replaced by X
+3
+4 +
+1
+32k−44 +ǫ for
+any k ≥ 2, which is weaker in the exponent by O(1/k2) but makes the
+k-dependence more clear. Theorem 1.1 improves the error term O(X
+6
+7+ǫ)
+from [Blo08], though we remark that the cusp form analogy still lags behind
+Bykovskii’s 2
+3 + ǫ exponent in the divisor function analogue. Exponents of
+size 1
+2 + ǫ are conjectured to hold in both problems.
+2. Outline of Paper
+As in [TT13, §4], we understand sums of the form �
+n≤X A(n2 + h) by
+studying the Dirichlet series Dh(s) defined in (1.2). In §3, we prove Proposi-
+tion 3.1, which relates Dh(s) to an inner product involving f(z), the Jacobi
+theta function, and an appropriate Poincar´e series P κ
+h (z, s). Spectral expan-
+sion of P κ
+h (z, s) in §4 then expresses Dh(s) as a sum of terms corresponding
+to the discrete, residual, and continuous spectra of the hyperbolic Laplacian.
+Our treatment of the discrete spectrum of half-integral weight Maass
+forms differs greatly from [TT13] and represents the main novelty of this
+work. We replace the use of weak estimates for the individual Fourier coef-
+ficients ρj(n) of Maass forms with strong bounds for their behavior in long
+averages over the spectrum of Maass forms and long averages over the coef-
+ficient n. The former is given by [Blo08], while the latter is new and refines
+ideas of [DFI02, §19] by incorporating uniform bounds for the Whittaker
+function. The derivation of these averages is carried out in §5.
+These Fourier coefficient estimates are applied in §6 to prove Theorem 6.1,
+a bound for the sum
+D :=
+�
+|tj|∼T
+|⟨y
+k
+2 + 1
+4fθ, µj⟩|2eπ|tj|,
+which averages over an orthonormal basis of (half-integral weight) Maass
+forms µj with spectral types |tj| ∈ [T, 2T]. As in [Blo08], we leverage the
+fact that f is holomorphic to write it as a linear combination of holomorphic
+Poincar´e series. Unlike [Blo08], however, these Poincar´e series are used to
+form shifted convolutions, instead of introducing Kloosterman-type sums.
+In §7, we use bounds for D to control the growth of Dh(s) with respect
+to | Im s| in vertical strips. Our main result Theorem 1.1 then follows by
+Perron’s formula and contour shifts.
+Acknowledgements
+The authors thank Stephen Lester, who introduced them to this problem,
+as well as Yiannis Petridis, whose suggestions led to a simplification of sec-
+tion §6. We also thank Thomas Hulse, whose involvement over many years
+has improved this paper and moreover its authors.
+The first author was supported in part by NSFC (No. 11901585). The
+second author was supported by the Simons Collaboration in Arithmetic Ge-
+ometry, Number Theory, and Computation via the Simons Foundation grant
+
+4
+CHAN IEONG KUAN, DAVID LOWRY-DUDA, AND ALEXANDER WALKER
+546235. The third author was supported by the Additional Funding Pro-
+gramme for Mathematical Sciences, delivered by EPSRC (EP/V521917/1)
+and the Heilbronn Institute for Mathematical Research.
+3. A Triple Inner Product
+For integral k ≥ 1, let Sk(Γ0(N), χ) denote the set of cusp forms on
+Γ0(N) which transform under the character χ · χk
+−1, where χ−1 = (−1
+· ).
+We assume without loss of generality that 4 | N.
+Once and for all, we
+fix a weight k modular form f(z) = � a(n)e(nz) ∈ Sk(Γ0(N), χ) and a
+positive integer h. Here and later, we use the common notation e(x) := e2πix.
+Let θ(z) = �
+n∈Z e(n2z) = �
+n≥0 r1(n)e(nz) denote the classical Jacobi
+theta function. The theta function is a modular form of weight 1
+2 on Γ0(4)
+(see [Shi73]), transforming via
+θ(γz) = ǫ−1
+d
+� c
+d
+�
+(cz + d)
+1
+2 θ(z),
+γ =
+�a
+b
+c
+d
+�
+∈ Γ0(4),
+in which ǫd = 1 for d ≡ 1 mod 4 and ǫd = i for d ≡ 3 mod 4. Let P κ
+h (z, s)
+denote the weight κ := k − 1
+2 twisted Poincar´e series on Γ0(N), defined by
+P κ
+h (z, s) :=
+�
+γ∈Γ∞\Γ0(N)
+χ(γ)Jθ(γ, z)−2κ Im(γz)se(hγz),
+(3.1)
+in which Jθ(γ, z) = ǫ−1
+d ( c
+d)(cz+d)
+1
+2/|cz+d|
+1
+2 is the normalized theta cocycle
+θ(γz)/θ(z). Then P κ
+h (z, w) and Im(z)
+k
+2 + 1
+4f(z)θ(z) transform identically un-
+der the action of Γ0(N), so the Petersson inner product ⟨y
+k
+2 + 1
+4fθ, P κ
+h (·, s)⟩
+is well-defined over Γ0(N). A standard unfolding argument gives
+⟨y
+k
+2 + 1
+4fθ, P κ
+h (·, s)⟩ =
+�
+Γ0(N)\H
+y
+k
+2 + 1
+4f(z)θ(z)P κ
+h (z, s) dxdy
+y2
+=
+� ∞
+0
+� 1
+0
+ys+ k
+2 − 3
+4 f(z)θ(z)e(hz)dxdy
+y
+=
+� ∞
+0
+� 1
+0
+ys+ k
+2 − 3
+4
+� �
+m1≥1
+a(m1)e2πim1x−2πm1y�
+×
+� �
+m2≥0
+r1(m2)e−2πim2x−2πm2y�
+e−2πihx−2πhy dxdy
+y
+.
+The x-integral extracts those terms with m1−m2−h = 0, and the remaining
+y-integral evaluates in terms of the gamma function:
+⟨y
+k
+2 + 1
+4 fθ, P κ
+h (·, s)⟩ =
+�
+m≥0
+r1(m)a(m + h)
+� ∞
+0
+ys+ k
+2 − 3
+4e−4π(m+h)y dy
+y
+= Γ(s + k
+2 − 3
+4)
+(4π)s+ k
+2 − 3
+4
+�
+m≥0
+r1(m)a(m + h)
+(m + h)s+ k
+2 − 3
+4
+.
+
+SUMS OF CUSP FORM COEFFICIENTS ALONG QUADRATIC SEQUENCES
+5
+Standard estimates show that this Dirichlet series converges absolutely for
+Re(s) > 3
+4. By rearranging, we obtain the following identity for the Dirichlet
+series Dh(s) introduced in (1.2).
+Proposition 3.1. Fix h > 0 and any f(z) = � a(n)e(nz) ∈ Sk(Γ0(N), χ).
+We have
+Dh(s) :=
+�
+m≥0
+r1(m)a(m + h)
+(m + h)s+ k
+2 − 3
+4
+= (4π)s+ k
+2 − 3
+4 ⟨y
+k
+2 + 1
+4fθ, P κ
+h (·, s)⟩
+Γ(s + k
+2 − 3
+4)
+in the region Re s > 3
+4.
+4. Spectral Expansion of the Poincar´e Series
+The series Dh(s) has a meromorphic continuation to all s ∈ C obtained
+through spectral expansion of the Poincar´e series P κ
+h (z, s) in Proposition 3.1.
+(See [Str08, §15.3] for a good general reference on spectral expansions for
+general weight and the shapes of each component.) As a weight κ = k − 1
+2
+object, this spectral expansion takes the form
+P κ
+h (z, s) =
+�
+j
+⟨P κ
+h (·, s), µj⟩µj(z) +
+�
+ℓ
+⟨P κ
+h (·, s), Rℓ⟩Rℓ(z)
++
+1
+4πi
+�
+a
+�
+( 1
+2)
+�
+P κ
+h (·, s), Eκ
+a (·, w; χ)
+�
+Eκ
+a (z, w; χ) dw,
+(4.1)
+in which {µj} denotes an orthonormal eigenbasis of weight κ Maass forms of
+level N and character χ·χk
+−1, {Rℓ} is a finite orthonormal basis of the resid-
+ual spectrum of weight κ and character χ·χk
+−1, and Eκ
+a (z, w; χ) is the weight
+κ twisted Eisenstein series with character χ · χk
+−1 attached to the cusp a of
+Γ0(N). We refer to the expressions at right in (4.1) as the discrete, residual,
+and continuous spectra, respectively. Inserting this spectral expansion into
+Dh(s) as presented in Proposition 3.1 gives a spectral expansion of the form
+Dh(s) = Σdisc(s) + Σres(s) + Σcont(s),
+which we now describe more fully.
+The automorphic forms in the discrete and continuous spectra have Fourier
+expansions, which take the form
+µj(z) =
+�
+n̸=0
+ρj(n)W nκ
+2|n| ,itj(4π|n|y)e(nx),
+(4.2)
+Eκ
+a (z, w; χ) = δ[a=∞]yw + ρa(0, w)y1−w
++
+�
+n̸=0
+ρa(n, w)W nκ
+2|n| ,w− 1
+2 (4π|n|y)e(nx),
+in which δ[·] is the Kronecker delta and Wη,ν(z) is the Whittaker function.
+By unfolding the Poincar´e series P κ
+h (z, s) and applying the integral for-
+mula [GR15, 7.621(3)] for the resulting y-integral, we evaluate ⟨P κ
+h (·, s), µj⟩
+
+6
+CHAN IEONG KUAN, DAVID LOWRY-DUDA, AND ALEXANDER WALKER
+and conclude that the discrete spectrum’s contribution towards Dh(s) equals
+Σdisc := (4π)
+k
+2 + 1
+4
+hs−1
+�
+j
+Γ(s − 1
+2 + itj)Γ(s − 1
+2 − itj)
+Γ(s − k
+2 + 1
+4)Γ(s + k
+2 − 3
+4) ρj(h)⟨y
+k
+2 + 1
+4 fθ, µj⟩. (4.3)
+We will establish in §7.1 that this series for Σdisc converges everywhere away
+from poles. Assuming this, Σdisc(s) defines a meromorphic function which
+is analytic in Re s > 1
+2 + supj|Im tj|. By the Shimura correspondence for
+Maass forms (see [KS93]), Σdisc(s) is analytic for Re s > 1
+2 + Θ
+2 , where Θ
+denotes the progress towards the Selberg eigenvalue conjecture as before.
+Similarly, we determine that the continuous spectrum’s contribution to-
+wards Dh(s) may be written
+Σcont(s) := (4π)
+k
+2 − 3
+4
+hs−1
+�
+a
+� ∞
+−∞
+Γ(s − 1
+2 + it)Γ(s − 1
+2 − it)
+Γ(s − k
+2 + 1
+4)Γ(s + k
+2 − 3
+4)
+(4.4)
+× ρa(h, 1
+2 + it)⟨y
+k
+2 + 1
+4 fθ, Eκ
+a (·, 1
+2 + it; χ)⟩ dt.
+We will prove in §7.3 that Σcont converges everywhere away from poles and
+therefore defines an analytic function of s in Re s > 1
+2. Through delicate
+contour shifting (as in [HH16, §4] or [HKLDW18, §3.3.2]), one may show
+that Σcont extends meromorphically to all Re s ∈ C, though we only consider
+Re s > 1
+2 in this work.
+We conclude this section by discussing the residual spectrum, which con-
+tributes a term of the form
+Σres(s) :=
+(4π)s+ k
+2 − 3
+4
+Γ(s + k
+2 − 3
+4)
+�
+ℓ
+⟨Rℓ, P κ
+h (·, s)⟩⟨y
+k
+2 + 1
+4 fθ, Rℓ⟩
+towards Dh(s). The material here follows [TT13, §3.5 and §4.7], with minor
+modifications.
+By the theory of raising operators, the weight κ = k− 1
+2 residual spectrum
+lifts from weight 1
+2 forms if k is odd and from weight 3
+2 forms is k is even.
+By [Duk88, §2], the residual spectrum does not appear in weight 3
+2, so we
+restrict to k odd for the rest of this section.
+Suppose initially that N/4 is square-free and odd. By [SS77], the weight
+1
+2 residual spectrum appears only when χ = (
+·
+N/4). In this case, the space
+is one-dimensional and spanned by the theta function
+y
+1
+4 θN(z) = y
+1
+4
+�
+n∈Z
+e(Nn2z
+4
+) = y
+1
+4 +
+�
+n̸=0
+(πNn2)− 1
+4W 1
+4 , 1
+4 (πNn2y)e(Nn2x
+4
+).
+Under these assumptions, we have
+Σres(s) :=
+(4π)s+ k
+2 − 3
+4
+Γ(s + k
+2 − 3
+4)⟨u, P κ
+h (·, s)⟩⟨y
+k
+2 + 1
+4 fθ, u⟩,
+in which u(z) denotes the L2-normalized lift of y
+1
+4 θN(z) to weight κ under
+the raising operators. As the weight η raising operator Rη = iy ∂
+∂x +y ∂
+∂y + η
+2
+
+SUMS OF CUSP FORM COEFFICIENTS ALONG QUADRATIC SEQUENCES
+7
+maps y1/4 to (η
+2 + 1
+4)y1/4 and W η
+2 , 1
+4 (4πmy)e(mx) to −W η+2
+2 , 1
+4(4πmy)e(mx),
+the (non-normalized) lift of y
+1
+4θN(z) to weight κ equals
+U(z) := y
+1
+4
+�
+1
+2≤η<κ
+η≡ 1
+2 mod 2
+(η
+2 + 1
+4) + (−1)
+k−1
+2
+�
+n̸=0
+(πNn2)− 1
+4 W κ
+2 , 1
+4 (πNn2y)e(Nn2x
+4
+).
+To relate U(z) and u(z), we consider the effect of the raising operators
+on L2 norms. Let µ denote any Maass cusp form of weight η and type ν
+on Γ0(N) and let Lη = −iy ∂
+∂x + y ∂
+∂y − η
+2 denote the weight η Maass low-
+ering operator. By combining [GH11, (3.9.4)] with the last offset equation
+on [GH11, p. 90], we produce
+∥Rηµ∥2 = ⟨Rηµ, Rηµ⟩ = ⟨µ, −Lη+2Rηµ⟩
+= ⟨µ, ∆ηµ⟩ + η
+2(1 + η
+2)⟨µ, µ⟩ =
+�1
+4 + ν2 + η
+2(1 + η
+2)
+�
+∥µ∥2.
+We note that this is analogous to [TT13, equation (3.20)].
+In the special case ν = i/4, we conclude that
+∥U∥2 = ∥y
+1
+4 θN∥2
+�
+1
+2≤η<κ
+η≡ 1
+2 mod 2
+(η
+2 + 1
+4)(η
+2 + 3
+4) = ∥y
+1
+4 θN∥2
+k−1
+2�
+j=1
+j(j − 1
+2),
+which implies that
+u(z) =
+y
+1
+4
+∥y
+1
+4 θN∥
+�
+1
+2≤η<κ
+η≡ 1
+2 mod 2
+(η
+2 + 1
+4)
+1
+2
+(η
+2 + 3
+4)
+1
+2
++ (−1)
+k−1
+2 d
+− 1
+2
+k
+∥y
+1
+4 θN∥
+�
+n̸=0
+W κ
+2 , 1
+4 (πNn2y)
+(πNn2)
+1
+4
+e(Nn2x
+4
+),
+in which dk = � k−1
+2
+j=1 j(j − 1
+2), cf. [TT13, §4.7].
+We now compute the two inner products which appear in Σres. To begin,
+we unfold the Poincar´e series and apply [GR15, 7.621(3)] to produce
+⟨u, P κ
+h (·, s)⟩ = 2(−1)
+k−1
+2 d
+− 1
+2
+k δ[Nn2=4h]
+(4πh)
+1
+4 ∥y
+1
+4θN∥
+� ∞
+0
+ys−1W κ
+2 , 1
+4(4πhy)e−2πhy dy
+y
+= (−1)
+k−1
+2 d
+− 1
+2
+k
+r1(4h
+N )Γ(s − 1
+4)Γ(s − 3
+4)
+(4πh)s− 3
+4∥y
+1
+4 θN∥Γ(s − κ
+2)
+.
+To compute ⟨y
+k
+2 + 1
+4 fθ, u⟩, we recognize u(z) as a multiple of the residue at
+w = 3
+4 of the Eisenstein series Eκ
+∞(z, w) of level N, weight κ, and trivial char-
+acter (i.e. (3.1), with h = 0 and χ trivial). We have Resw=3/4 Eκ
+∞(z, w) =
+
+8
+CHAN IEONG KUAN, DAVID LOWRY-DUDA, AND ALEXANDER WALKER
+u(z)d1/2
+k
+∥y
+1
+4 θN∥−1 from [TT13, §4.7], and hence
+⟨y
+k
+2 + 1
+4 fθ, u⟩ = d
+− 1
+2
+k
+∥y
+1
+4 θN∥ · Res
+w= 3
+4
+⟨y
+k
+2 + 1
+4 fθ, Eκ
+∞(z, w)⟩
+= d
+− 1
+2
+k
+∥y
+1
+4 θN∥ · Res
+w= 3
+4
+∞
+�
+n=1
+2a(n2)
+� ∞
+0
+yw+ k
+2 − 3
+4e−2πn2y dy
+y
+= 2d
+− 1
+2
+k
+∥y
+1
+4θN∥ · Res
+w= 3
+4
+Γ(w + k
+2 − 3
+4)
+(2π)w+ k
+2 − 3
+4
+∞
+�
+n=1
+a(n2)
+n2w+k− 3
+2
+.
+The Dirichlet series in the line above equals L(2w − 1
+2, Sym2f)/ζ(4w − 1),
+in which L(s, Sym2f) is the symmetric square L-function of f(z). Putting
+everything together, we derive the explicit formula
+Σres(s) = 2
+k
+2 Γ(k
+2)(−1)
+k−1
+2 r1(4h
+N )Γ(s − 1
+4)Γ(s − 3
+4)
+ζ(2)dkhs− 3
+4 Γ(s + k
+2 − 3
+4)Γ(s − κ
+2)
+· Res
+w=1 L(w, Sym2f)
+(4.5)
+= 2
+k
+2 √π(−1)
+k−1
+2 r1(4h
+N )
+ζ(2)Γ(k+1
+2 )hs− 3
+4
+·
+Γ(s − 1
+4)Γ(s − 3
+4)
+Γ(s + k
+2 − 3
+4)Γ(s − κ
+2) · Res
+w=1 L(w, Sym2f),
+in which the second line follows from the identity dk = Γ(k)/2k−1 and the
+gamma duplication formula. In conclusion, we note that Σres(s) is mero-
+morphic in C and analytic in Re s > 3
+4, with a simple pole at s = 3
+4 only
+when k is odd, f(z) is dihedral, χ = (
+·
+N/4), and r1(4h/N) ̸= 0.
+Remark 4.1. For N with generic square part and odd level k, the residual
+spectrum has a basis consisting of (finitely many) lifts of theta functions of
+the form
+y
+1
+4 θψ
+M(z) = y
+1
+4
+�
+n∈Z
+ψ(n)e(Mn2z),
+in which ψ is a character of conductor L such that ML2 | N
+4 and ψ·( ·
+M ) = χ.
+Since these lifts can all be recognized as residues of weight κ Eisenstein series
+on Γ0(N), analogous expressions for Σres exist for general N. In particular,
+Σres(s) = cf,h · (−1)
+k−1
+2 π
+1
+2
+hs− 3
+4
+·
+Γ(s − 1
+4)Γ(s − 3
+4)
+Γ(s + k
+2 − 3
+4)Γ(s − κ
+2),
+(4.6)
+in which cf,h = 0 unless the following conditions from [TT13, §4.7] are met:
+a. f(z) is a dihedral form of odd weight k;
+b. h | N
+4 (we assume h ≥ 0 throughout);
+c. If h has square-free part h0 and L is the conductor of χχ−1
+h , then
+h0L2 divides N/4.
+The non-obvious choice of normalization in (4.6) has been chosen so that
+the residue of Σres(s) at s = 3
+4 equals cf,h.
+
+SUMS OF CUSP FORM COEFFICIENTS ALONG QUADRATIC SEQUENCES
+9
+5. Averages for Fourier Coefficients
+To study the partial sums �
+n≤X A(n2+h), we require certain information
+about the growth of Dh(s) in vertical strips. To this end, we present here
+a few point-wise and on-average bounds involving the Fourier coefficients of
+half-integral weight Maass forms.
+To give context for our on-average results, we first quote a conditional
+point-wise result for the Fourier coefficients of a Maass form.
+Lemma 5.1 (cf. [Duk88], Theorem 5). Assume the generalized Lindel¨of
+hypothesis and the Ramanujan–Petersson conjecture.
+Let µj be an L2-
+normalized Maass form of (half-integral) weight κ on Γ0(N), with Fourier
+expansion of the form (4.2). If n is a fundamental discriminant, we have
+ρj(n) ≪κ,N,ǫ (1 + |tj|)− κ
+2 sgn(n)+ǫe
+π
+2 |tj||n|− 1
+2+ǫ
+for any ǫ > 0.
+The main content of [Duk88, Theorem 5] is an unconditional version of
+Lemma 5.1, though the tj-dependence in the unconditional bound is too
+weak for our applications. For adequate unconditional results, we require
+some amount of averaging. Our first average is a spectral second moment
+for half-integral weight Maass forms due to Blomer [Blo08].
+Proposition 5.2 (follows from [Blo08], Lemma 5). Let {µj} be an orthonor-
+mal basis of weight κ = k − 1
+2 Maass forms on Γ0(N) with character χ · χk
+−1
+and Fourier expansions of the form (4.2). For any T ≥ 1 and ǫ > 0,
+�
+|tj|≤T
+|ρj(n)|2(1 + |tj|)κ sgn(n)− 1
+2
+cosh(πtj)
+≪ǫ
+T
+3
+2
+|n| + T ǫ|n|− 1
+2 +ǫ.
+We note that Proposition 5.2 implies that Duke’s conditional result holds
+unconditionally in the long average over |tj| ∼ T when |n| ≪ T 2.
+We can also produce strong on-average results for ρj(m) in the m-aspect
+by refining the method leading to [DFI02, Lemma 19.3]. We show how to
+obtain the following m-average upper bound.
+Proposition 5.3. Let µj be an L2-normalized weight κ = k− 1
+2 Maass form
+on Γ0(N) with character χ · χk
+−1 and Fourier expansion (4.2). Then
+�
+m∼M
+|ρj(±m)|2 ≪κ,N (1 + |tj|)∓κ�
+1 + |tj|
+M
+�
+eπ|tj|
+for any M > 1 and any choice of sign ±.
+This result implies that Duke’s conditional result holds unconditionally
+in the long average over |m| ∼ M, provided |tj| ≪ M. To prove Proposi-
+tion 5.3, we require a strengthened form of [DFI02, Lemma 19.2], which in
+turn relies on the following uniform estimate for the Whittaker function.
+
+10
+CHAN IEONG KUAN, DAVID LOWRY-DUDA, AND ALEXANDER WALKER
+Lemma 5.4. For y > 0, η real, and t real with |t| ≥ 1, we have
+Wη,it(y) ≪η |t|η− 1
+2 e− π
+2 |t| · y
+1
+2
+uniformly in the interval 0 ≤ y ≤ 3
+2|t|.
+Proof. This result essentially follows from the Liouville–Green approxima-
+tion of the differential equation for Wη,it(y), as described in [Olv74, ch. 6].
+Our specific application to the Whittaker function is not new, and indeed
+appears in [Dun03].
+In particular, Lemma 5.4 follows from [Dun03, p. 210-211] by removing
+η-uniformity from the “s ≤ s+” case. (Note that µ−1/6 in [Dun03, (4.14)]
+should read µ1/6.) The restriction to y ≤ 3
+2|t| (as opposed to y ≈ 2|t|, where
+the Whittaker function stops oscillating) simplifies our expression further
+by avoiding complications near the “turning point” at s+.
+(Away from
+the turning point, we note that one may appeal to the simpler error analysis
+of [Olv74, ch. 6, §1-5] and avoid Dunster’s use of [Olv74, ch. 11] entirely.)
+□
+We now give a strengthened form of [DFI02, Lemma 19.2].
+Lemma 5.5. There exists a constant α > 0 depending only on κ for which
+� ∞
+α|tj|
+W± κ
+2 ,itj(4πy)2 dy
+y2 ≫κ |tj|±κ−1e−π|tj|,
+uniformly in |tj| ≥ 1.
+Proof. Let η = ± κ
+2. From [GR15, 7.611(4)] we derive
+� ∞
+0
+Wη,itj(4πy)2 dy
+y =
+π
+sin(2πitj) · ψ(1
+2 − η + itj) − ψ(1
+2 − η − itj)
+Γ(1
+2 − η + itj)Γ(1
+2 − η − itj) ,
+in which ψ(z) is the digamma function. We apply Stirling’s approximation
+and the asymptotic ψ(1
+2 −η+itj)−ψ(1
+2 −η−itj) = iπ+Oη(1/tj) to produce
+� ∞
+0
+Wη,itj(4πy)2 dy
+y = π|tj|2ηe−π|tj|(1 + Oη(t−1
+j )).
+To control the behavior of Wη,itj(4πy) near y = 0, we apply Lemma 5.4
+and integrate to obtain
+� α|tj|
+0
+Wη,itj(4πy)2 dy
+y ≪η α|tj|2ηe−π|tj|,
+uniformly in |tj| ≥ 1 and α ≤ 3/(8π). In particular, there exists a small
+constant α depending only on η for which
+� ∞
+α|tj|
+Wη,itj(4πy)2 dy
+y ≫η |tj|2ηe−π|tj|.
+(5.1)
+As in the proof of [DFI02, Lemma 19.2], integration by parts implies the
+existence of some β depending only on η for which (5.1) holds when the
+
+SUMS OF CUSP FORM COEFFICIENTS ALONG QUADRATIC SEQUENCES
+11
+domain of integration is restricted further to α|tj| ≤ y ≤ β|tj|. Then
+� ∞
+α|tj|
+Wη,itj(4πy)2 dy
+y2 ≥
+� β|tj|
+α|tj|
+Wη,itj(4πy)2 dy
+y2
+≫η |tj|−1
+� β|tj|
+α|tj|
+Wη,itj(4πy)2 dy
+y ≫η |tj|2η−1e−π|tj|,
+which completes the proof.
+□
+We now return to the proof of Proposition 5.3.
+Proof of Proposition 5.3. Our proof adapts the proof of [DFI02, Lemma
+19.3]. Parseval’s identity gives
+� 1
+0
+|µj(z)|2dx =
+�
+n̸=0
+|ρj(n)|2W κn
+2|n| ,itj(4π|n|y)2.
+Since every orbit {γz : z ∈ Γ0(N)} has ON(1+Y −1) points in [0, 1]×(Y, ∞),
+integrating over y ≥ Y produces
+1 + 1
+Y ≫N
+� ∞
+Y
+� 1
+0
+|µj(z)|2 dxdy
+y2
+=
+�
+n̸=0
+|ρj(n)|2
+� ∞
+Y
+W κn
+2|n| ,itj(4π|n|y)2 dy
+y2 .
+(5.2)
+Lemma 5.5 and a change of variables implies that
+� ∞
+Y
+W κn
+2|n| ,itj(4π|n|y)2 dy
+y2 ≫κ |n| · |tj|κ sgn(n)−1e−π|tj|,
+provided that |n|Y ≤ α|tj| and |tj| ≥ 1. We set Y = α|tj|M−1 and deduce
+from (5.2) that
+1 + M
+α|tj| ≫κ,N
+�
+|n|≤M
+|n| · |ρj(n)|2 · |tj|κ sgn(n)−1e−π|tj|,
+which proves the proposition in the case |tj| ≥ 1 after restricting to |n| ∼ M
+and simplifying.
+The complementary case |tj| ≤ 1 is far simpler. By setting Y = 1/(4πM)
+in (5.2) and changing variables y �→ y/(4π|n|), it suffices to prove that
+� ∞
+1
+W κn
+2|n| ,itj(y)2 dy
+y2 ≫κ 1,
+(5.3)
+uniformly in |tj| ≤ cκ. To prove this, we note that left-hand side of (5.3) is a
+continuous function of tj and thus attains a global minimum depending only
+on κ (and sgn(n)). Since the integrand is non-negative, this global minimum
+is non-negative since the Whittaker function is not identically zero.
+□
+
+12
+CHAN IEONG KUAN, DAVID LOWRY-DUDA, AND ALEXANDER WALKER
+6. An Average Involving Inner Products
+In addition to the coefficient bounds from section §5, we require estimates
+involving ⟨y
+k
+2 + 1
+4 fθ, µj⟩; specifically, we would like to estimate the sum
+D :=
+�
+|tj|∼T
+|⟨y
+k
+2 + 1
+4fθ, µj⟩|2eπ|tj|.
+(6.1)
+The main result in this section is the following bound for D.
+Theorem 6.1. We have D ≪f,ǫ (1 + T)2k+ 1
+2−√
+k(k−2)+ǫ for any T > 0,
+f ∈ Sk(Γ0(N), χ) of weight k ≥ 2, and ǫ > 0.
+For ease of comparison, we note that Theorem 6.1 implies that D ≪f,ǫ
+T k+ 3
+2+
+1
+2k−3 +ǫ+1 for all k ≥ 2. This is technically a weaker bound by O(1/k2)
+in the exponent, but it makes the behavior with respect to k more apparent.
+To prove Theorem 6.1, we represent f(z) as a finite sum of holomorphic
+Poincar´e series, which we use for unfolding. The resulting objects are then
+understood using Propositions 5.2 and 5.3.
+6.1. An upper bound for ⟨y
+k
+2 + 1
+4 fθ, µj⟩. The cusp space Sk(Γ0(N), χ) is
+finite-dimensional and spanned by Poincar´e series {Pm}m≥1 of the form
+Pm(z) =
+�
+γ∈Γ∞\Γ0(N)
+χ(γ)j(γ, z)−ke(mγz),
+in which j(γ, z) is the usual j-invariant.
+The Sturm bound ([Stu87], or
+see [Ste07, Corollary 9.19] for more direct exposition) implies that our span-
+ning set may restrict to m ≪k,N 1.
+Consider the inner product
+⟨y
+k
+2 + 1
+4Pmθ, µj⟩ =
+� ∞
+0
+� 1
+0
+Im(z)
+k
+2 + 1
+4 θ(z)µj(z)e(mz)dxdy
+y2
+=
+�
+m=n1+n2
+r1(n1)ρj(n2)
+� ∞
+0
+y
+k
+2 − 3
+4e−2π(n1+m)yW n2κ
+2|n2| ,itj(4π|n2|y)dy
+y .
+Let G(n1, n2, m) denote the final integral above.
+By [ODL+20, 13.23.4],
+G may be written in terms of the 2F1-hypergeometric function; using the
+Mellin–Barnes integral [ODL+20, 15.6.6], this implies
+G =
+
+
+
+
+
+
+
+
+
+
+
+1
+2πi
+�
+(Re w)
+Γ(κ
+2 + itj − w)Γ(κ
+2 − itj − w)Γ(w)
+(4πn2)
+k
+2 − 3
+4 Γ(1
+2 − w)
+�n2
+n1
+�w
+dw,
+n2 > 0;
+1
+2πi
+�
+(Re w)
+Γ(κ
+2 + itj − w)Γ(κ
+2 − itj − w)Γ(w)
+(4π|n2|)
+k
+2 − 3
+4Γ(k − w)
+�|n2|
+m
+�w
+dw,
+n2 < 0,
+for any Re w ∈ (0, κ
+2 − | Im tj|). The integrand decays exponentially outside
+of | Im w| ≤ |tj| by Stirling’s approximation. In particular, G(n1, n2, m) ≪κ
+|n2|Re w− k
+2 − 3
+4 for |tj| ≪ 1, which includes any potential exceptional eigen-
+values itj ∈ R. The case |tj| ≪ 1 therefore gives ⟨y
+k
+2 + 1
+4Pmθ, µj⟩ = Ok,N(1)
+
+SUMS OF CUSP FORM COEFFICIENTS ALONG QUADRATIC SEQUENCES
+13
+by Proposition 5.3 and dyadic subdivision (with Re w near enough to 0 to
+guarantee convergence of the sum).
+Otherwise, for |tj| ≥ 1, Stirling gives the estimate
+≪m,κ,Re w | Im w − tj|
+κ
+2 −Re w− 1
+2 | Im w + tj|
+κ
+2 −Re w− 1
+2
+×| Im w|2 Re w− 1
+2 −κ δ[n2<0]e−π|tj||n2|Re w− k
+2 + 3
+4
+for the integrand of G(n1, n2, m) on the interval | Im w| ≤ |tj|. In the case
+n2 > 0, integrating gives G(n1, n2, m) ≪k,N |tj|κ− 1
+2 e−π|tj| for any choice of
+Re w (in part because n2 ≤ m ≪k,N 1). For n2 < 0, we have instead
+G(n1, n2, m) ≪k,N,Re w
+�
+|tj|− 1
+2 + |tj|κ−2 Re w−1�
+e−π|tj||n2|Re w− k
+2 + 3
+4 ,
+valid for any Re w ∈ (0, κ
+2). When k ≥ 2 (as in Theorem 6.1), however, there
+is no benefit in taking Re w outside Re w ∈ (0, κ
+2 − 1
+4]. We conclude that
+⟨y
+k
+2 + 1
+4 Pmθ, µj⟩ ≪k,N,Re w |tj|κ− 1
+2 e−π|tj|
+�
+0≤n<√m
+|ρj(m − n2)|
+(6.2)
++ |tj|κ−2 Re w−1e−π|tj| �
+n>√m
+|ρj(m − n2)|
+|m − n2|
+k
+2 − 3
+4−Re w
+when |tj| ≥ 1, for any Re w ∈ (0, κ
+2 − 1
+4] = (0, k
+2 − 1
+2].
+6.2. An upper bound for D. When f(z) = Pm(z) and T ≪ 1, the esti-
+mate ⟨y
+k
+2 + 1
+4 Pmθ, µj⟩ = Ok,N(1) implies that the spectral sum D introduced
+in (6.1) satisfies D ≪k,N 1. Otherwise, for f(z) = Pm(z) and T ≥ 1, the
+inequality (6.2) implies that
+D ≪k,N,Rew T 2κ−1 �
+|tj|∼T
+� �
+n<√m
+|ρj(m − n2)|
+�2
+e−π|tj|
++ T 2κ−4 Re w−2 �
+|tj|∼T
+� �
+n>√m
+|ρj(m − n2)|
+|m − n2|
+k
+2 − 3
+4 −Re w
+�2
+e−π|tj|,
+where we omit the non-dominant cross-term.
+By Cauchy–Schwarz and Proposition 5.2, the contribution from n < √m
+satisfies the bound
+T 2κ−1√m
+�
+n<√m
+�
+|tj|∼T
+|ρj(m − n2)|2e−π|tj|
+≪ǫ T 2κ−1√m
+�
+n<√m
+� T 2−κ
+m − n2 + T 1−κ+ǫ(m − n2)− 1
+2+ǫ�
+≪m,κ T κ+1,
+which is admissible in Theorem 6.1 since m ≪k,N 1.
+For the terms with n > √m, we split the sum at an unspecified n for which
+n2 − m ∼ M = M(T). In the head Dhead corresponding to n2 − m < M, a
+
+14
+CHAN IEONG KUAN, DAVID LOWRY-DUDA, AND ALEXANDER WALKER
+worst-case bound over dyadic subintervals gives some M0 < M for which
+Dhead ≪ T 2κ−4 Re w−2(log M)2 �
+|tj|∼T
+�
+�
+n2−m∼M0
+|ρj(m − n2)|
+|m − n2|
+k
+2 − 3
+4−Re w
+�2
+e−π|tj|
+≪ T 2κ−4 Re w−2MǫM−k+2+2 Re w
+0
+�
+|tj|∼T
+�
+�
+n2−m∼M0
+|ρj(m − n2)|2
+cosh(πtj)
+�
+≪ T 2κ−4 Re w−2MǫM−k+2+2 Re w
+0
+�
+n2−m∼M0
+�
+T 2+κM−1
+0
++ T 1+κ+ǫM
+− 1
+2 +ǫ
+0
+�
+≪ T 3κ−4 Re w−1MǫM
+2 Re w−k+ 3
+2
+0
+�
+T + T ǫM
+1
+2 +ǫ
+0
+�
+,
+in which we’ve applied Proposition 5.2. Here and for the rest of this section,
+all implicit constants may depend on k, N, ǫ, and Re w (where that appears).
+Note that M0 depends on M, T, and Re w. To remove M0 and Re w and
+produce a bound which depends only on M and T, we vary Re w and find
+a worst-case M0 in each case.
+a. For Re w ≤ k
+2 −1, all M0-powers are non-positive and the worst-case
+M0 is M0 = 1. We find Dhead ≪ T 3κ−4 Re wMǫ. We optimize with
+Re w = k
+2 − 1 to produce Dhead ≪ T k+ 5
+2Mǫ.
+b. For Re w ≥ k
+2 − 3
+4, all M0-powers are non-negative and the worst-case
+M0 is M0 = M, so Dhead ≪ T 3κ−4 Re w−1+ǫM2 Re w−k+ 3
+2 +2ǫ(T +M
+1
+2 ).
+If M ≫ T 2, we benefit from taking Re w small; with Re w = k
+2 − 3
+4,
+we find Dhead ≪ T k+ 1
+2+ǫM
+1
+2+2ǫ. Conversely, if M ≪ T 2, we benefit
+from Re w = k
+2 − 1
+2, to produce Dhead ≪ T k+ 1
+2 +ǫM
+1
+2 +2ǫ (as before).
+c. For Re w ∈ [k
+2 − 1, k
+2 − 3
+4], the M0-powers have mixed sign. A gen-
+eral upper bound is Dhead ≪ T 3κ−4 Re w−1+ǫM2ǫ(T + M2 Re w−k+2).
+When T ≫ M2 Re w−k+2, we benefit from taking Re w large and op-
+timize with Re w = min(k
+2 − 3
+4, k
+2 − 1 + 1
+2 logM T) to find Dhead ≪
+T k+ 5
+2 −2 logM T+ǫM2ǫ when M ≫ T 2 and Dhead ≪ T k+ 3
+2+ǫM2ǫ when
+M ≪ T 2. Conversely, if T ≪ M2 Re w−k+2, we must have T ≪ M
+1
+2,
+which incentivizes Re w as small as possible, i.e. Re w = k
+2 − 1 +
+1
+2 logM T. We find Dhead ≪ T k+ 5
+2 −2 logM T+ǫM2ǫ as before.
+Of the three estimates above, (b) is minimal for M ≪ T 2 while (c) is minimal
+for M ≫ T 2.
+We now consider the tail Dtail in which n2−m > M. By Cauchy–Schwarz
+and Proposition 5.3, we have
+�
+n2−m∼M
+|ρj(m − n2)|
+|m − n2|
+k
+2 − 3
+4−Re w ≪ MRe w− k
+2 + 3
+4
+�
+�
+n2−m∼M
+|ρj(m − n2)|2� 1
+2 · M
+1
+4
+≪ MRe w− k
+2 +1� �
+ℓ∼M
+|ρj(−ℓ)|2� 1
+2≪ MRe w− k
+2 +1(1 + |tj|)
+κ
+2
+�
+1 + |tj|
+1
+2
+M
+1
+2
+�
+e
+π
+2 |tj|.
+
+SUMS OF CUSP FORM COEFFICIENTS ALONG QUADRATIC SEQUENCES
+15
+The same result holds for the sum over all n2−m > M by dyadic summation,
+provided Re w < k
+2 − 1. (Note that choice of Re w here is unrelated to our
+choice of Re w in Dhead.) We now compute
+Dtail ≪ T 2κ−4 Re w−2 �
+|tj|∼T
+�
+�
+n2−m>M
+|ρj(m − n2)|
+|m − n2|
+k
+2 − 3
+4 −Re w
+�2
+e−π|tj|
+≪ T 2κ−4 Re w−2 �
+|tj|∼T
+M2 Re w−k+2T κ�
+1 + T
+M
+�
+≪ T 3κ−4 Re wM2 Re w−k+2�
+1 + T
+M
+�
+,
+(6.3)
+by applying Proposition 5.3 and the Weyl law.
+Finally, we determine bounds for D. In the regime 1 ≪ M ≪ T 2, we are
+led by (6.3) to take Re w large; with Re w = k
+2 − 1 − ǫ, we produce
+D ≪ T k+ 1
+2+ǫM
+1
+2 +2ǫ + T k+ 5
+2+4ǫM−2ǫ(1 + T/M),
+which is optimized at M = T 2 to produce D ≪ T k+ 5
+2 +ǫ. Conversely, for
+M ≫ T 2, (6.3) incentivizes Re w small; with Re w = ǫ, we produce
+D ≪ T k+ 5
+2 −2 logM T+ǫM2ǫ + T 3κ−4ǫM2ǫ−k+2.
+In the case k = 2, the second term dominates and we produce D ≪ T
+9
+2 +ǫ
+by taking M = T 2. For k > 2, we take M = T 1+√
+k/(k−2), which satisfies
+M ≫ T 2 as desired. We conclude that
+D ≪ T 2k+ 1
+2 −√
+k(k−2)+ǫ
+for any ǫ > 0. Note that this agrees with D ≪ T
+9
+2+ǫ in the case k = 2. This
+completes the proof of Theorem 6.1 in the case f = Pm, and the extension
+to general f is straightforward.
+Combining Theorem 6.1 with Proposition 5.2 via Cauchy–Schwarz, we
+produce a useful average involving ρj(h)⟨y
+k
+2 + 1
+4fθ, µj⟩.
+Corollary 6.2. Fix h > 0 and k ≥ 2. For any T > 0 and ǫ > 0, we have
+�
+|tj|∼T
+|ρj(h)⟨y
+k
+2 + 1
+4 fθ, µj⟩| ≪f,ǫ (1 + T)
+k
+2 + 3
+2 − 1
+2
+√
+k(k−2)+ǫ.
+To facilitate comparison, we note that the upper bound in Corollary 6.2
+can be replaced by T 2+
+1
+4k−6 +ǫ + 1 for any k ≥ 2, which is weaker in the
+exponent by O(1/k2) but more readable.
+Remark 6.3. Theorem 6.1 and Corollary 6.2 are not sharp.
+Assuming
+the generalized Lindel¨of hypothesis and Ramanujan–Petersson conjecture,
+Lemma 5.1 implies that ρj(−ℓ) ≪ |tj|
+κ
+2 +ǫe
+π
+2 |tj||ℓ|− 1
+2 +ǫ for all ǫ > 0. Sta-
+tionary phase confirms that our estimate for G(n1, n2, m) is relatively sharp
+and suggests that the absolute values in (6.2) can be moved outside the sum
+
+16
+CHAN IEONG KUAN, DAVID LOWRY-DUDA, AND ALEXANDER WALKER
+n > √m. If the resulting sum demonstrates square-root cancellation, we
+would have
+⟨y
+k
+2 + 1
+4 fθ, µj⟩ ≪k,N,ǫ |tj|
+k
+2 − 3
+4+ǫe− π
+2 |tj|
+by combining Lemma 5.1 with (6.2) in the case Re w = k
+2 − 1
+2 −ǫ. This would
+imply that D ≪f,ǫ T k+ 1
+2+ǫ and that the sum in Corollary 6.2 is Of,ǫ(T
+3
+2+ǫ).
+7. Sharp Cutoff Result
+In this section, we apply Perron’s formula (cf. [Tit86, Lemma 3.12]) to
+Dh(s) to study the partial sums of A(n2 + h). By the definition of Dh(s)
+from (1.2) and Perron’s formula, we have
+�
+m2+h≤X2
+A(m2 + h) =
+1
+4πi
+� 1+ǫ+iT
+1+ǫ−iT
+Dh( s
+2 + 1
+4)Xs ds
+s + O
+�X1+ǫ
+T
+�
+(7.1)
+for fixed ǫ > 0 and any T > 1.
+To understand the integral, we replace Dh(s) with its spectral expansion
+Σdisc(s) + Σres(s) + Σcont(s) and shift the line of integration. To justify this
+shift, we must quantify the growth of Σdisc, Σres, and Σcont in vertical strips.
+7.1. Growth of Σdisc. Recall from (4.3) that the discrete spectral compo-
+nent of Dh(s) equals
+Σdisc := (4π)
+k
+2 + 1
+4
+hs−1
+�
+j
+Γ(s − 1
+2 + itj)Γ(s − 1
+2 − itj)
+Γ(s − k
+2 + 1
+4)Γ(s + k
+2 − 3
+4) ρj(h)⟨y
+k
+2 + 1
+4 fθ, µj⟩.
+On the line Re s = σ, Stirling’s approximation gives the estimate
+Σdisc ≪k
+h1−σ
+|s|2σ− 3
+2
+�
+j
+|ρj(h)⟨y
+k
+2 + 1
+4 fθ, µj⟩|
+× |s + itj|σ−1−Im tj|s − itj|σ−1+Im tjeπ|s|−π max(|s|,|tj|),
+showing that the mass of the discrete spectrum concentrates in |tj| < |s|.
+The contribution of Maass forms with |tj| ≪ 1 is Ok,N(|s|−1/2), since there
+are ON(1) Maass forms of bounded spectral type by the Weyl law.
+In the case |tj| ≥ 1, we perform dyadic subdivision based on the size of
+min(|s + itj|, |s − itj|) and determine that
+Σdisc(s) ≪f,h,ǫ |s|− 1
+2 + |s|
+1
+2−σ
+�
+0≤ℓ≤log2 |s|
+(2ℓ)σ−1(|s| − 2ℓ)
+k
+2 + 3
+2 − 1
+2
+√
+k(k−2)+ǫ
+≪f,h,ǫ |s|
+k
+2 +1− 1
+2
+√
+k(k−2)+ǫ + |s|
+k
+2 +2− 1
+2
+√
+k(k−2)−σ+ǫ
+by applying Corollary 6.2. (This implies that Σdisc ≪ |s|
+3
+2+
+1
+4k−6 +ǫ(1+|s|1−σ)
+for k ≥ 2, which is slightly weaker but easier to parse).
+7.2. Growth of Σres. The growth rate of the residual contribution Σres
+in vertical strips is obvious from Stirling’s approximation and the explicit
+formulas (4.5) and (4.6). We conclude that Σres(s) ≪f,h |s|−1/2.
+
+SUMS OF CUSP FORM COEFFICIENTS ALONG QUADRATIC SEQUENCES
+17
+7.3. Growth of Σcont. We recall from (4.4) that the continuous spectrum’s
+contribution towards Dh(s) in Re s > 1
+2 equals
+Σcont(s) := (4π)
+k
+2 − 3
+4
+hs−1
+�
+a
+� ∞
+−∞
+Γ(s − 1
+2 + it)Γ(s − 1
+2 − it)
+Γ(s − k
+2 + 1
+4)Γ(s + k
+2 − 3
+4)
+× ρa(h, 1
+2 + it)⟨y
+k
+2 + 1
+4fθ, Eκ
+a (·, 1
+2 + it; χ)⟩ dt,
+in which Eκ
+a (z, v) is the Eisenstein series of weight κ, character χ · χk
+−1, and
+level N at a and ρa(h, v) is its hth Fourier coefficient following (4.2).
+To bound Σcont, we need estimates for ρa(h, v) and ⟨y
+k
+2 + 1
+4 fθ, Eκ
+a (·, v; χ)⟩
+on the critical line Re v = 1
+2. These are given in the following lemmas.
+Lemma 7.1. Let χκ,h = (h(−1)κ− 1
+2
+·
+). For any ǫ > 0,
+ρa(h, 1
+2 + it) ≪h,κ,N,ǫ
+L(1
+2 + 2it, χχκ,h)
+L(2N)(1 + 4it, χ2)Γ(1
+2 + κ
+2 + it).
+Proof. This result follows from recognizing the coefficients of Eκ
+a as Dirichlet
+L-functions. The computations are tedious but very similar to the proofs of
+Proposition 1.2 and Corollary 1.3 of [GH85] (though the proofs there apply
+to a differently normalized Eisenstein series of level 4, restrict to coefficients
+with square-free m, and don’t evaluate the Archimedean integral).
+An alternative evaluation for general m and with our normalization is
+summarized in [LD17, (2.1)-(2.2)]. We note that the behavior of non-square-
+free coefficients m differ from those of square-free coefficients by a finite
+Dirichlet correction factor depending on m. The generalization to higher
+level and non-trivial character is analogous.
+□
+Lemma 7.2. For any ǫ > 0 and any singular cusp a,
+⟨y
+k
+2 + 1
+4 fθ, Eκ
+a (·, 1
+2 + it; χ)⟩ ≪f,ǫ (1 + |t|)
+k
+2 +ǫe− π
+2 |tj|.
+Proof. The inner product ⟨y
+k
+2 + 1
+4fθ, Eκ
+a (·, v; χ)⟩ can be written as a Rankin–
+Selberg integral. Writing Γa for the stabilizer of a in Γ0(N), we recall that
+Eκ
+a (z, v; χ) =
+�
+γ∈Γa\Γ0(N)
+χ(γ)Jθ(σ−1
+a γ, z)−2κ Im(σ−1
+a γz)v,
+in which σa is a scaling matrix for the cusp a. We take σ∞ = ( 1 0
+0 1 ) and
+otherwise use the specific scaling matrix
+σa =
+�
+a
+�
+[N, w2]
+0
+�
+[N, w2]
+1/(a
+�
+[N, w2])
+�
+for the cusp a = u
+w to agree with [DI83, (2.3)].
+A standard unfolding argument (following a change of variables z �→ σaz)
+then shows that
+⟨y
+k
+2 + 1
+4 fθ, Eκ
+a (·, v; χ)⟩ =
+��
+σ−1
+a
+(Γa\H)
+yv+ k
+2 + 1
+4 fa(z)θa(z)dxdy
+y2 ,
+
+18
+CHAN IEONG KUAN, DAVID LOWRY-DUDA, AND ALEXANDER WALKER
+in which θa = θ|σa and fa = f|σa. We also note that σ−1
+a (Γa\H) = Γ∞\H.
+As in the standard Rankin–Selberg construction, this double integral has
+the Dirichlet series representation
+��
+Γ∞\H
+yv+ k
+2 + 1
+4 fa(z)θa(z)dxdy
+y2
+= Γ(v + k
+2 − 3
+4)
+(4π)v+ k
+2 − 3
+4
+�
+n≥1
+aa(n)ra(n)
+nv+ k
+2 − 3
+4
+,
+(7.2)
+where aa(·) and ra(·) denote the Fourier coefficients of fa and θa, respectively.
+In the special case a = ∞ one can recognize the Dirichlet series in (7.2)
+in terms of the symmetric square L-function of f, so that
+⟨y
+k
+2 + 1
+4 fθ, Eκ
+∞(·, 1
+2 + it; χ)⟩ = 2Γ(κ
+2 − it)L(1
+2 − 2it, Sym2f)
+(4π)
+κ
+2 −itL(1 − 4it, χ2)
+,
+up to some factor addressing bad primes. In particular, in the case a = ∞,
+Lemma 7.2 follows from the Phragm´en–Lindel¨of convexity principle and
+Stirling’s approximation.
+More generally, Lemma 7.2 reduces to convexity for the symmetric square
+L-function attached to (a twist of) the cusp form fa. To see this, it suffices
+to show that θa has a Fourier expansion which resembles a twist of θ away
+from a finite set of exceptional primes p ≪N 1. This can be verified through
+explicit computation.
+For example, suppose that a = u
+w is Γ0(4)-equivalent to the infinite cusp.
+Then there exists some matrix γ ∈ Γ0(4) so that ∞ = γ · a, which we may
+write in the form γ = ( a
+b
+w −u ). By carefully tracking square roots in the
+relevant j-factors, we compute that
+θa(z) = −i ǫ−1
+a
+�−w
+a
+��
+N
+(N, w2)
+� 1
+4 �
+n∈Z
+e
+�−n2b
+u
+�
+e
+� n2N
+(N, w2)z
+�
+.
+This implies that the Dirichlet series in (7.2) equals a certain symmetric
+square L-function away from bad primes, so the lemma holds whenever a is
+Γ0(4)-equivalent to ∞.
+The casework for cusps which are Γ0(4)-equivalent to 0 or 1
+2 is suitably
+analogous, so we omit details.
+□
+Combining these two lemmas, we find that
+ρa(h, 1
+2 + it)⟨y
+k
+2 + 1
+4 fθ, Eκ
+a (·, 1
+2 + it; χ)⟩
+(7.3)
+≪f,h,ǫ (1 + |t|)
+1
+2− k
+2 +ǫ · (1 + |t|)
+k
+2 +ǫ ≪f,h,ǫ (1 + |t|)
+1
+2+ǫ.
+Consequently, for Re s = σ > 1
+2, we have
+Σcont ≪f,h,ǫ
+� ∞
+−∞
+����
+Γ(s − 1
+2 + it)Γ(s − 1
+2 − it)
+Γ(s − k
+2 + 1
+4)Γ(s + k
+2 − 3
+4)
+����|t|
+1
+2+ǫdt
+≪ |s|
+3
+2 −2σ
+� ∞
+−∞
+|s − it|σ−1|s + it|σ−1|t|
+1
+2+ǫe−π max(|Im s|,|t|)+π|Im s|dt.
+
+SUMS OF CUSP FORM COEFFICIENTS ALONG QUADRATIC SEQUENCES
+19
+The exponential terms effectively concentrate mass in |t| < |Im s|, so that
+Σcont(s) ≪f,h,ǫ |s|
+1
+2 −σ
+� |Im s|
+0
+|s − it|σ−1|t|
+1
+2 +ǫdt ≪f,h,ǫ |s|1−σ+ǫ + |s|1+ǫ,
+and hence Σcont(s) ≪f,h,ǫ |s|1+ǫ in Re s > 1
+2.
+Remark 7.3. The bound Σcont(s) ≪f,h,ǫ |s|1+ǫ suffices for our purposes
+but is by no means sharp. Under the generalized Lindel¨of hypothesis, the
+upper bound (7.3) improves to Of,h,ǫ((1+|t|)− 1
+2+ǫ) and it would follow that
+Σcont(s) ≪f,h,ǫ |s|ǫ in the half-plane Re s > 1
+2.
+7.4. Contour Shifting. The growth estimates from §7.1, §7.2, and §7.3
+imply that the growth of Dh(s) in vertical strips in Re s > 1
+2 is dominated
+by that of Σdisc. Hence Dh(s) ≪ |s|
+k
+2 +2− 1
+2
+√
+k(k−2)−Re s+ǫ in a fixed vertical
+strip in Re s > 1
+2, where here and throughout §7.4 all implicit constants are
+allowed to depend on f, h, ǫ, and Re s (where it appears).
+In particular, on the line Re s = 1
+2 + ǫ, it follows that
+Dh( s
+2 + 1
+4)Xs/s ≪ |s|
+k
+2 + 1
+2− 1
+2
+√
+k(k−2)+ǫX
+1
+2+ǫ.
+Note also that Dh( s
+2 + 1
+4)Xs/s ≪ X1+ǫ/|s| on the line Re s = 1 + ǫ, by
+absolute convergence of the Dirichlet series. In the vertical strip Re s ∈ (1
+2 +
+ǫ, 1+ǫ) between these estimates, Dh( s
+2 + 1
+4)Xs/s is meromorphic, with poles
+at most at s = 1 (from Σres) and each real s = 1
+2 ± 2itj corresponding to an
+exceptional eigenvalue (from Σdisc). The Weyl law implies that exceptional
+eigenvalues, if they exist, are limited in number by ON(1).
+Away from these finitely many poles, the convexity principle implies that
+Dh( s
+2 + 1
+4)Xs/s ≪ |s|
+k
+2 + 1
+2 − 1
+2
+√
+k(k−2)+ǫX
+1
+2+ǫ + X1+ǫ/|s| in the vertical strip
+Re s ∈ (1
+2 + ǫ, 1 + ǫ). We conclude from (4.3) and (4.6) that
+1
+4πi
+� 1+ǫ+iT
+1+ǫ−iT
+Dh( s
+2 + 1
+4)Xs ds
+s
+= cf,hX + RE +
+1
+4πi
+�
+1
+2 +ǫ+iT
+1
+2+ǫ−iT
+Dh( s
+2 + 1
+4)Xs ds
+s
++ O
+�X1+ǫ
+T
++ T
+k
+2 + 1
+2− 1
+2
+√
+k(k−2)+ǫX
+1
+2+ǫ�
+,
+(7.4)
+in which RE is the sum over possible residues arising from exceptional eigen-
+values, given explicitly by
+RE :=(4π)
+k
+2 + 1
+4 h
+1
+2 X
+1
+2
+�
+itj∈R
+(X2/h)itjΓ(2itj)ρj(h)⟨y
+k
+2 + 1
+4fθ, µj⟩
+(1
+2 + 2itj)Γ(3
+4 − k
+2 + itj)Γ(k
+2 − 1
+4 + itj)
++ (4π)
+k
+2 + 1
+4h
+1
+2 X
+1
+2
+�
+itj∈R
+(X2/h)−itjΓ(−2itj)ρj(h)⟨y
+k
+2 + 1
+4 fθ, µj⟩
+(1
+2 − 2itj)Γ(3
+4 − k
+2 − itj)Γ(k
+2 − 1
+4 − itj).
+
+20
+CHAN IEONG KUAN, DAVID LOWRY-DUDA, AND ALEXANDER WALKER
+The contribution of the continuous spectrum Σcont in Dh( s
+2 + 1
+4) on the line
+Re s = 1
+2+ǫ is O(X
+1
+2+ǫT 1+ǫ) following §7.3. For the components of Dh( s
+2+ 1
+4)
+coming from the residual and discrete spectra, we shift the vertical contour
+farther left, to the line Re s = ǫ. In Σdisc, this shift passes a line segment of
+non-exceptional spectral poles, which contributes a finite sum of residues R
+of the form
+2(4π)
+k
+2 + 1
+4 h
+1
+2 X
+1
+2 Re
+�
+�
+0≤tj≤2T
+(X2/h)itjΓ(2itj)ρj(h)⟨y
+k
+2 + 1
+4 fθ, µj⟩
+(1
+2 + 2itj)Γ(3
+4 − k
+2 + itj)Γ(k
+2 − 1
+4 + itj)
+�
+= (4π)
+k
+2 − 1
+4 h
+1
+2 X
+1
+2 Im
+�
+�
+0≤tj≤2T
+(4X2/h)itjρj(h)⟨y
+k
+2 + 1
+4 fθ, µj⟩
+tj
+�
+1 + Ok( 1
+tj )
+��
+.
+We treat R as an error term and apply Corollary 6.2 to conclude that
+R ≪ X
+1
+2
+�
+ℓ≤log2 T
+�
+|tj|∼2−ℓT
+|ρj(h)⟨y
+k
+2 + 1
+4fθ, µj⟩|
+|tj|
+(7.5)
+≪ X
+1
+2
+�
+ℓ≤log2 T
+(T/2ℓ)
+k
+2 + 1
+2− 1
+2
+√
+k(k−2)+ǫ ≪ X
+1
+2T
+k
+2 + 1
+2− 1
+2
+√
+k(k−2)+ǫ.
+Following (7.1), (7.4), (7.5), and the estimate O(X
+1
+2 +ǫT 1+ǫ) for the shifted
+continuous spectrum, we have
+�
+m2+h≤X2
+A(m2 + h) = cf,hX + RE +
+1
+4πi
+� ǫ+iT
+ǫ−iT
+(Σres + Σdisc)( s
+2 + 1
+4)Xs ds
+s
++ O
+�X1+ǫ
+T
++ T
+k
+2 + 1
+2− 1
+2
+√
+k(k−2)+ǫX
+1
+2 +ǫ
+�
+.
+The contour integral over Re s = ǫ is O(T
+k
+2 + 7
+4− 1
+2
+√
+k(k−2)+ǫXǫ) following
+the upper bounds Σdisc(s) ≪ |s|
+k
+2 + 7
+4− 1
+2
+√
+k(k−2)+ǫ and Σres(s) ≪ |s|− 1
+2 on the
+line Re s = 1
+4 + ǫ. By taking T = X1/(k+3−√
+k(k−2)), we optimize our collec-
+tive errors to size O(X1−1/(k+3−√
+k(k−2))+ǫ). Since this error term dominates
+RE ≪ X1/2+Θ ≪ X39/64 (by the comment following (4.3)), the potential
+contribution of RE may be ignored. This completes the proof of our main
+arithmetic result.
+Theorem 7.4. For k ≥ 2, h > 0, and any ǫ > 0, we have
+�
+m2+h≤X2
+A(m2 + h) = cf,hX + Of,h,ǫ
+�
+X
+1−
+1
+k+3−√
+k(k−2) +ǫ�
+,
+where cf,h = 0 in most cases following Remark 4.1.
+This result also holds with the more readable error term O(X
+3
+4 +
+1
+32k−44 +ǫ).
+
+SUMS OF CUSP FORM COEFFICIENTS ALONG QUADRATIC SEQUENCES
+21
+Remark 7.5. The heuristic and conditional improvements to Corollary 6.2
+noted in Remark 6.3 would imply that Σdisc ≪ |s|1+ǫ + |s|2−Re s+ǫ and that
+R ≪ X
+1
+2T
+1
+2+ǫ. In addition, our bound for the shifted continuous integral
+would improve to O(X
+1
+2+ǫT ǫ) under the generalized Lindel¨of hypothesis
+following Remark 7.3. Optimizing errors with T = X1/3 would improve the
+error in Theorem 7.4 to O(X
+2
+3+ǫ), which is comparable to Bykovskii’s work
+on the divisor function [Byk87]. Error terms of size O(X
+1
+2+ǫ) are conjectured
+to hold in both problems.
+References
+[Blo08] Valentin Blomer.
+Sums of Hecke eigenvalues over values of
+quadratic polynomials. Int. Math. Res. Not. IMRN, (16):Art.
+ID rnn059. 29, 2008.
+[Byk87] V. A. Bykovski˘ı. Spectral decompositions of certain automor-
+phic functions and their number-theoretic applications. Jour-
+nal of Soviet Mathematics, 36, 1987.
+[DFI02] W. Duke, J. B. Friedlander, and H. Iwaniec. The subconvexity
+problem for Artin L-functions. Invent. Math., 149(3):489–577,
+2002.
+[DI83] J.-M. Deshouillers and H. Iwaniec.
+Kloosterman sums and
+Fourier coefficients of cusp forms. Invent. Math., 70(2):219–
+288, 1982/83.
+[Duk88] W. Duke. Hyperbolic distribution problems and half-integral
+weight Maass forms. Invent. Math., 92(1):73–90, 1988.
+[Dun03] T. M. Dunster. Uniform asymptotic approximations for the
+Whittaker functions Mκ,iµ(z) and Wκ,iµ(z). Analysis and Ap-
+plications, 1(2):199–212, 2003.
+[GH85] Dorian Goldfeld and Jeffrey Hoffstein. Eisenstein series of 1
+2-
+integral weight and the mean value of real Dirichlet L-series.
+Invent. Math., 80(2):185–208, 1985.
+[GH11] Dorian Goldfeld and Joseph Hundley. Automorphic Represen-
+tations and L-Functions for the General Linear Group, vol-
+ume 1 of Cambridge Studies in Advanced Mathematics. Cam-
+bridge University Press, 2011.
+[GR15] I. S. Gradshteyn and I. M. Ryzhik. Table of integrals, series,
+and products.
+Elsevier/Academic Press, Amsterdam, eighth
+edition, 2015. Translated from the Russian, Translation edited
+and with a preface by Daniel Zwillinger and Victor Moll, Re-
+vised from the seventh edition [MR2360010].
+[HH16] Jeff Hoffstein and Thomas A. Hulse.
+Multiple Dirichlet se-
+ries and shifted convolutions. J. Number Theory, 161:457–533,
+2016. With an appendix by Andre Reznikov.
+[HKLDW18] Thomas A. Hulse, Chan Ieong Kuan, David Lowry-Duda, and
+Alexander Walker. Second moments in the generalized Gauss
+circle problem. Forum Math. Sigma, 6:Paper No. e24, 49, 2018.
+
+22
+CHAN IEONG KUAN, DAVID LOWRY-DUDA, AND ALEXANDER WALKER
+[Hoo63] Christopher Hooley. On the number of divisors of a quadratic
+polynomial. Acta Math., 110:97–114, 1963.
+[KS93] Svetlana Katok and Peter Sarnak. Heegner points, cycles and
+Maass forms. Israel J. Math., 84(1-2):193–227, 1993.
+[KS03] H. Kim and P. Sarnak. Refined estimates towards the Ramanu-
+jan and Selberg conjectures. J. Amer. Math. Soc., 16:175–181,
+2003.
+[LD17] David Lowry-Duda. On Some Variants of the Gauss Circle
+Problem. PhD thesis, Brown University, 2017.
+[ODL+20] F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I.
+Schneider, R. F. Boisvert, C. W. Clark, B. V. Saunders B. R.
+Mille and, H. S. Cohl, and eds. M. A. McClain. NIST Digi-
+tal Library of Mathematical Functions. http://dlmf.nist.gov/,
+Release 1.0.26 of 2020-03-15, 2020.
+[Olv74] Frank W. J. Olver. Asymptotics and special functions. Aca-
+demic Press New York, 1974.
+[PY19] Ian Petrow and Matthew P. Young. A generalized cubic mo-
+ment and the Petersson formula for newforms. Math. Ann.,
+373(1-2):287–353, 2019.
+[Sar84] Peter Sarnak. Additive number theory and Maass forms. In
+Number theory (New York, 1982), volume 1052 of Lecture
+Notes in Math., pages 286–309. Springer, Berlin, 1984.
+[Shi73] G. Shimura. On modular forms of half integral weight. Annals
+of Mathematics, 97(3):440–481, 1973.
+[SS77] J.-P. Serre and H. M. Stark. Modular forms of weight 1/2. In
+Modular functions of one variable, VI (Proc. Second Internat.
+Conf., Univ. Bonn, Bonn, 1976), Lecture Notes in Math., Vol.
+627, pages 27–67. Springer, Berlin, 1977.
+[Ste07] William A. Stein. Modular Forms, a Computational Approach.
+American Mathematical Society, 2007.
+[Str08] Fredrik Str¨omberg. Computation of Maass Waveforms with
+Nontrivial Multiplier Systems. Mathematics of Computation,
+77(264):2375–2416, 2008.
+[Stu87] Jacob Sturm. On the congruence of modular forms. In Number
+theory (New York, 1984–1985), volume 1240 of Lecture Notes
+in Math., pages 275–280. Springer, Berlin, 1987.
+[Tit86] E. C. Titchmarsh. The theory of the Riemann zeta-function.
+The Clarendon Press, Oxford University Press, New York, sec-
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+Brown.
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+Non-split sums of
+coefficients of GL(2)-automorphic forms.
+Israel J. Math.,
+195(2):677–723, 2013.
+
diff --git a/j9FKT4oBgHgl3EQfxC4a/content/tmp_files/load_file.txt b/j9FKT4oBgHgl3EQfxC4a/content/tmp_files/load_file.txt
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@@ -0,0 +1,698 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf,len=697
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='11901v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='NT]  27 Jan 2023  SUMS OF CUSP FORM COEFFICIENTS ALONG  QUADRATIC SEQUENCES  CHAN IEONG KUAN, DAVID LOWRY-DUDA, AND ALEXANDER WALKER  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Let f(z) = � A(n)n(k−1)/2e(nz) be a cusp form of weight  k ≥ 2 on Γ0(N) with character χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' By studying a certain shifted convo-  lution sum, we prove that �  n≤X A(n2 +h) = cf,hX +Of,h,ǫ(X  6k−8  8k−11 +ǫ)  for ǫ > 0, which improves a result of Blomer [Blo08] with error X6/7+ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Introduction  In [Hoo63], Hooley considers the average behavior of the divisor function  d(n) within a quadratic sequence and proves that  S(X) :=  �  n≤X  d(n2 + h) = chX log X + c′  hX + Oh,ǫ  �  X  8  9 log3 X  �  for constants ch, c′  h (when −h is non-square) using the theory of exponen-  tial sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Hooley’s error term was improved by Bykovskii [Byk87], who  uses the spectral theory of automorphic forms to study the generalized sum  �  n≤X σν(n2 + h), in which σν(n) = �  d|n dν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' In the case ν = 0, Bykovskii  obtains S(X) = chX log X + c′  hX + O(X2/3+ǫ) for any ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  An analogous question for the normalized coefficients of a GL(2) cusp  form was introduced by Blomer in [Blo08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Let f(z) = � a(n)e(nz) be a  cusp form on Sk(Γ0(N), χ) with weight k ≥ 4 and set A(n) = a(n)/n(k−1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  For any monic quadratic polynomial q(x) ∈ Z[x], Blomer proves  �  n≤X  A(q(n)) = cf,qX + Of,q,ǫ  �  X  6  7+ǫ�  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1)  for some constant cf,q which equals 0 in most but not all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Both Hooley and Bykovskii rely on the convolution identity σ0 = 1 ∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  As this has no analogue for cusp forms, Blomer instead proceeds by writing  f(z) as a sum of Poincar´e series, whose q(n)-th Fourier coefficients involve  sums of the form  �  c≥1  1  NcSχ(m, q(n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Nc)Jk−1  �4π  �  q(n)m  Nc  �  ,  in which Jk−1 is the J-Bessel function and Sχ is a twisted Kloosterman  sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Blomer evaluates a smooth version of the sum over n ≤ X using Pois-  son summation, which converts the sums Sχ(m, q(n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Nc) into half-integral  1  2  CHAN IEONG KUAN, DAVID LOWRY-DUDA, AND ALEXANDER WALKER  weight Kloosterman sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Blomer’s result then follows from cancellation in  the latter, as proved using a half-integral weight Kuznetsov formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Two alternative methods for treating the cusp form analogy are given by  Templier and Tsimerman in [TT13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' The first is inspired by earlier work by  Sarnak on d(n2 + h) in [Sar84], who relates the shifted convolution sum  �  n≥1  d(n2 + h)  (n2 + h)s  to the Petersson inner product ⟨Im(z)  1  4 θ(z)E(z, 1  2), Ph(z, s)⟩, where θ(z) is  a theta function, E(z, s) is a weight 0 real analytic Eisenstein series, and  Ph(z, s) is a half-integral weight Poincar´e series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' (Sarnak notes that this  connects S(X) to the spectrum of the half-integral weight Laplacian, going  no further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=') In [TT13, §4], similar methods are applied to study the shifted  convolution sum  Dh(s) :=  �  n≥0  r1(n)a(n + h)  (n + h)s+ k  2 − 3  4  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2)  in which rℓ(n) is the number of representations of n as a sum of ℓ squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Templier and Tsimerman give a meromorphic continuation of Dh(s) and  prove that Dh(s) grows polynomially in |Im s| in vertical strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  In par-  ticular, for any smooth function g(x) on R+ with Mellin transform �g(s)  satisfying �g(s) ≪ Γ(s) in vertical strips, [TT13, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='8] gives a constant c′  f,h  depending only on f(z) and h for which  �  n≥0  A(n2 + h)g  �n2 + h  X2  �  = c′  f,h �g(1  2)X + Oǫ  �X  1  2+Θ+ǫ  h  1  2Θ−δ  �  ,  in which Θ ≤  7  64 (due to [KS03]) denotes progress towards the Selberg  eigenvalue conjecture and δ ≤ 1  6 (due to [PY19]) denotes progress towards  the Ramanujan–Petersson conjecture for half-integral weight cusp forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Templier and Tsimerman give a second proof of their result for A(n2 +h)  using representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  This alternative framework allows for equal  treatment of holomorphic cusp forms and Maass cusp forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Here as before,  Templier–Tsimerman restrict to smoothed sums, so their results cannot be  directly compared to the sharp cutoff (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1) from [Blo08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  In this paper, we refine the shifted convolution sum technique described  in [TT13, §4] to produce the following sharp cutoff result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Let f(z) = � a(n)e(nz) be a cusp form in Sk(Γ0(N), χ) and  define A(n) = a(n)/n(k−1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' For k ≥ 2, h > 0, and any ǫ > 0, we have  �  n2+h≤X2  A(n2 + h) = cf,hX + Of,h,ǫ  �  X  1−  1  k+3−√  k(k−2) +ǫ�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  The constant cf,h is given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  SUMS OF CUSP FORM COEFFICIENTS ALONG QUADRATIC SEQUENCES  3  The error term in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1 may be replaced by X  3  4 +  1  32k−44 +ǫ for  any k ≥ 2, which is weaker in the exponent by O(1/k2) but makes the  k-dependence more clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1 improves the error term O(X  6  7+ǫ)  from [Blo08], though we remark that the cusp form analogy still lags behind  Bykovskii’s 2  3 + ǫ exponent in the divisor function analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Exponents of  size 1  2 + ǫ are conjectured to hold in both problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Outline of Paper  As in [TT13, §4], we understand sums of the form �  n≤X A(n2 + h) by  studying the Dirichlet series Dh(s) defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' In §3, we prove Proposi-  tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1, which relates Dh(s) to an inner product involving f(z), the Jacobi  theta function, and an appropriate Poincar´e series P κ  h (z, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Spectral expan-  sion of P κ  h (z, s) in §4 then expresses Dh(s) as a sum of terms corresponding  to the discrete, residual, and continuous spectra of the hyperbolic Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Our treatment of the discrete spectrum of half-integral weight Maass  forms differs greatly from [TT13] and represents the main novelty of this  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' We replace the use of weak estimates for the individual Fourier coef-  ficients ρj(n) of Maass forms with strong bounds for their behavior in long  averages over the spectrum of Maass forms and long averages over the coef-  ficient n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' The former is given by [Blo08], while the latter is new and refines  ideas of [DFI02, §19] by incorporating uniform bounds for the Whittaker  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' The derivation of these averages is carried out in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  These Fourier coefficient estimates are applied in §6 to prove Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1,  a bound for the sum  D :=  �  |tj|∼T  |⟨y  k  2 + 1  4fθ, µj⟩|2eπ|tj|,  which averages over an orthonormal basis of (half-integral weight) Maass  forms µj with spectral types |tj| ∈ [T, 2T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' As in [Blo08], we leverage the  fact that f is holomorphic to write it as a linear combination of holomorphic  Poincar´e series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Unlike [Blo08], however, these Poincar´e series are used to  form shifted convolutions, instead of introducing Kloosterman-type sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  In §7, we use bounds for D to control the growth of Dh(s) with respect  to | Im s| in vertical strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Our main result Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1 then follows by  Perron’s formula and contour shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Acknowledgements  The authors thank Stephen Lester, who introduced them to this problem,  as well as Yiannis Petridis, whose suggestions led to a simplification of sec-  tion §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' We also thank Thomas Hulse, whose involvement over many years  has improved this paper and moreover its authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  The first author was supported in part by NSFC (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' 11901585).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' The  second author was supported by the Simons Collaboration in Arithmetic Ge-  ometry, Number Theory, and Computation via the Simons Foundation grant  4  CHAN IEONG KUAN, DAVID LOWRY-DUDA, AND ALEXANDER WALKER  546235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' The third author was supported by the Additional Funding Pro-  gramme for Mathematical Sciences, delivered by EPSRC (EP/V521917/1)  and the Heilbronn Institute for Mathematical Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' A Triple Inner Product  For integral k ≥ 1, let Sk(Γ0(N), χ) denote the set of cusp forms on  Γ0(N) which transform under the character χ · χk  −1, where χ−1 = (−1  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  We assume without loss of generality that 4 | N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Once and for all, we  fix a weight k modular form f(z) = � a(n)e(nz) ∈ Sk(Γ0(N), χ) and a  positive integer h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Here and later, we use the common notation e(x) := e2πix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Let θ(z) = �  n∈Z e(n2z) = �  n≥0 r1(n)e(nz) denote the classical Jacobi  theta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' The theta function is a modular form of weight 1  2 on Γ0(4)  (see [Shi73]), transforming via  θ(γz) = ǫ−1  d  � c  d  �  (cz + d)  1  2 θ(z),  γ =  �a  b  c  d  �  ∈ Γ0(4),  in which ǫd = 1 for d ≡ 1 mod 4 and ǫd = i for d ≡ 3 mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Let P κ  h (z, s)  denote the weight κ := k − 1  2 twisted Poincar´e series on Γ0(N), defined by  P κ  h (z, s) :=  �  γ∈Γ∞\\Γ0(N)  χ(γ)Jθ(γ, z)−2κ Im(γz)se(hγz),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1)  in which Jθ(γ, z) = ǫ−1  d ( c  d)(cz+d)  1  2/|cz+d|  1  2 is the normalized theta cocycle  θ(γz)/θ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Then P κ  h (z, w) and Im(z)  k  2 + 1  4f(z)θ(z) transform identically un-  der the action of Γ0(N), so the Petersson inner product ⟨y  k  2 + 1  4fθ, P κ  h (·, s)⟩  is well-defined over Γ0(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' A standard unfolding argument gives  ⟨y  k  2 + 1  4fθ, P κ  h (·, s)⟩ =  �  Γ0(N)\\H  y  k  2 + 1  4f(z)θ(z)P κ  h (z, s) dxdy  y2  =  � ∞  0  � 1  0  ys+ k  2 − 3  4 f(z)θ(z)e(hz)dxdy  y  =  � ∞  0  � 1  0  ys+ k  2 − 3  4  � �  m1≥1  a(m1)e2πim1x−2πm1y�  ×  � �  m2≥0  r1(m2)e−2πim2x−2πm2y�  e−2πihx−2πhy dxdy  y  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  The x-integral extracts those terms with m1−m2−h = 0, and the remaining  y-integral evaluates in terms of the gamma function:  ⟨y  k  2 + 1  4 fθ, P κ  h (·, s)⟩ =  �  m≥0  r1(m)a(m + h)  � ∞  0  ys+ k  2 − 3  4e−4π(m+h)y dy  y  = Γ(s + k  2 − 3  4)  (4π)s+ k  2 − 3  4  �  m≥0  r1(m)a(m + h)  (m + h)s+ k  2 − 3  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  SUMS OF CUSP FORM COEFFICIENTS ALONG QUADRATIC SEQUENCES  5  Standard estimates show that this Dirichlet series converges absolutely for  Re(s) > 3  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' By rearranging, we obtain the following identity for the Dirichlet  series Dh(s) introduced in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Fix h > 0 and any f(z) = � a(n)e(nz) ∈ Sk(Γ0(N), χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  We have  Dh(s) :=  �  m≥0  r1(m)a(m + h)  (m + h)s+ k  2 − 3  4  = (4π)s+ k  2 − 3  4 ⟨y  k  2 + 1  4fθ, P κ  h (·, s)⟩  Γ(s + k  2 − 3  4)  in the region Re s > 3  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Spectral Expansion of the Poincar´e Series  The series Dh(s) has a meromorphic continuation to all s ∈ C obtained  through spectral expansion of the Poincar´e series P κ  h (z, s) in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  (See [Str08, §15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3] for a good general reference on spectral expansions for  general weight and the shapes of each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=') As a weight κ = k − 1  2  object, this spectral expansion takes the form  P κ  h (z, s) =  �  j  ⟨P κ  h (·, s), µj⟩µj(z) +  �  ℓ  ⟨P κ  h (·, s), Rℓ⟩Rℓ(z)  +  1  4πi  �  a  �  ( 1  2)  �  P κ  h (·, s), Eκ  a (·, w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' χ)  �  Eκ  a (z, w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' χ) dw,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1)  in which {µj} denotes an orthonormal eigenbasis of weight κ Maass forms of  level N and character χ·χk  −1, {Rℓ} is a finite orthonormal basis of the resid-  ual spectrum of weight κ and character χ·χk  −1, and Eκ  a (z, w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' χ) is the weight  κ twisted Eisenstein series with character χ · χk  −1 attached to the cusp a of  Γ0(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' We refer to the expressions at right in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1) as the discrete, residual,  and continuous spectra, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Inserting this spectral expansion into  Dh(s) as presented in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1 gives a spectral expansion of the form  Dh(s) = Σdisc(s) + Σres(s) + Σcont(s),  which we now describe more fully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  The automorphic forms in the discrete and continuous spectra have Fourier  expansions, which take the form  µj(z) =  �  n̸=0  ρj(n)W nκ  2|n| ,itj(4π|n|y)e(nx),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2)  Eκ  a (z, w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' χ) = δ[a=∞]yw + ρa(0, w)y1−w  +  �  n̸=0  ρa(n, w)W nκ  2|n| ,w− 1  2 (4π|n|y)e(nx),  in which δ[·] is the Kronecker delta and Wη,ν(z) is the Whittaker function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  By unfolding the Poincar´e series P κ  h (z, s) and applying the integral for-  mula [GR15, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='621(3)] for the resulting y-integral, we evaluate ⟨P κ  h (·, s), µj⟩  6  CHAN IEONG KUAN, DAVID LOWRY-DUDA, AND ALEXANDER WALKER  and conclude that the discrete spectrum’s contribution towards Dh(s) equals  Σdisc := (4π)  k  2 + 1  4  hs−1  �  j  Γ(s − 1  2 + itj)Γ(s − 1  2 − itj)  Γ(s − k  2 + 1  4)Γ(s + k  2 − 3  4) ρj(h)⟨y  k  2 + 1  4 fθ, µj⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3)  We will establish in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1 that this series for Σdisc converges everywhere away  from poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Assuming this, Σdisc(s) defines a meromorphic function which  is analytic in Re s > 1  2 + supj|Im tj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' By the Shimura correspondence for  Maass forms (see [KS93]), Σdisc(s) is analytic for Re s > 1  2 + Θ  2 , where Θ  denotes the progress towards the Selberg eigenvalue conjecture as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Similarly, we determine that the continuous spectrum’s contribution to-  wards Dh(s) may be written  Σcont(s) := (4π)  k  2 − 3  4  hs−1  �  a  � ∞  −∞  Γ(s − 1  2 + it)Γ(s − 1  2 − it)  Γ(s − k  2 + 1  4)Γ(s + k  2 − 3  4)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='4)  × ρa(h, 1  2 + it)⟨y  k  2 + 1  4 fθ, Eκ  a (·, 1  2 + it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' χ)⟩ dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  We will prove in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3 that Σcont converges everywhere away from poles and  therefore defines an analytic function of s in Re s > 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Through delicate  contour shifting (as in [HH16, §4] or [HKLDW18, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2]), one may show  that Σcont extends meromorphically to all Re s ∈ C, though we only consider  Re s > 1  2 in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  We conclude this section by discussing the residual spectrum, which con-  tributes a term of the form  Σres(s) :=  (4π)s+ k  2 − 3  4  Γ(s + k  2 − 3  4)  �  ℓ  ⟨Rℓ, P κ  h (·, s)⟩⟨y  k  2 + 1  4 fθ, Rℓ⟩  towards Dh(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' The material here follows [TT13, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='5 and §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='7], with minor  modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  By the theory of raising operators, the weight κ = k− 1  2 residual spectrum  lifts from weight 1  2 forms if k is odd and from weight 3  2 forms is k is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  By [Duk88, §2], the residual spectrum does not appear in weight 3  2, so we  restrict to k odd for the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Suppose initially that N/4 is square-free and odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' By [SS77], the weight  1  2 residual spectrum appears only when χ = ( N/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' In this case, the space  is one-dimensional and spanned by the theta function  y  1  4 θN(z) = y  1  4  �  n∈Z  e(Nn2z  4  ) = y  1  4 +  �  n̸=0  (πNn2)− 1  4W 1  4 , 1  4 (πNn2y)e(Nn2x  4  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Under these assumptions, we have  Σres(s) :=  (4π)s+ k  2 − 3  4  Γ(s + k  2 − 3  4)⟨u, P κ  h (·, s)⟩⟨y  k  2 + 1  4 fθ, u⟩,  in which u(z) denotes the L2-normalized lift of y  1  4 θN(z) to weight κ under  the raising operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' As the weight η raising operator Rη = iy ∂  ∂x +y ∂  ∂y + η  2  SUMS OF CUSP FORM COEFFICIENTS ALONG QUADRATIC SEQUENCES  7  maps y1/4 to (η  2 + 1  4)y1/4 and W η  2 , 1  4 (4πmy)e(mx) to −W η+2  2 , 1  4(4πmy)e(mx),  the (non-normalized) lift of y  1  4θN(z) to weight κ equals  U(z) := y  1  4  �  1  2≤η<κ  η≡ 1  2 mod 2  (η  2 + 1  4) + (−1)  k−1  2  �  n̸=0  (πNn2)− 1  4 W κ  2 , 1  4 (πNn2y)e(Nn2x  4  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  To relate U(z) and u(z), we consider the effect of the raising operators  on L2 norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Let µ denote any Maass cusp form of weight η and type ν  on Γ0(N) and let Lη = −iy ∂  ∂x + y ∂  ∂y − η  2 denote the weight η Maass low-  ering operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' By combining [GH11, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='4)] with the last offset equation  on [GH11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' 90], we produce  ∥Rηµ∥2 = ⟨Rηµ, Rηµ⟩ = ⟨µ, −Lη+2Rηµ⟩  = ⟨µ, ∆ηµ⟩ + η  2(1 + η  2)⟨µ, µ⟩ =  �1  4 + ν2 + η  2(1 + η  2)  �  ∥µ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  We note that this is analogous to [TT13, equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='20)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  In the special case ν = i/4, we conclude that  ∥U∥2 = ∥y  1  4 θN∥2  �  1  2≤η<κ  η≡ 1  2 mod 2  (η  2 + 1  4)(η  2 + 3  4) = ∥y  1  4 θN∥2  k−1  2�  j=1  j(j − 1  2),  which implies that  u(z) =  y  1  4  ∥y  1  4 θN∥  �  1  2≤η<κ  η≡ 1  2 mod 2  (η  2 + 1  4)  1  2  (η  2 + 3  4)  1  2  + (−1)  k−1  2 d  − 1  2  k  ∥y  1  4 θN∥  �  n̸=0  W κ  2 , 1  4 (πNn2y)  (πNn2)  1  4  e(Nn2x  4  ),  in which dk = � k−1  2  j=1 j(j − 1  2), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' [TT13, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  We now compute the two inner products which appear in Σres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' To begin,  we unfold the Poincar´e series and apply [GR15, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='621(3)] to produce  ⟨u, P κ  h (·, s)⟩ = 2(−1)  k−1  2 d  − 1  2  k δ[Nn2=4h]  (4πh)  1  4 ∥y  1  4θN∥  � ∞  0  ys−1W κ  2 , 1  4(4πhy)e−2πhy dy  y  = (−1)  k−1  2 d  − 1  2  k  r1(4h  N )Γ(s − 1  4)Γ(s − 3  4)  (4πh)s− 3  4∥y  1  4 θN∥Γ(s − κ  2)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  To compute ⟨y  k  2 + 1  4 fθ, u⟩, we recognize u(z) as a multiple of the residue at  w = 3  4 of the Eisenstein series Eκ  ∞(z, w) of level N, weight κ, and trivial char-  acter (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1), with h = 0 and χ trivial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' We have Resw=3/4 Eκ  ∞(z, w) =  8  CHAN IEONG KUAN, DAVID LOWRY-DUDA, AND ALEXANDER WALKER  u(z)d1/2  k  ∥y  1  4 θN∥−1 from [TT13, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='7], and hence  ⟨y  k  2 + 1  4 fθ, u⟩ = d  − 1  2  k  ∥y  1  4 θN∥ · Res  w= 3  4  ⟨y  k  2 + 1  4 fθ, Eκ  ∞(z, w)⟩  = d  − 1  2  k  ∥y  1  4 θN∥ · Res  w= 3  4  ∞  �  n=1  2a(n2)  � ∞  0  yw+ k  2 − 3  4e−2πn2y dy  y  = 2d  − 1  2  k  ∥y  1  4θN∥ · Res  w= 3  4  Γ(w + k  2 − 3  4)  (2π)w+ k  2 − 3  4  ∞  �  n=1  a(n2)  n2w+k− 3  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  The Dirichlet series in the line above equals L(2w − 1  2, Sym2f)/ζ(4w − 1),  in which L(s, Sym2f) is the symmetric square L-function of f(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Putting  everything together, we derive the explicit formula  Σres(s) = 2  k  2 Γ(k  2)(−1)  k−1  2 r1(4h  N )Γ(s − 1  4)Γ(s − 3  4)  ζ(2)dkhs− 3  4 Γ(s + k  2 − 3  4)Γ(s − κ  2)  Res  w=1 L(w, Sym2f)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='5)  = 2  k  2 √π(−1)  k−1  2 r1(4h  N )  ζ(2)Γ(k+1  2 )hs− 3  4 Γ(s − 1  4)Γ(s − 3  4)  Γ(s + k  2 − 3  4)Γ(s − κ  2) · Res  w=1 L(w, Sym2f),  in which the second line follows from the identity dk = Γ(k)/2k−1 and the  gamma duplication formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' In conclusion, we note that Σres(s) is mero-  morphic in C and analytic in Re s > 3  4, with a simple pole at s = 3  4 only  when k is odd, f(z) is dihedral, χ = ( N/4), and r1(4h/N) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' For N with generic square part and odd level k, the residual  spectrum has a basis consisting of (finitely many) lifts of theta functions of  the form  y  1  4 θψ  M(z) = y  1  4  �  n∈Z  ψ(n)e(Mn2z),  in which ψ is a character of conductor L such that ML2 | N  4 and ψ·( ·  M ) = χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Since these lifts can all be recognized as residues of weight κ Eisenstein series  on Γ0(N), analogous expressions for Σres exist for general N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' In particular,  Σres(s) = cf,h · (−1)  k−1  2 π  1  2  hs− 3  4 Γ(s − 1  4)Γ(s − 3  4)  Γ(s + k  2 − 3  4)Γ(s − κ  2),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='6)  in which cf,h = 0 unless the following conditions from [TT13, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='7] are met:  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' f(z) is a dihedral form of odd weight k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' h | N  4 (we assume h ≥ 0 throughout);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' If h has square-free part h0 and L is the conductor of χχ−1  h , then  h0L2 divides N/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  The non-obvious choice of normalization in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='6) has been chosen so that  the residue of Σres(s) at s = 3  4 equals cf,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  SUMS OF CUSP FORM COEFFICIENTS ALONG QUADRATIC SEQUENCES  9  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Averages for Fourier Coefficients  To study the partial sums �  n≤X A(n2+h), we require certain information  about the growth of Dh(s) in vertical strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' To this end, we present here  a few point-wise and on-average bounds involving the Fourier coefficients of  half-integral weight Maass forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  To give context for our on-average results, we first quote a conditional  point-wise result for the Fourier coefficients of a Maass form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' [Duk88], Theorem 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Assume the generalized Lindel¨of  hypothesis and the Ramanujan–Petersson conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Let µj be an L2-  normalized Maass form of (half-integral) weight κ on Γ0(N), with Fourier  expansion of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' If n is a fundamental discriminant, we have  ρj(n) ≪κ,N,ǫ (1 + |tj|)− κ  2 sgn(n)+ǫe  π  2 |tj||n|− 1  2+ǫ  for any ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  The main content of [Duk88, Theorem 5] is an unconditional version of  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1, though the tj-dependence in the unconditional bound is too  weak for our applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' For adequate unconditional results, we require  some amount of averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Our first average is a spectral second moment  for half-integral weight Maass forms due to Blomer [Blo08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2 (follows from [Blo08], Lemma 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Let {µj} be an orthonor-  mal basis of weight κ = k − 1  2 Maass forms on Γ0(N) with character χ · χk  −1  and Fourier expansions of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' For any T ≥ 1 and ǫ > 0,  �  |tj|≤T  |ρj(n)|2(1 + |tj|)κ sgn(n)− 1  2  cosh(πtj)  ≪ǫ  T  3  2  |n| + T ǫ|n|− 1  2 +ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  We note that Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2 implies that Duke’s conditional result holds  unconditionally in the long average over |tj| ∼ T when |n| ≪ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  We can also produce strong on-average results for ρj(m) in the m-aspect  by refining the method leading to [DFI02, Lemma 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' We show how to  obtain the following m-average upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Let µj be an L2-normalized weight κ = k− 1  2 Maass form  on Γ0(N) with character χ · χk  −1 and Fourier expansion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Then  �  m∼M  |ρj(±m)|2 ≪κ,N (1 + |tj|)∓κ�  1 + |tj|  M  �  eπ|tj|  for any M > 1 and any choice of sign ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  This result implies that Duke’s conditional result holds unconditionally  in the long average over |m| ∼ M, provided |tj| ≪ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' To prove Proposi-  tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3, we require a strengthened form of [DFI02, Lemma 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2], which in  turn relies on the following uniform estimate for the Whittaker function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  10  CHAN IEONG KUAN, DAVID LOWRY-DUDA, AND ALEXANDER WALKER  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' For y > 0, η real, and t real with |t| ≥ 1, we have  Wη,it(y) ≪η |t|η− 1  2 e− π  2 |t| · y  1  2  uniformly in the interval 0 ≤ y ≤ 3  2|t|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' This result essentially follows from the Liouville–Green approxima-  tion of the differential equation for Wη,it(y), as described in [Olv74, ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Our specific application to the Whittaker function is not new, and indeed  appears in [Dun03].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  In particular, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='4 follows from [Dun03, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' 210-211] by removing  η-uniformity from the “s ≤ s+” case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' (Note that µ−1/6 in [Dun03, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='14)]  should read µ1/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=') The restriction to y ≤ 3  2|t| (as opposed to y ≈ 2|t|, where  the Whittaker function stops oscillating) simplifies our expression further  by avoiding complications near the “turning point” at s+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  (Away from  the turning point, we note that one may appeal to the simpler error analysis  of [Olv74, ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' 6, §1-5] and avoid Dunster’s use of [Olv74, ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' 11] entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=')  □  We now give a strengthened form of [DFI02, Lemma 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' There exists a constant α > 0 depending only on κ for which  � ∞  α|tj|  W± κ  2 ,itj(4πy)2 dy  y2 ≫κ |tj|±κ−1e−π|tj|,  uniformly in |tj| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Let η = ± κ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' From [GR15, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='611(4)] we derive  � ∞  0  Wη,itj(4πy)2 dy  y =  π  sin(2πitj) · ψ(1  2 − η + itj) − ψ(1  2 − η − itj)  Γ(1  2 − η + itj)Γ(1  2 − η − itj) ,  in which ψ(z) is the digamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' We apply Stirling’s approximation  and the asymptotic ψ(1  2 −η+itj)−ψ(1  2 −η−itj) = iπ+Oη(1/tj) to produce  � ∞  0  Wη,itj(4πy)2 dy  y = π|tj|2ηe−π|tj|(1 + Oη(t−1  j )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  To control the behavior of Wη,itj(4πy) near y = 0, we apply Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='4  and integrate to obtain  � α|tj|  0  Wη,itj(4πy)2 dy  y ≪η α|tj|2ηe−π|tj|,  uniformly in |tj| ≥ 1 and α ≤ 3/(8π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' In particular, there exists a small  constant α depending only on η for which  � ∞  α|tj|  Wη,itj(4πy)2 dy  y ≫η |tj|2ηe−π|tj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1)  As in the proof of [DFI02, Lemma 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2], integration by parts implies the  existence of some β depending only on η for which (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1) holds when the  SUMS OF CUSP FORM COEFFICIENTS ALONG QUADRATIC SEQUENCES  11  domain of integration is restricted further to α|tj| ≤ y ≤ β|tj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Then  � ∞  α|tj|  Wη,itj(4πy)2 dy  y2 ≥  � β|tj|  α|tj|  Wη,itj(4πy)2 dy  y2  ≫η |tj|−1  � β|tj|  α|tj|  Wη,itj(4πy)2 dy  y ≫η |tj|2η−1e−π|tj|,  which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  □  We now return to the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Our proof adapts the proof of [DFI02, Lemma  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Parseval’s identity gives  � 1  0  |µj(z)|2dx =  �  n̸=0  |ρj(n)|2W κn  2|n| ,itj(4π|n|y)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Since every orbit {γz : z ∈ Γ0(N)} has ON(1+Y −1) points in [0, 1]×(Y, ∞),  integrating over y ≥ Y produces  1 + 1  Y ≫N  � ∞  Y  � 1  0  |µj(z)|2 dxdy  y2  =  �  n̸=0  |ρj(n)|2  � ∞  Y  W κn  2|n| ,itj(4π|n|y)2 dy  y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2)  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='5 and a change of variables implies that  � ∞  Y  W κn  2|n| ,itj(4π|n|y)2 dy  y2 ≫κ |n| · |tj|κ sgn(n)−1e−π|tj|,  provided that |n|Y ≤ α|tj| and |tj| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' We set Y = α|tj|M−1 and deduce  from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2) that  1 + M  α|tj| ≫κ,N  �  |n|≤M  |n| · |ρj(n)|2 · |tj|κ sgn(n)−1e−π|tj|,  which proves the proposition in the case |tj| ≥ 1 after restricting to |n| ∼ M  and simplifying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  The complementary case |tj| ≤ 1 is far simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' By setting Y = 1/(4πM)  in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2) and changing variables y �→ y/(4π|n|), it suffices to prove that  � ∞  1  W κn  2|n| ,itj(y)2 dy  y2 ≫κ 1,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3)  uniformly in |tj| ≤ cκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' To prove this, we note that left-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3) is a  continuous function of tj and thus attains a global minimum depending only  on κ (and sgn(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Since the integrand is non-negative, this global minimum  is non-negative since the Whittaker function is not identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  □  12  CHAN IEONG KUAN, DAVID LOWRY-DUDA, AND ALEXANDER WALKER  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' An Average Involving Inner Products  In addition to the coefficient bounds from section §5, we require estimates  involving ⟨y  k  2 + 1  4 fθ, µj⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' specifically, we would like to estimate the sum  D :=  �  |tj|∼T  |⟨y  k  2 + 1  4fθ, µj⟩|2eπ|tj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1)  The main result in this section is the following bound for D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' We have D ≪f,ǫ (1 + T)2k+ 1  2−√  k(k−2)+ǫ for any T > 0,  f ∈ Sk(Γ0(N), χ) of weight k ≥ 2, and ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  For ease of comparison, we note that Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1 implies that D ≪f,ǫ  T k+ 3  2+  1  2k−3 +ǫ+1 for all k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' This is technically a weaker bound by O(1/k2)  in the exponent, but it makes the behavior with respect to k more apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  To prove Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1, we represent f(z) as a finite sum of holomorphic  Poincar´e series, which we use for unfolding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' The resulting objects are then  understood using Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' An upper bound for ⟨y  k  2 + 1  4 fθ, µj⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' The cusp space Sk(Γ0(N), χ) is  finite-dimensional and spanned by Poincar´e series {Pm}m≥1 of the form  Pm(z) =  �  γ∈Γ∞\\Γ0(N)  χ(γ)j(γ, z)−ke(mγz),  in which j(γ, z) is the usual j-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  The Sturm bound ([Stu87], or  see [Ste07, Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='19] for more direct exposition) implies that our span-  ning set may restrict to m ≪k,N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Consider the inner product  ⟨y  k  2 + 1  4Pmθ, µj⟩ =  � ∞  0  � 1  0  Im(z)  k  2 + 1  4 θ(z)µj(z)e(mz)dxdy  y2  =  �  m=n1+n2  r1(n1)ρj(n2)  � ∞  0  y  k  2 − 3  4e−2π(n1+m)yW n2κ  2|n2| ,itj(4π|n2|y)dy  y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Let G(n1, n2, m) denote the final integral above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  By [ODL+20, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='4],  G may be written in terms of the 2F1-hypergeometric function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' using the  Mellin–Barnes integral [ODL+20, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='6], this implies  G =  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  1  2πi  �  (Re w)  Γ(κ  2 + itj − w)Γ(κ  2 − itj − w)Γ(w)  (4πn2)  k  2 − 3  4 Γ(1  2 − w)  �n2  n1  �w  dw,  n2 > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  1  2πi  �  (Re w)  Γ(κ  2 + itj − w)Γ(κ  2 − itj − w)Γ(w)  (4π|n2|)  k  2 − 3  4Γ(k − w)  �|n2|  m  �w  dw,  n2 < 0,  for any Re w ∈ (0, κ  2 − | Im tj|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' The integrand decays exponentially outside  of | Im w| ≤ |tj| by Stirling’s approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' In particular, G(n1, n2, m) ≪κ  |n2|Re w− k  2 − 3  4 for |tj| ≪ 1, which includes any potential exceptional eigen-  values itj ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' The case |tj| ≪ 1 therefore gives ⟨y  k  2 + 1  4Pmθ, µj⟩ = Ok,N(1)  SUMS OF CUSP FORM COEFFICIENTS ALONG QUADRATIC SEQUENCES  13  by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3 and dyadic subdivision (with Re w near enough to 0 to  guarantee convergence of the sum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Otherwise, for |tj| ≥ 1, Stirling gives the estimate  ≪m,κ,Re w | Im w − tj|  κ  2 −Re w− 1  2 | Im w + tj|  κ  2 −Re w− 1  2  ×| Im w|2 Re w− 1  2 −κ δ[n2<0]e−π|tj||n2|Re w− k  2 + 3  4  for the integrand of G(n1, n2, m) on the interval | Im w| ≤ |tj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' In the case  n2 > 0, integrating gives G(n1, n2, m) ≪k,N |tj|κ− 1  2 e−π|tj| for any choice of  Re w (in part because n2 ≤ m ≪k,N 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' For n2 < 0, we have instead  G(n1, n2, m) ≪k,N,Re w  �  |tj|− 1  2 + |tj|κ−2 Re w−1�  e−π|tj||n2|Re w− k  2 + 3  4 ,  valid for any Re w ∈ (0, κ  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' When k ≥ 2 (as in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1), however, there  is no benefit in taking Re w outside Re w ∈ (0, κ  2 − 1  4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' We conclude that  ⟨y  k  2 + 1  4 Pmθ, µj⟩ ≪k,N,Re w |tj|κ− 1  2 e−π|tj|  �  0≤n<√m  |ρj(m − n2)|  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2)  + |tj|κ−2 Re w−1e−π|tj| �  n>√m  |ρj(m − n2)|  |m − n2|  k  2 − 3  4−Re w  when |tj| ≥ 1, for any Re w ∈ (0, κ  2 − 1  4] = (0, k  2 − 1  2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' An upper bound for D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' When f(z) = Pm(z) and T ≪ 1, the esti-  mate ⟨y  k  2 + 1  4 Pmθ, µj⟩ = Ok,N(1) implies that the spectral sum D introduced  in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1) satisfies D ≪k,N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Otherwise, for f(z) = Pm(z) and T ≥ 1, the  inequality (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2) implies that  D ≪k,N,Rew T 2κ−1 �  |tj|∼T  � �  n<√m  |ρj(m − n2)|  �2  e−π|tj|  + T 2κ−4 Re w−2 �  |tj|∼T  � �  n>√m  |ρj(m − n2)|  |m − n2|  k  2 − 3  4 −Re w  �2  e−π|tj|,  where we omit the non-dominant cross-term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  By Cauchy–Schwarz and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2, the contribution from n < √m  satisfies the bound  T 2κ−1√m  �  n<√m  �  |tj|∼T  |ρj(m − n2)|2e−π|tj|  ≪ǫ T 2κ−1√m  �  n<√m  � T 2−κ  m − n2 + T 1−κ+ǫ(m − n2)− 1  2+ǫ�  ≪m,κ T κ+1,  which is admissible in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1 since m ≪k,N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  For the terms with n > √m, we split the sum at an unspecified n for which  n2 − m ∼ M = M(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' In the head Dhead corresponding to n2 − m < M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' a  14  CHAN IEONG KUAN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' DAVID LOWRY-DUDA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' AND ALEXANDER WALKER  worst-case bound over dyadic subintervals gives some M0 < M for which  Dhead ≪ T 2κ−4 Re w−2(log M)2 �  |tj|∼T  �  �  n2−m∼M0  |ρj(m − n2)|  |m − n2|  k  2 − 3  4−Re w  �2  e−π|tj|  ≪ T 2κ−4 Re w−2MǫM−k+2+2 Re w  0  �  |tj|∼T  �  �  n2−m∼M0  |ρj(m − n2)|2  cosh(πtj)  �  ≪ T 2κ−4 Re w−2MǫM−k+2+2 Re w  0  �  n2−m∼M0  �  T 2+κM−1  0  + T 1+κ+ǫM  − 1  2 +ǫ  0  �  ≪ T 3κ−4 Re w−1MǫM  2 Re w−k+ 3  2  0  �  T + T ǫM  1  2 +ǫ  0  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  in which we’ve applied Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Here and for the rest of this section,  all implicit constants may depend on k, N, ǫ, and Re w (where that appears).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Note that M0 depends on M, T, and Re w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' To remove M0 and Re w and  produce a bound which depends only on M and T, we vary Re w and find  a worst-case M0 in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' For Re w ≤ k  2 −1, all M0-powers are non-positive and the worst-case  M0 is M0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' We find Dhead ≪ T 3κ−4 Re wMǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' We optimize with  Re w = k  2 − 1 to produce Dhead ≪ T k+ 5  2Mǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' For Re w ≥ k  2 − 3  4, all M0-powers are non-negative and the worst-case  M0 is M0 = M, so Dhead ≪ T 3κ−4 Re w−1+ǫM2 Re w−k+ 3  2 +2ǫ(T +M  1  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  If M ≫ T 2, we benefit from taking Re w small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' with Re w = k  2 − 3  4,  we find Dhead ≪ T k+ 1  2+ǫM  1  2+2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Conversely, if M ≪ T 2, we benefit  from Re w = k  2 − 1  2, to produce Dhead ≪ T k+ 1  2 +ǫM  1  2 +2ǫ (as before).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' For Re w ∈ [k  2 − 1, k  2 − 3  4], the M0-powers have mixed sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' A gen-  eral upper bound is Dhead ≪ T 3κ−4 Re w−1+ǫM2ǫ(T + M2 Re w−k+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  When T ≫ M2 Re w−k+2, we benefit from taking Re w large and op-  timize with Re w = min(k  2 − 3  4, k  2 − 1 + 1  2 logM T) to find Dhead ≪  T k+ 5  2 −2 logM T+ǫM2ǫ when M ≫ T 2 and Dhead ≪ T k+ 3  2+ǫM2ǫ when  M ≪ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Conversely, if T ≪ M2 Re w−k+2, we must have T ≪ M  1  2,  which incentivizes Re w as small as possible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Re w = k  2 − 1 +  1  2 logM T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' We find Dhead ≪ T k+ 5  2 −2 logM T+ǫM2ǫ as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Of the three estimates above, (b) is minimal for M ≪ T 2 while (c) is minimal  for M ≫ T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  We now consider the tail Dtail in which n2−m > M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' By Cauchy–Schwarz  and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3, we have  �  n2−m∼M  |ρj(m − n2)|  |m − n2|  k  2 − 3  4−Re w ≪ MRe w− k  2 + 3  4  �  �  n2−m∼M  |ρj(m − n2)|2� 1  2 · M  1  4  ≪ MRe w− k  2 +1� �  ℓ∼M  |ρj(−ℓ)|2� 1  2≪ MRe w− k  2 +1(1 + |tj|)  κ  2  �  1 + |tj|  1  2  M  1  2  �  e  π  2 |tj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  SUMS OF CUSP FORM COEFFICIENTS ALONG QUADRATIC SEQUENCES  15  The same result holds for the sum over all n2−m > M by dyadic summation,  provided Re w < k  2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' (Note that choice of Re w here is unrelated to our  choice of Re w in Dhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=') We now compute  Dtail ≪ T 2κ−4 Re w−2 �  |tj|∼T  �  �  n2−m>M  |ρj(m − n2)|  |m − n2|  k  2 − 3  4 −Re w  �2  e−π|tj|  ≪ T 2κ−4 Re w−2 �  |tj|∼T  M2 Re w−k+2T κ�  1 + T  M  �  ≪ T 3κ−4 Re wM2 Re w−k+2�  1 + T  M  �  ,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3)  by applying Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3 and the Weyl law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Finally, we determine bounds for D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' In the regime 1 ≪ M ≪ T 2, we are  led by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3) to take Re w large;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' with Re w = k  2 − 1 − ǫ, we produce  D ≪ T k+ 1  2+ǫM  1  2 +2ǫ + T k+ 5  2+4ǫM−2ǫ(1 + T/M),  which is optimized at M = T 2 to produce D ≪ T k+ 5  2 +ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Conversely, for  M ≫ T 2, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3) incentivizes Re w small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' with Re w = ǫ, we produce  D ≪ T k+ 5  2 −2 logM T+ǫM2ǫ + T 3κ−4ǫM2ǫ−k+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  In the case k = 2, the second term dominates and we produce D ≪ T  9  2 +ǫ  by taking M = T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' For k > 2, we take M = T 1+√  k/(k−2), which satisfies  M ≫ T 2 as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' We conclude that  D ≪ T 2k+ 1  2 −√  k(k−2)+ǫ  for any ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Note that this agrees with D ≪ T  9  2+ǫ in the case k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' This  completes the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1 in the case f = Pm, and the extension  to general f is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Combining Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1 with Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2 via Cauchy–Schwarz, we  produce a useful average involving ρj(h)⟨y  k  2 + 1  4fθ, µj⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Fix h > 0 and k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' For any T > 0 and ǫ > 0, we have  �  |tj|∼T  |ρj(h)⟨y  k  2 + 1  4 fθ, µj⟩| ≪f,ǫ (1 + T)  k  2 + 3  2 − 1  2  √  k(k−2)+ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  To facilitate comparison, we note that the upper bound in Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2  can be replaced by T 2+  1  4k−6 +ǫ + 1 for any k ≥ 2, which is weaker in the  exponent by O(1/k2) but more readable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1 and Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2 are not sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Assuming  the generalized Lindel¨of hypothesis and Ramanujan–Petersson conjecture,  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1 implies that ρj(−ℓ) ≪ |tj|  κ  2 +ǫe  π  2 |tj||ℓ|− 1  2 +ǫ for all ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Sta-  tionary phase confirms that our estimate for G(n1, n2, m) is relatively sharp  and suggests that the absolute values in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2) can be moved outside the sum  16  CHAN IEONG KUAN, DAVID LOWRY-DUDA, AND ALEXANDER WALKER  n > √m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' If the resulting sum demonstrates square-root cancellation, we  would have  ⟨y  k  2 + 1  4 fθ, µj⟩ ≪k,N,ǫ |tj|  k  2 − 3  4+ǫe− π  2 |tj|  by combining Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1 with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2) in the case Re w = k  2 − 1  2 −ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' This would  imply that D ≪f,ǫ T k+ 1  2+ǫ and that the sum in Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2 is Of,ǫ(T  3  2+ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Sharp Cutoff Result  In this section, we apply Perron’s formula (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' [Tit86, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='12]) to  Dh(s) to study the partial sums of A(n2 + h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' By the definition of Dh(s)  from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2) and Perron’s formula, we have  �  m2+h≤X2  A(m2 + h) =  1  4πi  � 1+ǫ+iT  1+ǫ−iT  Dh( s  2 + 1  4)Xs ds  s + O  �X1+ǫ  T  �  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1)  for fixed ǫ > 0 and any T > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  To understand the integral, we replace Dh(s) with its spectral expansion  Σdisc(s) + Σres(s) + Σcont(s) and shift the line of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' To justify this  shift, we must quantify the growth of Σdisc, Σres, and Σcont in vertical strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Growth of Σdisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Recall from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3) that the discrete spectral compo-  nent of Dh(s) equals  Σdisc := (4π)  k  2 + 1  4  hs−1  �  j  Γ(s − 1  2 + itj)Γ(s − 1  2 − itj)  Γ(s − k  2 + 1  4)Γ(s + k  2 − 3  4) ρj(h)⟨y  k  2 + 1  4 fθ, µj⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  On the line Re s = σ, Stirling’s approximation gives the estimate  Σdisc ≪k  h1−σ  |s|2σ− 3  2  �  j  |ρj(h)⟨y  k  2 + 1  4 fθ, µj⟩|  × |s + itj|σ−1−Im tj|s − itj|σ−1+Im tjeπ|s|−π max(|s|,|tj|),  showing that the mass of the discrete spectrum concentrates in |tj| < |s|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  The contribution of Maass forms with |tj| ≪ 1 is Ok,N(|s|−1/2), since there  are ON(1) Maass forms of bounded spectral type by the Weyl law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  In the case |tj| ≥ 1, we perform dyadic subdivision based on the size of  min(|s + itj|, |s − itj|) and determine that  Σdisc(s) ≪f,h,ǫ |s|− 1  2 + |s|  1  2−σ  �  0≤ℓ≤log2 |s|  (2ℓ)σ−1(|s| − 2ℓ)  k  2 + 3  2 − 1  2  √  k(k−2)+ǫ  ≪f,h,ǫ |s|  k  2 +1− 1  2  √  k(k−2)+ǫ + |s|  k  2 +2− 1  2  √  k(k−2)−σ+ǫ  by applying Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' (This implies that Σdisc ≪ |s|  3  2+  1  4k−6 +ǫ(1+|s|1−σ)  for k ≥ 2, which is slightly weaker but easier to parse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Growth of Σres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' The growth rate of the residual contribution Σres  in vertical strips is obvious from Stirling’s approximation and the explicit  formulas (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='5) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' We conclude that Σres(s) ≪f,h |s|−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  SUMS OF CUSP FORM COEFFICIENTS ALONG QUADRATIC SEQUENCES  17  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Growth of Σcont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' We recall from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='4) that the continuous spectrum’s  contribution towards Dh(s) in Re s > 1  2 equals  Σcont(s) := (4π)  k  2 − 3  4  hs−1  �  a  � ∞  −∞  Γ(s − 1  2 + it)Γ(s − 1  2 − it)  Γ(s − k  2 + 1  4)Γ(s + k  2 − 3  4)  × ρa(h, 1  2 + it)⟨y  k  2 + 1  4fθ, Eκ  a (·, 1  2 + it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' χ)⟩ dt,  in which Eκ  a (z, v) is the Eisenstein series of weight κ, character χ · χk  −1, and  level N at a and ρa(h, v) is its hth Fourier coefficient following (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  To bound Σcont, we need estimates for ρa(h, v) and ⟨y  k  2 + 1  4 fθ, Eκ  a (·, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' χ)⟩  on the critical line Re v = 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' These are given in the following lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Let χκ,h = (h(−1)κ− 1  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' For any ǫ > 0,  ρa(h, 1  2 + it) ≪h,κ,N,ǫ  L(1  2 + 2it, χχκ,h)  L(2N)(1 + 4it, χ2)Γ(1  2 + κ  2 + it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' This result follows from recognizing the coefficients of Eκ  a as Dirichlet  L-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' The computations are tedious but very similar to the proofs of  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2 and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3 of [GH85] (though the proofs there apply  to a differently normalized Eisenstein series of level 4, restrict to coefficients  with square-free m, and don’t evaluate the Archimedean integral).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  An alternative evaluation for general m and with our normalization is  summarized in [LD17, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' We note that the behavior of non-square-  free coefficients m differ from those of square-free coefficients by a finite  Dirichlet correction factor depending on m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' The generalization to higher  level and non-trivial character is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  □  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' For any ǫ > 0 and any singular cusp a,  ⟨y  k  2 + 1  4 fθ, Eκ  a (·, 1  2 + it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' χ)⟩ ≪f,ǫ (1 + |t|)  k  2 +ǫe− π  2 |tj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' The inner product ⟨y  k  2 + 1  4fθ, Eκ  a (·, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' χ)⟩ can be written as a Rankin–  Selberg integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Writing Γa for the stabilizer of a in Γ0(N), we recall that  Eκ  a (z, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' χ) =  �  γ∈Γa\\Γ0(N)  χ(γ)Jθ(σ−1  a γ, z)−2κ Im(σ−1  a γz)v,  in which σa is a scaling matrix for the cusp a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' We take σ∞ = ( 1 0  0 1 ) and  otherwise use the specific scaling matrix  σa =  �  a  �  [N, w2]  0  �  [N, w2]  1/(a  �  [N, w2])  �  for the cusp a = u  w to agree with [DI83, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  A standard unfolding argument (following a change of variables z �→ σaz)  then shows that  ⟨y  k  2 + 1  4 fθ, Eκ  a (·, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' χ)⟩ =  ��  σ−1  a  (Γa\\H)  yv+ k  2 + 1  4 fa(z)θa(z)dxdy  y2 ,  18  CHAN IEONG KUAN, DAVID LOWRY-DUDA, AND ALEXANDER WALKER  in which θa = θ|σa and fa = f|σa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' We also note that σ−1  a (Γa\\H) = Γ∞\\H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  As in the standard Rankin–Selberg construction, this double integral has  the Dirichlet series representation  ��  Γ∞\\H  yv+ k  2 + 1  4 fa(z)θa(z)dxdy  y2  = Γ(v + k  2 − 3  4)  (4π)v+ k  2 − 3  4  �  n≥1  aa(n)ra(n)  nv+ k  2 − 3  4  ,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2)  where aa(·) and ra(·) denote the Fourier coefficients of fa and θa, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  In the special case a = ∞ one can recognize the Dirichlet series in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2)  in terms of the symmetric square L-function of f, so that  ⟨y  k  2 + 1  4 fθ, Eκ  ∞(·, 1  2 + it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' χ)⟩ = 2Γ(κ  2 − it)L(1  2 − 2it, Sym2f)  (4π)  κ  2 −itL(1 − 4it, χ2)  ,  up to some factor addressing bad primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' In particular, in the case a = ∞,  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2 follows from the Phragm´en–Lindel¨of convexity principle and  Stirling’s approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  More generally, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2 reduces to convexity for the symmetric square  L-function attached to (a twist of) the cusp form fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' To see this, it suffices  to show that θa has a Fourier expansion which resembles a twist of θ away  from a finite set of exceptional primes p ≪N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' This can be verified through  explicit computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  For example, suppose that a = u  w is Γ0(4)-equivalent to the infinite cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Then there exists some matrix γ ∈ Γ0(4) so that ∞ = γ · a, which we may  write in the form γ = ( a  b  w −u ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' By carefully tracking square roots in the  relevant j-factors, we compute that  θa(z) = −i ǫ−1  a  �−w  a  ��  N  (N, w2)  � 1  4 �  n∈Z  e  �−n2b  u  �  e  � n2N  (N, w2)z  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  This implies that the Dirichlet series in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2) equals a certain symmetric  square L-function away from bad primes, so the lemma holds whenever a is  Γ0(4)-equivalent to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  The casework for cusps which are Γ0(4)-equivalent to 0 or 1  2 is suitably  analogous, so we omit details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  □  Combining these two lemmas, we find that  ρa(h, 1  2 + it)⟨y  k  2 + 1  4 fθ, Eκ  a (·, 1  2 + it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' χ)⟩  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3)  ≪f,h,ǫ (1 + |t|)  1  2− k  2 +ǫ · (1 + |t|)  k  2 +ǫ ≪f,h,ǫ (1 + |t|)  1  2+ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Consequently, for Re s = σ > 1  2, we have  Σcont ≪f,h,ǫ  � ∞  −∞  ����  Γ(s − 1  2 + it)Γ(s − 1  2 − it)  Γ(s − k  2 + 1  4)Γ(s + k  2 − 3  4)  ����|t|  1  2+ǫdt  ≪ |s|  3  2 −2σ  � ∞  −∞  |s − it|σ−1|s + it|σ−1|t|  1  2+ǫe−π max(|Im s|,|t|)+π|Im s|dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  SUMS OF CUSP FORM COEFFICIENTS ALONG QUADRATIC SEQUENCES  19  The exponential terms effectively concentrate mass in |t| < |Im s|, so that  Σcont(s) ≪f,h,ǫ |s|  1  2 −σ  � |Im s|  0  |s − it|σ−1|t|  1  2 +ǫdt ≪f,h,ǫ |s|1−σ+ǫ + |s|1+ǫ,  and hence Σcont(s) ≪f,h,ǫ |s|1+ǫ in Re s > 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' The bound Σcont(s) ≪f,h,ǫ |s|1+ǫ suffices for our purposes  but is by no means sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Under the generalized Lindel¨of hypothesis, the  upper bound (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3) improves to Of,h,ǫ((1+|t|)− 1  2+ǫ) and it would follow that  Σcont(s) ≪f,h,ǫ |s|ǫ in the half-plane Re s > 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Contour Shifting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' The growth estimates from §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2, and §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3  imply that the growth of Dh(s) in vertical strips in Re s > 1  2 is dominated  by that of Σdisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Hence Dh(s) ≪ |s|  k  2 +2− 1  2  √  k(k−2)−Re s+ǫ in a fixed vertical  strip in Re s > 1  2, where here and throughout §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='4 all implicit constants are  allowed to depend on f, h, ǫ, and Re s (where it appears).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  In particular, on the line Re s = 1  2 + ǫ, it follows that  Dh( s  2 + 1  4)Xs/s ≪ |s|  k  2 + 1  2− 1  2  √  k(k−2)+ǫX  1  2+ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Note also that Dh( s  2 + 1  4)Xs/s ≪ X1+ǫ/|s| on the line Re s = 1 + ǫ, by  absolute convergence of the Dirichlet series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' In the vertical strip Re s ∈ (1  2 +  ǫ, 1+ǫ) between these estimates, Dh( s  2 + 1  4)Xs/s is meromorphic, with poles  at most at s = 1 (from Σres) and each real s = 1  2 ± 2itj corresponding to an  exceptional eigenvalue (from Σdisc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' The Weyl law implies that exceptional  eigenvalues, if they exist, are limited in number by ON(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Away from these finitely many poles, the convexity principle implies that  Dh( s  2 + 1  4)Xs/s ≪ |s|  k  2 + 1  2 − 1  2  √  k(k−2)+ǫX  1  2+ǫ + X1+ǫ/|s| in the vertical strip  Re s ∈ (1  2 + ǫ, 1 + ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' We conclude from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='6) that  1  4πi  � 1+ǫ+iT  1+ǫ−iT  Dh( s  2 + 1  4)Xs ds  s  = cf,hX + RE +  1  4πi  �  1  2 +ǫ+iT  1  2+ǫ−iT  Dh( s  2 + 1  4)Xs ds  s  + O  �X1+ǫ  T  + T  k  2 + 1  2− 1  2  √  k(k−2)+ǫX  1  2+ǫ�  ,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='4)  in which RE is the sum over possible residues arising from exceptional eigen-  values, given explicitly by  RE :=(4π)  k  2 + 1  4 h  1  2 X  1  2  �  itj∈R  (X2/h)itjΓ(2itj)ρj(h)⟨y  k  2 + 1  4fθ, µj⟩  (1  2 + 2itj)Γ(3  4 − k  2 + itj)Γ(k  2 − 1  4 + itj)  + (4π)  k  2 + 1  4h  1  2 X  1  2  �  itj∈R  (X2/h)−itjΓ(−2itj)ρj(h)⟨y  k  2 + 1  4 fθ, µj⟩  (1  2 − 2itj)Γ(3  4 − k  2 − itj)Γ(k  2 − 1  4 − itj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  20  CHAN IEONG KUAN, DAVID LOWRY-DUDA, AND ALEXANDER WALKER  The contribution of the continuous spectrum Σcont in Dh( s  2 + 1  4) on the line  Re s = 1  2+ǫ is O(X  1  2+ǫT 1+ǫ) following §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' For the components of Dh( s  2+ 1  4)  coming from the residual and discrete spectra, we shift the vertical contour  farther left, to the line Re s = ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' In Σdisc, this shift passes a line segment of  non-exceptional spectral poles, which contributes a finite sum of residues R  of the form  2(4π)  k  2 + 1  4 h  1  2 X  1  2 Re  �  �  0≤tj≤2T  (X2/h)itjΓ(2itj)ρj(h)⟨y  k  2 + 1  4 fθ, µj⟩  (1  2 + 2itj)Γ(3  4 − k  2 + itj)Γ(k  2 − 1  4 + itj)  �  = (4π)  k  2 − 1  4 h  1  2 X  1  2 Im  �  �  0≤tj≤2T  (4X2/h)itjρj(h)⟨y  k  2 + 1  4 fθ, µj⟩  tj  �  1 + Ok( 1  tj )  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  We treat R as an error term and apply Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2 to conclude that  R ≪ X  1  2  �  ℓ≤log2 T  �  |tj|∼2−ℓT  |ρj(h)⟨y  k  2 + 1  4fθ, µj⟩|  |tj|  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='5)  ≪ X  1  2  �  ℓ≤log2 T  (T/2ℓ)  k  2 + 1  2− 1  2  √  k(k−2)+ǫ ≪ X  1  2T  k  2 + 1  2− 1  2  √  k(k−2)+ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Following (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='4), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='5), and the estimate O(X  1  2 +ǫT 1+ǫ) for the shifted  continuous spectrum, we have  �  m2+h≤X2  A(m2 + h) = cf,hX + RE +  1  4πi  � ǫ+iT  ǫ−iT  (Σres + Σdisc)( s  2 + 1  4)Xs ds  s  + O  �X1+ǫ  T  + T  k  2 + 1  2− 1  2  √  k(k−2)+ǫX  1  2 +ǫ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  The contour integral over Re s = ǫ is O(T  k  2 + 7  4− 1  2  √  k(k−2)+ǫXǫ) following  the upper bounds Σdisc(s) ≪ |s|  k  2 + 7  4− 1  2  √  k(k−2)+ǫ and Σres(s) ≪ |s|− 1  2 on the  line Re s = 1  4 + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' By taking T = X1/(k+3−√  k(k−2)), we optimize our collec-  tive errors to size O(X1−1/(k+3−√  k(k−2))+ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Since this error term dominates  RE ≪ X1/2+Θ ≪ X39/64 (by the comment following (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3)), the potential  contribution of RE may be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' This completes the proof of our main  arithmetic result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' For k ≥ 2, h > 0, and any ǫ > 0, we have  �  m2+h≤X2  A(m2 + h) = cf,hX + Of,h,ǫ  �  X  1−  1  k+3−√  k(k−2) +ǫ�  ,  where cf,h = 0 in most cases following Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  This result also holds with the more readable error term O(X  3  4 +  1  32k−44 +ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  SUMS OF CUSP FORM COEFFICIENTS ALONG QUADRATIC SEQUENCES  21  Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' The heuristic and conditional improvements to Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='2  noted in Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3 would imply that Σdisc ≪ |s|1+ǫ + |s|2−Re s+ǫ and that  R ≪ X  1  2T  1  2+ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' In addition, our bound for the shifted continuous integral  would improve to O(X  1  2+ǫT ǫ) under the generalized Lindel¨of hypothesis  following Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Optimizing errors with T = X1/3 would improve the  error in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='4 to O(X  2  3+ǫ), which is comparable to Bykovskii’s work  on the divisor function [Byk87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Error terms of size O(X  1  2+ǫ) are conjectured  to hold in both problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  References  [Blo08] Valentin Blomer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Sums of Hecke eigenvalues over values of  quadratic polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' IMRN, (16):Art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  ID rnn059.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' 29, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  [Byk87] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Bykovski˘ı.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Spectral decompositions of certain automor-  phic functions and their number-theoretic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Jour-  nal of Soviet Mathematics, 36, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  [DFI02] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Duke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Friedlander, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Iwaniec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
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+page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
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+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Deshouillers and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Iwaniec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Kloosterman sums and  Fourier coefficients of cusp forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
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+page_content=' Duke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Hyperbolic distribution problems and half-integral  weight Maass forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Dunster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Uniform asymptotic approximations for the  Whittaker functions Mκ,iµ(z) and Wκ,iµ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Analysis and Ap-  plications, 1(2):199–212, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
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+page_content=' Eisenstein series of 1  2-  integral weight and the mean value of real Dirichlet L-series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
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+page_content='  [GH11] Dorian Goldfeld and Joseph Hundley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Automorphic Represen-  tations and L-Functions for the General Linear Group, vol-  ume 1 of Cambridge Studies in Advanced Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Cam-  bridge University Press, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  [GR15] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Gradshteyn and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
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+page_content=' Ryzhik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Table of integrals, series,  and products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Elsevier/Academic Press, Amsterdam, eighth  edition, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Translated from the Russian, Translation edited  and with a preface by Daniel Zwillinger and Victor Moll, Re-  vised from the seventh edition [MR2360010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  [HH16] Jeff Hoffstein and Thomas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Hulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Multiple Dirichlet se-  ries and shifted convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Number Theory, 161:457–533,  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' With an appendix by Andre Reznikov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  [HKLDW18] Thomas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Hulse, Chan Ieong Kuan, David Lowry-Duda, and  Alexander Walker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Second moments in the generalized Gauss  circle problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Forum Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Sigma, 6:Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' e24, 49, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  22  CHAN IEONG KUAN, DAVID LOWRY-DUDA, AND ALEXANDER WALKER  [Hoo63] Christopher Hooley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' On the number of divisors of a quadratic  polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
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+page_content=' Heegner points, cycles and  Maass forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
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+page_content=' Sarnak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Refined estimates towards the Ramanu-  jan and Selberg conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
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+page_content='  [LD17] David Lowry-Duda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' On Some Variants of the Gauss Circle  Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' PhD thesis, Brown University, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Olde Daalhuis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
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+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Schneider, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Boisvert, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Saunders B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
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+page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
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+page_content='  The Clarendon Press, Oxford University Press, New York, sec-  ond edition, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content=' Edited and with a preface by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
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+page_content='  [TT13] Nicolas Templier and Jacob Tsimerman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
+page_content='  Non-split sums of  coefficients of GL(2)-automorphic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FKT4oBgHgl3EQfxC4a/content/2301.11901v1.pdf'}
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+Long-range quenched bond disorder in the bi-dimensional Potts
+model
+Francesco Chippari
+Sorbonne Universit´e & CNRS, UMR 7589, LPTHE, F-75005, Paris, France
+e-mail: fchippari@lpthe.jussieu.fr
+Marco Picco
+Sorbonne Universit´e & CNRS, UMR 7589, LPTHE, F-75005, Paris, France
+e-mail: picco@lpthe.jussieu.fr
+Raoul Santachiara
+Paris-Saclay Universit´e & CNRS, UMR 8626, LPTMS, 91405, Saclay, France
+e-mail: raoul.santachiara@gmail.com
+(Dated: February 1, 2023)
+ABSTRACT
+We study the bi-dimensional q-Potts model with long-range bond corre-
+lated disorder. Similarly to [1], we implement a disorder bimodal distribution
+by coupling the Potts model to auxiliary spin-variables, which are correlated
+with a power-law decaying function. The universal behaviour of different ob-
+servables, especially the thermal and the order-parameter critical exponents,
+are computed by Monte-Carlo techniques for q = 1, 2, 3-Potts models for dif-
+ferent values of the power-law decaying exponent a.
+On the basis of our
+conclusions, which are in agreement with previous theoretical and numerical
+results for q = 1 and q = 2, we can conjecture the phase diagram for q ∈ [1, 4].
+In particular, we establish that the system is driven to a fixed point at finite
+or infinite long-range disorder depending on the values of q and a. Finally, we
+discuss the role of the higher cumulants of the disorder distribution. This is
+done by drawning the auxiliary spin-variables from different statistical mod-
+els. While the main features of the phase diagram depend only on the first
+and second cumulant, we argue, for the infinite disorder fixed point, that
+certain universal effects are affected by the higher cumulants of the disorder
+distribution.
+PACS numbers: 75.50.Lk, 05.50.+q, 64.60.Fr
+Contents
+1
+Introduction
+2
+2
+The long-range bond disordered q-Potts model
+4
+2.1
+Implementation of disorder . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+2.2
+Pure Potts model (r=1)
+. . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+1
+arXiv:2301.13727v1  [cond-mat.stat-mech]  31 Jan 2023
+
+2.3
+Short-range disordered Potts (a ≥ 2)
+. . . . . . . . . . . . . . . . . . . .
+8
+2.4
+Long-Range disordered Potts (a < 2) . . . . . . . . . . . . . . . . . . . .
+8
+2.5
+The infinite disorder point (r = ∞): the q-colored critical pure percolation
+9
+3
+Phase diagrams from Monte-Carlo measures of q = 1, 2, 3-Potts
+10
+3.1
+The infinite disorder point (r = ∞): LRp and Bp fixed points
+. . . . . .
+11
+3.2
+The q = 1 phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . .
+12
+3.3
+The q = 2 phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . .
+14
+3.4
+The q = 3 phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . .
+18
+3.5
+Thermal behaviour for the q = 3-Potts model with long-range correlated
+disorder
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+20
+3.6
+The effects of higher disorder cumulants
+. . . . . . . . . . . . . . . . . .
+21
+4
+Conclusions
+24
+5
+Acknowledgments
+25
+A Disorder distributions
+25
+A.1 n-replicated Ising model
+. . . . . . . . . . . . . . . . . . . . . . . . . . .
+25
+A.2 Fractional Gaussian free fields . . . . . . . . . . . . . . . . . . . . . . . .
+26
+1
+Introduction
+The q-Potts model [2] is a lattice spin model where the spins take one of q-possible values
+and interact via nearest neighbours couplings. Through a geometrical reformulation [3],
+the model admits a natural continuation to real values of q. When the couplings take the
+same positive value, the two-dimensional Potts model has a second-order ferromagnetic-
+paramagnetic phase transition for q ∈ [1, 4] [2]. Here, we consider the bond disordered
+q-Potts model, where the couplings have random fluctuations around a given positive
+value. In particular, we are interested in how the critical properties of this model depend
+on the fluctuations (disorder) distribution. Part of the answer to this problem is provided
+by the Harris criterion [4] and by some generalizations of it [5,6]. These criteria determine
+the relevance of the disorder on the basis of the correlation properties of its distribution.
+The most studied example is the uncorrelated disorder where the couplings fluctuations
+are independent and equally distributed. In this case the Harris criterion states that
+the disorder modifies the critical properties of the pure system if its thermal exponent
+νP is greater than one, νP > 1 [4]. This condition is satisfied by the q-Potts model for
+q > 2. When the disorder is relevant, a situation of particular interest is the one where
+the system still undergoes a continuous phase transition but it is described by a different
+stable renormalization group (RG) fixed point. It is now well established that this is the
+situation for the two-dimensional Potts model with uncorrelated disorder. For q > 2,
+there is a stable RG fixed point [7–9], henceforth referred to as the Short-Range (SR)
+2
+
+point, which determines the critical properties of the model. The thermal νSR and the
+order-parameter βSR critical exponents have been computed via an RG computation of
+a perturbed conformal field theory [8] and their values match with different numerical
+tests [9–11].
+In this paper we consider the case where the couplings {J(x)}, at each position x, are
+drawn from an homogenous and isotropic distribution, with first cumulant, E[J(x)] =
+E[J].
+We focus in particular on distributions whose second cumulant, g(|x − y|) =
+E[J(x)J(y)] − E[J]2, decreases as a power-law for large distances, g(|x|) ∝ (r − 1)2|x|−a
+for |x| ≫ 1. The parameter a is taken positive, a > 0. The value r parametrizes the
+disorder strength and it is taken greater than one, r ∈ [1, ∞], see Section 2.1. For r = 1,
+one recovers the pure Potts model, see Section 2.2.
+The r = ∞ corresponds to the
+infinite disorder point and it is discussed in detail in Section 2.5. The numerical set-up
+to implement these distributions follows closely the one introduced in [1], where the pure
+Potts model is coupled to the Ashkin-Teller one. Most of the results presented here are
+obtained by using a replicated Ising model instead, see Section 2.1, which differs from
+the one of [1] in the cumulants of order higher than two. We have verified that our main
+findings on the phase diagram of these models do not depend on the higher cumulants
+of these distributions.
+The extended Harris criterion [5,6] determines in which region of the parameters (q, a)
+the disorder is relevant and when its long-range properties prevail over the short-range
+ones. When a > 2 the critical behaviour of the system is expected to be the same as the
+one with uncorrelated disorder. The regions (q < 2, a < 2/νP) and (q > 2, a < 2/νSR),
+where the disorder is relevant and dominated by the long-range correlations, are much
+less understood, especially for general values of q.
+For q = 1, as we explain in Section 3.2, the pure Potts model and the Potts with
+bimodal short range disorder correspond, for any value of the disorder strength r, to
+the Bernoulli bond percolation model [12], where the edges are independently activated
+(pure percolation). The long-range disordered q = 1-Potts model coincides instead with
+a one-parameter (the disorder strength r) family of long-range bond percolation model,
+where the activation of two distant edges is correlated by a power-law decreasing function
+with exponent a. The d-dimensional long-range percolation was studied in [6] by using
+a one-loop order RG computation with a double expansion in ϵ, ϵ = 6 − d and in δ,
+δ = 4 − a. It was found that the long-range nature of the correlation drives the system
+to a new stable RG point, that, in the following, we refer to as the LRp (Long-Range
+percolation) point. Even if these theoretical predictions have validity near dimension
+six and for a close to 4, they have motivated a series of numerical works [13–17] on
+some d = 2 long-range percolation models with a > 0. These models were defined by
+using the level sets of fractional Gaussian free fields with Hurst exponent H = −a/2.
+As the thermal exponent of the pure q = 1-Potts is νP = 4/3, the long-range disorder
+is relevant for a < 3/2. The general agreement in [13–17] was that, for a < 3/2, the
+percolation transition is described by a new LRp point. In our approach, we recover
+the critical behaviour studied in [13–17] at the r = ∞ point of a Potts model coupled
+with a fractional Gaussian field. We will show that, irrespective to the value of r, these
+long-range percolation models are described, for a > 3/2, by the Bernoulli percolation
+(Bp) point and, for a < 3/2, by the LRp point. This can be rephrased by saying that, for
+each a, there exists an unique fixed point which is stable in the direction of decreasing
+3
+
+r. We observed that, by coupling instead the Potts model to n−replicated Ising models,
+the numerical investigations of the LRp type points becomes more precise, in particular
+for small values of a. We will argue in Section 3.6 that some universal properties of the
+LRp points can depend on the higher cumulants of the disorder distribution.
+The q = 2 case corresponds to the bond disordered Ising model and it has been studied
+quite intensively in the past. Here the cross-over exponent is 2/νP = 2. For a ≥ 2, the SR
+and the pure (P) point coincide and the disorder effects generate logarithmic corrections
+to the scaling relations [18]. For 1 ≲ a < 2, one finds a stable long-range (LR) point,
+whose existence has been proven in [19] and in [20] by using the Ising massive free fermion
+representation. Numerical tests of these predictions have been found in [21] for a = 1.
+In [20] it was also observed that LR point looses its stability for a ≲ 1, and the question
+whether the system undergoes a smeared phase transition or whether it flows to an infinite
+disorder fixed point remained open. In [1, 22], the q = 2 (and q = 4) Potts long-range
+disordered model were investigated for values of a smaller than one, 0 < a < 1. It was
+observed there that, for all this values of a, the Monte-Carlo measures of β/ν were very
+close to the LRp point value.
+Our aim here is to provide a complete Potts phase diagram in the parametric region
+(q, a) ∈ (0, 4] × (0, ∞). This is achieved by considering the disordered q = 1, 2, 3-Potts
+models for different values of a. These three values of q are indeed representative of
+the three regions where the short-range disorder is respectively irrelevant, marginal and
+relevant.
+Our results are summarised in Fig. (1). We establish that for all values of q > 1
+there is a region where the LR point exists and it is stable in the direction of decreasing
+r. Moreover, for each q ∈ [1, 4], there is a value a∗(q) at which the LR point becomes
+unstable. For smaller values of a, a < a∗(q), the model is described by the LRp point,
+which is the fixed point of the Potts model with infinite disorder.
+2
+The long-range bond disordered q-Potts model
+The model is defined by the partition function:
+Z({J<ij>}) =
+�
+{si}
+exp
+��
+<ij>
+J<ij>δsi,sj
+�
+,
+(2.1)
+where the spin si, living on the lattice vertex i, takes q possible states, si = {1, · · · , q}.
+The < ij > identifies the edge connecting the neighbouring sites i and j and the δk,l is the
+Kronecker delta. The set of couplings {J<ij>} is made of random variables drawn from
+a given distribution, the average over which will be indicated by the notation E[· · · ].
+We extract the critical exponent of the model (2.1) by measuring the properties of
+its Fortuyn-Kasteleyn (FK) clusters [3, 23]. The FK clusters are the connected sets of
+bonds that enter in the geometric representation of the Potts partition function. From
+4
+
+1
+1.5
+2
+2.5
+3
+3.5
+4
+0
+1
+2
+3
+q
+a
+Bp
+LRp
+Short-Range
+(SR, finite disorder)
+Irrelevant
+(P)
+Long-Range
+(LR, finite disorder)
+Long-Range percolation
+(LRp, Infinite disorder)
+Figure 1: Phase diagram of the disordered bi-dimensional q-Potts model for (q, a) ∈
+(0, 4] × (0, ∞).
+Eq. (2.1), we get:
+Z ({J<ij>}) =
+�
+{si}
+�
+<ij>
+�
+1 + (eJ<ij> − 1)δsi,sj
+�
+=
+�
+G
+q# FK clusters(G)
+� �
+<ij>∈G
+eJ<ij> − 1
+�
+=
+�
+G
+�
+<ij>∈G
+�
+1 − e−J<ij>�
+�
+<ij>/∈G
+�
+e−J<ij>�
+q# FK clusters(G),
+(2.2)
+where the sum is over the set G of activated edges or bonds. Note that in the last equality
+we discard a global factor �
+<ij>
+eJ<ij>. A bond connects two equal spins with probability
+p<ij> = 1 − e−J<ij>.
+We will consider self-averaging observables, such as the thermal (ν) and the order-
+parameter (β) critical exponents. More specifically, we will compute, for q = 1, 2, 3 the
+ratio β/ν from the measure of the FK fractal dimension df using:
+df = 2 − β
+ν ,
+(2.3)
+see Section 3. The exponent ν, instead, is extracted, for q = 3, from the FK clusters
+wrapping probability, defined in Eq. (3.10). The exponent ν determines the finite size
+scaling of the probability defined in Eq. (3.11). We consider, in Section 3.3, the Ising
+5
+
+(q = 2) spin-spin correlation function, which is a non self-averaging observable. We will
+measure the multifractal behaviour of this quantity, see Eq. (3.8), and compare with
+recent predictions [20].
+2.1
+Implementation of disorder
+In our simulations, we implement a quenched bi-modal disorder where the couplings can
+take randomly two values, J<ij> = J1 or J<ij> = J2, with equal probability. The values
+of J1 and J2 are not taken independently but they are chosen to satisfy the following
+self-dual condition:
+(eJ1 − 1)(eJ2 − 1) = q.
+(2.4)
+The self-dual line Eq. (2.4) corresponds to a line of critical points separating a paramag-
+netic phase and a ferromagnetic phase [24].
+In order to simulate this disorder, we associate to each vertex of the square lattice a
+random variable σi = {−1, 1} and we set:
+J<ij>(R) = J<ij>(B) = J1 + J2
+2
++ σi
+J1 − J2
+2
+,
+(2.5)
+where, the J<ij>(R) and J<ij>(B) are the couplings associated respectively to the edge on
+the right and to the edge on the bottom of a given site i. The Fig. (2) illustrates the
+connection between the σi and the J<ij> configurations as given by the Eq. (2.5).
+Figure 2: A particular configuration {σi} is shown. Blue (red) filled circles are the sites
+corresponding to σ = 1 (σ = −1). The corresponding set of couplings J<ij>, associated
+to the edges, follow from the Eq. (2.5). In the Figure, the J<ij> = J1 (J<ij> = J2)
+couplings are associated to the blue (red) colored edges.
+In principle there are other possibles choices: for instance one could introduce, instead,
+an edge random variable σ<ij> and impose J<ij> = (J1+J2)/2+σ<ij>(J1−J2)/2. All these
+different set-ups are expected to be equivalent as far universal properties are concerned.
+We have chosen simply the one that was more convenient for our numerical simulations.
+6
+
+The disorder distribution is then fixed by the {σi} one. Here we consider the distri-
+butions that are characterized by a vanishing first moment:
+E [σi] = 0,
+(2.6)
+which forces the couplings to take values J1 or J2 with the same probability. The couplings
+then fluctuate around the value (J1 + J2)/2:
+J<ij> = E[J<ij>] + δJ<ij>,
+E[J<ij>] = J1 + J2
+2
+.
+(2.7)
+We can then consider a {σi} distribution whose second cumulant has a power-law decay-
+ing:
+E [σi σk] ∼ |i − k|−a
+for |i − k| ≫ 1.
+(2.8)
+This, of course, correlates the fluctuations of the couplings at distant edges:
+E
+�
+δJ<ij>(X) δJ<kl>(Y )
+�
+= (J1 − J2)2
+4
+E [σi σk]
+∼ |i − k|−a
+for |i − k| ≫ 1,
+(2.9)
+where X, Y = R, B.
+Once we have chosen the {σi} distribution and satisfied the relation Eq. (2.4), the
+phase diagram of the q-Potts model Eq. (2.1) depends only on the ratio r,
+r = J1
+J2
+∈ [1, ∞],
+(2.10)
+that parametrizes the disorder strength. We assumed, without loss of generality, that
+J2 ≤ J1. In our simulations we will analyse the behaviour of the system by varying r.
+The stability of the RG fixed points are therefore probed with respect to this direction
+while the relation Eq. (2.4) makes the system stay at the critical temperature.
+2.2
+Pure Potts model (r=1)
+The pure case corresponds to r = 1 where the couplings take all the same value, see
+Eq. (2.10). On the square lattice, the corresponding critical point P is located at J1 =
+J2 = Jc = log (1 + √q).
+In the continuum limit, the P point is described by the Potts conformal field theory
+whose energy spectrum is known since longtime [25]. In these last years a bootstrap
+approach [26–30] has been proposed for studying the cluster connectivity properties,
+which finally was used in [31–33]. This provides the exact Potts correlation functions
+and a complete characterization of the symmetry representation of its states.
+The thermal and order-parameter critical exponent νP and βP of the pure Potts
+model have been determined for any value of q ∈ [0, 4], see for instance [34] and references
+therein. In Table 1, we give the values of νP and of βP/νP for q = 1, 2, 3:
+7
+
+q
+νP
+βP /νP
+1
+4/3
+5/48
+2
+1
+1/8
+3
+5/6
+2/15
+Table 1: Pure critical exponents for different q values.
+2.3
+Short-range disordered Potts (a ≥ 2)
+When a ≥ 2 the disorder is effectively uncorrelated. In Fourier space, the contribution
+of the power-law tail |x|−a is of order O(ka−2) and, in the long wavelength limit k → 0,
+is finite for a ≥ 2. At large distances, the properties of a long-range distribution are the
+same as the one of a delta function.
+The phase diagram of the uncorrelated disorder is quite well understood. For q < 2 the
+P point is the only stable RG point: the weak disorder does not modify the universality
+of the pure model. For q > 2 instead, the system is driven to a stable SR point. For
+q = 2 the disorder is marginal (the P and SR points coincide) and generate universal
+logarithmic corrections to the P universality class [18]. We report in Table 2 the critical
+exponents for q = 3, computed perturbatively in [7,8]:
+q
+νSR
+βSR/νSR
+3
+∼ 1.02
+∼ 0.134655
+Table 2: Short-Range critical exponents for q = 3.
+In the case of uncorrelated disorder, one can show by simple power counting that only
+the first two cumulants of the distributions determine the long-distance behaviour of the
+system [8].
+2.4
+Long-Range disordered Potts (a < 2)
+For a < 2 one has to take into account the power-law decay of the disorder distribution.
+As mentioned in the Introduction, according to the extended Harris criterion [5, 6], for
+(q < 2, a < 2/νP) and for (q > 2, a < 2/νSR), the long-range part of the distribution is
+relevant and dominates the short-range one. In [5,6], it was also observed that, when it
+exists, the LR point has to be stable with respect to an additional sub-dominant term
+to E[δJ(x)δJ(y)], E[δJ(x)δJ(y)] → E[δJ(x)δJ(y)] + w0|x − y|−b, with b > a. Reiterating
+8
+
+the extended Harris criterion, one expects that when b > a then b > 2/νLR, i.e. the per-
+turbation is irrelevant. Otherwise, when b < a, the additional b-term becomes dominant
+and therefore b < 2/νLR. This in turn brings to the strong conjecture that:
+νLR = 2
+a,
+for any q.
+(2.11)
+The above relation has been proven by perturbative RG computation in different long-
+range disordered models [5, 6, 19, 20] at one or two-loops order.
+It is now also quite
+established that the Eq. (2.11) should be valid at all perturbation orders [35, 36]. In
+Section 3.5, we will test the relation Eq. (2.11) for q = 3.
+2.5
+The infinite disorder point (r = ∞): the q-colored critical
+pure percolation
+The r → ∞ point is the only point, besides the pure one (r = 1), where exact results can
+be given for any q. From the Eq. (2.4), in the limit r → ∞ one has J1 → ∞ and J2 → 0
+with rJ2 ∼ − log(J2) → ∞. In particular:
+E [δJ<ij>δJ<kl>] = (r − 1)2J2
+2
+4
+E [σiσk] ∼ (rJ2)2E [σiσk] .
+(2.12)
+The r = ∞ limit corresponds then to the maximal amount of disorder we can consider.
+In this limit, the Eq. (2.2) greatly simplifies. Indeed, with probability one, there is a
+bond on an edge with coupling J1 (1 − e−J1 = 1) and probability zero of having a bond
+on a edge with J2 (1−e−J2 = 0). So, once a configuration of σi (and consequentely of the
+couplings J<ij>) is drawn, only the graph where the edges < ij > associated to J1 are
+activated, contributes to the partion sum, see Fig. (3). The J2 edges are not activated.
+Figure 3: Given the configuration of σi and therefore of J<ij> shown in Fig. (2), we show
+the only random cluster graph G in (2.2) that does not vanish in the limit r → ∞. Notice
+that the σi = 1 cluster, formed of blue spins, are different from the ones of the FK bonds
+(black lines)
+9
+
+The disorder average free-energy, for instance, can be written as:
+E [log Z({J<ij>}] = log q × E [# (J1-clusters)] ∼ log q × E [# (σ = +1-clusters)] (2.13)
+where ∼ means that the scaling behaviour is the same. The disordered Potts model at
+r = ∞ is then strictly related to the site percolation model based on the σ clusters. In
+particular, the probability distribution of the r = ∞ FK clusters is, besides a trivial
+dependence on q, the same as the one of the σ clusters. In our construction Eq. (2.5),
+the < ij(R) > and < ij(B) > are always activated or not activated at the same time. At
+the lattice level, the r = ∞ FK clusters are therefore different from the σ clusters, as
+illustrated in Fig. (3). However their universal behaviour, such as their fractal dimension,
+as well as the correlation lenght exponent ν, remains the same, as discussed in the
+following Section 3.1. A more quantitative argument can be provided for a = 1/4, where
+the σ are Ising spins at the critical point. In fact, one can observe that the two type of
+clusters differ only on the boundaries. For an Ising cluster of linear size R, the number of
+spins on the boundary scales as ≃ Rdb
+f with db
+f = 11/8 = 1.3751, which is much smaller
+than the number of spins in the cluster that scales as Rdf with df = 187/96 ≃ 1.948.
+Then, the FK clusters for a = 0.25 are expected to have the same fractal dimension
+df = 2 − 5/96 as the Ising spin clusters.
+When a > 2, the system is short-range and it is described by the Bernoulli critical
+point Bp [38]. For a < 2, one applies the extended Harris criterion using νBp = 4/3: for
+a > 2/νBp = 3/2 the system remains in the Bp point while a < 3/2, there is the new
+LRp fixed point already mentioned in the Introduction. In Section 3.1 we tested this
+behaviour and we computed the exponent βLRp/νLRp.
+We insist on the fact that the LRp exists at r = ∞ for any q and its critical exponents
+depend on a but do not depend on q. As we will discuss in the following, see Fig.s (10,
+12), the LRp point is for q > 1 distinct from the LR point, which is the finite disorder
+fixed point. The LRp point exhanges stability with the LR one when a is greater than a
+particular value a∗(q), a > a∗(q).
+3
+Phase diagrams from Monte-Carlo measures of q =
+1, 2, 3-Potts
+We present here the numerical results according to which we established the phase di-
+agram of Fig. (1).
+Simulations were done on square lattices with periodic boundary
+conditions. If not stated otherwise, we average over one million samples of disorder. For
+each value of q and a, we determined first the average autocorrelation time τa,q(L). Next
+the thermal average is done, for each sample of disorder, by runing 100 × τa,q(L) updates
+after the same number of updates for thermalisation. If not stated otherwise, the results
+presented in the following are obtained by using the disorder distribution described in
+Section A.1.
+For each value of n = {1, · · · , 8}, corresponding to values a given in Eq. (A.3), we
+generate the coupling configuration J<ij>, see Eq. (2.5).
+We then construct the FK
+1Note that the Ising spin boundaries are described by the CLE3 loops and db
+f = 11/8 is the fractal
+dimension of the CLE3 loops [37]
+10
+
+clusters by connecting equal spins with the probability p<ij> = 1−e−J<ij>, see Eq. (2.2).
+Given a lattice of linear size L, we compute, for each coupling configuration, the (thermal
+averaged) number of spins M({J<ij>}, L) of the largest FK cluster. We average on the
+coupling distribution,
+M(L) = E [M(L, {J<ij>})] ,
+(3.1)
+to extract the fractal dimension df by the scaling
+M(L) ≃ Ldf.
+(3.2)
+More specifically, recalling the relation Eq. (2.3), we use Eq. (3.2) to compute an effective
+magnetic dimension:
+β
+ν (L) = − log
+� m(L)
+m(L/2)
+�
+/ log [2] ,
+(3.3)
+where m(L) = M(L)/L2 is the average magnetization. The above data, computed for
+different q, a and r are the ones shown in the plots β/ν vs. L below. The ratio β/ν is
+then evaluated in the scaling limit:
+β
+ν = lim
+L→∞
+β
+ν (L) .
+(3.4)
+3.1
+The infinite disorder point (r = ∞):
+LRp and Bp fixed
+points
+We collect here results for the q-colored percolation model, found at the infinite disorder
+point r = ∞, see Section 2.5. We recall that the critical exponents at the infinite disorder
+point do not depend on the value of q.
+In Fig. (4), we show the effective exponent β/ν(L) as a function of L for a =
+{0.25, · · · , 2.00}. The corresponding scaling limit results βLRp/νLRp, which are obtained
+by a best fit of all our data, including a sub-leading correction, are reported in Table 3.
+a
+βLRp/νLRp
+0.25
+0.0522(4)
+0.5
+0.086(2)
+0.75
+0.103(2)
+1, 1.25, 1.5, 1.75, 2
+∼ 0.105
+Table 3: Best fit of the critical exponent βLRp/νLRp for different values of a. Notice that
+for a = {1.5, 1.75, 2} the system is described by the Bp point where βBp /νBp = 5/48 ∼
+0.105. However, for a ≥ 1, the numerical simulations cannot distinguish between the LRp
+point and the Bp point.
+As discussed in Section 2.5, for a = 0.25, the βLRp/νLRp must correspond to the
+fractal dimension of the Ising clusters. The latter has been considered in [39, 40] and
+argued to be:
+βLRp
+νLRp = 2 − 187
+96 = 5
+96 ∼ 0.0521.
+(3.5)
+11
+
+Figure 4: The r = ∞ long-range disorder: β/ν vs. L for the values of a shown in the
+caption.
+From the Table 3, one can see that the agreement to the value 5/96, shown as a dashed
+line in Fig. (4), is very good.
+We remind that according to the Harris criterion, one expects that, for a < 3/2,
+βLRp/νLRp < βBp/νBp.
+As a matter of fact, the difference between βLRp/νLRp and
+βBp/νBp becomes very small for increasing a and the numerical simulations cannot ef-
+fectively distinguish between these two fixed points. In fact, for a = 1 and a = 1.25, it
+seems to be even slightly larger than 5/48. A similar results was also observed in [15].
+But, since the finite size correction are important, it is difficult to really predict the
+asymptotic value.
+We consider in the following Potts observables at finite value of disorder r, where the
+scaling behaviour is in general expected to depend on q.
+3.2
+The q = 1 phase diagram
+We consider here the phase diagram of the q = 1 Potts model. For a > 2 the disorder is
+effectively short-range and irrelevant, see Fig. (1). The fact that a short-range disorder
+is irrelevant can also be explained by a simple lattice argument. On the critical line
+Eq. (2.4), for r = 1, J1 = J2 = Jc = log (1 +
+√
+2) = log 2, thus we are exactly at the Bp
+critical point, p = 1 − e−Jc = pc = 1/2. By adding disorder, meaning by setting r > 1,
+the dual line Eq. (2.4) fixes, at q = 1, the relation 1 − e−J1 − e−J2 = 0. Recalling that
+each bond J1 and J2 occurs with equal probability, the density of activated bond remains
+12
+
+0.11
+0.1
+0.09
+Bp
+0.08
+LRp
+a=0.25
+a=0.50
+0.07
+a=0.75
+a=1.00
+a=1.25
+a=1.50
+0.06
+a=1.75
+a=2.00
+0.05
+10
+100
+1000p = 1
+2(1−e−J1)+ 1
+2(1−e−J2) = 1/2. So, we can conclude that, for any value of r, we stay
+in the Bp point. For 3/2 = 2/νBp < a < 2, the long-range disorder remains irrelevant,
+and the system is still described by the Bp point.
+The effective exponent β/ν(L) is measured for various values of disorder strength r.
+For a < 3/2 our numerical measures shows clearly that the systems flows from the Bp
+point to the LRp point, located at r = ∞. The LRp point and its scaling exponents have
+been discussed in the Section 2.5 and in Section. 3.1.
+We present first the model with a = 1/4. In the left part of Fig. (5), β/ν(L) is shown
+for r = 1, 2, 5, 10, 100 and ∞ as a function of the size L. We observe that this exponent,
+for all values of r > 1, goes towards the value of r = ∞, with a crossover function of the
+disorder. The corresponding expected value 5/96, see [39], is shown as a blue dashed line.
+The r = 1 corresponds to Bernoulli percolation whose scaling limit value βBp/νBp = 5/48
+is shown as a red dashed-dotted line. Thus, for any amount of disorder, the large distance
+behaviour is controlled by an infinite disorder fixed point. In the right part of Fig. (5),
+Figure 5: q = 1 Potts model with a = 0.25. The left panel shows β/ν vs. L while the
+right panel shows the wrapping probability Pw(L) vs. L.
+we show Pw(L), see Eq. (3.10). For r = 1, it converges towards the exact value for the
+Bernoulli percolation [41]:
+Bernoulli clusters:
+lim
+L→∞ Pw(L) = 0.69046
+(3.6)
+which is shown as a red dashed-dotted line. For any value r > 1, it goes to the same
+value as for LRp point with a crossover function of the disorder. In addition to the fractal
+dimension, we can verify that also the wrapping probability supports the fact that, as
+stated above, the FK clusters behave, at the LRp point, as the Ising spin ones. For spin
+clusters this probability can be computed from the results in [42] and is:
+Ising clusters:
+lim
+L→∞ Pw(L) = 0.515884 .
+(3.7)
+This is shown as a blue dashed line in the right part of Fig. (5). The agreement is perfect.
+13
+
+a=0.25
+a=0.25
+0.2
+0.7
+Bp
+LRp
+0.68
+0.18
+r=1
+r=2
+0.66
+0.16
+r=5
+r=10
+0.64
+Bernoulli Clusters
+r=100
+0.14
+Ising Clusters
+r=inf
+0.62
+r=1
+R"0.6
+r=2
+≥0.12
+r=5
+r=10
+0.58
+0.1
+r=100
+0.56
+r=inf
+0.08
+0.54
+0.06
+0.52
+0.04
+0.5
+10
+100
+10
+100
+1000Coming back to the β/ν exponent, similar behaviours are observed for the others
+values of a. In Fig. (6), we show β/ν(L) for a = 0.75 and a = 1.5. Again, for all values
+of r, the large size limit is the one reported in Table 3.
+Figure 6: β/ν vs. L for the q = 1 Potts model with long-range disorder with a = 0.75 in
+the left panel and with a = 1.50 in the right panel.
+We can therefore summarize the numerical findings for q = 1 in Fig. (7).
+3.3
+The q = 2 phase diagram
+We consider here in detail the q = 2 case. In the short-range regime a ≥ 2, the disorder
+is marginal and the P and SR points coincide. When a∗(2) < a < 2 we can numerically
+probe the existence of the LR point, predicted in [19,20]. Here, it will correspond to a
+fixed point at finite disorder r∗
+LR, 1 < r∗
+LR < ∞. We estimate a∗(2) to be in the interval
+1/2 < a∗(2) < 3/4, but the determination of its exact value goes beyond the aims of this
+paper. When a < a∗(2) there are just two fixed points, the P and the LRp (r = ∞) ones.
+In this case, the LRp becomes the stable one and we observe a flow from the P point to
+the LRp point. These different behaviours are shown in Fig. (8). In particular:
+• In the left panel, we show the measured values of β/ν(L) for a = 1/4. We clearly
+observe that for all values of disorder, the effective magnetic exponent goes to the
+same value as the infinite disorder one, βLRp/νLRp = 5/96 ∼ 0.0521, see Table 3.
+We obtain a similar behaviour for a = 0.5, with β/ν(L) → 0.086 in the L → ∞
+limit, again very close to the value βLRp/νLRp reported in Table 3.
+• In the middle panel, we show our data for a = 1 and we observe three different
+large size limits: i) for r = 1 the magnetic exponent is the one of the short-range
+Ising, βP/νP = 1/8 = 0.125; ii) for r = ∞ the magnetic exponent is close to the
+value βBp/νBp = 5/48. As already mentioned for the case of infinite disorder, see
+Section 3.1, for a ≃ 1, it is difficult to distinguish the LRp point from the Bp point;
+iii) for any finite disorder, there is an unique large size limit, limL→∞ β/ν(L) =
+14
+
+a=0.75
+a=1.50
+0.135
+Bp
+0.13
+=
+r=2
+r=5
+0.125
+r=10
+r=100
+0.12
+r=inf 
+0.115
+0.11
+0.105
+0.1
+0.095
+0.09
+10
+100
+10
+100q = 1
+a
+1/r
+1
+Bernoulli percolation
+a = 3
+2, LRp = Bp
+Bp
+Fig. (6) right
+Fig. (6) left
+a = 3
+4, LRp
+Bp
+Fig. (5)
+a = 1
+4, LRp
+Bp
+Figure 7: Fixed point stability, obtained via the measurements of the β/ν critical expo-
+nent, for q = 1. The red zone represents a unique fixed point, the Bp one. There is no
+flow while tuning r. The arrows describe a flow between two fixed points. The red line
+represents again the Bp fixed point and the blue line is a line of fixed points, the LRp
+ones.
+βLR/νLR ≃ 0.115 − 0.120. This corresponds to the LR fixed point. The curves
+converge to the curve with a finite disorder r ≃ 10, that is the value in which we
+observe the smallest corrections.
+We observed a similar behaviour for a = 0.75 and a = 1.25 with the scaling limit
+limL→∞ β/ν(L) ≃ 0.108 and limL→∞ β/ν(L) ≃ 0.122 respectively.
+• The right panel shows the measured values of β/ν(L) for a = 1.75. In that case,
+the measurements for finite disorder also converge to a single value βLR/νLR which
+is very close to the P=SR value βP/νP = 1/8. For all values of a ≥ 3/2, due to
+strong finite size corrections, we find values close to the P=SR values.
+Since, for values of 3/2 ≤ a < 2, the LR point is close to the P fixed point and we
+can not really distinguish via the β/ν exponent these two points, we will then consider
+another way of checking the existence of the LR point. We consider the moments of the
+spin-spin correlation function,
+E[< s(0)s(x) >n] ≃ x−ηn .
+(3.8)
+15
+
+Figure 8: q = 2 Potts model with long-range disorder: β/ν vs. L for a = 0.25 on the
+left, a = 1.00 on the middle and a = 1.75 on the right.
+and we are interested in extracting the exponents ηn. While η1 = 2β/ν gives access to
+the same exponent that we computed above via the FK clusters dimensions, ηn for n > 1
+are new exponents. The authors of [20] were able to compute η2 for 0.995 ≤ a < 2. They
+obtained, at the lowest order in disorder, the following:
+η2 = 1
+2 − (2 − a)
+4
++ O((2 − a)2).
+(3.9)
+In this computation the lower range, a = 0.995, is obtained as a limit of the stability of
+their perturbative fixed point at the second order.
+In Table 4, we report our measured values of η1 and η2 for various values of a. Note
+that the values for η1 can be compared to the values of β/ν previously obtained. One
+can verify that the two ways of measuring the same exponent give totally consistent
+results. For each value a, the measurements have been done for a fixed value of disorder
+r chosen in order to minimise corrections to scaling in the measurement of the fractal
+dimension. The last column of the table contains the prediction of Eq. (3.9) at the first
+order, ηp
+2. Our results are also shown in Fig. (9). The agreement is good between the
+measured η2 and the predicted one ηp
+2 for 1.5 ≤ a ≤ 2.0. For smaller values of a, the
+deviation is larger but also the second order correction ≃ (a−2)2. Quite interestingly, for
+a ≤ 0.5, η1 ≃ η2. For the r = ∞ disorder point, we recall that the random FK clusters
+(and the spin variables) are completely determined by the disorder configurations and
+the thermal fluctuations are frozen, see Section 2.5. This implies that the distribution
+function S(p12(x)) of the probability p12 that the spin s(0) and s(x) belong to the same
+16
+
+a=0.25
+a=1.00
+a=1.75
+Bp
+r=1
+0.18
+LRp
+r=2
+r=5
+r=10
+0.16
+r=100
+r=inf 
+0.14
+0.12
+0.1
+0.08
+0.06
+0.04
+10
+100
+10
+100
+10
+100a
+η1(= 2β/ν)
+η2
+r
+ηp
+2
+0.25
+0.100
+0.101
+100
+0.0625
+0.50
+0.167
+0.172
+100
+0.125
+0.75
+0.209
+0.243
+10
+0.1875
+1.00
+0.233
+0.298(2)
+10
+0.25
+1.25
+0.242(2)
+0.351(2)
+5
+0.3125
+1.50
+0.246
+0.389
+5
+0.375
+1.75
+0.251
+0.444
+2
+0.4375
+2.00
+0.250
+0.491
+1
+0.50
+Table 4: η1 and η2 measured with the disorder r for 0.25 ≤ a < 2. ηp
+2 = 1/2 − (2 − a)/4
+is the predicted value of Eq. (3.9) at the first order. Errors bars on the measured values
+η1 and η2 are shown in parenthesis and are smaller than one on the last digit otherwise.
+FK cluster, has the form S(y) = a0δy,0 + a1δy,1, with a0 + a1 = 1. This in turn implies
+η1 = η2. So the fact that, at finite r, where the spin degrees of freedom are not frozen,
+we obtain η1 ∼ η2 is a very non-trivial test of the fact that the infinite disorder point
+describes the physics of the system for these values of a. The conclusions of our analysis
+Figure 9: q = 2 Potts model with long-range disorder: η1, η2 and ηp
+2 vs a.
+are summarised in Fig. (10).
+17
+
+0.55
+0.5
+0.45
+0.4
+No.35
+n2t
+TU
+0.3
+.0.25
+.
+0.2
+0.15
+n2p
+n2
+0.1
+2 ni
+ni
+中
+0.05
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+aq = 2
+a
+1/r
+1
+LR
+LR
+a∗(2)
+a = 2
+Fig.(8) right
+a = 7
+4, Bp
+P
+a = 3
+2, LRp = Bp
+a = 1, LRp
+P
+Fig.(8) middle
+Fig.(8) left
+a = 1
+4, LRp
+P
+Figure 10: Fixed point stability, obtained via the measurements of the β/ν critical expo-
+nent, for q = 2. The dashed line represents the crossover with the purely SR physics at
+a = 2. The arrows describe a flow between two fixed points. The orange line represents
+the P (unique) fixed point, the red one the Bp (unique) fixed point, while the blue and
+magenta lines are lines of fixed points, respectively the LRp and the LR ones. a∗(2)
+indicates the crossover between the LR and LRp points and we located it slightly below
+a = 0.75.
+3.4
+The q = 3 phase diagram
+In the short-range regime a > 2, the disorder is now relevant and the critical properties
+are described by the SR, which is located at finite r∗
+SR, r∗
+SR = 6.08(12) [43]. The two
+fixed points, SR for 1 < r < ∞ and Bp for r = ∞ are observed for all values of q up to
+infinite [38, 44, 45]. The r = 1 point corresponds to the P point for q ∈ [1, 4] and to a
+first order phase transition for q > 4.
+When a < 2, the long-range disorder is relevant for a < 2/νSR ∼ 1.96. We show
+below that our numerical simulations support the existence of a stable LR that is located
+at finite 1 < r∗
+SR < r∗
+LR. We observed, moreover, that the LR point tends to merge with
+the LRp one, located at a critical value a∗(3) slightly smaller than 0.75.
+Our numerical findings for q = 3 are shown in Fig. (11). Again, we obtain three
+different behaviours:
+18
+
+Figure 11: q = 3 Potts model with long-range disorder: β/ν vs. L for a = 0.25 on the
+left, a = 1.00 on the middle and a = 2.00 on the right.
+• In the left panel, we show β/ν for a = 1/4. For all values of disorder r > 1, in the
+large size limit, the effective magnetic exponent goes to the value βLRp/νLRp = 5/96
+while it goes to the value of the pure q = 3-Potts model βP/νP = 2/15 for r = 1.
+We observe a similar behaviour for a = 0.5 with βLRp/νLRp ≃ 0.09.
+• In the middle panel, we show results for a = 1, with three different large size limits:
+i) for r = 1, the magnetic exponent is the one of the pure q = 3-Potts model
+βP/νP = 2/15; ii) for r = ∞, the magnetic exponent goes towards a value ≃ 5/48.
+Notice that this is the same as the one mention in the section above for q = 2 and
+a = 1: we recall that at this point the exponents do not depend on q; iii) for any
+finite disorder, a unique limit is obtained with βLR/νLR ∼ 0.12 corresponding to a
+LR fixed point with finite disorder r ≃ 10.
+A similar behaviour is observed for a = 0.75, 1.25 with βLR/νLR ∼ 0.11 and
+βLR/νLR ∼ 0.13 respectively.
+• In the right panel, we show results for a = 1.75 . For a = 1.50 and a = 1.75, β/ν is
+compatible, at large distances, with either with the P and the SR values, reported
+in Table 1 and in Table 2, which are very close. Our measurements therefore do
+not allow to discriminate between these two values, but according to the arguments
+discussed in Section 2.4, we conjecture the LR point will merge with the SR point
+at a ∼ 1.96, as illustrated in Fig. (12).
+In conclusion, for a < a∗(3) ≃ 3/4, we find that the system flows to the LRp point,
+19
+
+a=0.25
+a=1.00
+a=1.75
+Bp
+0.18
+LRp
+r=5
+r=10
+0.16
+r=100
+r=inf
+0.14
+0.12
+0.1
+0.08
+0.06
+0.04
+10
+10
+10while for a∗(3) < a < 1.96, we find the LR at some intermediate value of disorder with
+βBp/νBp < βLR/νLR < βSR/νSR.
+Our findings are summarized in Fig. (12).
+q = 3
+a
+1/r
+1
+LR
+LR
+LR = SR
+a = 1.96
+Fig. (11) right
+a = 7
+4, Bp
+P
+a = 3
+2, LRp = Bp
+Fig. (11) middle
+a = 1, LRp
+P
+a∗(3)
+a = 1
+4, LRp
+P
+Fig. (11) left
+Figure 12: Fixed point stability, obtained via the measurements of the β/ν critical expo-
+nent, for q = 3. The dashed line represents the crossover with the purely SR physics
+at a = 1.96. The arrows describe a flow between two fixed points. The orange line
+represents the P (unique) fixed point, the red line the Bp (unique) fixed point and the
+cyan one the SR (unique) fixed point. The blue and magenta lines are lines of fixed
+points, respectively the LRp and the LR ones. a∗(3) represents the crossover between
+the LR and LRp points and we located it slightly below a = 0.75.
+3.5
+Thermal behaviour for the q = 3-Potts model with long-
+range correlated disorder
+In this section, to test Eq. 2.11, we present results for the measurement of the exponent ν
+as a function of a. A convenient way to measure it is to consider the FK cluster wrapping
+probability
+Pw(r, J1, L) = Prob. of having a wrapping FK cluster on a lattice of size L.
+(3.10)
+20
+
+This is a quantity which is constant at the critical point for a fixed value r = J2/J1. J1
+is coupled to the thermal behaviour and we expect:
+Pw(r, J1(1 + ϵ), L) = f(ϵL1/ν)
+≃ f(0) + f ′(0)ϵL1/ν + f ′′(0)
+2
+ϵ2L2/ν + f ′′′(0)
+3!
+ϵ3L3/ν + · · ·
+(3.11)
+for a small ϵ, which measures the deviation from the critical point. Pw(r, J1, L) should
+also depend on r and one expect also corrections to scaling. These dependencies can be
+ignored if we consider the derivative
+P′
+w(ϵ) = Pw(r, J1(1 + ϵ), L) − Pw(r, J1(1 − ϵ), L)
+2ϵ
+≃ f ′(0)L1/ν + f ′′′(0)
+3!
+ϵ2L3/ν + · · · .
+(3.12)
+We first attempted to compute 1/ν by considering P′
+w at the first order, but the results
+were not convergent for large L. This is due to the fact that the errors on the measure-
+ment grow like the inverse of ϵ. While averaging over one million samples of disorder
+configurations, we found that errors bars allow us to decrease ϵ only down to 0.01. For
+this value, the second correction can not be neglected. We then considered
+˜
+P′
+w(ϵ) = 4P′
+w(ϵ) − P′
+w(2ϵ)
+3
+= f ′(0)L1/ν(1 + O(ϵ4L4/ν))
+(3.13)
+In Fig. (13), we show the results for the effective ν as a function of L obtained from
+a two point fit of our data to the form of Eq. (3.13) for a ≥ 0.75 and also the value 2/a
+as a dotted line. For each value of a, we choose the value of r which seems closer to the
+critical point with the least finite size corrections. For a ≥ 1.25, we used the data for
+r = 5, while for a = 0.75 and 1.00, we used the data for r = 10. Note that for a = 0.75,
+we still have large error bars even if for this value of a, we averaged over 10 millions
+samples of disorder configurations (the same statistics is used for a = 1.00, with much
+smaller error bars, for a > 1, we averaged over one million samples). For a = 0.75, we
+also observe in Fig. (13) that there are strong finite size corrections. For smaller values
+of a, we were not able to obtain convergent results. Note that this corresponds to values
+of a for which we expect that the critical point is at an infinite disorder.
+In conclusion, we observe that the value of ν increase slowly as one decreases a. For
+1 ≤ a < 2, ν is close to the prediction in Eq. (2.11). For a = 0.75, it seems to be bigger
+than expected with ν ≃ 3.5, but in this case, we observe strong finite size corrections.
+For smaller value of a, we can not measure ν.
+3.6
+The effects of higher disorder cumulants
+Let us come back to the value q = 1 where the disordered Potts model becomes a long-
+range percolation model.
+Without passing via a Potts model, a long-range percolation model can be directly
+defined by using the level sets of a fractional Gaussian free field (fGFF) with negative
+Hurst exponent H = −a/2 [13–17].
+21
+
+Figure 13: ν vs. L for the various values of a shown in the caption.
+In our approach, we can use the fGFF to generate a long-range disorder distribution,
+see Appendix A.2. In doing so for the q = 1 Potts model, we obtain a one-parameter (r)
+family of Gaussian long-range percolation models. In our simulations, we have verified
+that, analogously to what explained in Section 3.2, all these models flow to the point
+r = ∞, see Fig. (14). At this point, our long-range percolation models are expected to
+have the same critical properties as the models studied in [13–17]. More specifically, as
+we used toric boundary conditions, our model coincides at r = ∞ to the one studied
+in [17].
+Our approach allows then to compare directly the LRp points obtained using different
+distributions, in particular non-Gaussian ones. The disorder distribution generated by
+the n-replicated Ising model is an instance of such distributions. In the following, we
+use the suffix I (G) for the disorder distribution generated by n-replicated Ising model
+(fractional Gaussian free field).
+For the two distributions, the Gaussian one based on the fGFF and the non-Gaussian
+one based on the Ising model, Eq. (2.11) gives the same percolation thermal exponent,
+ν = 2/a = 8.
+The situation however is different for the exponent β. Let us discuss more in detail
+the case a = 1/4, that we have analyzed more carefully in our simulations. In Table 5
+are reported the r = ∞ results of our Monte-Carlo measures of
+�
+βLRp/νLRp�
+I, obtained
+by using the n-replicated Ising distribution, see Appendix A.1, and of
+�
+βLRp/νLRp�
+G
+obtained with the fGFF distribution, see Appendix A.2. In the same table we have also
+put the measures obtained in [15] that can be then directly compared.
+22
+
+a=0.75
+3.5
+a=1.00
+F
+a=1.25
+3
+a=1.50
+上
+a=1.75
+2.5
+2
+1.5
+10Figure 14: β/ν vs. L at q = 1 with a = 0.25 and disorder generated by using fGFF, see
+Appendix A.2.
+a
+�
+βLRp/νLRp�
+I
+�
+βLRp/νLRp�
+G
+�
+βLRp/νLRp�
+G [15]
+0.25
+0.0522(4)
+0.0721(9)
+0.0640(4)
+Table 5: Comparison of βLRp/νLRp for Gaussian and non-Gaussian disorders.
+The exponents
+�
+βLRp/νLRp�
+G and
+�
+βLRp/νLRp�
+I obtained respectively with the fGFF
+and with the Ising model seems different. Note that the numerical result
+�
+βLRp/νLRp�
+I
+agrees perfectly with the Ising model: as argued earlier, for the non-Gaussian disorders
+with n = 1-Ising copies, the FK clusters have the same fractal dimensions as the Ising
+spin clusters.
+The above results can be explained by noticing that, differently from the short-range
+disorder, in a long-range disorder, the terms generated by the higher cumulants of the
+distribution can be relevant. We can conclude that the higher cumulants do not change
+the existence and the stability of an LRp point but can modify certain critical exponents.
+As far the exponent βLRp/νLRp is concerned, one can doubt that the differences seen
+in the simulations are just due to the numerical precision.
+This is particularly true
+when a is increasing, see Table 3. However, we can mention the measure of another
+universal property which is clearly different between the (LRp)G and the (LRp)I and
+therefore supports the conclusion above. Indeed in [17] and in [46] the torus two-point
+connectivity pG
+12(r) and pI
+12(r) of, respectively, the fGFF level sets and the Ising clusters
+23
+
+a=0.25
+LRp
+0.14
+Bp
+r=1
+0.13
+r=2
+r=5 
+0.12
+r=10 μ
+r=100 
+r=inf 
+队
+0.1
+0.09
+0.08
+10
+100
+1000
+Lwere considered. It was argued that:
+pG
+12(r) = aG
+0 r
+−2
+�βLRp
+νLRp
+�
+G
+�
+1 + cG � r
+L
+�1.875
++ o
+�� r
+L
+�4��
+(3.14)
+pI
+12(r) = aI
+0 r
+−2
+�βLRp
+νLRp
+�
+I ×
+�
+1 + cI � r
+L
+�
++ o
+� r
+L
+��
+,
+(3.15)
+where aG
+0 and aI
+0 are non-universal constants. The cG and cI are universal quantities that
+depend on the ratio Lh/Lv between the horizontal and the vertical torus size and on the
+structure constants of the eventual CFT behind, see [47]. The cI is known and it has
+been computed in [46] while cG, investigated in [17], is not known exactly.
+The exponent x of the sub-leading term (r/L)x is computed on the assumption that
+exists a local CFT describing the critical point. This gives [47]:
+x = 2 − 1
+ν .
+(3.16)
+The exponents in Eq. (3.15) are then obtained by using ν = 1/8 (x = 1.875) for the
+Gaussian distribution and ν = 1 (x = 1) for the Ising one. For the Ising case, where the
+existence of a CFT behind is well established, the form for pI
+12 has to be considered exact.
+For the fGFF, the CFT predictions (3.15), with ν = 2/a, were tested numerically only for
+values of a > 1 [17]. For pG
+12 we can extend the results of [17] to a = 1/4, by assuming an
+analytical a−dependence on the critical exponents for all values of a < 3/2. So one can
+see that the sub-leading term exponents depend strongly on the disorder distribution.
+4
+Conclusions
+We considered the long-range disordered two-dimensional q-Potts model. The disorder is
+implemented by a bimodal distribution whose long-range nature is induced by auxiliary
+{σi} spin degrees of freedom to which the Potts model is coupled, see Eq. (2.5). We have
+considered different {σi} distributions, the Ising distribution described in Appendix A.1
+and the fGFF distribution described in Appendix A.2.
+These distributions have the
+same first cumulant, see Eq. (2.6), and a second cumulant with the same long distance
+power-law decay, see Eq.(2.8).
+By Monte-Carlo techniques, we studied the fractal dimension of the q = 1, 2, 3-Potts
+FK clusters for different values of the power-law exponent a. The measures were taken
+at the self-dual point, see Eq. (2.4), for different values of the strength disorder r. The
+outcome of these analysis are resumed in the phase diagram Fig. (1), which represents
+the main finding of this work.
+We considered first the infinite disorder (r = ∞) fixed point LRp, at which the thermal
+fluctuations of the Potts degrees of freedom are frozen, and the disorder averages coincide
+with the ones of a long-range bond percolation model. The LRp critical exponents do not
+depend on q. The measured value of βLRP/νLRp are reported in Table 3. We observed
+that, by using the Ising distribution, the Monte-Carlo measures on the LRp point are
+24
+
+more precise than the corresponding ones for the fGFF. In particular we could probe
+with good precisions the physics of the LRp point for small values of a, the regime where
+the fGFF methods are more difficult to implement, see [17].
+For q = 1, we observed that, for a < 3/2, the LRp is attractive while for a ≥ 3/2
+the system is described by the Bernoulli critical point Bp. For q = 2, 3 we establish the
+existence of the LR point, which is a long range critical point at finite disorder. We
+observed that the LR point and the LRp point exchange stability at a certain value a∗(q)
+which depend on q.
+The above results are supported by the study of other observables than the FK
+fractal dimension. For q = 2 we measured the multi-fractal behaviour of the spin-spin
+correlation, see Table 4. The results are consistent with the theoretical findings of [20] in
+their region of validity, especially for 1.5 < a < 2. For lower values of a, where theoretical
+predicitons are missing, our results rule out the occurence of a softening of the transition
+and support instead the fact that the system is driven to the LRp point. For q = 3
+we mesure the thermal νLR exponent by measuring the wrapping probability of the FK
+clusters. The results are shown in Fig. (13) and are consistent, for a wide range of values
+of a, with the theoretical prediction of Eq. (2.11).
+We verified that the above mentioned results, in particular the form of the phase
+diagram, are valid for different disorder distributions. On the other hand, we observed
+some universal effects of the higher cumulants at the LRp point. By focusing on the
+a = 1/4 value, we showed that the exponent βLRp is different between a Gaussian and a
+non-Gaussian distribution, see Table 5. Finally we pointed out that clear effects of the
+higher cumulants are expected for the universal finite size effects of the FK connectivities,
+see Eq.(3.15).
+5
+Acknowledgments
+We are grateful to Nina Javerzat with whom this project started. We thank Pierre Le
+Doussal for useful discussions.
+A
+Disorder distributions
+A.1
+n-replicated Ising model
+In most of the simulations presented here we chose a particular distributions for dis-
+order.
+Let us consider n independent critical Ising models with spins σ(a) ∈ {0, 1},
+a = {1, · · · , n}. The variables σi in Eq. (2.5) are given by:
+σi =
+�
+a=1
+σ(a)
+i ,
+(A.1)
+and their statistical properties are determined by the correlation functions of the the n
+independent Ising models. For the first two moments, we have:
+E [σi] = 0,
+E [σi σj] ∼ |i − j|−n
+4 for |i − j| ≫ 1,
+(A.2)
+25
+
+where we used the fact that the critical Ising spin-spin correlation function
+�
+σ(a)
+i σ(a)
+i
+�
+∼
+|i − j|−1/4. Comparing with Eq. (2.8), we have:
+a = n
+4.
+(A.3)
+The convenience of this choice is that the n-Ising models are very simple, and fast, to be
+simulated. The price to pay is that we cannot vary a continuously. The above distribution
+is not Gaussian because the critical Ising point is not a Gaussian point.
+A.2
+Fractional Gaussian free fields
+The other one is related to the percolation point of the level sets of a fractional Gaussian
+random field with negative Hurst exponent H = −a/2.
+Let us, now, consider an L × L square lattice, whose vertices i have coordinates
+i = (ih, iv) and a set of random variables φi that defines a discrete fractional Gaussian
+free field:
+φi =
+�
+�
+�
+�
+�
+�
+�
+L
+�
+(l,m)̸=(0,0)
+cl,m
+Sl,m
+exp
+�2π
+L (l ih + m iv)
+�
+, if (l, m) ̸= (0, 0)
+c0,0, if (l, m) = (0, 0)
+(A.4)
+where Sl,m =
+����2 cos
+�2πl
+L
+�
++ 2 cos
+�2πm
+L
+�
+− 4
+����
+1 − a/2
+2
+and the cl,m are independent
+random Gaussian variables, i.e.
+cl.m ∈ N(0, 1).
+Using the properties of the Fourier
+transform, the covariance E[φiφj] has the following asymptotic limit:
+E[φi φj] ∼ |i − j|−a for |i − j| ≫ 1.
+(A.5)
+The σi variables are defined by:
+σi =
+�
+0,
+if φi > 0
+1
+if φi < 0,
+(A.6)
+The fact that E[σi] = 0 implies the level 0. One can notice that the clusters built with the
+{σi} variables are not at the percolation treshold. Indeed, this is spin-percolation physics
+and it is known [17] that it has a critical level greater than 0. These {σi} variables are
+only needed to built the bond distribution as explained in Section 2.1. One can show,
+instead, that the FK-cluster built on these bonds, are at the percolation transition. Thus,
+one has:
+E[σi σj] ∼ |i − j|−a for |i − j| ≫ 1,
+(A.7)
+and a long-range correlated disorder.
+26
+
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+
diff --git a/kNFST4oBgHgl3EQfIDgO/content/tmp_files/load_file.txt b/kNFST4oBgHgl3EQfIDgO/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,1448 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf,len=1447
+page_content='Long-range quenched bond disorder in the bi-dimensional Potts  model  Francesco Chippari  Sorbonne Universit´e & CNRS, UMR 7589, LPTHE, F-75005, Paris, France  e-mail: fchippari@lpthe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='jussieu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='fr  Marco Picco  Sorbonne Universit´e & CNRS, UMR 7589, LPTHE, F-75005, Paris, France  e-mail: picco@lpthe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='jussieu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='fr  Raoul Santachiara  Paris-Saclay Universit´e & CNRS, UMR 8626, LPTMS, 91405, Saclay, France  e-mail: raoul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='santachiara@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='com  (Dated: February 1, 2023)  ABSTRACT  We study the bi-dimensional q-Potts model with long-range bond corre-  lated disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Similarly to [1], we implement a disorder bimodal distribution  by coupling the Potts model to auxiliary spin-variables, which are correlated  with a power-law decaying function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The universal behaviour of different ob-  servables, especially the thermal and the order-parameter critical exponents,  are computed by Monte-Carlo techniques for q = 1, 2, 3-Potts models for dif-  ferent values of the power-law decaying exponent a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  On the basis of our  conclusions, which are in agreement with previous theoretical and numerical  results for q = 1 and q = 2, we can conjecture the phase diagram for q ∈ [1, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  In particular, we establish that the system is driven to a fixed point at finite  or infinite long-range disorder depending on the values of q and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Finally, we  discuss the role of the higher cumulants of the disorder distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' This is  done by drawning the auxiliary spin-variables from different statistical mod-  els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' While the main features of the phase diagram depend only on the first  and second cumulant, we argue, for the infinite disorder fixed point, that  certain universal effects are affected by the higher cumulants of the disorder  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  PACS numbers: 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='Lk, 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='+q, 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='Fr  Contents  1  Introduction  2  2  The long-range bond disordered q-Potts model  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1  Implementation of disorder .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
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+page_content='  6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='2  Pure Potts model (r=1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
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+page_content='  7  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
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+page_content='1  The infinite disorder point (r = ∞): LRp and Bp fixed points  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
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+page_content='  18  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5  Thermal behaviour for the q = 3-Potts model with long-range correlated  disorder  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
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+page_content='  20  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='6  The effects of higher disorder cumulants  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
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+page_content='  21  4  Conclusions  24  5  Acknowledgments  25  A Disorder distributions  25  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1 n-replicated Ising model  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
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+page_content='  25  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='2 Fractional Gaussian free fields .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
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+page_content='  26  1  Introduction  The q-Potts model [2] is a lattice spin model where the spins take one of q-possible values  and interact via nearest neighbours couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Through a geometrical reformulation [3],  the model admits a natural continuation to real values of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' When the couplings take the  same positive value, the two-dimensional Potts model has a second-order ferromagnetic-  paramagnetic phase transition for q ∈ [1, 4] [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Here, we consider the bond disordered  q-Potts model, where the couplings have random fluctuations around a given positive  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In particular, we are interested in how the critical properties of this model depend  on the fluctuations (disorder) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Part of the answer to this problem is provided  by the Harris criterion [4] and by some generalizations of it [5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' These criteria determine  the relevance of the disorder on the basis of the correlation properties of its distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  The most studied example is the uncorrelated disorder where the couplings fluctuations  are independent and equally distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In this case the Harris criterion states that  the disorder modifies the critical properties of the pure system if its thermal exponent  νP is greater than one, νP > 1 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' This condition is satisfied by the q-Potts model for  q > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' When the disorder is relevant, a situation of particular interest is the one where  the system still undergoes a continuous phase transition but it is described by a different  stable renormalization group (RG) fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' It is now well established that this is the  situation for the two-dimensional Potts model with uncorrelated disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For q > 2,  there is a stable RG fixed point [7–9], henceforth referred to as the Short-Range (SR)  2  point, which determines the critical properties of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The thermal νSR and the  order-parameter βSR critical exponents have been computed via an RG computation of  a perturbed conformal field theory [8] and their values match with different numerical  tests [9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  In this paper we consider the case where the couplings {J(x)}, at each position x, are  drawn from an homogenous and isotropic distribution, with first cumulant, E[J(x)] =  E[J].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  We focus in particular on distributions whose second cumulant, g(|x − y|) =  E[J(x)J(y)] − E[J]2, decreases as a power-law for large distances, g(|x|) ∝ (r − 1)2|x|−a  for |x| ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The parameter a is taken positive, a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The value r parametrizes the  disorder strength and it is taken greater than one, r ∈ [1, ∞], see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For r = 1,  one recovers the pure Potts model, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  The r = ∞ corresponds to the  infinite disorder point and it is discussed in detail in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The numerical set-up  to implement these distributions follows closely the one introduced in [1], where the pure  Potts model is coupled to the Ashkin-Teller one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Most of the results presented here are  obtained by using a replicated Ising model instead, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1, which differs from  the one of [1] in the cumulants of order higher than two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' We have verified that our main  findings on the phase diagram of these models do not depend on the higher cumulants  of these distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  The extended Harris criterion [5,6] determines in which region of the parameters (q, a)  the disorder is relevant and when its long-range properties prevail over the short-range  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' When a > 2 the critical behaviour of the system is expected to be the same as the  one with uncorrelated disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The regions (q < 2, a < 2/νP) and (q > 2, a < 2/νSR),  where the disorder is relevant and dominated by the long-range correlations, are much  less understood, especially for general values of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  For q = 1, as we explain in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='2, the pure Potts model and the Potts with  bimodal short range disorder correspond, for any value of the disorder strength r, to  the Bernoulli bond percolation model [12], where the edges are independently activated  (pure percolation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The long-range disordered q = 1-Potts model coincides instead with  a one-parameter (the disorder strength r) family of long-range bond percolation model,  where the activation of two distant edges is correlated by a power-law decreasing function  with exponent a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The d-dimensional long-range percolation was studied in [6] by using  a one-loop order RG computation with a double expansion in ϵ, ϵ = 6 − d and in δ,  δ = 4 − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' It was found that the long-range nature of the correlation drives the system  to a new stable RG point, that, in the following, we refer to as the LRp (Long-Range  percolation) point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Even if these theoretical predictions have validity near dimension  six and for a close to 4, they have motivated a series of numerical works [13–17] on  some d = 2 long-range percolation models with a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' These models were defined by  using the level sets of fractional Gaussian free fields with Hurst exponent H = −a/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  As the thermal exponent of the pure q = 1-Potts is νP = 4/3, the long-range disorder  is relevant for a < 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The general agreement in [13–17] was that, for a < 3/2, the  percolation transition is described by a new LRp point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In our approach, we recover  the critical behaviour studied in [13–17] at the r = ∞ point of a Potts model coupled  with a fractional Gaussian field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' We will show that, irrespective to the value of r, these  long-range percolation models are described, for a > 3/2, by the Bernoulli percolation  (Bp) point and, for a < 3/2, by the LRp point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' This can be rephrased by saying that, for  each a, there exists an unique fixed point which is stable in the direction of decreasing  3  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' We observed that, by coupling instead the Potts model to n−replicated Ising models,  the numerical investigations of the LRp type points becomes more precise, in particular  for small values of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' We will argue in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='6 that some universal properties of the  LRp points can depend on the higher cumulants of the disorder distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  The q = 2 case corresponds to the bond disordered Ising model and it has been studied  quite intensively in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Here the cross-over exponent is 2/νP = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For a ≥ 2, the SR  and the pure (P) point coincide and the disorder effects generate logarithmic corrections  to the scaling relations [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For 1 ≲ a < 2, one finds a stable long-range (LR) point,  whose existence has been proven in [19] and in [20] by using the Ising massive free fermion  representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Numerical tests of these predictions have been found in [21] for a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  In [20] it was also observed that LR point looses its stability for a ≲ 1, and the question  whether the system undergoes a smeared phase transition or whether it flows to an infinite  disorder fixed point remained open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In [1, 22], the q = 2 (and q = 4) Potts long-range  disordered model were investigated for values of a smaller than one, 0 < a < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' It was  observed there that, for all this values of a, the Monte-Carlo measures of β/ν were very  close to the LRp point value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  Our aim here is to provide a complete Potts phase diagram in the parametric region  (q, a) ∈ (0, 4] × (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' This is achieved by considering the disordered q = 1, 2, 3-Potts  models for different values of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' These three values of q are indeed representative of  the three regions where the short-range disorder is respectively irrelevant, marginal and  relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  Our results are summarised in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' We establish that for all values of q > 1  there is a region where the LR point exists and it is stable in the direction of decreasing  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Moreover, for each q ∈ [1, 4], there is a value a∗(q) at which the LR point becomes  unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For smaller values of a, a < a∗(q), the model is described by the LRp point,  which is the fixed point of the Potts model with infinite disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  2  The long-range bond disordered q-Potts model  The model is defined by the partition function:  Z({J<ij>}) =  �  {si}  exp  ��  <ij>  J<ij>δsi,sj  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1)  where the spin si, living on the lattice vertex i, takes q possible states, si = {1, · · · , q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  The < ij > identifies the edge connecting the neighbouring sites i and j and the δk,l is the  Kronecker delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The set of couplings {J<ij>} is made of random variables drawn from  a given distribution, the average over which will be indicated by the notation E[· · · ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  We extract the critical exponent of the model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1) by measuring the properties of  its Fortuyn-Kasteleyn (FK) clusters [3, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The FK clusters are the connected sets of  bonds that enter in the geometric representation of the Potts partition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' From  4  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5  4  0  1  2  3  q  a  Bp  LRp  Short-Range  (SR, finite disorder)  Irrelevant  (P)  Long-Range  (LR, finite disorder)  Long-Range percolation  (LRp, Infinite disorder)  Figure 1: Phase diagram of the disordered bi-dimensional q-Potts model for (q, a) ∈  (0, 4] × (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1), we get:  Z ({J<ij>}) =  �  {si}  �  <ij>  �  1 + (eJ<ij> − 1)δsi,sj  �  =  �  G  q# FK clusters(G)  � �  <ij>∈G  eJ<ij> − 1  �  =  �  G  �  <ij>∈G  �  1 − e−J<ij>�  �  <ij>/∈G  �  e−J<ij>�  q# FK clusters(G),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='2)  where the sum is over the set G of activated edges or bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Note that in the last equality  we discard a global factor �  <ij>  eJ<ij>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' A bond connects two equal spins with probability  p<ij> = 1 − e−J<ij>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  We will consider self-averaging observables, such as the thermal (ν) and the order-  parameter (β) critical exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' More specifically, we will compute, for q = 1, 2, 3 the  ratio β/ν from the measure of the FK fractal dimension df using:  df = 2 − β  ν ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='3)  see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The exponent ν, instead, is extracted, for q = 3, from the FK clusters  wrapping probability, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The exponent ν determines the finite size  scaling of the probability defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' We consider, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='3, the Ising  5  (q = 2) spin-spin correlation function, which is a non self-averaging observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' We will  measure the multifractal behaviour of this quantity, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='8), and compare with  recent predictions [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1  Implementation of disorder  In our simulations, we implement a quenched bi-modal disorder where the couplings can  take randomly two values, J<ij> = J1 or J<ij> = J2, with equal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The values  of J1 and J2 are not taken independently but they are chosen to satisfy the following  self-dual condition:  (eJ1 − 1)(eJ2 − 1) = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='4)  The self-dual line Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='4) corresponds to a line of critical points separating a paramag-  netic phase and a ferromagnetic phase [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  In order to simulate this disorder, we associate to each vertex of the square lattice a  random variable σi = {−1, 1} and we set:  J<ij>(R) = J<ij>(B) = J1 + J2  2  + σi  J1 − J2  2  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5)  where, the J<ij>(R) and J<ij>(B) are the couplings associated respectively to the edge on  the right and to the edge on the bottom of a given site i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2) illustrates the  connection between the σi and the J<ij> configurations as given by the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  Figure 2: A particular configuration {σi} is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Blue (red) filled circles are the sites  corresponding to σ = 1 (σ = −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The corresponding set of couplings J<ij>, associated  to the edges, follow from the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In the Figure, the J<ij> = J1 (J<ij> = J2)  couplings are associated to the blue (red) colored edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  In principle there are other possibles choices: for instance one could introduce, instead,  an edge random variable σ<ij> and impose J<ij> = (J1+J2)/2+σ<ij>(J1−J2)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' All these  different set-ups are expected to be equivalent as far universal properties are concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  We have chosen simply the one that was more convenient for our numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  6  The disorder distribution is then fixed by the {σi} one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Here we consider the distri-  butions that are characterized by a vanishing first moment:  E [σi] = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='6)  which forces the couplings to take values J1 or J2 with the same probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The couplings  then fluctuate around the value (J1 + J2)/2:  J<ij> = E[J<ij>] + δJ<ij>,  E[J<ij>] = J1 + J2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='7)  We can then consider a {σi} distribution whose second cumulant has a power-law decay-  ing:  E [σi σk] ∼ |i − k|−a  for |i − k| ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='8)  This, of course, correlates the fluctuations of the couplings at distant edges:  E  �  δJ<ij>(X) δJ<kl>(Y )  �  = (J1 − J2)2  4  E [σi σk]  ∼ |i − k|−a  for |i − k| ≫ 1,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='9)  where X, Y = R, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  Once we have chosen the {σi} distribution and satisfied the relation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='4), the  phase diagram of the q-Potts model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1) depends only on the ratio r,  r = J1  J2  ∈ [1, ∞],  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='10)  that parametrizes the disorder strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' We assumed, without loss of generality, that  J2 ≤ J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In our simulations we will analyse the behaviour of the system by varying r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  The stability of the RG fixed points are therefore probed with respect to this direction  while the relation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='4) makes the system stay at the critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='2  Pure Potts model (r=1)  The pure case corresponds to r = 1 where the couplings take all the same value, see  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' On the square lattice, the corresponding critical point P is located at J1 =  J2 = Jc = log (1 + √q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  In the continuum limit, the P point is described by the Potts conformal field theory  whose energy spectrum is known since longtime [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In these last years a bootstrap  approach [26–30] has been proposed for studying the cluster connectivity properties,  which finally was used in [31–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' This provides the exact Potts correlation functions  and a complete characterization of the symmetry representation of its states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  The thermal and order-parameter critical exponent νP and βP of the pure Potts  model have been determined for any value of q ∈ [0, 4], see for instance [34] and references  therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In Table 1, we give the values of νP and of βP/νP for q = 1, 2, 3:  7  q  νP  βP /νP  1  4/3  5/48  2  1  1/8  3  5/6  2/15  Table 1: Pure critical exponents for different q values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='3  Short-range disordered Potts (a ≥ 2)  When a ≥ 2 the disorder is effectively uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In Fourier space, the contribution  of the power-law tail |x|−a is of order O(ka−2) and, in the long wavelength limit k → 0,  is finite for a ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' At large distances, the properties of a long-range distribution are the  same as the one of a delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  The phase diagram of the uncorrelated disorder is quite well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For q < 2 the  P point is the only stable RG point: the weak disorder does not modify the universality  of the pure model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For q > 2 instead, the system is driven to a stable SR point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For  q = 2 the disorder is marginal (the P and SR points coincide) and generate universal  logarithmic corrections to the P universality class [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' We report in Table 2 the critical  exponents for q = 3, computed perturbatively in [7,8]:  q  νSR  βSR/νSR  3  ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='02  ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='134655  Table 2: Short-Range critical exponents for q = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  In the case of uncorrelated disorder, one can show by simple power counting that only  the first two cumulants of the distributions determine the long-distance behaviour of the  system [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='4  Long-Range disordered Potts (a < 2)  For a < 2 one has to take into account the power-law decay of the disorder distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  As mentioned in the Introduction, according to the extended Harris criterion [5, 6], for  (q < 2, a < 2/νP) and for (q > 2, a < 2/νSR), the long-range part of the distribution is  relevant and dominates the short-range one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In [5,6], it was also observed that, when it  exists, the LR point has to be stable with respect to an additional sub-dominant term  to E[δJ(x)δJ(y)], E[δJ(x)δJ(y)] → E[δJ(x)δJ(y)] + w0|x − y|−b, with b > a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Reiterating  8  the extended Harris criterion, one expects that when b > a then b > 2/νLR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' the per-  turbation is irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Otherwise, when b < a, the additional b-term becomes dominant  and therefore b < 2/νLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' This in turn brings to the strong conjecture that:  νLR = 2  a,  for any q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='11)  The above relation has been proven by perturbative RG computation in different long-  range disordered models [5, 6, 19, 20] at one or two-loops order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  It is now also quite  established that the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='11) should be valid at all perturbation orders [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5, we will test the relation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='11) for q = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5  The infinite disorder point (r = ∞): the q-colored critical  pure percolation  The r → ∞ point is the only point, besides the pure one (r = 1), where exact results can  be given for any q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' From the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='4), in the limit r → ∞ one has J1 → ∞ and J2 → 0  with rJ2 ∼ − log(J2) → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In particular:  E [δJ<ij>δJ<kl>] = (r − 1)2J2  2  4  E [σiσk] ∼ (rJ2)2E [σiσk] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='12)  The r = ∞ limit corresponds then to the maximal amount of disorder we can consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  In this limit, the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='2) greatly simplifies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Indeed, with probability one, there is a  bond on an edge with coupling J1 (1 − e−J1 = 1) and probability zero of having a bond  on a edge with J2 (1−e−J2 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' So, once a configuration of σi (and consequentely of the  couplings J<ij>) is drawn, only the graph where the edges < ij > associated to J1 are  activated, contributes to the partion sum, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The J2 edges are not activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  Figure 3: Given the configuration of σi and therefore of J<ij> shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2), we show  the only random cluster graph G in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='2) that does not vanish in the limit r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Notice  that the σi = 1 cluster, formed of blue spins, are different from the ones of the FK bonds  (black lines)  9  The disorder average free-energy, for instance, can be written as:  E [log Z({J<ij>}] = log q × E [# (J1-clusters)] ∼ log q × E [# (σ = +1-clusters)] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='13)  where ∼ means that the scaling behaviour is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The disordered Potts model at  r = ∞ is then strictly related to the site percolation model based on the σ clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In  particular, the probability distribution of the r = ∞ FK clusters is, besides a trivial  dependence on q, the same as the one of the σ clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In our construction Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5),  the < ij(R) > and < ij(B) > are always activated or not activated at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' At  the lattice level, the r = ∞ FK clusters are therefore different from the σ clusters, as  illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' However their universal behaviour, such as their fractal dimension,  as well as the correlation lenght exponent ν, remains the same, as discussed in the  following Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' A more quantitative argument can be provided for a = 1/4, where  the σ are Ising spins at the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In fact, one can observe that the two type of  clusters differ only on the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For an Ising cluster of linear size R, the number of  spins on the boundary scales as ≃ Rdb  f with db  f = 11/8 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='3751, which is much smaller  than the number of spins in the cluster that scales as Rdf with df = 187/96 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='948.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  Then, the FK clusters for a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25 are expected to have the same fractal dimension  df = 2 − 5/96 as the Ising spin clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  When a > 2, the system is short-range and it is described by the Bernoulli critical  point Bp [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For a < 2, one applies the extended Harris criterion using νBp = 4/3: for  a > 2/νBp = 3/2 the system remains in the Bp point while a < 3/2, there is the new  LRp fixed point already mentioned in the Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1 we tested this  behaviour and we computed the exponent βLRp/νLRp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  We insist on the fact that the LRp exists at r = ∞ for any q and its critical exponents  depend on a but do not depend on q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' As we will discuss in the following, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='s (10,  12), the LRp point is for q > 1 distinct from the LR point, which is the finite disorder  fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The LRp point exhanges stability with the LR one when a is greater than a  particular value a∗(q), a > a∗(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  3  Phase diagrams from Monte-Carlo measures of q =  1, 2, 3-Potts  We present here the numerical results according to which we established the phase di-  agram of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  Simulations were done on square lattices with periodic boundary  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' If not stated otherwise, we average over one million samples of disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For  each value of q and a, we determined first the average autocorrelation time τa,q(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Next  the thermal average is done, for each sample of disorder, by runing 100 × τa,q(L) updates  after the same number of updates for thermalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' If not stated otherwise, the results  presented in the following are obtained by using the disorder distribution described in  Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  For each value of n = {1, · · · , 8}, corresponding to values a given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='3), we  generate the coupling configuration J<ij>, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  We then construct the FK  1Note that the Ising spin boundaries are described by the CLE3 loops and db  f = 11/8 is the fractal  dimension of the CLE3 loops [37]  10  clusters by connecting equal spins with the probability p<ij> = 1−e−J<ij>, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  Given a lattice of linear size L, we compute, for each coupling configuration, the (thermal  averaged) number of spins M({J<ij>}, L) of the largest FK cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' We average on the  coupling distribution,  M(L) = E [M(L, {J<ij>})] ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1)  to extract the fractal dimension df by the scaling  M(L) ≃ Ldf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='2)  More specifically, recalling the relation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='3), we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='2) to compute an effective  magnetic dimension:  β  ν (L) = − log  � m(L)  m(L/2)  �  / log [2] ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='3)  where m(L) = M(L)/L2 is the average magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The above data, computed for  different q, a and r are the ones shown in the plots β/ν vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' L below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The ratio β/ν is  then evaluated in the scaling limit:  β  ν = lim  L→∞  β  ν (L) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='4)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1  The infinite disorder point (r = ∞):  LRp and Bp fixed  points  We collect here results for the q-colored percolation model, found at the infinite disorder  point r = ∞, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' We recall that the critical exponents at the infinite disorder  point do not depend on the value of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (4), we show the effective exponent β/ν(L) as a function of L for a =  {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25, · · · , 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='00}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The corresponding scaling limit results βLRp/νLRp, which are obtained  by a best fit of all our data, including a sub-leading correction, are reported in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  a  βLRp/νLRp  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='0522(4)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='086(2)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='103(2)  1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75, 2  ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='105  Table 3: Best fit of the critical exponent βLRp/νLRp for different values of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Notice that  for a = {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75, 2} the system is described by the Bp point where βBp /νBp = 5/48 ∼  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' However, for a ≥ 1, the numerical simulations cannot distinguish between the LRp  point and the Bp point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  As discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5, for a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25, the βLRp/νLRp must correspond to the  fractal dimension of the Ising clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The latter has been considered in [39, 40] and  argued to be:  βLRp  νLRp = 2 − 187  96 = 5  96 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='0521.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5)  11  Figure 4: The r = ∞ long-range disorder: β/ν vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' L for the values of a shown in the  caption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  From the Table 3, one can see that the agreement to the value 5/96, shown as a dashed  line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (4), is very good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  We remind that according to the Harris criterion, one expects that, for a < 3/2,  βLRp/νLRp < βBp/νBp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  As a matter of fact, the difference between βLRp/νLRp and  βBp/νBp becomes very small for increasing a and the numerical simulations cannot ef-  fectively distinguish between these two fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In fact, for a = 1 and a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25, it  seems to be even slightly larger than 5/48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' A similar results was also observed in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  But, since the finite size correction are important, it is difficult to really predict the  asymptotic value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  We consider in the following Potts observables at finite value of disorder r, where the  scaling behaviour is in general expected to depend on q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='2  The q = 1 phase diagram  We consider here the phase diagram of the q = 1 Potts model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For a > 2 the disorder is  effectively short-range and irrelevant, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The fact that a short-range disorder  is irrelevant can also be explained by a simple lattice argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' On the critical line  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='4), for r = 1, J1 = J2 = Jc = log (1 +  √  2) = log 2, thus we are exactly at the Bp  critical point, p = 1 − e−Jc = pc = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' By adding disorder, meaning by setting r > 1,  the dual line Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='4) fixes, at q = 1, the relation 1 − e−J1 − e−J2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Recalling that  each bond J1 and J2 occurs with equal probability, the density of activated bond remains  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='09  Bp  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='08  LRp  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='07  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='00  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='06  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75  a=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='05  10  100  1000p = 1  2(1−e−J1)+ 1  2(1−e−J2) = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' So, we can conclude that, for any value of r, we stay  in the Bp point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For 3/2 = 2/νBp < a < 2, the long-range disorder remains irrelevant,  and the system is still described by the Bp point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  The effective exponent β/ν(L) is measured for various values of disorder strength r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  For a < 3/2 our numerical measures shows clearly that the systems flows from the Bp  point to the LRp point, located at r = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The LRp point and its scaling exponents have  been discussed in the Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5 and in Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  We present first the model with a = 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In the left part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (5), β/ν(L) is shown  for r = 1, 2, 5, 10, 100 and ∞ as a function of the size L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' We observe that this exponent,  for all values of r > 1, goes towards the value of r = ∞, with a crossover function of the  disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The corresponding expected value 5/96, see [39], is shown as a blue dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  The r = 1 corresponds to Bernoulli percolation whose scaling limit value βBp/νBp = 5/48  is shown as a red dashed-dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Thus, for any amount of disorder, the large distance  behaviour is controlled by an infinite disorder fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In the right part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (5),  Figure 5: q = 1 Potts model with a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The left panel shows β/ν vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' L while the  right panel shows the wrapping probability Pw(L) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  we show Pw(L), see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For r = 1, it converges towards the exact value for the  Bernoulli percolation [41]:  Bernoulli clusters:  lim  L→∞ Pw(L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='69046  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='6)  which is shown as a red dashed-dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For any value r > 1, it goes to the same  value as for LRp point with a crossover function of the disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In addition to the fractal  dimension, we can verify that also the wrapping probability supports the fact that, as  stated above, the FK clusters behave, at the LRp point, as the Ising spin ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For spin  clusters this probability can be computed from the results in [42] and is:  Ising clusters:  lim  L→∞ Pw(L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='515884 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='7)  This is shown as a blue dashed line in the right part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The agreement is perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  13  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='7  Bp  LRp  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='68  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='18  r=1  r=2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='66  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='16  r=5  r=10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='64  Bernoulli Clusters  r=100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='14  Ising Clusters  r=inf  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='62  r=1  R"0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='6  r=2  ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='12  r=5  r=10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1  r=100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='56  r=inf  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='54  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5  10  100  10  100  1000Coming back to the β/ν exponent, similar behaviours are observed for the others  values of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (6), we show β/ν(L) for a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75 and a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Again, for all values  of r, the large size limit is the one reported in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  Figure 6: β/ν vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' L for the q = 1 Potts model with long-range disorder with a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75 in  the left panel and with a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='50 in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  We can therefore summarize the numerical findings for q = 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='3  The q = 2 phase diagram  We consider here in detail the q = 2 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In the short-range regime a ≥ 2, the disorder  is marginal and the P and SR points coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' When a∗(2) < a < 2 we can numerically  probe the existence of the LR point, predicted in [19,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Here, it will correspond to a  fixed point at finite disorder r∗  LR, 1 < r∗  LR < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' We estimate a∗(2) to be in the interval  1/2 < a∗(2) < 3/4, but the determination of its exact value goes beyond the aims of this  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' When a < a∗(2) there are just two fixed points, the P and the LRp (r = ∞) ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  In this case, the LRp becomes the stable one and we observe a flow from the P point to  the LRp point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' These different behaviours are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In particular:  In the left panel, we show the measured values of β/ν(L) for a = 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' We clearly  observe that for all values of disorder, the effective magnetic exponent goes to the  same value as the infinite disorder one, βLRp/νLRp = 5/96 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='0521, see Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  We obtain a similar behaviour for a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5, with β/ν(L) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='086 in the L → ∞  limit, again very close to the value βLRp/νLRp reported in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  In the middle panel, we show our data for a = 1 and we observe three different  large size limits: i) for r = 1 the magnetic exponent is the one of the short-range  Ising, βP/νP = 1/8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='125;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' ii) for r = ∞ the magnetic exponent is close to the  value βBp/νBp = 5/48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' As already mentioned for the case of infinite disorder, see  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1, for a ≃ 1, it is difficult to distinguish the LRp point from the Bp point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  iii) for any finite disorder, there is an unique large size limit, limL→∞ β/ν(L) =  14  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='135  Bp  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='13  =  r=2  r=5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='125  r=10  r=100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='12  r=inf  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='115  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='105  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='095  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='09  10  100  10  100q = 1  a  1/r  1  Bernoulli percolation  a = 3  2, LRp = Bp  Bp  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (6) right  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (6) left  a = 3  4, LRp  Bp  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (5)  a = 1  4, LRp  Bp  Figure 7: Fixed point stability, obtained via the measurements of the β/ν critical expo-  nent, for q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The red zone represents a unique fixed point, the Bp one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' There is no  flow while tuning r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The arrows describe a flow between two fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The red line  represents again the Bp fixed point and the blue line is a line of fixed points, the LRp  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  βLR/νLR ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='115 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' This corresponds to the LR fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The curves  converge to the curve with a finite disorder r ≃ 10, that is the value in which we  observe the smallest corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  We observed a similar behaviour for a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75 and a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25 with the scaling limit  limL→∞ β/ν(L) ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='108 and limL→∞ β/ν(L) ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='122 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  The right panel shows the measured values of β/ν(L) for a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In that case,  the measurements for finite disorder also converge to a single value βLR/νLR which  is very close to the P=SR value βP/νP = 1/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For all values of a ≥ 3/2, due to  strong finite size corrections, we find values close to the P=SR values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  Since, for values of 3/2 ≤ a < 2, the LR point is close to the P fixed point and we  can not really distinguish via the β/ν exponent these two points, we will then consider  another way of checking the existence of the LR point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' We consider the moments of the  spin-spin correlation function,  E[< s(0)s(x) >n] ≃ x−ηn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='8)  15  Figure 8: q = 2 Potts model with long-range disorder: β/ν vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' L for a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25 on the  left, a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='00 on the middle and a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75 on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  and we are interested in extracting the exponents ηn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' While η1 = 2β/ν gives access to  the same exponent that we computed above via the FK clusters dimensions, ηn for n > 1  are new exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The authors of [20] were able to compute η2 for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='995 ≤ a < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' They  obtained, at the lowest order in disorder, the following:  η2 = 1  2 − (2 − a)  4  + O((2 − a)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='9)  In this computation the lower range, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='995, is obtained as a limit of the stability of  their perturbative fixed point at the second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  In Table 4, we report our measured values of η1 and η2 for various values of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Note  that the values for η1 can be compared to the values of β/ν previously obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' One  can verify that the two ways of measuring the same exponent give totally consistent  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For each value a, the measurements have been done for a fixed value of disorder  r chosen in order to minimise corrections to scaling in the measurement of the fractal  dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The last column of the table contains the prediction of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='9) at the first  order, ηp  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Our results are also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The agreement is good between the  measured η2 and the predicted one ηp  2 for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5 ≤ a ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For smaller values of a, the  deviation is larger but also the second order correction ≃ (a−2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Quite interestingly, for  a ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5, η1 ≃ η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For the r = ∞ disorder point, we recall that the random FK clusters  (and the spin variables) are completely determined by the disorder configurations and  the thermal fluctuations are frozen, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' This implies that the distribution  function S(p12(x)) of the probability p12 that the spin s(0) and s(x) belong to the same  16  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='00  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75  Bp  r=1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='18  LRp  r=2  r=5  r=10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='16  r=100  r=inf  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='04  10  100  10  100  10  100a  η1(= 2β/ν)  η2  r  ηp  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='101  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='0625  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='167  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='172  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='125  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='209  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='243  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1875  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='233  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='298(2)  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='242(2)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='351(2)  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='3125  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='246  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='389  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='375  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='251  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='444  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='4375  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='250  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='491  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='50  Table 4: η1 and η2 measured with the disorder r for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25 ≤ a < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' ηp  2 = 1/2 − (2 − a)/4  is the predicted value of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='9) at the first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Errors bars on the measured values  η1 and η2 are shown in parenthesis and are smaller than one on the last digit otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  FK cluster, has the form S(y) = a0δy,0 + a1δy,1, with a0 + a1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' This in turn implies  η1 = η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' So the fact that, at finite r, where the spin degrees of freedom are not frozen,  we obtain η1 ∼ η2 is a very non-trivial test of the fact that the infinite disorder point  describes the physics of the system for these values of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The conclusions of our analysis  Figure 9: q = 2 Potts model with long-range disorder: η1, η2 and ηp  2 vs a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  are summarised in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='4  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='35  n2t  TU  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='15  n2p  n2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1  2 ni  ni  中  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='8  2  aq = 2  a  1/r  1  LR  LR  a∗(2)  a = 2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (8) right  a = 7  4, Bp  P  a = 3  2, LRp = Bp  a = 1, LRp  P  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (8) middle  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (8) left  a = 1  4, LRp  P  Figure 10: Fixed point stability, obtained via the measurements of the β/ν critical expo-  nent, for q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The dashed line represents the crossover with the purely SR physics at  a = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The arrows describe a flow between two fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The orange line represents  the P (unique) fixed point, the red one the Bp (unique) fixed point, while the blue and  magenta lines are lines of fixed points, respectively the LRp and the LR ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' a∗(2)  indicates the crossover between the LR and LRp points and we located it slightly below  a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='4  The q = 3 phase diagram  In the short-range regime a > 2, the disorder is now relevant and the critical properties  are described by the SR, which is located at finite r∗  SR, r∗  SR = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='08(12) [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The two  fixed points, SR for 1 < r < ∞ and Bp for r = ∞ are observed for all values of q up to  infinite [38, 44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The r = 1 point corresponds to the P point for q ∈ [1, 4] and to a  first order phase transition for q > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  When a < 2, the long-range disorder is relevant for a < 2/νSR ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' We show  below that our numerical simulations support the existence of a stable LR that is located  at finite 1 < r∗  SR < r∗  LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' We observed, moreover, that the LR point tends to merge with  the LRp one, located at a critical value a∗(3) slightly smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  Our numerical findings for q = 3 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Again, we obtain three  different behaviours:  18  Figure 11: q = 3 Potts model with long-range disorder: β/ν vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' L for a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25 on the  left, a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='00 on the middle and a = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='00 on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  In the left panel, we show β/ν for a = 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For all values of disorder r > 1, in the  large size limit, the effective magnetic exponent goes to the value βLRp/νLRp = 5/96  while it goes to the value of the pure q = 3-Potts model βP/νP = 2/15 for r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  We observe a similar behaviour for a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5 with βLRp/νLRp ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  In the middle panel, we show results for a = 1, with three different large size limits:  i) for r = 1, the magnetic exponent is the one of the pure q = 3-Potts model  βP/νP = 2/15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' ii) for r = ∞, the magnetic exponent goes towards a value ≃ 5/48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  Notice that this is the same as the one mention in the section above for q = 2 and  a = 1: we recall that at this point the exponents do not depend on q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' iii) for any  finite disorder, a unique limit is obtained with βLR/νLR ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='12 corresponding to a  LR fixed point with finite disorder r ≃ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  A similar behaviour is observed for a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25 with βLR/νLR ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='11 and  βLR/νLR ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='13 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  In the right panel, we show results for a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='50 and a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75, β/ν is  compatible, at large distances, with either with the P and the SR values, reported  in Table 1 and in Table 2, which are very close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Our measurements therefore do  not allow to discriminate between these two values, but according to the arguments  discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='4, we conjecture the LR point will merge with the SR point  at a ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='96, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  In conclusion, for a < a∗(3) ≃ 3/4, we find that the system flows to the LRp point,  19  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='00  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75  Bp  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='18  LRp  r=5  r=10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='16  r=100  r=inf  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='04  10  10  10while for a∗(3) < a < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='96, we find the LR at some intermediate value of disorder with  βBp/νBp < βLR/νLR < βSR/νSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  Our findings are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  q = 3  a  1/r  1  LR  LR  LR = SR  a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='96  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (11) right  a = 7  4, Bp  P  a = 3  2, LRp = Bp  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (11) middle  a = 1, LRp  P  a∗(3)  a = 1  4, LRp  P  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (11) left  Figure 12: Fixed point stability, obtained via the measurements of the β/ν critical expo-  nent, for q = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The dashed line represents the crossover with the purely SR physics  at a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The arrows describe a flow between two fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The orange line  represents the P (unique) fixed point, the red line the Bp (unique) fixed point and the  cyan one the SR (unique) fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The blue and magenta lines are lines of fixed  points, respectively the LRp and the LR ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' a∗(3) represents the crossover between  the LR and LRp points and we located it slightly below a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5  Thermal behaviour for the q = 3-Potts model with long-  range correlated disorder  In this section, to test Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='11, we present results for the measurement of the exponent ν  as a function of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' A convenient way to measure it is to consider the FK cluster wrapping  probability  Pw(r, J1, L) = Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' of having a wrapping FK cluster on a lattice of size L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='10)  20  This is a quantity which is constant at the critical point for a fixed value r = J2/J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' J1  is coupled to the thermal behaviour and we expect:  Pw(r, J1(1 + ϵ), L) = f(ϵL1/ν)  ≃ f(0) + f ′(0)ϵL1/ν + f ′′(0)  2  ϵ2L2/ν + f ′′′(0)  3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  ϵ3L3/ν + · · ·  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='11)  for a small ϵ, which measures the deviation from the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Pw(r, J1, L) should  also depend on r and one expect also corrections to scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' These dependencies can be  ignored if we consider the derivative  P′  w(ϵ) = Pw(r, J1(1 + ϵ), L) − Pw(r, J1(1 − ϵ), L)  2ϵ  ≃ f ′(0)L1/ν + f ′′′(0)  3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  ϵ2L3/ν + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='12)  We first attempted to compute 1/ν by considering P′  w at the first order, but the results  were not convergent for large L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' This is due to the fact that the errors on the measure-  ment grow like the inverse of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' While averaging over one million samples of disorder  configurations, we found that errors bars allow us to decrease ϵ only down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For  this value, the second correction can not be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' We then considered  ˜  P′  w(ϵ) = 4P′  w(ϵ) − P′  w(2ϵ)  3  = f ′(0)L1/ν(1 + O(ϵ4L4/ν))  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='13)  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (13), we show the results for the effective ν as a function of L obtained from  a two point fit of our data to the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='13) for a ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75 and also the value 2/a  as a dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For each value of a, we choose the value of r which seems closer to the  critical point with the least finite size corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For a ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25, we used the data for  r = 5, while for a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='00, we used the data for r = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Note that for a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75,  we still have large error bars even if for this value of a, we averaged over 10 millions  samples of disorder configurations (the same statistics is used for a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='00, with much  smaller error bars, for a > 1, we averaged over one million samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75, we  also observe in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (13) that there are strong finite size corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For smaller values  of a, we were not able to obtain convergent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Note that this corresponds to values  of a for which we expect that the critical point is at an infinite disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  In conclusion, we observe that the value of ν increase slowly as one decreases a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For  1 ≤ a < 2, ν is close to the prediction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75, it seems to be bigger  than expected with ν ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5, but in this case, we observe strong finite size corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  For smaller value of a, we can not measure ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='6  The effects of higher disorder cumulants  Let us come back to the value q = 1 where the disordered Potts model becomes a long-  range percolation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  Without passing via a Potts model, a long-range percolation model can be directly  defined by using the level sets of a fractional Gaussian free field (fGFF) with negative  Hurst exponent H = −a/2 [13–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  21  Figure 13: ν vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' L for the various values of a shown in the caption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  In our approach, we can use the fGFF to generate a long-range disorder distribution,  see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In doing so for the q = 1 Potts model, we obtain a one-parameter (r)  family of Gaussian long-range percolation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In our simulations, we have verified  that, analogously to what explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='2, all these models flow to the point  r = ∞, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' At this point, our long-range percolation models are expected to  have the same critical properties as the models studied in [13–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' More specifically, as  we used toric boundary conditions, our model coincides at r = ∞ to the one studied  in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  Our approach allows then to compare directly the LRp points obtained using different  distributions, in particular non-Gaussian ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The disorder distribution generated by  the n-replicated Ising model is an instance of such distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In the following, we  use the suffix I (G) for the disorder distribution generated by n-replicated Ising model  (fractional Gaussian free field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  For the two distributions, the Gaussian one based on the fGFF and the non-Gaussian  one based on the Ising model, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='11) gives the same percolation thermal exponent,  ν = 2/a = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  The situation however is different for the exponent β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Let us discuss more in detail  the case a = 1/4, that we have analyzed more carefully in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In Table 5  are reported the r = ∞ results of our Monte-Carlo measures of  �  βLRp/νLRp�  I, obtained  by using the n-replicated Ising distribution, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1, and of  �  βLRp/νLRp�  G  obtained with the fGFF distribution, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In the same table we have also  put the measures obtained in [15] that can be then directly compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  22  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='00  F  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25  3  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='50  上  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5  10Figure 14: β/ν vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' L at q = 1 with a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25 and disorder generated by using fGFF, see  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  a  �  βLRp/νLRp�  I  �  βLRp/νLRp�  G  �  βLRp/νLRp�  G [15]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='0522(4)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='0721(9)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='0640(4)  Table 5: Comparison of βLRp/νLRp for Gaussian and non-Gaussian disorders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  The exponents  �  βLRp/νLRp�  G and  �  βLRp/νLRp�  I obtained respectively with the fGFF  and with the Ising model seems different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Note that the numerical result  �  βLRp/νLRp�  I  agrees perfectly with the Ising model: as argued earlier, for the non-Gaussian disorders  with n = 1-Ising copies, the FK clusters have the same fractal dimensions as the Ising  spin clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  The above results can be explained by noticing that, differently from the short-range  disorder, in a long-range disorder, the terms generated by the higher cumulants of the  distribution can be relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' We can conclude that the higher cumulants do not change  the existence and the stability of an LRp point but can modify certain critical exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  As far the exponent βLRp/νLRp is concerned, one can doubt that the differences seen  in the simulations are just due to the numerical precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  This is particularly true  when a is increasing, see Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' However, we can mention the measure of another  universal property which is clearly different between the (LRp)G and the (LRp)I and  therefore supports the conclusion above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Indeed in [17] and in [46] the torus two-point  connectivity pG  12(r) and pI  12(r) of, respectively, the fGFF level sets and the Ising clusters  23  a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='25  LRp  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='14  Bp  r=1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='13  r=2  r=5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='12  r=10 μ  r=100  r=inf  队  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='08  10  100  1000  Lwere considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' It was argued that:  pG  12(r) = aG  0 r  −2  �βLRp  νLRp  �  G  �  1 + cG � r  L  �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='875  + o  �� r  L  �4��  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='14)  pI  12(r) = aI  0 r  −2  �βLRp  νLRp  �  I ×  �  1 + cI � r  L  �  + o  � r  L  ��  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='15)  where aG  0 and aI  0 are non-universal constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The cG and cI are universal quantities that  depend on the ratio Lh/Lv between the horizontal and the vertical torus size and on the  structure constants of the eventual CFT behind, see [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The cI is known and it has  been computed in [46] while cG, investigated in [17], is not known exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  The exponent x of the sub-leading term (r/L)x is computed on the assumption that  exists a local CFT describing the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' This gives [47]:  x = 2 − 1  ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='16)  The exponents in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='15) are then obtained by using ν = 1/8 (x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='875) for the  Gaussian distribution and ν = 1 (x = 1) for the Ising one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For the Ising case, where the  existence of a CFT behind is well established, the form for pI  12 has to be considered exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  For the fGFF, the CFT predictions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='15), with ν = 2/a, were tested numerically only for  values of a > 1 [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For pG  12 we can extend the results of [17] to a = 1/4, by assuming an  analytical a−dependence on the critical exponents for all values of a < 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' So one can  see that the sub-leading term exponents depend strongly on the disorder distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  4  Conclusions  We considered the long-range disordered two-dimensional q-Potts model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The disorder is  implemented by a bimodal distribution whose long-range nature is induced by auxiliary  {σi} spin degrees of freedom to which the Potts model is coupled, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' We have  considered different {σi} distributions, the Ising distribution described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1  and the fGFF distribution described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  These distributions have the  same first cumulant, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='6), and a second cumulant with the same long distance  power-law decay, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  By Monte-Carlo techniques, we studied the fractal dimension of the q = 1, 2, 3-Potts  FK clusters for different values of the power-law exponent a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The measures were taken  at the self-dual point, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='4), for different values of the strength disorder r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The  outcome of these analysis are resumed in the phase diagram Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (1), which represents  the main finding of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  We considered first the infinite disorder (r = ∞) fixed point LRp, at which the thermal  fluctuations of the Potts degrees of freedom are frozen, and the disorder averages coincide  with the ones of a long-range bond percolation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The LRp critical exponents do not  depend on q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The measured value of βLRP/νLRp are reported in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' We observed  that, by using the Ising distribution, the Monte-Carlo measures on the LRp point are  24  more precise than the corresponding ones for the fGFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' In particular we could probe  with good precisions the physics of the LRp point for small values of a, the regime where  the fGFF methods are more difficult to implement, see [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  For q = 1, we observed that, for a < 3/2, the LRp is attractive while for a ≥ 3/2  the system is described by the Bernoulli critical point Bp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For q = 2, 3 we establish the  existence of the LR point, which is a long range critical point at finite disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' We  observed that the LR point and the LRp point exchange stability at a certain value a∗(q)  which depend on q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  The above results are supported by the study of other observables than the FK  fractal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For q = 2 we measured the multi-fractal behaviour of the spin-spin  correlation, see Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The results are consistent with the theoretical findings of [20] in  their region of validity, especially for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5 < a < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For lower values of a, where theoretical  predicitons are missing, our results rule out the occurence of a softening of the transition  and support instead the fact that the system is driven to the LRp point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For q = 3  we mesure the thermal νLR exponent by measuring the wrapping probability of the FK  clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (13) and are consistent, for a wide range of values  of a, with the theoretical prediction of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  We verified that the above mentioned results, in particular the form of the phase  diagram, are valid for different disorder distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' On the other hand, we observed  some universal effects of the higher cumulants at the LRp point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' By focusing on the  a = 1/4 value, we showed that the exponent βLRp is different between a Gaussian and a  non-Gaussian distribution, see Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Finally we pointed out that clear effects of the  higher cumulants are expected for the universal finite size effects of the FK connectivities,  see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  5  Acknowledgments  We are grateful to Nina Javerzat with whom this project started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' We thank Pierre Le  Doussal for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  A  Disorder distributions  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1  n-replicated Ising model  In most of the simulations presented here we chose a particular distributions for dis-  order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  Let us consider n independent critical Ising models with spins σ(a) ∈ {0, 1},  a = {1, · · · , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The variables σi in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5) are given by:  σi =  �  a=1  σ(a)  i ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1)  and their statistical properties are determined by the correlation functions of the the n  independent Ising models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' For the first two moments, we have:  E [σi] = 0,  E [σi σj] ∼ |i − j|−n  4 for |i − j| ≫ 1,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='2)  25  where we used the fact that the critical Ising spin-spin correlation function  �  σ(a)  i σ(a)  i  �  ∼  |i − j|−1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Comparing with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='8), we have:  a = n  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='3)  The convenience of this choice is that the n-Ising models are very simple, and fast, to be  simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The price to pay is that we cannot vary a continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' The above distribution  is not Gaussian because the critical Ising point is not a Gaussian point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='2  Fractional Gaussian free fields  The other one is related to the percolation point of the level sets of a fractional Gaussian  random field with negative Hurst exponent H = −a/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  Let us, now, consider an L × L square lattice, whose vertices i have coordinates  i = (ih, iv) and a set of random variables φi that defines a discrete fractional Gaussian  free field:  φi =  �  �  �  �  �  �  �  L  �  (l,m)̸=(0,0)  cl,m  Sl,m  exp  �2π  L (l ih + m iv)  �  , if (l, m) ̸= (0, 0)  c0,0, if (l, m) = (0, 0)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='4)  where Sl,m =  ����2 cos  �2πl  L  �  + 2 cos  �2πm  L  �  − 4  ����  1 − a/2  2  and the cl,m are independent  random Gaussian variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='m ∈ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  Using the properties of the Fourier  transform, the covariance E[φiφj] has the following asymptotic limit:  E[φi φj] ∼ |i − j|−a for |i − j| ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='5)  The σi variables are defined by:  σi =  �  0,  if φi > 0  1  if φi < 0,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='6)  The fact that E[σi] = 0 implies the level 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' One can notice that the clusters built with the  {σi} variables are not at the percolation treshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Indeed, this is spin-percolation physics  and it is known [17] that it has a critical level greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' These {σi} variables are  only needed to built the bond distribution as explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' One can show,  instead, that the FK-cluster built on these bonds, are at the percolation transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
+page_content=' Thus,  one has:  E[σi σj] ∼ |i − j|−a for |i − j| ≫ 1,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFST4oBgHgl3EQfIDgO/content/2301.13727v1.pdf'}
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+ 
+ 
+ 
+ 
+Self-assembled neuromorphic networks at self-organized criticality in Ag-hBN platform 
+ 
+Authors 
+Ankit Rao1, Sooraj Sanjay1, Majid Ahmadi2, Anirudh Venugopalrao1, Navakanta Bhat,1 Bart Kooi2,3, Srinivasan 
+Raghavan1, Pavan Nukala1* 
+ 
+Affiliations  
+1Centre for Nano Science and Engineering, Indian Institute of Science, Bengaluru 560012, 2Zernike Institute 
+for Advanced Materials, University of Groningen, Groningen, the Netherlands, 3CogniGron center, University 
+of Groningen, Groningen, 9747 AG, The Netherlands. Corresponding Author. * E-mail: pnukala@iisc.ac.in 
+Keywords: hexagonal Boron Nitride, Avalanche Dynamics, self-organized criticality, neuromorphic networks 
+ 
+Abstract 
+Networks and systems which exhibit brain-like behavior can analyze information from intrinsically noisy and 
+unstructured data with very low power consumption. Such characteristics arise due to the critical nature and 
+complex interconnectivity of the brain and its neuronal network. We demonstrate that a system comprising of 
+multilayer hexagonal Boron Nitride (hBN) films contacted with Silver (Ag), that can uniquely host two different 
+self-assembled networks, which are self-organized at criticality (SOC). This system shows bipolar resistive 
+switching between high resistance (HRS) and low resistance states (LRS). In the HRS, Ag clusters (nodes) 
+intercalate in the van der Waals gaps of hBN forming a network of tunnel junctions, whereas the LRS contains 
+a network of Ag filaments. The temporal avalanche dynamics in both these states exhibit power-law scaling, 
+long-range temporal correlation, and SOC. These networks can be tuned from one to another with voltage as a 
+control parameter. For the first time, different neuron-like networks are realized in a single CMOS compatible, 
+2D materials platform. 
+ 
+ 
+1. Introduction 
+The efficient computational ability of the brain in tasks such as classification and pattern recognition has 
+inspired the identification and implementation of novel computing architectures for energy efficient and parallel 
+computing needs.[1–3] Neuromorphic hardware architectures such as crossbar memristive arrays are assembled 
+bottom up, where individual elements of the array emulate the functionality of modular units of the nervous 
+system i.e., a synapse or a neuron.[4–7] Using these highly organized architectures, it is difficult to capture the 
+complexity of a biological brain, which is a self-assembled network of ~1011 neurons and ~1015 synapses (in 
+humans).[8] These interconnected neuronal networks form a dynamical system, and in a state-of-rest, exhibit 
+self-organized criticality (SOC).[9,10] In addition to neuronal cortices, SOC has also been observed in many other 
+physical systems and phenomena such as earthquakes[11], magnetic interactions[12], ferroelastic and ferroelectric 
+
+materials (Barkhausan’s noise)[13,14], and charge density waves[15]. A system self-organized at critical point, 
+when perturbed triggers successive events known as avalanches, or in other words emits ‘crackling noise’.[16] 
+The spatiotemporal dynamics of such avalanches follow power-law scaling with long range temporal 
+correlations and scale-invariant behavior. [17–20] It has been proposed that operating at SOC allows the network 
+to efficiently process and transport information with large scale connectivity.[21,22] A pertinent question then is 
+whether one can engineer CMOS compatible dynamical system of self-assembled materials networks at SOC 
+and be able to compute with these top-down neuromorphic architectures (TDNA). 
+In this context, self-assembled nanoscale networks comprising of nanowire networks[23] or nanoparticle 
+complexes[24] are being recently explored.  However, TDNA on CMOS compatible and large area scalable 
+platforms are desirable for any future neuromorphic computing applications.  Here, we demonstrate two 
+different types of networks poised at SOC, both hosted in a robust 2D material platform, i.e., hBN deposited on 
+Cu substrates with Ag as the top electrode. The choice of high quality hBN as an insulator in conjunction with 
+fast diffusing Ag as the top electrode, allows us to uniquely stabilize, a) percolative tunnel junction network of 
+Ag intercalates and b) Ag filamentary network. The 2D material platform also allows for heterogenous 
+incorporation across different device architectures.[25] 
+ 
+ 
+ 
+Figure 1. Avalanche Dynamics in Ag-hBN system. A) Schematic of Ag-hBN-Cu (MIM) device. Voltage is 
+applied to Ag, and Cu substrate is ground. B) SEM image of the top view of the devices. C) Representative 
+current data (blue) when sampled at a constant voltage (red) as a function of time in HRS state, showing 
+stochastic spiking events and the differential conductance data representing the event size ∆G (red) expressed 
+in quanta of mean conductance, G0. The current levels observed are in nA. D) Representative current and 
+differential conductance data in the LRS state. The current levels observed are in mA. 
+ 
+ 
+ 
+
+Ag
+Current (nA)
+50
+0.10
+(V)
+Itage
+25
+0.05
+Vol
+hBN
+0.5
+10.00
+(Go)
+0.0
+AG
+Cu
+HRS
+0.5
+0
+25
+50
+75
+D
+Time (sec)
+1.0
+Current (mA)
+Voltage (V)
+3.00
+0.5 
+Ag
+2.25
+0.0
+AG (G)
+0.4
+0.0
+0.4
+hBN
+0
+25
+50
+75
+50um
+Time (sec) 
+2. Results 
+2.1. Dual state switching of Ag networks 
+ 
+The growth of large area (upto 6 sq inches) multilayer hBN (~10nm thick) on Cu was carried out by chemical 
+vapor deposition process as described in the methods section which has been previously reported.[26] Metal-
+insulator-metal devices arrays (25x25) were fabricated with Ag (50 um) as the top electrode, hBN as the 
+insulator and Cu substrate as the bottom electrode (Figure 1A, 1B and S1A). The measurements were performed 
+on 3 different samples with 10 devices measured on each sample, all of which showed similar behavior. Cu was 
+ground, and voltage was applied to Ag electrode. I-V sweeps revealed bipolar and bistable resistive switching 
+(Figure S1B), after an initial forming step (Figure S1C). In all these devices HRS to LRS transition occurs at 
+Vset= 0.9V (±0.15V), and LRS to HRS transition at Vreset= -0.6V (±0.1V). To characterize the spatio-temporal 
+avalanches in both HRS and LRS, we performed voltage sampling measurements i.e., measured instantaneous 
+current, by applying a constant voltage bias. Representative device stabilized in HRS and LRS was tested at 
+small perturbing voltages of 0.1 V and 0.6 V respectively. Current response in HRS (Figure 1C) shows irregular 
+spikes of various amplitudes, ranging from 5 nA to 50 nA. Response in LRS also shows irregular spikes of a 
+much greater amplitude in the range of 2.2 mA to 3 mA (Figure 1D).  The statistical behavior and correlations 
+of these spikes can be understood through order parameters that characterize them, such as change in 
+conductance, ∆G (Go) (Figure 1C and 1D), inter event/spike interval (IEI) and autocorrelation function of IEI 
+(ACF).  
+ 
+Figure 2. Diffusion based State Transition in the Ag-hBN system. Log scale plot of the behavior observed. 
+Different colors indicate different stages of device, HRS (blue), Transition from HRS to LRS (Yellow), LRS 
+(Purple). JMAK model fit for the transition is shown in red solid line. 
+
+A
+0.8
+Log
+10-3
+10-4
+0.6
+Current (A)
+10-5
+Voltage (V)
+10-6.
+0.4
+10-7,
+10-8.
+HRS
+0.2
+10-9
+10-10
+TRANSITION
+LRS
+10-11
+0.0
+0
+50
+100
+150
+200
+Time (sec) 
+At voltages close to, yet smaller than the set voltage, devices in HRS undergo a phase transition to LRS in 
+measurable time scales. For e.g., a representative device prepared in HRS and held at 0.6 V (Vset= 0.9V) exhibits 
+nano ampere current fluctuations upto 70 sec in HRS, followed by a transition to LRS and milliampere current 
+fluctuations in LRS (Figure 2, linear scale Figure S1D). The average current fits well to John-Avrami-Mehl-
+Kolmogorov kinetics in the transition region[27] as per Eq (1),  
+𝑖𝑇 = 𝑎𝑡𝑛−1𝑒(−𝑏𝑡𝑛)  
+ 
+(1) 
+where 𝑎 is a constant, 𝑏 is a composite parameter which includes the kinetic rates of nucleation and growth, and 
+the parameter 𝑛 is the Avrami exponent, where 𝑛 = 𝑑 + 1, d being the dimension of the system. From our fits, 
+we estimate n~2 (Figure S2A), likely suggesting a 1D growth process consistent with the formation of Ag 
+filaments through long-range diffusion.[28] Subsequent large amplitude current fluctuations in LRS suggest 
+sporadic events of breakage and reformation of the conducting filaments.  
+The spiking rates observed in HRS and LRS are dependent on the sampling rates. For the sampling rates of 10 
+ms, the measured spiking rates for the representative device are in the range of 15-20 spikes per second in HRS 
+and 25-30 spikes per second in LRS. Furthermore, spiking rates are regulated in both HRS and LRS with respect 
+to the voltage applied (0.05V to 0.5V; Figure S2C). Such rate coding allows for encoding the device with voltage 
+dependent spike rate information for stochastic device applications.[29] Finally, for the devices prepared in LRS, 
+upon the application of negative voltage close to the RESET voltage (-0.6V), there is a discrete and abrupt 
+transition to HRS state as seen in Figure S2B, followed by spiking behavior in the HRS. Notice that under 
+negative bias there is no spiking behavior in LRS. 
+2.2 Long Range Temporal Correlation and Avalanche dynamics 
+To better envisage the avalanche dynamics, the current spiking pattern is binarized by classifying those spikes 
+which exceeded a threshold limit as events and rest as not. The top panel event train in Figure 3A (HRS) and 
+3E (LRS) contains data acquired for 100 s, while the temporal scales are magnified 5 and 20 times for the 
+subsequent panels. The event train patterns in all the panels look qualitatively self-similar and are independent 
+of the sampling rate. The switching activity in Figure 1C and 1D (binarized subsections in Figure 3A and 3E 
+respectively) exhibits event bursts with varying range of ∆G values. These bursts indicate a consecutive 
+correlated activity distributed across the network, rather than isolated local events.  To quantify the correlation 
+and self-similarity behavior, we investigate the distribution of event sizes (∆G), inter-event interval (IEI) and 
+the autocorrelation function of the IEIs (ACF).  
+
+ 
+Figure 3. Long Range Temporal Correlation (LRTC) and Power Law Scaling. The distribution plots shown 
+in (A to D) are for the device in HRS state and (E to H) in LRS state. (A and E) The binarized patterns of the 
+switching activity in the Ag-hBN-Cu device, plotted for varying time scales. These patterns exhibit self-similar 
+temporal behavior. (B and F) The probability density of change in the conductance of the device (∆G), plotted 
+for the avalanche patterns in HRS (B) and LRS (F) states. (C and G) The power law distributions for the Inter 
+event intervals (IEIs) are demonstrated as the probability distribution functions (PDF). (D and H) 
+Autocorrelation function of IEIs in both HRS and LRS, both of which show a slow power law decay.  
+ 
+Figure 3B and 3F shows the probability distribution functions (PDF) of event sizes in HRS and LRS. The P(∆G) 
+in both states follows a power law distribution spanning 3 orders of magnitude with exponents of ~2.5±0.1 
+(HRS) and ~2.8±0.1 (LRS). The PDF of the inter-event intervals (IEIs) (Figure 3C and 3G) also follow a power-
+law distribution with exponent, γ in the function, (P(t) ~ t-γ) ~1.54±0.05 in HRS and 1.72±0.05 in LRS. This is 
+a quantitative proof of self-similar, scale free nature of the events (qualitatively also observed in Figure 1A and 
+1E). Self-similarity is a signature of temporal correlations which are observed in electrophysiological signals 
+from brain cortex and are an outcome of fractal correlations.[30] The γ in HRS is similar to the values obtained 
+in nanoparticles systems exhibiting percolating tunnel junction networks[19] and the γ in LRS is similar to the 
+values in nanowire networks[23]. These values are also in similar range as those observed in neuronal 
+cortices.[31,32] The ACFs in both HRS and LRS (Figure 3D and 3H) exhibit a slow power law decay over several 
+decades. Another order parameter of interest to understand the correlation between successive events is inter-
+spike interval (ISIn), defined as the time between rising edges of consecutive spikes. Figure S4A (in HRS) and 
+Figure S4C (in LRS) show a tight scatter of ISIn and ISIn+1 about the mean ISI. The observed ACF and the ISIn 
+behavior is a result of long-range temporal correlations (LRTC) among events. Finally, as is the case with any 
+correlated noise, the event power spectrum follows a power law with frequency as 1/fβ. β is 0.91 in HRS (Figure 
+S4B) and 0.82 in LRS (Figure S4D) in our representative device.   
+
+101
+105
+1o sec
+t-1.54
+10-2
+104
+4G-2.59
+(-0.19
+PDF
+103
+I..
+10-3
+2 sec
+102
+10-1
+10-6
+10-5
+10-4
+10-3
+10-1
+100
+101
+100
+101
+AG (Q-1)
+IEI (s)
+Time (sec)
+Time
+0.5 sec
+101
+101
+10-1,
+10 sec
+100
+4G-2.87
+t-1.72
+PDF
+t-0.09
+10-1
+10-2
+10-1
+2 sec
+10-2
+10-2
+10-1
+10°
+10-2
+10-1
+10°
+100
+101
+AG (Q-1)
+IEI (s)
+Time (sec)
+Time 
+Figure 4. Self-organized criticality. The distribution plots in A to D are for the HRS whereas plots E to H are 
+for LRS which exhibit self-organized behavior across numerous temporal scales. (A and E) The probability 
+distribution function for the size of an avalanche P(S) in both cases follows a power law with similar exponent 
+values of τ ~1.95. (B and F) Distribution of duration of avalanche P(T) also follows power laws in both states 
+with exponent values of α~2.6 (HRS) and α~2.45 (LRS) with linear binning. (C and G) The average avalanche 
+size per unit time bin <S>(T) also exhibits power law scaling with exponent 1/συz ~ 1.5 (HRS) and 1/συz ~ 1.4 
+(LRS). (D and H) The mean avalanche shapes for avalanches of various Ts (different colors) collapse onto a 
+universal parabolic scaling function (black solid line). The exponent values obtained fulfill the crackling 
+relationship (Eq 5) which establishes the existence of critical avalanche dynamics. In (D) avalanche duration, 
+T=4 (Purple), 4 (Teal), 5 (Orange), 12 (Brown), 15 (Red), 24 (Blue)], and in (H) T= 4 (Red). 5 (Green), 6 (Teal), 
+10 (Green), 32 (Orange)] 
+ 
+2.3 Self-organized criticality observation 
+An avalanche is defined as the sequence of successive event occurrences between two empty time bins. The 
+size of an avalanche (S) in such an occurrence is the total number of events in the avalanche, time of avalanche 
+(T) is the number of time bins and the (<S>(T)) is defined as the average size of the avalanche of a duration T. 
+In a critical system, the probability distribution functions of S, T and <S>(T) follow power-law scaling as 
+described in Eq (2), Eq (3) and Eq (4) respectively.[33] The distributions are defined as follows: 
+𝑃(𝑆) ~ 𝑆−𝜏  
+ 
+(2) 
+𝑃(𝑇) ~ 𝑇−𝛼   
+(3) 
+< 𝑆 > (𝑇) ~ 𝑇1/𝜎𝜐𝑧  (4) 
+A signature of avalanche activity which exhibits SOC is when the power law exponents obtained agree with the 
+crackling relationship as shown in Eq (5).[34]  
+1
+𝜎𝜐𝑧 = 
+𝛼−1
+𝜏−1   
+(5) 
+
+10°
+103
+10°
+10-1
+10-1
+@ 102
+E
+E 103
+<S>(
+T1.59
+S-1.98
+F(t/T)
+10-3.
+PDF 
+T-2.61
+2.
+10-2
+10-4
+101
+10-3
+10-5
+:
+10-6
+100
+10-4
+100
+101
+102
+10°
+101
+102
+103
+0.0
+0.5
+1.0
+s
+100
+101
+102
+T
+t/T
+-
+T
+10°
+103
+4
+100
+10-1
+@ 102
+10-1
+S-1.93
+E
+T-2.43
+T1.38
+10-3
+<S>(
+10-4
+101
+10-3
+10-5
+10-6
+104
+10°
+10°
+101
+102
+103
+100
+101
+102
+100
+101
+102
+0.0
+0.5
+1.0
+s
+T
+T
+t/TEq (5) distinguishes a system poised at SOC from other random walk processes that also yield power laws and 
+exponents.  In both HRS and LRS as shown in Figure 4, the switching activity exhibits power law dependence 
+for P(S), P(T) and <S>(T) with power law exponents τ, α and 1/συz respectively. In HRS, τ ≈ 1.96±0.05 (Figure 
+4A) and α≈2.61±0.05 (Figure 4B), which gives 1/συz ≈ 1.67±0.1 from Eq (5). This is in close agreement with 
+1/συz ≈ 1.59±0.05 independently estimated from <S>(T) (Figure 4C). Similar agreement is observed in LRS 
+case, with τ ≈ 1.93±0.05 (Figure 4F) and α≈2.43±0.05 (Figure 4G), which gives 1/συz ≈ 1.52±0.1 from Eq (5), 
+which is also in agreement with 1/συz ≈ 1.38±0.05 obtained from <S>(T) (Figure 4H). Table S1 summarizes 
+the exponents observed in various order parameters in both HRS and LRS and compares it with the 
+corresponding exponents in neuronal cortices and other systems. This validation of the crackling noise 
+relationship proves that our system is poised at SOC in both HRS and LRS.  
+Another crucial feature of a system at criticality is that the average shape of the avalanches exhibits universal 
+scaling or shape collapse.[34] The universal scaled shape is obtained by normalizing the duration of avalanche 
+with respect to the largest avalanche duration, T, and size of the avalanche with respect to the average size of 
+the avalanche of duration as described in Eq (6) and (7). [35] 
+𝑠(𝑡, 𝑇) ∝ 𝑇( 1
+𝜎𝜈𝑧−1) 𝐹 (
+𝑡
+𝑇)      (6) 
+< 𝑆 > (𝑇) = ∫ 𝑠(𝑡, 𝑇)𝑑𝑡
+𝑇
+0
+    (7) 
+Here, 𝑠(𝑡, 𝑇) is the mean number of spikes (s) at a time t in an avalanche with duration T and 𝐹 (
+𝑡
+𝑇) is the shape 
+of the avalanche when plotted for a scaled duration (t/T). We perform this shape collapse analysis for all the 
+avalanches of different times T and observe universal parabolic scaling that has also been reported in neuronal 
+avalanches.[35,36] The shape collapse analysis yields another independent estimate of the 1/συz which is 1.5±0.05 
+for HRS (Figure 4D) and 1.4±0.05 for LRS (Figure 4I). These values are similar to those obtained from Eq (5) 
+and provides another independent verification of 1/συz. The exponents 1/συz, α and τ do not show any 
+significant variation with voltage in both HRS and LRS (Figure S5). The close agreement of 1/συz obtained 
+independently from the crackling relationship estimates, shape collapse analysis and Eq (5) provides robust 
+evidence of SOC in HRS and LRS in the Ag-hBN system. 
+The network structure of HRS and LRS can be understood from the cross-sectional TEM analysis shown in 
+Figure 5. Cross-sectional lamellae of representative devices prepared in pristine state, HRS (after 10 cycles), 
+and LRS (after 20 cycles) were made using focused ion beam technique. Figure 5A shows HRTEM image of 
+pristine device with high quality crystalline two-dimensional layers of hBN grown on Cu substrate and 
+contacted with Ag. In HRS, the HAADF STEM (Figure 5D) and EDS analysis (Figure 5F) of the selected region 
+shows dispersion of Ag intercalates inside the hBN matrix (also see Figure S6A-E). The HRTEM data and 
+analysis further shows lattice expansion from 0.35 nm in pristine state (Figure 5B) to 0.39 nm in HRS (Figure 
+5C), which is a consequence of intercalation of Ag inside the h-BN matrix. Similar expansion resulting from 
+intercalation has been observed in other 2D systems.[37] HAADF STEM and EDS analysis of the device in the 
+LRS state shows formation of multiple Ag filaments across the h-BN film (filamentary networks, Figure 5F,  
+5G and Figure S6 I-J).  
+ 
+ 
+ 
+ 
+
+3. Discussion 
+A simple model proposed hereunder allows us to explain all the above results in our system. HRS and LRS 
+form physically two different types of networks, both poised at SOC. It is important to point out that such a 
+unique behavior occurs due to a combination of fast diffusing species in Ag (top electrode) and CVD grown 
+high quality robust hBN matrix. We confirmed the uniqueness of this structure by contacting hBN with standard 
+top contacts used in memristive devices such as Ni and Au, both of which did not exhibit any spiking activity 
+(Figure S7A and S7B).  
+Pristine hBN electroforms to an LRS beyond a threshold voltage through the formation of Ag filaments as 
+shown in Figure S1C. Ag is known to be highly diffusive in nature and the ionization potential for Ag to Ag+ 
+transition was reported to be 2-2.5 eV lower than other metals such as Au, Cu, and Ni.[38] This allows Ag to 
+ionize at relatively lower electrochemical potentials and migrate owing to its comparatively small ionic size 
+through the hBN matrix to the Cu electrode.[39] Here nanoclusters are nucleated which grow in the form of 
+filaments. Following electroforming, the device is cycled between LRS and HRS several times.  
+ 
+ 
+Figure 5. Transmission Electron Microscopy Analysis of Ag-hBN system. A) Cross-sectional high-
+resolution transmission electron microscopy (HRTEM) image of the as grown (pristine) hBN on Cu contacted 
+with Ag. B) HR-TEM of the hBN stack in pristine state exhibiting interplanar distance of 0.35 nm. (Inset) FFT 
+of the hBN region showing layered structure. C) HR-TEM of the region after multiple cycles and spiking in 
+HRS state, the resulting interplanar distance is 0.39 nm. (Inset) FFT of the hBN region exhibiting ring patterns 
+due to presence of Ag particles along with the layered structure pattern.  D) HAADF-STEM image in the HRS 
+state of the Ag-hBN-Cu stack. E) EDS map of the area reveals presence of Ag clusters further confirming the 
+ionic movement of Ag inside the hBN matrix. F) HAADF-STEM image in the LRS state shows formation of 
+the Ag filament due to large scale diffusion of Ag under continual cycling to the LRS state. G) EDS map of Ag 
+in the area shown in (F). 
+
+Pristine
+Ag
+Ag
+Ag
+0.35nm
+hBN
+hBN
+0.39 nm
+hBN
+Cu
+3nm
+3nm
+5nm
+Cu
+Pristine
+Cu
+HRS
+HAADF
+Ag
+Ag
+HAADF
+Ag
+Ag
+hBN
+hBN
+LRS
+Cu
+HRS
+HRS
+5nm
+cu
+LRS
+5nm
+5nm
+5nmHRS, as evidenced from TEM data, contains Ag nanoclusters which intercalate into the van der Waals gaps of 
+the hBN matrix (schematic in Figure 6A). The high crystalline quality of hBN planes (low defect density), 
+means that while intralayer diffusion of Ag can be fast, the interlayer diffusion, mediated by defects such as 
+point defects, stacking faults and grain boundaries is a slower process. This allows for intralayer clustering, and 
+formation of a stable network of Ag intercalates (nodes). Intercalation of species such as Ag is well reported in 
+other 2D systems too.[37,40,41] 
+The conduction in the HRS state (2 orders higher than the pristine state) is predominantly mediated by electron 
+tunneling across these nodes, with hBN acting as tunnel barrier between interlayer nodes, and vacuum as the 
+barrier between intralayer nodes. In such a self-assembled tunnel network of Ag nodes, application of a small 
+perturbing voltage creates fluctuations in the clusters/nodes, dynamically changing the spacing between them, 
+thus giving rise to correlated nanoampere fluctuations in tunnel current (Figure 1C). These events are polarity 
+insensitive and happen both in positive and negative biasing voltages (Figure 1C, Figure S2B). This network 
+self-organizes to criticality, characterized by scale free avalanches analogous to sand pile at a critical slope, and 
+corresponding avalanches created upon a small perturbation to maintain the critical slope. 
+ 
+Fig. 6. Avalanche Mechanism A) The Ag acts as the top electrode, which is biased, and the Cu substrate acts 
+as the bottom electrode which is the common terminal. The device schematic when cycled and rested in HRS 
+state showing the Ag (grey) gets dispersed in the hBN matrix. Schematic of the tunneling current between the 
+Ag clusters which are formed between the hBN layers. B) The device schematic when cycled and rested in LRS 
+state showing the formation of filamentary networks. 3D representation of the filamentary networks formed 
+within the hBN matrix. 
+
+Ag
+HRS
+hBN
+cu
+LRSThe interlayer migration of Ag, which is a slower process at low voltages, dictates the process of filament 
+formation, and thus the transition between HRS and LRS. The kinetics of this transition can be non-linearly 
+enhanced by increasing the voltage, a phenomenon often referred to as voltage-time dilemma in resistive 
+memory.[42] Thus, by operating our representative devices at voltages of 0.6 V (Vset=0.9 V), an HRS to LRS 
+transition is recorded over the background of current fluctuations as shown in Figure 2A. The kinetics fit JMAK 
+model, with the dimension of the phase transition ~1, strongly supporting filament formation. LRS is thus a 
+network of multiple Ag filaments which are further directly imaged using HAADF-STEM (Figure 5G, Figure 
+S6 I, J). Application of a negative reset voltage beyond a threshold, results in electrochemical dissolution of the 
+filaments, giving rise to the Ag intercalated network structure of the HRS, and bipolar memristive switching. 
+The milliampere current fluctuations in LRS occur when a small positive voltage is applied on the system. This 
+can be explained as a competing series of filament breaking and mending events, with Joule heating and 
+corresponding random surface ion diffusion causing the filaments to break, and ionic migration and redox 
+reactions at the electrodes (electrochemistry) supporting their mending. These events also show long range 
+temporal correlations, and scale free nature. However, when a negative bias is applied in such filamentary 
+networks (LRS), both electrochemistry and Joule heating cause the filaments to break, and thus our devices did 
+not show any spiking behavior (Figure S2B). We also observe diffusion based self-stabilizing behavior under 
+application of long-range pulsed stimulus (Figure S3).  
+Both the tunnel networks (HRS) and the filamentary networks (LRS) are poised at self-organized criticality, 
+showing a universality behavior, parametrized by critical exponents of various order parameters. In the HRS 
+the values of τ, α, 1/συz correspond to the values discovered in nanoparticle percolative tunnel networks as seen 
+in Table S1, suggesting not only that these networks belong to the same universality class, but also that the 
+underlying dictating mechanism is the formation and dissociation of percolative paths of tunneling currents. In 
+LRS, however τ, α and 1/συz show marginally different values, probably indicating that this could belong to a 
+different universality class. The exponents obtained are similar in values to those in the nanowire networks 
+which are filamentary in nature.  The critical exponents observed in both these classes are different from the 
+well-known universality classes such as mean-field, propagating front, short-range models in various 
+dimensions, established, for e.g., in studies on magnetic Barkhausan’s noise or charge density waves.[13–15] To 
+comment any further requires rigorous modeling of these types of networks. Our work opens a fundamental 
+problem in physics of identifying various universality classes of different types of neuromorphic networks. It is 
+our belief that correlating the universality classes with their computational performance will be a crucial step in 
+the development of top-down neuromorphic architectures.  
+ 
+4. Conclusion 
+In this work, we demonstrate a unique CMOS friendly platform which hosts neuronal-like networks of two 
+different universality classes (percolative tunnel junction networks and filamentary networks), both poised at 
+SOC. These networks can be transformed between each other through electrical biasing, aided by field 
+dependent migration kinetics of Ag in the hBN matrix. Such observations are also first of their kind in any 2D 
+material platform. The distinct advantage of our large area CVD grown hBN platform is that it can be transferred 
+to any substrate, including Si, allowing for the implementation of integrated systems on large-area devices. 
+Furthermore, such 2D platforms allows for engineering new structures via heterogenous layer stacking for future 
+applications in neuromorphic computing. As an outlook, we envision arrays of these devices (multielectrode 
+arrays) which can emulate spatiotemporal dynamics similar to those observed in neuronal cortices and can 
+mimic brain-like operations. To compute using these platforms, approaches such as reservoir computing will 
+be a good starting point.[43,44] 
+
+5. Materials and Methods 
+ 
+5.1 Growth and Device Fabrication 
+The growth process of multilayer hBN on Cu was carried out using BH3NH3 (Sigma Aldrich, 97%, 682098) 
+as the precursor in a custom built hot-wall CVD reactor. We employed a temperature and pressure-controlled 
+vaporizer setup to control the flux of the precursor entering the reactor to grow multilayer hBN. A 1 cm x 6 cm 
+Cu foil (Alfa Aesar, 99.8%, 46986) of 25 µm thickness was normally used. The foil was washed in dilute 
+hydrochloric acid for 5 min to eliminate the oxide layer on the surface along with a deionized (DI) water rinse. 
+Successive cleaning steps were conducted in acetone and isopropyl alcohol for 5 minutes in each solvent for 
+eliminating the organic contaminations on the surface. The foil was then immediately inserted into the reactor 
+compartment at a permanent location. The compartment was then pumped to the base pressure flushed with 200 
+sccm of H2 for 10 min and then ramped up to the growth temperature which was usually between 1025°C and 
+1050°C. The details of the growth process can be found in the previous work. Ag contacts (50 um) were 
+thermally evaporated on the as-grown hBN-Cu with the aid of stencil masks. The electrical measurements for 
+the metal-insulator-metal stack of Ag-hBN-Cu were carried out as seen in Figure 1B. 
+ 
+5.2 Measurements  
+DC probe station (Agilent Device Analyzer B1500A) was used for electrical characterization. Voltage was 
+applied on the top electrode (Ag) and the bottom contact of Cu was held at ground potential. The primary 
+method of measurements adopted is application of constant DC voltages within the I-V-t (current-voltage-time) 
+sampling module. This allows for study of current reconfigurations of the local switches in the device leading 
+to spikes and thus, the analysis of avalanche dynamics. These measurements are carried out at multiple sampling 
+rates, such 100 us, 1 ms and primarily at 10 ms sampling interval. Larger sampling interval allows for large data 
+set and extended time sampling which helps to circumvent substantial breaks in the power law distributions. 
+The data obtained for all the sampling intervals demonstrate similar data in quality and quantity, with similar 
+power law exponents upon analysis. Thus, the results obtained are indifferent to the sampling rate. In this work, 
+we show data from ten devices on a single sample, the data and analysis of further 3 samples and 30 devices are 
+consistent with the DC and pulsed measurements performed. Scanning Electron Microscopy was performed 
+using Zeiss Ultra 55 FESEM, Zeiss. Transmission Electron Microscopy was performed Dual-aberration 
+corrected Thermofisher Themis microscopes operated at 300kV. The cross-sectional TEM of the sample was 
+prepared using focused ion beam (FIB) Helios G4-UX.  
+ 
+5.3 Analysis 
+The analysis of the electrical data has been carried out by methods similar to those used to analyze multi 
+electrode array measurements in neural cortices in neuroscience and biological studies. The data analysis 
+routines are borrowed from ref [35], with minor modifications. Thresholding method is used to define changes 
+in the conductance value as per the current signal when exceeds a threshold value. The threshold value is 
+calculated according to the mean range analysis so as to not alter the significance of IEI distributions and 
+autocorrelation function. The maximum likelihood estimation (MLE) approach is incorporated to estimate the 
+power-law exponents in the spiking distributions. The MLE fitting algorithm estimates the best fit for the 
+distribution. The MLE estimators for the distributions are calculated according to the Kolmogorov Smirnov 
+(KS) test which identifies the variance from the power-law fits. We accept the fit for a KS-statistic (p) of the 
+
+data that was less than the KS-statistic found for at least 20% of the power-law models (pt) (i.e., p ≥ pt = 0.2). 
+For the autocorrelation function, the standard linear regression method is used to obtain the exponents. For 
+shape collapse analysis, a polynomial fit using least squares fitting is used to obtain the curvature scaling 
+parameter which minimizes the variance. 
+ 
+Acknowledgements 
+This work was partly carried out at Micro and Nano Characterization Facility (MNCF), and National 
+Nanofabrication Center (NNfC) located at CeNSE, IISc Bengaluru, funded by NPMAS-DRDO and MCIT, 
+MeitY, Government of India; and benefitted from all the help and support from the staff. P.N. acknowledges 
+Start-up grant from IISc, Infosys Young Researcher award, and DST-starting research grant SRG/2021/000285. 
+The authors acknowledge funding support from the Ministry of Human Resource Development (MHRD) 
+through NIEIN project, from Ministry of Electronics and Information Technology (MeitY) and Department of 
+Science and Technology (DST) through NNetRA and the Thematic Unit of Excellence for Nano Science and 
+Technology project from DST Nano Mission. A.R. acknowledges SERB-ITS Travel Grant for providing travel 
+support. The authors thank Prof. Beatriz Noheda (RUG Groningen), Mart Salverda (RUG Groningen) and 
+Ruben Hamming-Green (RUG Groningen) for their valuable support in ex-situ performing electrical 
+measurements prior to making TEM samples.  
+ 
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+DOI 10.1038/s41467-021-24260-z. 
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+Arnold, S. A. Brown, ACS Appl. Mater. Interfaces 2021, 13, 52861. 
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+Neuromorphic Comput. Eng. 2021, 1, 014003. 
+ 
+ 
+ 
+
+Supporting Information  
+ 
+ 
+Self-assembled neuromorphic networks at self-organized criticality in Ag-hBN platform 
+Ankit Rao1, Sooraj Sanjay1, Majid Ahmadi2, Anirudh Venugopalrao1, Navakanta Bhat,1 Bart Kooi2,3, Srinivasan 
+Raghavan1, Pavan Nukala1* 
+ 
+Affiliations  
+1Centre for Nano Science and Engineering, Indian Institute of Science, Bengaluru 560012, 2Zernike Institute 
+for Advanced Materials, University of Groningen, Groningen, the Netherlands, 3CogniGron center, University 
+of Groningen, Groningen, 9747 AG, The Netherlands. Corresponding Author. * E-mail: pnukala@iisc.ac.in 
+ 
+  
+ 
+ 
+ 
+ 
+ 
+This file includes: 
+ 
+Figures S1 to S7 
+Table S1 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
+ 
+ 
+Figure S1. A) Large Area Ag-hBN-Cu device. B) Resistive switching behavior observed in the device at SET 
+voltage (0.9V) and RESET voltage (0.6V). C) Electroforming process for the Ag-hBN device with forming 
+voltage (Vf) of ~4V. D) Current-time plot in linear scale at a constant voltage of 0.6V. The temporal behavior 
+of device exhibiting transition from HRS to LRS due to transport of ions under constant electric field. (Inset) 
+The HRS pattern observed scaled to the current levels of the device in nA. 
+ 
+ 
+
+B
+1E-02
+ (A)
+Current
+1E-04
+1E-06
+HRSSpiking
+1E-08
+LRSSpiking
+ConstantTransition
+Voltage
+1E-10
+-1.2
+-0.6
+0.0
+0.6
+1.2
+Voltage (V)
+D
+0.8
+8
+Linear
+1E-031
+6
+0.6
+Current (A)
+Current (mA)
+1E-05
+(V)
+LRS
+Voltage(
+01020304050607080
+10.0
+0.4
+Time (sec)
+1E-07
+2
+1E-09
+0.2
+HRS
+1E-11
+0
+0.0
+0
+2
+4
+6
+8
+0
+50
+100
+150
+200
+Voltage (V)
+Time (sec) 
+Figure S2. A) Avrami exponent determination from the ln(-ln(1-x) vs ln(t) curve. B) Time series plot of current 
+under constant negative bias (-0.6V) resulting in filamentary breakage and reversal phenomenon. C) Rate 
+coding. Controlling spiking rate as a function of input voltage. Spiking rate can be used to encode information 
+for a specific input voltage.  
+ 
+
+-2 
+c
+0.05v
+-3-
+0.00
+-4 +
+93.00μ
+In(-In(1-Ilt)
+-5.
+Slope = 2.12
+186.00μ
+-6
+0.1V
+-7 .
+0.00
+-8 -
+110.00μ
+-9.
+220.00μ
+(A)
+0.2V
+-10.
+10
+12
+14
+16
+18
+20
+0.00
+In(t)
+220.00μ
+0.0
+1m
+440.00μ
+100m
+0.3V
+-0.2
+0.00
+10μ
+610.00μ
+1μ
+Current
+Itage
+100n
+1.22m
+-0.6 
+0.5V
+10n
+Vol
+0.00
+1n:
+-0.8
+610.00μ
+100p
+1.22m
+10p:
+-1.0
+0
+20
+40
+60
+80
+100
+0
+20
+40
+60
+80
+100
+120
+Time (sec)
+Time (sec) 
+Figure S3. Diffusion based self-stabilizing behavior under application of long-range pulsed stimulus. (Top 
+Panel) Under negative voltage stimulus (-0.6V), when the system is at higher conductance levels, the filaments 
+tend to break and the return to lower conductance levels. (Middle Panel) Pulsing at a voltage (0.6V) lower than 
+the SET voltage allows the system to stay in a stable configuration with small temporal change in conductance. 
+(Bottom Panel) Under pulsing at voltage (0.8V) closer to SET point, the system conductance increases due to 
+diffusion under application of higher fields.  
+ 
+Long Range Pulsed Stimulus 
+ 
+When negative voltage pulses in LRS state are applied (-0.6V), the system tends to switch from a high 
+conductance level and the return to low conductance level as the filaments tend to break. When the device is in 
+HRS state, the pulsing is applied at a voltage (0.6V) lower than the SET voltage which allows the system to 
+stay in a stable configuration as small temporal changes in conductance are observed.  Under pulsing at voltage 
+(0.8V) closer to SET point, the system conductance increases due to diffusion under application of higher fields. 
+
+20
+0.0
+Voltage (V)
+-0.4
+0.
+-0.8
+0
+20
+40
+60
+80
+100
+3
+0.8
+Voltage (V)
+G(Go)
+2
+0.4
+wm
+0.0
+0
+0
+20
+40
+60
+80
+100
+12
+0.8
+Voltage (V)
+8
+G(Go)
+0.4
+4
+0
+0.0
+0
+20
+40
+60
+80
+100
+Time (s) 
+Figure S4. (A and C) Correlation between successive inter-spike intervals (ISIn) for HRS and LRS cases. (B 
+and D) Power Distribution (A2/Hz) as a function of frequency (Hz) plot for determination of 1/f noise and β 
+exponent value for HRS and LRS.  
+ 
+ 
+
+100
+6
+10-1
+5
+n+1
+10-2
+Power
+4
+10-3
+Dul
+10-4
+3
+1/ fβ
+β= 0.91
+10-5,
+2
+10-6
+2
+3
+4
+5
+6
+10-2
+10-1
+10°
+101
+In(ISIn)
+Frequency
+101
+4.0
+10°
+In(ISIn+1)
+3.5
+10-1
+Power
+10-2.
+1 / fβ
+3.0
+10-3
+β = 0.82
+10-4.
+2.5
+10-5
+2.5
+3.0
+3.5
+4.0
+10-2
+10-1
+100
+101
+102
+In(ISIn)
+Frequency 
+Figure S5. Variation of estimations of 1/συz as a function of voltage from <S>(T) vs T, (α-1)/(τ-1) and shape 
+collapse analysis. Trend of critical exponents (α, τ) as a function of voltage for HRS (A) and LRS (B).  
+ 
+ 
+
+2.0
+2.0
+<S>(T)
+<S>(T)
+-(α-1)/(t-1)
+α
+- (α-1)/(t-1)
+ Shape collapse
+α
+Shape collapse
+4
+1/ovz
+1.5
+1.5
+1/ovz
+1%
+1.0
+2
+1.0
+2
+HRS
+LRS
+0.0
+0.2
+0.4
+0.4
+0.6
+0.8
+Voltage (V)
+Voltage (V) 
+ 
+Figure S6. (A to E) HAADF-STEM EDS mapping of Ag-hBN-Cu stack in the HRS state. F) HRTEM image 
+of the cross section from same sample after 10 switching cycles and rested in HRS state. G) Magnified area 
+from F) (white region) which suggests the presence of an intercalated planar cluster of Ag between two hBN 
+layers. H) The Fast Fourier Transform (FFT) of the regions pristine (green) and defective (red) from G) confirms 
+change in the crystallinity of the structure due to Ag intercalation.  I, J) HAADF-STEM (I) EDS (J) mapping of 
+the Ag-hBN-Cu in the LRS state showing formation of Ag filaments. 
+ 
+ 
+ 
+
+HAADF
+cu
+B
+N
+A
+B
+C
+D
+E
+10nm
+10 nm
+10nm
+10nm
+10nm
+HRS
+HRS
+hBN
+5nm
+cu
+1nm
+HAADF
+10nm
+10nmTable S1. Critical Exponents  
+  
+ 
+∆G 
+IEI 
+τ 
+α 
+1/συz 
+HRS 
+2.59±0.05 
+1.54±0.05 
+1.98±0.1 
+2.61±0.1 
+1.59±0.03 
+LRS 
+2.87±0.05 
+1.72±0.05 
+1.93±0.1 
+2.43±0.1 
+1.38±0.03 
+Nanoparticle 
+Network 
+2.59 
+1.93 
+2.05±0.1 
+2.66±0.1 
+1.55±0.03 
+Nanowire 
+Network 
+- 
+- 
+2±0.1 
+2.3±0.1 
+1.3±0.03 
+Rat cortices 
+- 
+- 
+1.74 
+1.96 
+1.22 
+Neuronal 
+cortices 
+- 
+- 
+1.69 
+1.92 
+1.56 
+ 
+(*Error Analysis - This standard deviation is used as a measure of the error of the exponent of the original fit, 
+and is estimated from the routines presented in Ref 36) 
+ 
+ 
+
+ 
+ 
+Figure S7. A) Ni and B) Au as the top electrode. No spiking behavior is observed with Ni and Au as the top 
+electrode. 
+ 
+ 
+ 
+ 
+
+1.1E-3
+B
+1.1E-2
+1.0E-3
+1.1E-2
+9.0E-4
+Au
+Current (
+8.0E-4 .
+1.0E-2
+!N
+7.0E-4
+9.5E-3
+6.0E-4 -
+5.0E-4
+9.0E-3
+FO
+10
+20
+30
+40
+50
+0
+10
+20
+30
+40
+50
+60
+Time (sec)
+Time
+e (sec)
\ No newline at end of file
diff --git a/lNAzT4oBgHgl3EQfpv3p/content/tmp_files/load_file.txt b/lNAzT4oBgHgl3EQfpv3p/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..b45f029168e54d4b430d419b1132a6198c211398
--- /dev/null
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf,len=1031
+page_content='Self-assembled neuromorphic networks at self-organized criticality in Ag-hBN platform  Authors  Ankit Rao1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Sooraj Sanjay1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Majid Ahmadi2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Anirudh Venugopalrao1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Navakanta Bhat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='1 Bart Kooi2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Srinivasan  Raghavan1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Pavan Nukala1*  Affiliations  1Centre for Nano Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Indian Institute of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Bengaluru 560012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' 2Zernike Institute  for Advanced Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' University of Groningen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Groningen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' the Netherlands,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' 3CogniGron center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' University  of Groningen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Groningen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' 9747 AG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Corresponding Author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' * E-mail: pnukala@iisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='in  Keywords: hexagonal Boron Nitride, Avalanche Dynamics, self-organized criticality, neuromorphic networks  Abstract  Networks and systems which exhibit brain-like behavior can analyze information from intrinsically noisy and  unstructured data with very low power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Such characteristics arise due to the critical nature and  complex interconnectivity of the brain and its neuronal network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' We demonstrate that a system comprising of  multilayer hexagonal Boron Nitride (hBN) films contacted with Silver (Ag), that can uniquely host two different  self-assembled networks, which are self-organized at criticality (SOC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' This system shows bipolar resistive  switching between high resistance (HRS) and low resistance states (LRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' In the HRS, Ag clusters (nodes)  intercalate in the van der Waals gaps of hBN forming a network of tunnel junctions, whereas the LRS contains  a network of Ag filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The temporal avalanche dynamics in both these states exhibit power-law scaling,  long-range temporal correlation, and SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' These networks can be tuned from one to another with voltage as a  control parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' For the first time, different neuron-like networks are realized in a single CMOS compatible,  2D materials platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Introduction  The efficient computational ability of the brain in tasks such as classification and pattern recognition has  inspired the identification and implementation of novel computing architectures for energy efficient and parallel  computing needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [1–3] Neuromorphic hardware architectures such as crossbar memristive arrays are assembled  bottom up, where individual elements of the array emulate the functionality of modular units of the nervous  system i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=', a synapse or a neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [4–7] Using these highly organized architectures, it is difficult to capture the  complexity of a biological brain, which is a self-assembled network of ~1011 neurons and ~1015 synapses (in  humans).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [8] These interconnected neuronal networks form a dynamical system, and in a state-of-rest, exhibit  self-organized criticality (SOC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [9,10] In addition to neuronal cortices, SOC has also been observed in many other  physical systems and phenomena such as earthquakes[11], magnetic interactions[12], ferroelastic and ferroelectric  materials (Barkhausan’s noise)[13,14], and charge density waves[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' A system self-organized at critical point,  when perturbed triggers successive events known as avalanches, or in other words emits ‘crackling noise’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [16]  The spatiotemporal dynamics of such avalanches follow power-law scaling with long range temporal  correlations and scale-invariant behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [17–20] It has been proposed that operating at SOC allows the network  to efficiently process and transport information with large scale connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [21,22] A pertinent question then is  whether one can engineer CMOS compatible dynamical system of self-assembled materials networks at SOC  and be able to compute with these top-down neuromorphic architectures (TDNA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  In this context, self-assembled nanoscale networks comprising of nanowire networks[23] or nanoparticle  complexes[24] are being recently explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' However, TDNA on CMOS compatible and large area scalable  platforms are desirable for any future neuromorphic computing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Here, we demonstrate two  different types of networks poised at SOC, both hosted in a robust 2D material platform, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=', hBN deposited on  Cu substrates with Ag as the top electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The choice of high quality hBN as an insulator in conjunction with  fast diffusing Ag as the top electrode, allows us to uniquely stabilize, a) percolative tunnel junction network of  Ag intercalates and b) Ag filamentary network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The 2D material platform also allows for heterogenous  incorporation across different device architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [25]  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Avalanche Dynamics in Ag-hBN system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' A) Schematic of Ag-hBN-Cu (MIM) device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Voltage is  applied to Ag, and Cu substrate is ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' B) SEM image of the top view of the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' C) Representative  current data (blue) when sampled at a constant voltage (red) as a function of time in HRS state, showing  stochastic spiking events and the differential conductance data representing the event size ∆G (red) expressed  in quanta of mean conductance, G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The current levels observed are in nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' D) Representative current and  differential conductance data in the LRS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The current levels observed are in mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  Ag  Current (nA)  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='10  (V)  Itage  25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='05  Vol  hBN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='00  (Go)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  AG  Cu  HRS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='5  0  25  50  75  D  Time (sec)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  Current (mA)  Voltage (V)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='5  Ag  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  AG (G)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='4  hBN  0  25  50  75  50um  Time (sec)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Results  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Dual state switching of Ag networks  The growth of large area (upto 6 sq inches) multilayer hBN (~10nm thick) on Cu was carried out by chemical  vapor deposition process as described in the methods section which has been previously reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [26] Metal-  insulator-metal devices arrays (25x25) were fabricated with Ag (50 um) as the top electrode, hBN as the  insulator and Cu substrate as the bottom electrode (Figure 1A, 1B and S1A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The measurements were performed  on 3 different samples with 10 devices measured on each sample, all of which showed similar behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Cu was  ground, and voltage was applied to Ag electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' I-V sweeps revealed bipolar and bistable resistive switching  (Figure S1B), after an initial forming step (Figure S1C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' In all these devices HRS to LRS transition occurs at  Vset= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='9V (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='15V), and LRS to HRS transition at Vreset= -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='6V (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='1V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' To characterize the spatio-temporal  avalanches in both HRS and LRS, we performed voltage sampling measurements i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=', measured instantaneous  current, by applying a constant voltage bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Representative device stabilized in HRS and LRS was tested at  small perturbing voltages of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='1 V and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='6 V respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Current response in HRS (Figure 1C) shows irregular  spikes of various amplitudes, ranging from 5 nA to 50 nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Response in LRS also shows irregular spikes of a  much greater amplitude in the range of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='2 mA to 3 mA (Figure 1D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The statistical behavior and correlations  of these spikes can be understood through order parameters that characterize them, such as change in  conductance, ∆G (Go) (Figure 1C and 1D), inter event/spike interval (IEI) and autocorrelation function of IEI  (ACF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Diffusion based State Transition in the Ag-hBN system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Log scale plot of the behavior observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  Different colors indicate different stages of device, HRS (blue), Transition from HRS to LRS (Yellow), LRS  (Purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' JMAK model fit for the transition is shown in red solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='8  Log  10-3  10-4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='6  Current (A)  10-5  Voltage (V)  10-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='4  10-7,  10-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  HRS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='2  10-9  10-10  TRANSITION  LRS  10-11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  0  50  100  150  200  Time (sec)  At voltages close to, yet smaller than the set voltage, devices in HRS undergo a phase transition to LRS in  measurable time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' For e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=', a representative device prepared in HRS and held at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='6 V (Vset= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='9V) exhibits  nano ampere current fluctuations upto 70 sec in HRS, followed by a transition to LRS and milliampere current  fluctuations in LRS (Figure 2, linear scale Figure S1D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The average current fits well to John-Avrami-Mehl-  Kolmogorov kinetics in the transition region[27] as per Eq (1),  𝑖𝑇 = 𝑎𝑡𝑛−1𝑒(−𝑏𝑡𝑛)  (1)  where 𝑎 is a constant, 𝑏 is a composite parameter which includes the kinetic rates of nucleation and growth, and  the parameter 𝑛 is the Avrami exponent, where 𝑛 = 𝑑 + 1, d being the dimension of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' From our fits,  we estimate n~2 (Figure S2A), likely suggesting a 1D growth process consistent with the formation of Ag  filaments through long-range diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [28] Subsequent large amplitude current fluctuations in LRS suggest  sporadic events of breakage and reformation of the conducting filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  The spiking rates observed in HRS and LRS are dependent on the sampling rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' For the sampling rates of 10  ms, the measured spiking rates for the representative device are in the range of 15-20 spikes per second in HRS  and 25-30 spikes per second in LRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Furthermore, spiking rates are regulated in both HRS and LRS with respect  to the voltage applied (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='05V to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='5V;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Figure S2C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Such rate coding allows for encoding the device with voltage  dependent spike rate information for stochastic device applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [29] Finally, for the devices prepared in LRS,  upon the application of negative voltage close to the RESET voltage (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='6V), there is a discrete and abrupt  transition to HRS state as seen in Figure S2B, followed by spiking behavior in the HRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Notice that under  negative bias there is no spiking behavior in LRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='2 Long Range Temporal Correlation and Avalanche dynamics  To better envisage the avalanche dynamics, the current spiking pattern is binarized by classifying those spikes  which exceeded a threshold limit as events and rest as not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The top panel event train in Figure 3A (HRS) and  3E (LRS) contains data acquired for 100 s, while the temporal scales are magnified 5 and 20 times for the  subsequent panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The event train patterns in all the panels look qualitatively self-similar and are independent  of the sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The switching activity in Figure 1C and 1D (binarized subsections in Figure 3A and 3E  respectively) exhibits event bursts with varying range of ∆G values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' These bursts indicate a consecutive  correlated activity distributed across the network, rather than isolated local events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' To quantify the correlation  and self-similarity behavior, we investigate the distribution of event sizes (∆G), inter-event interval (IEI) and  the autocorrelation function of the IEIs (ACF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Long Range Temporal Correlation (LRTC) and Power Law Scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The distribution plots shown  in (A to D) are for the device in HRS state and (E to H) in LRS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' (A and E) The binarized patterns of the  switching activity in the Ag-hBN-Cu device, plotted for varying time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' These patterns exhibit self-similar  temporal behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' (B and F) The probability density of change in the conductance of the device (∆G), plotted  for the avalanche patterns in HRS (B) and LRS (F) states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' (C and G) The power law distributions for the Inter  event intervals (IEIs) are demonstrated as the probability distribution functions (PDF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' (D and H)  Autocorrelation function of IEIs in both HRS and LRS, both of which show a slow power law decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  Figure 3B and 3F shows the probability distribution functions (PDF) of event sizes in HRS and LRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The P(∆G)  in both states follows a power law distribution spanning 3 orders of magnitude with exponents of ~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='1  (HRS) and ~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='1 (LRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The PDF of the inter-event intervals (IEIs) (Figure 3C and 3G) also follow a power-  law distribution with exponent, γ in the function, (P(t) ~ t-γ) ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='54±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='05 in HRS and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='72±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='05 in LRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' This is  a quantitative proof of self-similar, scale free nature of the events (qualitatively also observed in Figure 1A and  1E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Self-similarity is a signature of temporal correlations which are observed in electrophysiological signals  from brain cortex and are an outcome of fractal correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [30] The γ in HRS is similar to the values obtained  in nanoparticles systems exhibiting percolating tunnel junction networks[19] and the γ in LRS is similar to the  values in nanowire networks[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' These values are also in similar range as those observed in neuronal  cortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [31,32] The ACFs in both HRS and LRS (Figure 3D and 3H) exhibit a slow power law decay over several  decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Another order parameter of interest to understand the correlation between successive events is inter-  spike interval (ISIn), defined as the time between rising edges of consecutive spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Figure S4A (in HRS) and  Figure S4C (in LRS) show a tight scatter of ISIn and ISIn+1 about the mean ISI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The observed ACF and the ISIn  behavior is a result of long-range temporal correlations (LRTC) among events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Finally, as is the case with any  correlated noise, the event power spectrum follows a power law with frequency as 1/fβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' β is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='91 in HRS (Figure  S4B) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='82 in LRS (Figure S4D) in our representative device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  101  105  1o sec  t-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='54  10-2  104  4G-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='59  (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='19  PDF  103  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='.  10-3  2 sec  102  10-1  10-6  10-5  10-4  10-3  10-1  100  101  100  101  AG (Q-1)  IEI (s)  Time (sec)  Time  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='5 sec  101  101  10-1,  10 sec  100  4G-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='87  t-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='72  PDF  t-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='09  10-1  10-2  10-1  2 sec  10-2  10-2  10-1  10°  10-2  10-1  10°  100  101  AG (Q-1)  IEI (s)  Time (sec)  Time  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Self-organized criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The distribution plots in A to D are for the HRS whereas plots E to H are  for LRS which exhibit self-organized behavior across numerous temporal scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' (A and E) The probability  distribution function for the size of an avalanche P(S) in both cases follows a power law with similar exponent  values of τ ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' (B and F) Distribution of duration of avalanche P(T) also follows power laws in both states  with exponent values of α~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='6 (HRS) and α~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='45 (LRS) with linear binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' (C and G) The average avalanche  size per unit time bin <S>(T) also exhibits power law scaling with exponent 1/συz ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='5 (HRS) and 1/συz ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='4  (LRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' (D and H) The mean avalanche shapes for avalanches of various Ts (different colors) collapse onto a  universal parabolic scaling function (black solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The exponent values obtained fulfill the crackling  relationship (Eq 5) which establishes the existence of critical avalanche dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' In (D) avalanche duration,  T=4 (Purple), 4 (Teal), 5 (Orange), 12 (Brown), 15 (Red), 24 (Blue)], and in (H) T= 4 (Red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' 5 (Green), 6 (Teal),  10 (Green), 32 (Orange)]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='3 Self-organized criticality observation  An avalanche is defined as the sequence of successive event occurrences between two empty time bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The  size of an avalanche (S) in such an occurrence is the total number of events in the avalanche, time of avalanche  (T) is the number of time bins and the (<S>(T)) is defined as the average size of the avalanche of a duration T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  In a critical system, the probability distribution functions of S, T and <S>(T) follow power-law scaling as  described in Eq (2), Eq (3) and Eq (4) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [33] The distributions are defined as follows:  𝑃(𝑆) ~ 𝑆−𝜏  (2)  𝑃(𝑇) ~ 𝑇−𝛼  (3)  < 𝑆 > (𝑇) ~ 𝑇1/𝜎𝜐𝑧  (4)  A signature of avalanche activity which exhibits SOC is when the power law exponents obtained agree with the  crackling relationship as shown in Eq (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [34]  1  𝜎𝜐𝑧 =  𝛼−1  𝜏−1  (5)  10°  103  10°  10-1  10-1  @ 102  E  E 103  <S>(  T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='59  S-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='98  F(t/T)  10-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  PDF  T-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='61  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  10-2  10-4  101  10-3  10-5  :  10-6  100  10-4  100  101  102  10°  101  102  103  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  s  100  101  102  T  t/T T  10°  103  4  100  10-1  @ 102  10-1  S-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='93  E  T-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='43  T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='38  10-3  <S>(  10-4  101  10-3  10-5  10-6  104  10°  10°  101  102  103  100  101  102  100  101  102  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  s  T  T  t/TEq (5) distinguishes a system poised at SOC from other random walk processes that also yield power laws and  exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' In both HRS and LRS as shown in Figure 4, the switching activity exhibits power law dependence  for P(S), P(T) and <S>(T) with power law exponents τ, α and 1/συz respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' In HRS, τ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='96±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='05 (Figure  4A) and α≈2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='61±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='05 (Figure 4B), which gives 1/συz ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='67±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='1 from Eq (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' This is in close agreement with  1/συz ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='59±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='05 independently estimated from <S>(T) (Figure 4C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Similar agreement is observed in LRS  case, with τ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='93±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='05 (Figure 4F) and α≈2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='43±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='05 (Figure 4G), which gives 1/συz ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='52±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='1 from Eq (5),  which is also in agreement with 1/συz ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='38±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='05 obtained from <S>(T) (Figure 4H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Table S1 summarizes  the exponents observed in various order parameters in both HRS and LRS and compares it with the  corresponding exponents in neuronal cortices and other systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' This validation of the crackling noise  relationship proves that our system is poised at SOC in both HRS and LRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  Another crucial feature of a system at criticality is that the average shape of the avalanches exhibits universal  scaling or shape collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [34] The universal scaled shape is obtained by normalizing the duration of avalanche  with respect to the largest avalanche duration, T, and size of the avalanche with respect to the average size of  the avalanche of duration as described in Eq (6) and (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [35]  𝑠(𝑡, 𝑇) ∝ 𝑇( 1  𝜎𝜈𝑧−1) 𝐹 (  𝑡  𝑇)      (6)  < 𝑆 > (𝑇) = ∫ 𝑠(𝑡, 𝑇)𝑑𝑡  𝑇  0  (7)  Here, 𝑠(𝑡, 𝑇) is the mean number of spikes (s) at a time t in an avalanche with duration T and 𝐹 (  𝑡  𝑇) is the shape  of the avalanche when plotted for a scaled duration (t/T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' We perform this shape collapse analysis for all the  avalanches of different times T and observe universal parabolic scaling that has also been reported in neuronal  avalanches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [35,36] The shape collapse analysis yields another independent estimate of the 1/συz which is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='05  for HRS (Figure 4D) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='05 for LRS (Figure 4I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' These values are similar to those obtained from Eq (5)  and provides another independent verification of 1/συz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The exponents 1/συz, α and τ do not show any  significant variation with voltage in both HRS and LRS (Figure S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The close agreement of 1/συz obtained  independently from the crackling relationship estimates, shape collapse analysis and Eq (5) provides robust  evidence of SOC in HRS and LRS in the Ag-hBN system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  The network structure of HRS and LRS can be understood from the cross-sectional TEM analysis shown in  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Cross-sectional lamellae of representative devices prepared in pristine state, HRS (after 10 cycles),  and LRS (after 20 cycles) were made using focused ion beam technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Figure 5A shows HRTEM image of  pristine device with high quality crystalline two-dimensional layers of hBN grown on Cu substrate and  contacted with Ag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' In HRS, the HAADF STEM (Figure 5D) and EDS analysis (Figure 5F) of the selected region  shows dispersion of Ag intercalates inside the hBN matrix (also see Figure S6A-E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The HRTEM data and  analysis further shows lattice expansion from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='35 nm in pristine state (Figure 5B) to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='39 nm in HRS (Figure  5C), which is a consequence of intercalation of Ag inside the h-BN matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Similar expansion resulting from  intercalation has been observed in other 2D systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [37] HAADF STEM and EDS analysis of the device in the  LRS state shows formation of multiple Ag filaments across the h-BN film (filamentary networks, Figure 5F,  5G and Figure S6 I-J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Discussion  A simple model proposed hereunder allows us to explain all the above results in our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' HRS and LRS  form physically two different types of networks, both poised at SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' It is important to point out that such a  unique behavior occurs due to a combination of fast diffusing species in Ag (top electrode) and CVD grown  high quality robust hBN matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' We confirmed the uniqueness of this structure by contacting hBN with standard  top contacts used in memristive devices such as Ni and Au, both of which did not exhibit any spiking activity  (Figure S7A and S7B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  Pristine hBN electroforms to an LRS beyond a threshold voltage through the formation of Ag filaments as  shown in Figure S1C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Ag is known to be highly diffusive in nature and the ionization potential for Ag to Ag+  transition was reported to be 2-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='5 eV lower than other metals such as Au, Cu, and Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [38] This allows Ag to  ionize at relatively lower electrochemical potentials and migrate owing to its comparatively small ionic size  through the hBN matrix to the Cu electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [39] Here nanoclusters are nucleated which grow in the form of  filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Following electroforming, the device is cycled between LRS and HRS several times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Transmission Electron Microscopy Analysis of Ag-hBN system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' A) Cross-sectional high-  resolution transmission electron microscopy (HRTEM) image of the as grown (pristine) hBN on Cu contacted  with Ag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' B) HR-TEM of the hBN stack in pristine state exhibiting interplanar distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='35 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' (Inset) FFT  of the hBN region showing layered structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' C) HR-TEM of the region after multiple cycles and spiking in  HRS state, the resulting interplanar distance is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='39 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' (Inset) FFT of the hBN region exhibiting ring patterns  due to presence of Ag particles along with the layered structure pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' D) HAADF-STEM image in the HRS  state of the Ag-hBN-Cu stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' E) EDS map of the area reveals presence of Ag clusters further confirming the  ionic movement of Ag inside the hBN matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' F) HAADF-STEM image in the LRS state shows formation of  the Ag filament due to large scale diffusion of Ag under continual cycling to the LRS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' G) EDS map of Ag  in the area shown in (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  Pristine  Ag  Ag  Ag  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='35nm  hBN  hBN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='39 nm  hBN  Cu  3nm  3nm  5nm  Cu  Pristine  Cu  HRS  HAADF  Ag  Ag  HAADF  Ag  Ag  hBN  hBN  LRS  Cu  HRS  HRS  5nm  cu  LRS  5nm  5nm  5nmHRS, as evidenced from TEM data, contains Ag nanoclusters which intercalate into the van der Waals gaps of  the hBN matrix (schematic in Figure 6A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The high crystalline quality of hBN planes (low defect density),  means that while intralayer diffusion of Ag can be fast, the interlayer diffusion, mediated by defects such as  point defects, stacking faults and grain boundaries is a slower process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' This allows for intralayer clustering, and  formation of a stable network of Ag intercalates (nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Intercalation of species such as Ag is well reported in  other 2D systems too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [37,40,41]  The conduction in the HRS state (2 orders higher than the pristine state) is predominantly mediated by electron  tunneling across these nodes, with hBN acting as tunnel barrier between interlayer nodes, and vacuum as the  barrier between intralayer nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' In such a self-assembled tunnel network of Ag nodes, application of a small  perturbing voltage creates fluctuations in the clusters/nodes, dynamically changing the spacing between them,  thus giving rise to correlated nanoampere fluctuations in tunnel current (Figure 1C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' These events are polarity  insensitive and happen both in positive and negative biasing voltages (Figure 1C, Figure S2B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' This network  self-organizes to criticality, characterized by scale free avalanches analogous to sand pile at a critical slope, and  corresponding avalanches created upon a small perturbation to maintain the critical slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Avalanche Mechanism A) The Ag acts as the top electrode, which is biased, and the Cu substrate acts  as the bottom electrode which is the common terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The device schematic when cycled and rested in HRS  state showing the Ag (grey) gets dispersed in the hBN matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Schematic of the tunneling current between the  Ag clusters which are formed between the hBN layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' B) The device schematic when cycled and rested in LRS  state showing the formation of filamentary networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' 3D representation of the filamentary networks formed  within the hBN matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  Ag  HRS  hBN  cu  LRSThe interlayer migration of Ag, which is a slower process at low voltages, dictates the process of filament  formation, and thus the transition between HRS and LRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The kinetics of this transition can be non-linearly  enhanced by increasing the voltage, a phenomenon often referred to as voltage-time dilemma in resistive  memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [42] Thus, by operating our representative devices at voltages of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='6 V (Vset=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='9 V), an HRS to LRS  transition is recorded over the background of current fluctuations as shown in Figure 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The kinetics fit JMAK  model, with the dimension of the phase transition ~1, strongly supporting filament formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' LRS is thus a  network of multiple Ag filaments which are further directly imaged using HAADF-STEM (Figure 5G, Figure  S6 I, J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Application of a negative reset voltage beyond a threshold, results in electrochemical dissolution of the  filaments, giving rise to the Ag intercalated network structure of the HRS, and bipolar memristive switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  The milliampere current fluctuations in LRS occur when a small positive voltage is applied on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' This  can be explained as a competing series of filament breaking and mending events, with Joule heating and  corresponding random surface ion diffusion causing the filaments to break, and ionic migration and redox  reactions at the electrodes (electrochemistry) supporting their mending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' These events also show long range  temporal correlations, and scale free nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' However, when a negative bias is applied in such filamentary  networks (LRS), both electrochemistry and Joule heating cause the filaments to break, and thus our devices did  not show any spiking behavior (Figure S2B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' We also observe diffusion based self-stabilizing behavior under  application of long-range pulsed stimulus (Figure S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  Both the tunnel networks (HRS) and the filamentary networks (LRS) are poised at self-organized criticality,  showing a universality behavior, parametrized by critical exponents of various order parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' In the HRS  the values of τ, α, 1/συz correspond to the values discovered in nanoparticle percolative tunnel networks as seen  in Table S1, suggesting not only that these networks belong to the same universality class, but also that the  underlying dictating mechanism is the formation and dissociation of percolative paths of tunneling currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' In  LRS, however τ, α and 1/συz show marginally different values, probably indicating that this could belong to a  different universality class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The exponents obtained are similar in values to those in the nanowire networks  which are filamentary in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The critical exponents observed in both these classes are different from the  well-known universality classes such as mean-field, propagating front, short-range models in various  dimensions, established, for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=', in studies on magnetic Barkhausan’s noise or charge density waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [13–15] To  comment any further requires rigorous modeling of these types of networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Our work opens a fundamental  problem in physics of identifying various universality classes of different types of neuromorphic networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' It is  our belief that correlating the universality classes with their computational performance will be a crucial step in  the development of top-down neuromorphic architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Conclusion  In this work, we demonstrate a unique CMOS friendly platform which hosts neuronal-like networks of two  different universality classes (percolative tunnel junction networks and filamentary networks), both poised at  SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' These networks can be transformed between each other through electrical biasing, aided by field  dependent migration kinetics of Ag in the hBN matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Such observations are also first of their kind in any 2D  material platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The distinct advantage of our large area CVD grown hBN platform is that it can be transferred  to any substrate, including Si, allowing for the implementation of integrated systems on large-area devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  Furthermore, such 2D platforms allows for engineering new structures via heterogenous layer stacking for future  applications in neuromorphic computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' As an outlook, we envision arrays of these devices (multielectrode  arrays) which can emulate spatiotemporal dynamics similar to those observed in neuronal cortices and can  mimic brain-like operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' To compute using these platforms, approaches such as reservoir computing will  be a good starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' [43,44]  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Materials and Methods  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='1 Growth and Device Fabrication  The growth process of multilayer hBN on Cu was carried out using BH3NH3 (Sigma Aldrich, 97%, 682098)  as the precursor in a custom built hot-wall CVD reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' We employed a temperature and pressure-controlled  vaporizer setup to control the flux of the precursor entering the reactor to grow multilayer hBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' A 1 cm x 6 cm  Cu foil (Alfa Aesar, 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='8%, 46986) of 25 µm thickness was normally used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The foil was washed in dilute  hydrochloric acid for 5 min to eliminate the oxide layer on the surface along with a deionized (DI) water rinse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  Successive cleaning steps were conducted in acetone and isopropyl alcohol for 5 minutes in each solvent for  eliminating the organic contaminations on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The foil was then immediately inserted into the reactor  compartment at a permanent location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The compartment was then pumped to the base pressure flushed with 200  sccm of H2 for 10 min and then ramped up to the growth temperature which was usually between 1025°C and  1050°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The details of the growth process can be found in the previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Ag contacts (50 um) were  thermally evaporated on the as-grown hBN-Cu with the aid of stencil masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The electrical measurements for  the metal-insulator-metal stack of Ag-hBN-Cu were carried out as seen in Figure 1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='2 Measurements  DC probe station (Agilent Device Analyzer B1500A) was used for electrical characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Voltage was  applied on the top electrode (Ag) and the bottom contact of Cu was held at ground potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The primary  method of measurements adopted is application of constant DC voltages within the I-V-t (current-voltage-time)  sampling module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' This allows for study of current reconfigurations of the local switches in the device leading  to spikes and thus, the analysis of avalanche dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' These measurements are carried out at multiple sampling  rates, such 100 us, 1 ms and primarily at 10 ms sampling interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Larger sampling interval allows for large data  set and extended time sampling which helps to circumvent substantial breaks in the power law distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  The data obtained for all the sampling intervals demonstrate similar data in quality and quantity, with similar  power law exponents upon analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Thus, the results obtained are indifferent to the sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' In this work,  we show data from ten devices on a single sample, the data and analysis of further 3 samples and 30 devices are  consistent with the DC and pulsed measurements performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Scanning Electron Microscopy was performed  using Zeiss Ultra 55 FESEM, Zeiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Transmission Electron Microscopy was performed Dual-aberration  corrected Thermofisher Themis microscopes operated at 300kV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The cross-sectional TEM of the sample was  prepared using focused ion beam (FIB) Helios G4-UX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='3 Analysis  The analysis of the electrical data has been carried out by methods similar to those used to analyze multi  electrode array measurements in neural cortices in neuroscience and biological studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The data analysis  routines are borrowed from ref [35], with minor modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Thresholding method is used to define changes  in the conductance value as per the current signal when exceeds a threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The threshold value is  calculated according to the mean range analysis so as to not alter the significance of IEI distributions and  autocorrelation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The maximum likelihood estimation (MLE) approach is incorporated to estimate the  power-law exponents in the spiking distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The MLE fitting algorithm estimates the best fit for the  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The MLE estimators for the distributions are calculated according to the Kolmogorov Smirnov  (KS) test which identifies the variance from the power-law fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' We accept the fit for a KS-statistic (p) of the  data that was less than the KS-statistic found for at least 20% of the power-law models (pt) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=', p ≥ pt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  For the autocorrelation function, the standard linear regression method is used to obtain the exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' For  shape collapse analysis, a polynomial fit using least squares fitting is used to obtain the curvature scaling  parameter which minimizes the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  Acknowledgements  This work was partly carried out at Micro and Nano Characterization Facility (MNCF), and National  Nanofabrication Center (NNfC) located at CeNSE, IISc Bengaluru, funded by NPMAS-DRDO and MCIT,  MeitY, Government of India;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' and benefitted from all the help and support from the staff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' acknowledges  Start-up grant from IISc, Infosys Young Researcher award, and DST-starting research grant SRG/2021/000285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  The authors acknowledge funding support from the Ministry of Human Resource Development (MHRD)  through NIEIN project, from Ministry of Electronics and Information Technology (MeitY) and Department of  Science and Technology (DST) through NNetRA and the Thematic Unit of Excellence for Nano Science and  Technology project from DST Nano Mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' acknowledges SERB-ITS Travel Grant for providing travel  support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The authors thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Beatriz Noheda (RUG Groningen), Mart Salverda (RUG Groningen) and  Ruben Hamming-Green (RUG Groningen) for their valuable support in ex-situ performing electrical  measurements prior to making TEM samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' Kumaran, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  Wierstra, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Legg, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' Hussain, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Srinivasa, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Lu, Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content='  [5]  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Sourikopoulos, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Hedayat, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Loyez, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Danneville, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Hoel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Mercier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Cappy, Front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' 1993, 70, 355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' Nieus, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Massobrio, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Martinoia, PLoS One 2009, 4, e6482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' Pasquale, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Martinoia, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' 2015, 5, 10578.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Kuntz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Sethna, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' B - Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Matter Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' 2000, 62, 11699.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' Mallinson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Galli, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Acharya, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Bose, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Arnold, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Bones, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Brown, Neuromorphic  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' 2022, 2, 024009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' Marshall, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' Bennett, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Ripp, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' Physiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' 2016, 7,  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  [36]  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Friedman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Ito, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Brinkman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' DeVille, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Dahmen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' Butler, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' Sheremetyeva, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Niedzielski, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Tristant, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Liang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Kerstetter, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' Meunier, 2D  Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' 2021, 8, 025031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' Kostko, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Morgner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' Issendorff, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' 2020, 12, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Miao, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Dang, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Energy Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' Menzel, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Böttger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Wimmer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' 2015, 25, 6306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' Pedretti, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Montano, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' Hashemkhani, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Boarino, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Ielmini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' Hochstetter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Diaz-Alvarez, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Shine, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' Anirudh Venugopalrao1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content=' Pavan Nukala1*  Affiliations  1Centre for Nano Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Indian Institute of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Bengaluru 560012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' 2Zernike Institute  for Advanced Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' University of Groningen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Groningen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' the Netherlands,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' 3CogniGron center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' University  of Groningen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Groningen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' 9747 AG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Corresponding Author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' * E-mail: pnukala@iisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='in  This file includes:  Figures S1 to S7  Table S1  Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' A) Large Area Ag-hBN-Cu device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' B) Resistive switching behavior observed in the device at SET  voltage (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='9V) and RESET voltage (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='6V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' C) Electroforming process for the Ag-hBN device with forming  voltage (Vf) of ~4V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' D) Current-time plot in linear scale at a constant voltage of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='6V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' The temporal behavior  of device exhibiting transition from HRS to LRS due to transport of ions under constant electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' (Inset)  The HRS pattern observed scaled to the current levels of the device in nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  B  1E-02  (A)  Current  1E-04  1E-06  HRSSpiking  1E-08  LRSSpiking  ConstantTransition  Voltage  1E-10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='2  Voltage (V)  D  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='8  8  Linear  1E-031  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='6  Current (A)  Current (mA)  1E-05  (V)  LRS  Voltage(  01020304050607080  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='4  Time (sec)  1E-07  2  1E-09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='2  HRS  1E-11  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  0  2  4  6  8  0  50  100  150  200  Voltage (V)  Time (sec)  Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' A) Avrami exponent determination from the ln(-ln(1-x) vs ln(t) curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' B) Time series plot of current  under constant negative bias (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='6V) resulting in filamentary breakage and reversal phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' C) Rate  coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Controlling spiking rate as a function of input voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Spiking rate can be used to encode information  for a specific input voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  2  c  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='05v  3-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='00  4 +  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='00μ  In(-In(1-Ilt)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  Slope = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='12  186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='00μ  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='1V  7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='00  8 -  110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='00μ  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='00μ  (A)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='2V  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  10  12  14  16  18  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='00  In(t)  220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='00μ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  1m  440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+page_content='0  0  20  40  60  80  100  0  20  40  60  80  100  120  Time (sec)  Time (sec)  Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Diffusion based self-stabilizing behavior under application of long-range pulsed stimulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' (Top  Panel) Under negative voltage stimulus (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='6V), when the system is at higher conductance levels, the filaments  tend to break and the return to lower conductance levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' (Middle Panel) Pulsing at a voltage (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='6V) lower than  the SET voltage allows the system to stay in a stable configuration with small temporal change in conductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  (Bottom Panel) Under pulsing at voltage (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='8V) closer to SET point, the system conductance increases due to  diffusion under application of higher fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  Long Range Pulsed Stimulus  When negative voltage pulses in LRS state are applied (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='6V), the system tends to switch from a high  conductance level and the return to low conductance level as the filaments tend to break.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' When the device is in  HRS state, the pulsing is applied at a voltage (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='6V) lower than the SET voltage which allows the system to  stay in a stable configuration as small temporal changes in conductance are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Under pulsing at voltage  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='8V) closer to SET point, the system conductance increases due to diffusion under application of higher fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  Voltage (V)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='8  0  20  40  60  80  100  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='8  Voltage (V)  G(Go)  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='4  wm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  0  0  20  40  60  80  100  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='8  Voltage (V)  8  G(Go)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='4  4  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  0  20  40  60  80  100  Time (s)  Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' (A and C) Correlation between successive inter-spike intervals (ISIn) for HRS and LRS cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' (B  and D) Power Distribution (A2/Hz) as a function of frequency (Hz) plot for determination of 1/f noise and β  exponent value for HRS and LRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  100  6  10-1  5  n+1  10-2  Power  4  10-3  Dul  10-4  3  1/ fβ  β= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='91  10-5,  2  10-6  2  3  4  5  6  10-2  10-1  10°  101  In(ISIn)  Frequency  101  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  10°  In(ISIn+1)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='5  10-1  Power  10-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  1 / fβ  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  10-3  β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='82  10-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='5  10-5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  10-2  10-1  100  101  102  In(ISIn)  Frequency  Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Variation of estimations of 1/συz as a function of voltage from <S>(T) vs T, (α-1)/(τ-1) and shape  collapse analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Trend of critical exponents (α, τ) as a function of voltage for HRS (A) and LRS (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  <S>(T)  <S>(T)  (α-1)/(t-1)  α  (α-1)/(t-1)  Shape collapse  α  Shape collapse  4  1/ovz  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='5  1/ovz  1%  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  2  HRS  LRS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='8  Voltage (V)  Voltage (V)  Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' (A to E) HAADF-STEM EDS mapping of Ag-hBN-Cu stack in the HRS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' F) HRTEM image  of the cross section from same sample after 10 switching cycles and rested in HRS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' G) Magnified area  from F) (white region) which suggests the presence of an intercalated planar cluster of Ag between two hBN  layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' H) The Fast Fourier Transform (FFT) of the regions pristine (green) and defective (red) from G) confirms  change in the crystallinity of the structure due to Ag intercalation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' I, J) HAADF-STEM (I) EDS (J) mapping of  the Ag-hBN-Cu in the LRS state showing formation of Ag filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  HAADF  cu  B  N  A  B  C  D  E  10nm  10 nm  10nm  10nm  10nm  HRS  HRS  hBN  5nm  cu  1nm  HAADF  10nm  10nmTable S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' Critical Exponents  ∆G  IEI  τ  α  1/συz  HRS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='59±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='54±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='98±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='61±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='59±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='03  LRS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='87±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='72±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='93±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='43±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='38±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='03  Nanoparticle  Network  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='59  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='93  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='66±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='55±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='03  Nanowire  Network 2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='03  Rat cortices 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='74  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='96  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='22  Neuronal  cortices 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='69  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='92  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='56  (*Error Analysis - This standard deviation is used as a measure of the error of the exponent of the original fit,  and is estimated from the routines presented in Ref 36)  Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' A) Ni and B) Au as the top electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content=' No spiking behavior is observed with Ni and Au as the top  electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='1E-3  B  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='1E-2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0E-3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='1E-2  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0E-4  Au  Current (  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
+page_content='0E-4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfpv3p/content/2301.01619v1.pdf'}
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+Enabling surrogate-assisted evolutionary
+reinforcement learning via policy embedding∗
+Lan Tang1, Xiaxi Li2†, Jinyuan Zhang1, Guiying Li1,3, Peng Yang1,4‡ (�), and
+Ke Tang1,3
+1 Guangdong Provincial Key Laboratory of Brain-Inspired Intelligent Computation,
+Department of Computer Science and Engineering, Southern University of Science
+and Technology, Shenzhen 518055, China
+2 Faculty of engineering, Shenzhen MSU-BIT university, Shenzhen 518172, China
+3 Research Institute of Trustworthy Autonomous Systems, Southern University of
+Science and Technology, Shenzhen 518055, China
+4 Department of Statistics and Data Science, Southern University of Science and
+Technology, Shenzhen 518055, China
+Abstract. Evolutionary Reinforcement Learning (ERL) that applying
+Evolutionary Algorithms (EAs) to optimize the weight parameters of
+Deep Neural Network (DNN) based policies has been widely regarded as
+an alternative to traditional reinforcement learning methods. However,
+the evaluation of the iteratively generated population usually requires a
+large amount of computational time and can be prohibitively expensive,
+which may potentially restrict the applicability of ERL. Surrogate is of-
+ten used to reduce the computational burden of evaluation in EAs. Unfor-
+tunately, in ERL, each individual of policy usually represents millions of
+weights parameters of DNN. This high-dimensional representation of pol-
+icy has introduced a great challenge to the application of surrogates into
+ERL to speed up training. This paper proposes a PE-SAERL Framework
+to at the first time enable surrogate-assisted evolutionary reinforcement
+learning via policy embedding (PE). Empirical results on 5 Atari games
+show that the proposed method can perform more efficiently than the
+four state-of-the-art algorithms. The training process is accelerated up to
+7x on tested games, comparing to its counterpart without the surrogate
+and PE.
+Keywords: Reinforcement learning · Evolutionary algorithms · Surro-
+gates.
+∗ This work was supported partly by the National Natural Science Foundation of
+China (Grants 62272210 and 62106099), partly by the Guangdong Provincial Key
+Laboratory (Grant 2020B121201001), partly by the Program for Guangdong Intro-
+ducing Innovative and Entrepreneurial Teams (Grant 2017ZT07X386), and partly
+by the Stable Support Plan Program of Shenzhen Natural Science Fund (Grant
+20200925154942002).
+† Lan Tang and Xiaxi Li contributed equally to this work.
+‡ Corresponding author: Peng Yang
+E-mail address: yangp@sustech.edu.cn
+arXiv:2301.13374v1  [cs.NE]  31 Jan 2023
+
+2
+L. Tang and X. Li et al.
+1
+Introduction
+In recent years, Evolutionary Algorithms (EAs) have been successfully applied to
+exploratively optimize the parameters of the Deep Neural Network (DNN) based
+policy for challenging reinforcement learning tasks, i.e., video games [17,29],
+Robotics [3], and computer vision [5]. The resultant Evolutionary Reinforcement
+Learning (ERL) [16] has been widely regarded as an alternative to traditional
+reinforcement learning methods [17], due to the ability of EAs on exploration,
+noise resistance, and parallel acceleration. In ERL, a diverse population of poli-
+cies is generated by EA and is evaluated via interactions with the environment.
+The fitness value of each individual is considered as the total returns (e.g., cu-
+mulative rewards with discount factor 1.0) across the entire episode.
+However, there is a notable issue with ERL [16] : the evaluation of the pop-
+ulation usually requires a large amount of computational time and can be pro-
+hibitively expensive. First, ERL finds the optimal policy by iteratively searching
+in a population-based manner. This manner leads to the number of evaluation
+times of individual policies being numerous. Second, each real fitness evaluation
+can be computationally time-consuming due to the complex mechanism of the
+reinforcement learning simulators. These unfavorable characteristics can poten-
+tially restrict the application of ERL in real-world problems.
+Actually, the computationally expensive problems have been extensively stud-
+ied in the EA community for decades. Many Surrogate-Assisted Evolutionary
+Algorithms (SAEAs) have been proposed for expensive problems. In typical
+SAEAs, surrogate models are trained to replace the expensive fitness function for
+evaluation [22]. Based on this idea, this paper focus on how to apply surrogates
+into ERL to speed up training while keeping the ERL effective. And we call this
+framework as Surrogate-Assisted ERL (SAERL). In this paper, we argue that
+the existing SAEA approaches can hardly be directly applied to ERL. The reason
+is that each individual in ERL is usually a Deep Neural Network (DNN) based
+policy with millions of weights. This requires the surrogates accepting very high
+dimensional input vectors to represent the parameters of individual policies. This
+leads to three technical issues: First, the surrogate model can be complex and
+computationally expensive. Second, the training of the surrogate model can be
+very difficult due to the lack of specific training data. Third, the surrogate model
+may encounter the difficulty of distinguishing different weight vectors of policies
+in high-dimensional space. Therefore, the existing surrogate approaches may fail
+to accelerate the training of ERL. Few exceptions [6,20,24] employ the surro-
+gates in the training phases, but they are not for high-dimensional DNN-based
+policies.
+To eliminate the above issues, we first propose to project the surrogate model
+to work in a low dimensional space. A policy embedding (PE) module is proposed
+for the projection, and a new framework PE-SAERL is proposed. Specifically,
+before a high-dimensional policy vector is input into the surrogate model, the PE
+module first encodes the vector to a lower-dimensional space by preserving its
+major features. The existing surrogate models are thus able to be applied to the
+low-dimensional embedded input vectors for computationally cheap evaluation.
+
+Enabling surrogate-assisted ERL via policy embedding
+3
+After the surrogate evaluation, the embedded policies are decoded to the original
+high-dimensional space for further processing of ERL, as shown in Fig. 1.
+RL problems
+Evaluation
+Population
+Policy 1
+Policy 2
+...
+...
+...
+Policy n
+Surrogate 
+Model
+Surrogate-
+assisted 
+Evaluation
+EA module
+Encoder
+Decoder
+Fig. 1. The overview of the PE-SAERL framework.
+The PE module is essentially a bidirectional mapper of both high-dimensional
+and low-dimensional spaces. Also note that, the focus of this paper is the first
+time discussing how to technically enable the SAERL. Hence, we are not focusing
+on sophisticated mappper designing. We restrict the complexity of the mapper
+used in this paper for better demonstrating that the PE module is useful for the
+SAERL framework.
+Specifically, we choose the random embedding method [15] as the PE mod-
+ule here. This PE basically maps between the high-dimensional space and some
+randomly selected lower dimensions while is able to preserving the major fea-
+tures of the original space [15]. We believe other learning-based PE module can
+further improve the effectiveness of PE-SAERL, and this technically simple PE
+of random embedding can be viewed as an appropriate baseline for further in-
+vestigations.
+To instantiated the PE-SAERL for empirical studies, the EA and the surro-
+gate model should be specified. In this work, we choose Negatively Correlated
+Search (NCS) [21] as the EA module and the Fuzzy Classification Pre-Selection
+(FCPS) [33] module as surrogate module. We should emphasize that generally
+any EA and its surrogate-assisted version can be adopted in the PE-SAERL
+framework for optimizing policies. Trade-offs will be considered between the
+
+4
+L. Tang and X. Li et al.
+computing complexity and the search effectiveness. As a result, we get an in-
+stantiated method called PE-FCPS-NCS. The empirical studies on 5 Atari games
+successfully verify that ERL with surrogate acceleration is highly competitive to
+traditional DRLs.
+The remainder of this paper is organized as follows. Section 2 reviews the
+related works. Section 3 details the proposed method. Section 4 presents the
+empirical studies on Atari games. Finally, the conclusion will be drawn in Section
+5.
+2
+Related Works
+This section briefly reviews the literature in the fields of surrogate-assisted evo-
+lutionary algorithms and evolutionary reinforcement learning.
+2.1
+Surrogate-Assisted Evolutionary Algorithms
+Many surrogate modelling approaches have been introduced to SAEAs over the
+past few years. According to whether the fitness function values of candidate
+solutions in the optimization process is directly predicted, it can be generally
+divided into two categories : absolute fitness models and relative fitness models
+[22].
+Absolute fitness models are commonly used in SAEAs [22]. They aim at pre-
+dicting new candidate solutions’ fitness values by approximating the real fitness
+function, including Polynomial Regression (PR), Gaussian Process regression
+(a.k.a. Kriging model), Support Vector Regression (SVR), Radial Basis Func-
+tion (RBF), Neural Network(NN) etc. Kriging [19,20] is a popular surrogate
+model in SAEAs because it can estimate the uncertainty as well as the fitness.
+SVR [11] is an effective surrogate model which has some practical applications,
+such as the optimization of railway wind barriers, but its training process will
+be expensive when the dimensionality of the problem is very high. RBF models
+[34] have also been used in SAEAs to solve many real-world problems.
+Relative fitness models focus on providing the relative rank or preference
+of candidate solutions rather than their absolute fitness values [22]. Wang et
+al. in [14] proposed a hierarchical clustering algorithm by grouping the patient
+data into different numbers of clusters to reduce the amount of computation
+time. Subsequently, Wang et al. in [23] employed an artificial neural network as
+a classifier to predict the dominance relationship between candidate solutions
+and reference solutions. Zhou et al. in [32] used a fuzzy clustering method to
+extract the weight vectors, which works by constructing new objectives as linear
+combinations of the original Many-objective optimization problems. In addition,
+Zhang et al. in [33,31] applied a Fuzzy-KNN method to filter out ’unpromising’
+individuals before the actual evaluation.
+
+Enabling surrogate-assisted ERL via policy embedding
+5
+2.2
+Evolutionary reinforcement learning
+ERL has been developed for years [26], and we noticed that ERL attracts increas-
+ing attentions recently with the DNN-based policies. As early as 1999, David et
+al. in [26] pointed out in the literature that evolutionary algorithm is a class of
+classical algorithms that search the space of policies for solving reinforcement
+learning tasks. Salimans et al. in [17] from OpenAI proposed to use a simplified
+natural evolution strategies to directly optimize the weights of policy neural net-
+works, where the natural gradients was used to update a parameterized search
+distribution in the direction of higher expected fitness. It proved that Evolu-
+tionary Algorithm (EA) is competitive for policy neural networks search in deep
+RL [17]. Subsequently, Chrabaszcz et al. in [4] compared a simper and basic
+canonical ES algorithm with OpenAI ES further confirmes that the power of
+ES-based algorithms for policy search may rival that of the gradient-based al-
+gorithms. Recently, Yang et al. [29] proposed the CCNCS method, which uses a
+novel search method called Negatively Correlated Search [21,27,28] by explicitly
+measuring the diversity among different search processes to drive them to the re-
+gions with uncertainty. Another related work is a hybrid algorithm [9] combining
+gradient-based and derivative-free optimization.
+Related to using surrogates in ERL, Wang et al. in [24] proposed Surrogate-
+assisted Controller (SC) applied on it to alleviate the computational burden of
+evaluation. Nevertheless, the core of SC is to replace the critic model, which
+aims to evaluate the action not the policy. In this regard, it does not follow
+the framework of SAEA and thus is quite different from the SAERL framework
+discussed in this work. In addition, there exists a few studies on applying sur-
+rogates to enhance EAs on RL tasks, such as Evolutionary Surrogate-assisted
+Prescription (ESP) [6], surrogate model-based Neuroevolution (SMB-NE) [20].
+However, these methods do not take the DNN into acount, and does not face
+the high-dimensional difficulties as is in this work.
+As can be seen from the literature, ERL is a extremely promising method for
+solving RL tasks, but the evaluation process is prohibitively expensive. Using the
+surrogate model can solve the problems of low sampling efficiency and expensive
+evaluation in traditional SAEAs. However, considering that each individual in
+ERL usually involves millions of parameters and the commonly surrogate models
+(i.e., Kriging [19], kNN [31]) by calculating the genotype distance may have no
+way to extract effective features, the direct application of surrogates into ERL to
+speed up training may face inevitably failure. This article extends the surrogate
+model to ERL via policy embedding to speed up training.
+3
+Methodology
+This section first introduces the proposed framework consisting of three main
+modules. Then we briefly describe random embedding employed by the PE mod-
+ule. Finally a concrete algorithm called PE-FCPS-NCS is described in detail.
+
+6
+L. Tang and X. Li et al.
+3.1
+PE-SAERL Framework
+Intuitively, we intend to apply surrogates into ERL to speed up training by
+replacing a part of the real simulations with more computationally cheap surro-
+gates. Importantly, the PE module is designed to address the three key technical
+issues faced by directly applying the surrogates to millions of dimensions, as men-
+tioned previously. PE-SAERL Framework fully follows the PE-SAERL flowchart
+depicted in Fig. 1.
+Specifically, the PE-SAERL Framework is roughly divided into 3 modules,
+which are ERL module, PE module and SA module.
+• ERL module: ERL iteratively search for the optimal policy in a population-
+based manner with EAs. Each individual in the population is represented
+as a vector of all connection weights of the neural network policy. At the
+beginning of training, λ search processes are initialized in parallel. At each
+iteration, each of them separately preserves a individual xi, and generates an
+offspring individual x′
+i by crossover or variation operators. Eventually, under
+the effect of environmental selection, the individual with higher fitness value
+as new parent individual into the next generation, As depicted in Fig. 2.
+• PE module: The PE module is essentially a bidirectional mapper of both
+high-dimensional and low-dimensional spaces. It generally consists of two
+steps, i.e., encoding and decoding. Encoding extracts a low-dimensional vec-
+tor representation from a high-dimensional input sentence, and decoding
+generates a correct high-dimensional target translation from that represen-
+tation.
+• SA module: The SA module is used to speed up training by partially replac-
+ing expensive ground-truth fitness evaluations. The surrogate model uses
+all individuals in the population and their corresponding true fitness values
+as historical data to train the model, and then uses the trained model to
+evaluate candidate solutions and select the optimal solution y∗ from them.
+3.2
+PE module : Random Embedding
+Random Embedding refers to a dimensional reduction technique with techni-
+cally simple and desirable theoretical property, which project data into low-
+dimensional spaces with a randomly generated matrix [15,25,30]. The implicit
+assumption is that, for the SAERL framework, the fitness of candidate solutions
+are only affected by a few dimensions instead of all. This assumption is techni-
+cally reasonable for the DNN-based policy since DNN often preserves redundent
+weights and can be effectively prunned [10,7,8].
+Based on this assumption, given a high-dimensional function with low S-
+efficient dimension (i.e., de ≪ D) and a random embedding matrix A ∈ RD×d,
+for any x ∈ RD, there must exist y ∈ Rd such that f(Ay) = f(x) with probability
+1. That is, random embedding enables us to optimize the lower-dimensional
+function g(y) = f(Ay) in Rd instead of optimizing the original high-dimensional
+
+Enabling surrogate-assisted ERL via policy embedding
+7
+���1
+������
+⋯
+λ search processes
+���1,1
+’
+���1,���
+’
+⋯
+������,1
+’
+������,���
+’
+⋯
+���1
+������
+Mutation
+⋯
+���1
+������
+⋯
+Encoding
+��� = ������
+���∗
+���∗
+Best predicted 
+fitness value
+���1
+’
+������
+’
+Decoding
+���1
+∗
+������
+∗
+Best fitness 
+value
+⋯
+⋯
+⋯
+⋯
+⋯
+���∗
+Global best
+Fig. 2. The parallel architecture and main processes of the PE-SAERL framework.
+f(x) in RD, while the function value is still evaluated in the original solution
+space. For S-efficient dimension definition and provable completeness guarantee,
+please refer to [15].
+Specifically, at encoding steps, a random embedding matrix A ∈ RD×d is
+generated drawn from N(0, 1). For each solution x, there must be a unique y
+corresponding to it, according to x = Ay. On the embedding space, the surrogate
+preselects the best solution from M candidate solutions. The final candidate is
+then decoded to the original high-dimensional solution space.
+3.3
+PE-FCPS-NCS: a concrete algorithm
+As mentioned above, the PE module provides the possibility for SA to be used in
+ERL. To instantiated the PE-SAERL for empirical studies, we choose Negatively
+Correlated Search (NCS) [21] as the EA module and the Fuzzy Classification Pre-
+Selection (FCPS) [33] as the surrogate module. The pseudo-code for the concrete
+algorithm is given in Algorithm 1.
+Some components in Algorithm 1 are explained as follows.
+* Initialization: A set of policies {x1, x2, . . . , xλ} are initialized uniformly and
+first evaluated with respect to the RL simulator f in lines 1 ∼ 2.
+* FCPS section: In Line 5, M candidate policies are sampled by means of
+a Gaussian operator. In Line 8, the high-rank policies with the maximal
+membership degree belongs to ”promising” class are chosen out. And one is
+randomly selected from the ”promising” high-rank policies as the candidate
+y∗.
+
+8
+L. Tang and X. Li et al.
+Algorithm 1 PE-FCPS-NCS
+Input: Number of sub-process λ; RL simulator f; S-Effective embedding dimension d;
+Origin policy dimension D; Evaluation limitation max steps; Number of candidate
+policies M.
+Output: BestFound policy x∗.
+1: Initialize λ policies xi ∈ RD uniformly, i = 1, · · · , λ.
+2: Evaluate the λ policies with respect to the objective function f.
+3: repeat
+4:
+for i = 1 to λ do
+5:
+Sample
+candidate
+policies
+{x′
+i,1, · · · , x′
+i,M}
+from
+the
+distribution
+pi ∼ N(xi, Σi);
+6:
+Generate a random matrices A ∈ RD×d with N(0, I);
+7:
+Encoding with yi,j = A−1 · x′
+i,j where j = 1, · · · , M;
+8:
+Choose a policy y∗
+i with maximal membership degree belongs to ”promising”
+class by a fuzzy classifier model m;
+9:
+Let x′
+i = A · y∗
+i ;
+10:
+Evaluate f(x′
+i) as the qualities of x′
+i;
+11:
+Calculate d(pi) and d(p′
+i), where p′
+i ∼ N(x′
+i, Σi);
+12:
+if f(x′
+i) + ϕ · d(p′
+i) > f(xi) + ϕ · d(pi) then
+13:
+xi = x′
+i; and pi = p′
+i;
+14:
+end if
+15:
+Update Σi according to the 1/5 successful rule;
+16:
+end for
+17: until steps passed ≥ max steps
+* PT section: A Gaussian random matrix is generated in line 6 and applied
+in line 7 to encode the candidate policies from the original high-dimensional
+policy space into a low-dimensional effective subspace. Then, the policy space
+is decoded in line 9.
+* NCS section: Considering both the quality and the diversity, i.e., f(x′
+i) + ϕ · d(p′
+i)
+and f(xi) + ϕ · d(pi), the following steps are carried out. In line 10, the can-
+didate policy is evaluated with respect to f. And the diversity value of the
+current and candidate policies are also calculated as a basis for environmen-
+tal selection in line 11. In Lines 12 ∼ 14, the better one between the current
+and candidate policy is selected into the next generation. Finally, the step
+size is updated as described in line 15.
+* Stopping condition: In Line 17, when the time budget has run out, i.e., the
+number of times the policy interacts with the environment exceeds the given
+maximum number max steps, the best solution x∗ ever found is output.
+4
+Experiments
+In this experiment, we aim to answer the following questions: (1) Does the sur-
+rogate model really improve ERL compared to baselines? (2) Does the surrogate
+model really speed up training compared to original NCS? (3) Is the proposed
+method sensitive to parameters?
+
+Enabling surrogate-assisted ERL via policy embedding
+9
+4.1
+Experiments settings
+Environment We apply the algorithm to a series of Atari 2600 games imple-
+mented in The Arcade Learning Environment (ALE) [1]. The Atari 2600 is a
+challenging RL testbed that provides agents with high-dimensional visual input
+and a diverse and interesting set of tasks, including obstacle avoidance (e.g. Free-
+way), shooting (e.g. Beamrider), two-player (e.g. DoubleDunk) and other types.
+These types of games provide agents with significantly different environmen-
+tal settings and ways of interacting, thus enabling testing of RL methods with
+different tasks in maximizing long-term rewards. All these tasks are packaged
+according to the standard OpenAI Gym API [2] and are simulated friendly.
+Baselines Four state-of-the-art algorithms, namely A3C [12], PPO [18], CES
+[4], and NCS [21], are used as the major baselines and follow all original hy-
+perparameter settings. Among them, PPO and A3C are popular gradient-based
+methods that use traditional backpropagation to train networks. CES is a re-
+cently proposed ERL that utilizes a canonical evolution strategy to directly
+optimize the weights of policy neural networks. Besides, to show how PE-SA
+promotes NCS, we directly apply NCS as a baseline to Atari games.
+Performance metric The quality of the policy is measured by the testing
+score, i.e., the average score over 30 repetitions of the game without the frame
+limitations. More specifically, the testing score refers to the cumulative reward
+of the policy in an episode, and the episode refers to the time when there is
+no action in the random ”noop” frame (sample in the interval 0 to 30) at the
+beginning of playing until the end signal is received, as mentioned by Machado
+et al. [1]. Technically, to prevent the agent from falling into a ”dead situation”,
+the frames of one episode is limited to a very large value (i.e. 100,000 frames).
+Other protocols All the baselines share the policy network.We directly adopt
+the policy network proposed by Mnih et al. [13], which consists of three convolu-
+tional layers and two fully connected layers. The network architecture is shown
+in Table 1, which involves nearly 1.7 million connection weights that need to be
+optimized. As suggested by Mnih et al. [13], the raw observations are resized to
+84x84, and the skipping frame is set to 4. The time budget is set to the con-
+sumed total game frames allowed in each training phase. For ERL methods (i.e.,
+CES, NCS), The total game frames are set to 0.1B. For gradient-based methods
+(i.e., PPO), the time budget is set to 0.04B for fairness, as it works via back-
+propagation and the ratio of consumed frames on the same CPU-cored hardware
+is 2.5 [29]. Table 2 shows the hyper-parameters of PE-FCPS-NCS in this work,
+which are common across all environments.
+4.2
+Results and analysis
+Comparisons with baselines For each game, we train a neural network policy
+model and do 30 repeated testings of the trained model to obtain a average test-
+
+10
+L. Tang and X. Li et al.
+Table 1. The network architecture of the policy.
+Input size
+Output size Kernel size Stride #filters activation
+Conv1 4 × 84 × 84
+32 × 20 × 20 8 × 8
+4
+32
+ReLU
+Conv2 32 × 20 × 204 64 × 9 × 9
+4 × 4
+2
+64
+ReLU
+Conv3 64 × 9 × 9
+64 × 7 × 7
+3 × 3
+1
+64
+ReLU
+Fc1
+64 × 7 × 7
+512
+−
+−
+−
+ReLU
+Fc2
+512
+#Action
+−
+−
+−
+−
+Table 2. The major hyperparameter settings of PE-FCPS-NCS. The hyperparameters
+consists of three parts : NCS [21], FCPS [33], and random embedding [15].
+Parameter Value
+Remark
+λ
+6
+The number of processes, that is, the number of populations
+d
+100
+Effective subspace dimension of random embedding
+epoch
+5
+The update interval of covariance matrix
+r
+1.2
+Learning rate for covariance matrix
+M
+3
+The number of candidate solutions
+ϕ
+1.0
+The trade-off factor between quality and correlation
+[l, h]
+[-0.1,0.1] Search space boundaries in low-dimensional spaces
+ing score as performance. To eliminate bias, we repeat the process three times,
+and the testing scores of each algorithm on 5 games are shown in Table 3. The
+Average row shows the average performance of the three executions where the
+best performance is marked in bold, and the Percent row indicates the aver-
+age performance as a percentage of the best performance. It can be seen that
+PE-FCPS-NCS generally performs the best among all compared algorithms. In
+particular, in the Freeway and Bowling games, the best performances of base-
+lines are only 65.23% and 69.49% compared to our proposed method. In the
+two-player game (DoubleDunk), a high score with a positive score is preferred,
+and a negative score implies agent failure. However, all algorithms score are
+negative, but relatively speaking, our algorithm performed best. Note that in
+the Alien game, PE-FCPS-NCS, while better than traditional gradient-based
+reinforcement learning algorithms (A3C, PPO) and evolution-based algorithms
+(CES), performs worse than the original NCS algorithm. In some specific tasks,
+the application of surrogate may encourage exploration in a worse direction.
+Computational time analysis Intuitively, we need to verify whether the sur-
+rogate model can speed up training. More specifically, we compare the training
+time consumed by NCS and PE-FCPS-NCS to reach a specified score. This score
+is set to the test performance achieved by the NCS method after training for 10
+million game frames. The computational time for the PE-FCPS-NCS method to
+reach that score is then counted. Fig. 3 shows the computational time costs (in
+minutes) of NCS and PE-FCPS-NCS on three games (i.e., BeamRider, Bowling,
+Freeway). The test performance achieved by the NCS method is 656, 34 and
+
+Enabling surrogate-assisted ERL via policy embedding
+11
+Table 3. The performance of PE-FCPS-NCS and four baselines on 5 Atari games.
+Priority is given to higher average scores in all games.
+Game
+Performance A3C
+PPO
+CES
+NCS
+PE-FCPS-NCS
+Time Budget
+40M
+40M
+0.1B
+0.1B
+0.1B
+Alien
+Average
+766.00 638.20 561.80 1043.40 825.68
+Percent
+73.41% 61.17% 53.84% 100.00% 79.13%
+BeamRider
+Average
+633.80 600.80 433.10 703.80
+724.80
+Percent
+87.44% 82.89% 59.75% 97.10%
+100.00%
+Bowling
+Average
+0
+23.40
+25
+64.53
+92.87
+Percent
+0.00%
+25.20% 26.92% 69.49%
+100.00%
+DoubleDunk Average
+-2
+-3.1
+-3.7
+-1.3
+-0.60
+Percent
+54.78% 19.33% 0.00%
+77.34%
+100.00%
+Freeway
+Average
+0.00
+14.8
+14.2
+16.7
+22.69
+Percent
+0.00%
+65.23% 62.58% 73.60%
+100.00%
+14.9, respectively. From Fig. 3, it can be seen that the computational time cost
+of the surrogate-assisted NCS is quite superior to the original NCS. In particu-
+lar, in the BeamRider game, the time consumption of the original NCS is nearly
+7 times that of PE-FCPS-NCS. In addition, the insignificant difference in the
+Bowling game is mainly because the number of game frames consumed by each
+iteration is extremely large, which is nearly 4 times that of Freeway. Considering
+that some additional computational burden is introduced, e.g., the training of
+the surrogate model and the computation of the embedding, it suffices to show
+that PE-FCPS-NCS is promising not only for its solution performance but also
+in accelerated training.
+Sensitivity analysis We vary the parameter value over a wide range and re-
+evaluate to perform parameter sensitivity analysis. More specifically, we choose
+the number of candidate policies involved in pre-selection in the surrogate model
+as the hyperparameter, and set it to four different values as [3, 5, 10, 100]. We also
+select three games (i.e., BeamRider, Bowling, Freeway) for sensitivity analysis.
+As shown in Fig. 4, the performance curves did not change drastically with the
+perturbation of the parameter, which means that PE-FCPS-NCS is usually not
+very sensitive to this important parameter.
+5
+Conclusion
+This paper studies how to effectively enable surrogate-assisted evolutionary re-
+inforcement learning to speed up training. First, To apply surrogate to preselect
+millions of connection weights of neural network policies, this paper employs
+the PE-SAERL Framework to scale up surrogate for large-scale search space.
+Next, we propose a concrete algorithm based on the framework, which called
+PE-FCPS-NCS. Then, empirical studies are conducted on 5 Atari games to ver-
+ify the proposed method. Empirical results show that the proposed method can
+
+12
+L. Tang and X. Li et al.
+BeamRider
+Bowling
+Freeway
+Games
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Time cost(m)
+208
+76
+116
+30
+60
+28
+Computational time analysis
+NCS
+PE-FCPS-NCS
+Fig. 3. The training time costs (in minutes) of NCS and PE-FCPS-NCS on three games
+(i.e., BeamRider, Bowling, Freeway) to reach a specified score that achieved by the NCS
+method after training for 10 million game frames.
+n=3
+n=5
+n=10
+n=100
+The number of candidate solutions
+0
+100
+200
+300
+400
+500
+600
+700
+Performance
+Hyperparameter sensitivity analysis
+BeamRider
+Bowling
+Freeway
+Fig. 4. The performance curves as the number of candidate policies vary in [3, 5, 10,
+100].
+
+Enabling surrogate-assisted ERL via policy embedding
+13
+perform more efficiently than the four state-of-the-art algorithms (i.e., PPO,
+A3C, CES, NCS), while effectively accelerating training compared to the orig-
+inal NCS. Last, this paper studies the parameters sensitivity of the proposed
+method.
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf,len=713
+page_content='Enabling surrogate-assisted evolutionary  reinforcement learning via policy embedding∗  Lan Tang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Xiaxi Li2†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Jinyuan Zhang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Guiying Li1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Peng Yang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='4‡ (�),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' and  Ke Tang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='3  1 Guangdong Provincial Key Laboratory of Brain-Inspired Intelligent Computation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Department of Computer Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Southern University of Science  and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Shenzhen 518055,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' China  2 Faculty of engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Shenzhen MSU-BIT university,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Shenzhen 518172,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' China  3 Research Institute of Trustworthy Autonomous Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Southern University of  Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Shenzhen 518055,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' China  4 Department of Statistics and Data Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Southern University of Science and  Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Shenzhen 518055,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' China  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Evolutionary Reinforcement Learning (ERL) that applying  Evolutionary Algorithms (EAs) to optimize the weight parameters of  Deep Neural Network (DNN) based policies has been widely regarded as  an alternative to traditional reinforcement learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' However,  the evaluation of the iteratively generated population usually requires a  large amount of computational time and can be prohibitively expensive,  which may potentially restrict the applicability of ERL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Surrogate is of-  ten used to reduce the computational burden of evaluation in EAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Unfor-  tunately, in ERL, each individual of policy usually represents millions of  weights parameters of DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' This high-dimensional representation of pol-  icy has introduced a great challenge to the application of surrogates into  ERL to speed up training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' This paper proposes a PE-SAERL Framework  to at the first time enable surrogate-assisted evolutionary reinforcement  learning via policy embedding (PE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Empirical results on 5 Atari games  show that the proposed method can perform more efficiently than the  four state-of-the-art algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' The training process is accelerated up to  7x on tested games, comparing to its counterpart without the surrogate  and PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Keywords: Reinforcement learning · Evolutionary algorithms · Surro-  gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  ∗ This work was supported partly by the National Natural Science Foundation of  China (Grants 62272210 and 62106099), partly by the Guangdong Provincial Key  Laboratory (Grant 2020B121201001), partly by the Program for Guangdong Intro-  ducing Innovative and Entrepreneurial Teams (Grant 2017ZT07X386), and partly  by the Stable Support Plan Program of Shenzhen Natural Science Fund (Grant  20200925154942002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  † Lan Tang and Xiaxi Li contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  ‡ Corresponding author: Peng Yang  E-mail address: yangp@sustech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='cn  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='13374v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='NE]  31 Jan 2023  2  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Tang and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  1  Introduction  In recent years, Evolutionary Algorithms (EAs) have been successfully applied to  exploratively optimize the parameters of the Deep Neural Network (DNN) based  policy for challenging reinforcement learning tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=', video games [17,29],  Robotics [3], and computer vision [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' The resultant Evolutionary Reinforcement  Learning (ERL) [16] has been widely regarded as an alternative to traditional  reinforcement learning methods [17], due to the ability of EAs on exploration,  noise resistance, and parallel acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' In ERL, a diverse population of poli-  cies is generated by EA and is evaluated via interactions with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  The fitness value of each individual is considered as the total returns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=', cu-  mulative rewards with discount factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='0) across the entire episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  However, there is a notable issue with ERL [16] : the evaluation of the pop-  ulation usually requires a large amount of computational time and can be pro-  hibitively expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' First, ERL finds the optimal policy by iteratively searching  in a population-based manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' This manner leads to the number of evaluation  times of individual policies being numerous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Second, each real fitness evaluation  can be computationally time-consuming due to the complex mechanism of the  reinforcement learning simulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' These unfavorable characteristics can poten-  tially restrict the application of ERL in real-world problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Actually, the computationally expensive problems have been extensively stud-  ied in the EA community for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Many Surrogate-Assisted Evolutionary  Algorithms (SAEAs) have been proposed for expensive problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' In typical  SAEAs, surrogate models are trained to replace the expensive fitness function for  evaluation [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Based on this idea, this paper focus on how to apply surrogates  into ERL to speed up training while keeping the ERL effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' And we call this  framework as Surrogate-Assisted ERL (SAERL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' In this paper, we argue that  the existing SAEA approaches can hardly be directly applied to ERL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' The reason  is that each individual in ERL is usually a Deep Neural Network (DNN) based  policy with millions of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' This requires the surrogates accepting very high  dimensional input vectors to represent the parameters of individual policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' This  leads to three technical issues: First, the surrogate model can be complex and  computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Second, the training of the surrogate model can be  very difficult due to the lack of specific training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Third, the surrogate model  may encounter the difficulty of distinguishing different weight vectors of policies  in high-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Therefore, the existing surrogate approaches may fail  to accelerate the training of ERL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Few exceptions [6,20,24] employ the surro-  gates in the training phases, but they are not for high-dimensional DNN-based  policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  To eliminate the above issues, we first propose to project the surrogate model  to work in a low dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' A policy embedding (PE) module is proposed  for the projection, and a new framework PE-SAERL is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Specifically,  before a high-dimensional policy vector is input into the surrogate model, the PE  module first encodes the vector to a lower-dimensional space by preserving its  major features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' The existing surrogate models are thus able to be applied to the  low-dimensional embedded input vectors for computationally cheap evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Enabling surrogate-assisted ERL via policy embedding  3  After the surrogate evaluation, the embedded policies are decoded to the original  high-dimensional space for further processing of ERL, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  RL problems  Evaluation  Population  Policy 1  Policy 2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Policy n  Surrogate  Model  Surrogate-  assisted  Evaluation  EA module  Encoder  Decoder  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' The overview of the PE-SAERL framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  The PE module is essentially a bidirectional mapper of both high-dimensional  and low-dimensional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Also note that, the focus of this paper is the first  time discussing how to technically enable the SAERL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Hence, we are not focusing  on sophisticated mappper designing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' We restrict the complexity of the mapper  used in this paper for better demonstrating that the PE module is useful for the  SAERL framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Specifically, we choose the random embedding method [15] as the PE mod-  ule here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' This PE basically maps between the high-dimensional space and some  randomly selected lower dimensions while is able to preserving the major fea-  tures of the original space [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' We believe other learning-based PE module can  further improve the effectiveness of PE-SAERL, and this technically simple PE  of random embedding can be viewed as an appropriate baseline for further in-  vestigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  To instantiated the PE-SAERL for empirical studies, the EA and the surro-  gate model should be specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' In this work, we choose Negatively Correlated  Search (NCS) [21] as the EA module and the Fuzzy Classification Pre-Selection  (FCPS) [33] module as surrogate module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' We should emphasize that generally  any EA and its surrogate-assisted version can be adopted in the PE-SAERL  framework for optimizing policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Trade-offs will be considered between the  4  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Tang and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  computing complexity and the search effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' As a result, we get an in-  stantiated method called PE-FCPS-NCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' The empirical studies on 5 Atari games  successfully verify that ERL with surrogate acceleration is highly competitive to  traditional DRLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Section 2 reviews the  related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Section 3 details the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Section 4 presents the  empirical studies on Atari games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Finally, the conclusion will be drawn in Section  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  2  Related Works  This section briefly reviews the literature in the fields of surrogate-assisted evo-  lutionary algorithms and evolutionary reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='1  Surrogate-Assisted Evolutionary Algorithms  Many surrogate modelling approaches have been introduced to SAEAs over the  past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' According to whether the fitness function values of candidate  solutions in the optimization process is directly predicted, it can be generally  divided into two categories : absolute fitness models and relative fitness models  [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Absolute fitness models are commonly used in SAEAs [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' They aim at pre-  dicting new candidate solutions’ fitness values by approximating the real fitness  function, including Polynomial Regression (PR), Gaussian Process regression  (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Kriging model), Support Vector Regression (SVR), Radial Basis Func-  tion (RBF), Neural Network(NN) etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Kriging [19,20] is a popular surrogate  model in SAEAs because it can estimate the uncertainty as well as the fitness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  SVR [11] is an effective surrogate model which has some practical applications,  such as the optimization of railway wind barriers, but its training process will  be expensive when the dimensionality of the problem is very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' RBF models  [34] have also been used in SAEAs to solve many real-world problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Relative fitness models focus on providing the relative rank or preference  of candidate solutions rather than their absolute fitness values [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Wang et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' in [14] proposed a hierarchical clustering algorithm by grouping the patient  data into different numbers of clusters to reduce the amount of computation  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Subsequently, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' in [23] employed an artificial neural network as  a classifier to predict the dominance relationship between candidate solutions  and reference solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' in [32] used a fuzzy clustering method to  extract the weight vectors, which works by constructing new objectives as linear  combinations of the original Many-objective optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' In addition,  Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' in [33,31] applied a Fuzzy-KNN method to filter out ’unpromising’  individuals before the actual evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Enabling surrogate-assisted ERL via policy embedding  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='2  Evolutionary reinforcement learning  ERL has been developed for years [26], and we noticed that ERL attracts increas-  ing attentions recently with the DNN-based policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' As early as 1999, David et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' in [26] pointed out in the literature that evolutionary algorithm is a class of  classical algorithms that search the space of policies for solving reinforcement  learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Salimans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' in [17] from OpenAI proposed to use a simplified  natural evolution strategies to directly optimize the weights of policy neural net-  works, where the natural gradients was used to update a parameterized search  distribution in the direction of higher expected fitness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' It proved that Evolu-  tionary Algorithm (EA) is competitive for policy neural networks search in deep  RL [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Subsequently, Chrabaszcz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' in [4] compared a simper and basic  canonical ES algorithm with OpenAI ES further confirmes that the power of  ES-based algorithms for policy search may rival that of the gradient-based al-  gorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Recently, Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' [29] proposed the CCNCS method, which uses a  novel search method called Negatively Correlated Search [21,27,28] by explicitly  measuring the diversity among different search processes to drive them to the re-  gions with uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Another related work is a hybrid algorithm [9] combining  gradient-based and derivative-free optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Related to using surrogates in ERL, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' in [24] proposed Surrogate-  assisted Controller (SC) applied on it to alleviate the computational burden of  evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Nevertheless, the core of SC is to replace the critic model, which  aims to evaluate the action not the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' In this regard, it does not follow  the framework of SAEA and thus is quite different from the SAERL framework  discussed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' In addition, there exists a few studies on applying sur-  rogates to enhance EAs on RL tasks, such as Evolutionary Surrogate-assisted  Prescription (ESP) [6], surrogate model-based Neuroevolution (SMB-NE) [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  However, these methods do not take the DNN into acount, and does not face  the high-dimensional difficulties as is in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  As can be seen from the literature, ERL is a extremely promising method for  solving RL tasks, but the evaluation process is prohibitively expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Using the  surrogate model can solve the problems of low sampling efficiency and expensive  evaluation in traditional SAEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' However, considering that each individual in  ERL usually involves millions of parameters and the commonly surrogate models  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=', Kriging [19], kNN [31]) by calculating the genotype distance may have no  way to extract effective features, the direct application of surrogates into ERL to  speed up training may face inevitably failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' This article extends the surrogate  model to ERL via policy embedding to speed up training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  3  Methodology  This section first introduces the proposed framework consisting of three main  modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Then we briefly describe random embedding employed by the PE mod-  ule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Finally a concrete algorithm called PE-FCPS-NCS is described in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  6  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Tang and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='1  PE-SAERL Framework  Intuitively, we intend to apply surrogates into ERL to speed up training by  replacing a part of the real simulations with more computationally cheap surro-  gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Importantly, the PE module is designed to address the three key technical  issues faced by directly applying the surrogates to millions of dimensions, as men-  tioned previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' PE-SAERL Framework fully follows the PE-SAERL flowchart  depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Specifically, the PE-SAERL Framework is roughly divided into 3 modules,  which are ERL module, PE module and SA module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  ERL module: ERL iteratively search for the optimal policy in a population-  based manner with EAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Each individual in the population is represented  as a vector of all connection weights of the neural network policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' At the  beginning of training, λ search processes are initialized in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' At each  iteration, each of them separately preserves a individual xi, and generates an  offspring individual x′  i by crossover or variation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Eventually, under  the effect of environmental selection, the individual with higher fitness value  as new parent individual into the next generation, As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  PE module: The PE module is essentially a bidirectional mapper of both  high-dimensional and low-dimensional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' It generally consists of two  steps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=', encoding and decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Encoding extracts a low-dimensional vec-  tor representation from a high-dimensional input sentence, and decoding  generates a correct high-dimensional target translation from that represen-  tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  SA module: The SA module is used to speed up training by partially replac-  ing expensive ground-truth fitness evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' The surrogate model uses  all individuals in the population and their corresponding true fitness values  as historical data to train the model, and then uses the trained model to  evaluate candidate solutions and select the optimal solution y∗ from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='2  PE module : Random Embedding  Random Embedding refers to a dimensional reduction technique with techni-  cally simple and desirable theoretical property, which project data into low-  dimensional spaces with a randomly generated matrix [15,25,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' The implicit  assumption is that, for the SAERL framework, the fitness of candidate solutions  are only affected by a few dimensions instead of all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' This assumption is techni-  cally reasonable for the DNN-based policy since DNN often preserves redundent  weights and can be effectively prunned [10,7,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Based on this assumption, given a high-dimensional function with low S-  efficient dimension (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=', de ≪ D) and a random embedding matrix A ∈ RD×d,  for any x ∈ RD, there must exist y ∈ Rd such that f(Ay) = f(x) with probability  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' That is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' random embedding enables us to optimize the lower-dimensional  function g(y) = f(Ay) in Rd instead of optimizing the original high-dimensional  Enabling surrogate-assisted ERL via policy embedding  7  ���1  ������  ⋯  λ search processes  ���1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='1  ’  ���1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='���  ’  ⋯  ������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='1  ’  ������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='���  ’  ⋯  ���1  ������  Mutation  ⋯  ���1  ������  ⋯  Encoding  ��� = ������  ���∗  ���∗  Best predicted  fitness value  ���1  ’  ������  ’  Decoding  ���1  ∗  ������  ∗  Best fitness  value  ⋯  ⋯  ⋯  ⋯  ⋯  ���∗  Global best  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' The parallel architecture and main processes of the PE-SAERL framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  f(x) in RD, while the function value is still evaluated in the original solution  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' For S-efficient dimension definition and provable completeness guarantee,  please refer to [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Specifically, at encoding steps, a random embedding matrix A ∈ RD×d is  generated drawn from N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' For each solution x, there must be a unique y  corresponding to it, according to x = Ay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' On the embedding space, the surrogate  preselects the best solution from M candidate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' The final candidate is  then decoded to the original high-dimensional solution space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='3  PE-FCPS-NCS: a concrete algorithm  As mentioned above, the PE module provides the possibility for SA to be used in  ERL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' To instantiated the PE-SAERL for empirical studies, we choose Negatively  Correlated Search (NCS) [21] as the EA module and the Fuzzy Classification Pre-  Selection (FCPS) [33] as the surrogate module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' The pseudo-code for the concrete  algorithm is given in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Some components in Algorithm 1 are explained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Initialization: A set of policies {x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' , xλ} are initialized uniformly and  first evaluated with respect to the RL simulator f in lines 1 ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  FCPS section: In Line 5, M candidate policies are sampled by means of  a Gaussian operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' In Line 8, the high-rank policies with the maximal  membership degree belongs to ”promising” class are chosen out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' And one is  randomly selected from the ”promising” high-rank policies as the candidate  y∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  8  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Tang and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Algorithm 1 PE-FCPS-NCS  Input: Number of sub-process λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' RL simulator f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' S-Effective embedding dimension d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Origin policy dimension D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Evaluation limitation max steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Number of candidate  policies M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Output: BestFound policy x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  1: Initialize λ policies xi ∈ RD uniformly, i = 1, · · · , λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  2: Evaluate the λ policies with respect to the objective function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  3: repeat  4:  for i = 1 to λ do  5:  Sample  candidate  policies  {x′  i,1, · · · , x′  i,M}  from  the  distribution  pi ∼ N(xi, Σi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  6:  Generate a random matrices A ∈ RD×d with N(0, I);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  7:  Encoding with yi,j = A−1 · x′  i,j where j = 1, · · · , M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  8:  Choose a policy y∗  i with maximal membership degree belongs to ”promising”  class by a fuzzy classifier model m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  9:  Let x′  i = A · y∗  i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  10:  Evaluate f(x′  i) as the qualities of x′  i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  11:  Calculate d(pi) and d(p′  i), where p′  i ∼ N(x′  i, Σi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  12:  if f(x′  i) + ϕ · d(p′  i) > f(xi) + ϕ · d(pi) then  13:  xi = x′  i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' and pi = p′  i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  14:  end if  15:  Update Σi according to the 1/5 successful rule;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  16:  end for  17: until steps passed ≥ max steps  PT section: A Gaussian random matrix is generated in line 6 and applied  in line 7 to encode the candidate policies from the original high-dimensional  policy space into a low-dimensional effective subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Then, the policy space  is decoded in line 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  NCS section: Considering both the quality and the diversity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=', f(x′  i) + ϕ · d(p′  i)  and f(xi) + ϕ · d(pi), the following steps are carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' In line 10, the can-  didate policy is evaluated with respect to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' And the diversity value of the  current and candidate policies are also calculated as a basis for environmen-  tal selection in line 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' In Lines 12 ∼ 14, the better one between the current  and candidate policy is selected into the next generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Finally, the step  size is updated as described in line 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Stopping condition: In Line 17, when the time budget has run out, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=', the  number of times the policy interacts with the environment exceeds the given  maximum number max steps, the best solution x∗ ever found is output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  4  Experiments  In this experiment, we aim to answer the following questions: (1) Does the sur-  rogate model really improve ERL compared to baselines?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' (2) Does the surrogate  model really speed up training compared to original NCS?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' (3) Is the proposed  method sensitive to parameters?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Enabling surrogate-assisted ERL via policy embedding  9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='1  Experiments settings  Environment We apply the algorithm to a series of Atari 2600 games imple-  mented in The Arcade Learning Environment (ALE) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' The Atari 2600 is a  challenging RL testbed that provides agents with high-dimensional visual input  and a diverse and interesting set of tasks, including obstacle avoidance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Free-  way), shooting (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Beamrider), two-player (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' DoubleDunk) and other types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  These types of games provide agents with significantly different environmen-  tal settings and ways of interacting, thus enabling testing of RL methods with  different tasks in maximizing long-term rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' All these tasks are packaged  according to the standard OpenAI Gym API [2] and are simulated friendly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Baselines Four state-of-the-art algorithms, namely A3C [12], PPO [18], CES  [4], and NCS [21], are used as the major baselines and follow all original hy-  perparameter settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Among them, PPO and A3C are popular gradient-based  methods that use traditional backpropagation to train networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' CES is a re-  cently proposed ERL that utilizes a canonical evolution strategy to directly  optimize the weights of policy neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Besides, to show how PE-SA  promotes NCS, we directly apply NCS as a baseline to Atari games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Performance metric The quality of the policy is measured by the testing  score, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=', the average score over 30 repetitions of the game without the frame  limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' More specifically, the testing score refers to the cumulative reward  of the policy in an episode, and the episode refers to the time when there is  no action in the random ”noop” frame (sample in the interval 0 to 30) at the  beginning of playing until the end signal is received, as mentioned by Machado  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Technically, to prevent the agent from falling into a ”dead situation”,  the frames of one episode is limited to a very large value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' 100,000 frames).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Other protocols All the baselines share the policy network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='We directly adopt  the policy network proposed by Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' [13], which consists of three convolu-  tional layers and two fully connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' The network architecture is shown  in Table 1, which involves nearly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='7 million connection weights that need to be  optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' As suggested by Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' [13], the raw observations are resized to  84x84, and the skipping frame is set to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' The time budget is set to the con-  sumed total game frames allowed in each training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' For ERL methods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=',  CES, NCS), The total game frames are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' For gradient-based methods  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=', PPO), the time budget is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='04B for fairness, as it works via back-  propagation and the ratio of consumed frames on the same CPU-cored hardware  is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='5 [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Table 2 shows the hyper-parameters of PE-FCPS-NCS in this work,  which are common across all environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='2  Results and analysis  Comparisons with baselines For each game, we train a neural network policy  model and do 30 repeated testings of the trained model to obtain a average test-  10  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Tang and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' The network architecture of the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Input size  Output size Kernel size Stride #filters activation  Conv1 4 × 84 × 84  32 × 20 × 20 8 × 8  4  32  ReLU  Conv2 32 × 20 × 204 64 × 9 × 9  4 × 4  2  64  ReLU  Conv3 64 × 9 × 9  64 × 7 × 7  3 × 3  1  64  ReLU  Fc1  64 × 7 × 7  512  −  −  −  ReLU  Fc2  512  #Action  −  −  −  −  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' The major hyperparameter settings of PE-FCPS-NCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' The hyperparameters  consists of three parts : NCS [21], FCPS [33], and random embedding [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Parameter Value  Remark  λ  6  The number of processes, that is, the number of populations  d  100  Effective subspace dimension of random embedding  epoch  5  The update interval of covariance matrix  r  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='2  Learning rate for covariance matrix  M  3  The number of candidate solutions  ϕ  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='0  The trade-off factor between quality and correlation  [l, h]  [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='1] Search space boundaries in low-dimensional spaces  ing score as performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' To eliminate bias, we repeat the process three times,  and the testing scores of each algorithm on 5 games are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' The  Average row shows the average performance of the three executions where the  best performance is marked in bold, and the Percent row indicates the aver-  age performance as a percentage of the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' It can be seen that  PE-FCPS-NCS generally performs the best among all compared algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' In  particular, in the Freeway and Bowling games, the best performances of base-  lines are only 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='23% and 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='49% compared to our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' In the  two-player game (DoubleDunk), a high score with a positive score is preferred,  and a negative score implies agent failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' However, all algorithms score are  negative, but relatively speaking, our algorithm performed best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Note that in  the Alien game, PE-FCPS-NCS, while better than traditional gradient-based  reinforcement learning algorithms (A3C, PPO) and evolution-based algorithms  (CES), performs worse than the original NCS algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' In some specific tasks,  the application of surrogate may encourage exploration in a worse direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Computational time analysis Intuitively, we need to verify whether the sur-  rogate model can speed up training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' More specifically, we compare the training  time consumed by NCS and PE-FCPS-NCS to reach a specified score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' This score  is set to the test performance achieved by the NCS method after training for 10  million game frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' The computational time for the PE-FCPS-NCS method to  reach that score is then counted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' 3 shows the computational time costs (in  minutes) of NCS and PE-FCPS-NCS on three games (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=', BeamRider, Bowling,  Freeway).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' The test performance achieved by the NCS method is 656, 34 and  Enabling surrogate-assisted ERL via policy embedding  11  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' The performance of PE-FCPS-NCS and four baselines on 5 Atari games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Priority is given to higher average scores in all games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Game  Performance A3C  PPO  CES  NCS  PE-FCPS-NCS  Time Budget  40M  40M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='1B  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='1B  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='1B  Alien  Average  766.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='00 638.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='20 561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='80 1043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='40 825.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='68  Percent  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='41% 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='17% 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='84% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='00% 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='13%  BeamRider  Average  633.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='80 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='80 433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='10 703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='80  724.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='80  Percent  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='44% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='89% 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='75% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='10%  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='00%  Bowling  Average  0  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='40  25  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='53  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='87  Percent  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
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+page_content='23% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='58% 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='60%  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='00%  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='9, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' 3, it can be seen that the computational time cost  of the surrogate-assisted NCS is quite superior to the original NCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' In particu-  lar, in the BeamRider game, the time consumption of the original NCS is nearly  7 times that of PE-FCPS-NCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' In addition, the insignificant difference in the  Bowling game is mainly because the number of game frames consumed by each  iteration is extremely large, which is nearly 4 times that of Freeway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Considering  that some additional computational burden is introduced, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=', the training of  the surrogate model and the computation of the embedding, it suffices to show  that PE-FCPS-NCS is promising not only for its solution performance but also  in accelerated training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Sensitivity analysis We vary the parameter value over a wide range and re-  evaluate to perform parameter sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' More specifically, we choose  the number of candidate policies involved in pre-selection in the surrogate model  as the hyperparameter, and set it to four different values as [3, 5, 10, 100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' We also  select three games (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=', BeamRider, Bowling, Freeway) for sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' 4, the performance curves did not change drastically with the  perturbation of the parameter, which means that PE-FCPS-NCS is usually not  very sensitive to this important parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  5  Conclusion  This paper studies how to effectively enable surrogate-assisted evolutionary re-  inforcement learning to speed up training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' First, To apply surrogate to preselect  millions of connection weights of neural network policies, this paper employs  the PE-SAERL Framework to scale up surrogate for large-scale search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Next, we propose a concrete algorithm based on the framework, which called  PE-FCPS-NCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Then, empirical studies are conducted on 5 Atari games to ver-  ify the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Empirical results show that the proposed method can  12  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Tang and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  BeamRider  Bowling  Freeway  Games  0  25  50  75  100  125  150  175  200  Time cost(m)  208  76  116  30  60  28  Computational time analysis  NCS  PE-FCPS-NCS  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' The training time costs (in minutes) of NCS and PE-FCPS-NCS on three games  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=', BeamRider, Bowling, Freeway) to reach a specified score that achieved by the NCS  method after training for 10 million game frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  n=3  n=5  n=10  n=100  The number of candidate solutions  0  100  200  300  400  500  600  700  Performance  Hyperparameter sensitivity analysis  BeamRider  Bowling  Freeway  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' The performance curves as the number of candidate policies vary in [3, 5, 10,  100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='  Enabling surrogate-assisted ERL via policy embedding  13  perform more efficiently than the four state-of-the-art algorithms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=', PPO,  A3C, CES, NCS), while effectively accelerating training compared to the orig-  inal NCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Last, this paper studies the parameters sensitivity of the proposed  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
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+page_content=' Information Sciences 509, 343–355 (Jan  2020)  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Zhou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=', Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
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+page_content=': Fuzzy-classification assisted solution pre-  selection in evolutionary optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Proceedings of the AAAI Conference on  Artificial Intelligence 33(01), 2403–2410 (Jul 2019)  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Østerg˚ard, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=', Jensen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=', Maagaard, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
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+page_content=' : A comparison of six metamodeling  techniques applied to building performance simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
+page_content=' Applied Energy 211, 89–  103 (Feb 2018)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFQT4oBgHgl3EQfozZu/content/2301.13374v1.pdf'}
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+EQUI-VOCAL: Synthesizing Queries for Compositional Video
+Events from Limited User Interactions
+Technical Report
+Enhao Zhang1, Maureen Daum1, Dong He1,
+Magdalena Balazinska1, Brandon Haynes2, and Ranjay Krishna1
+1University of Washington,
+{enhaoz, mdaum, donghe, magda, ranjay}@cs.washington.edu
+2Microsoft Gray Systems Lab,
+brandon.haynes@microsoft.com
+ABSTRACT
+We introduce EQUI-VOCAL: a new system that automatically syn-
+thesizes queries over videos from limited user interactions. The
+user only provides a handful of positive and negative examples
+of what they are looking for. EQUI-VOCAL utilizes these initial
+examples and additional ones collected through active learning to
+efficiently synthesize complex user queries. Our approach enables
+users to find events without database expertise, with limited label-
+ing effort, and without declarative specifications or sketches. Core
+to EQUI-VOCAL’s design is the use of spatio-temporal scene graphs
+in its data model and query language and a novel query synthesis
+approach that works on large and noisy video data. Our system
+outperforms two baseline systems—in terms of F1 score, synthesis
+time, and robustness to noise—and can flexibly synthesize complex
+queries that the baselines do not support.
+1
+INTRODUCTION
+Video data is increasingly becoming a prized commodity. Inexpen-
+sive large-scale video storage and advances in machine learning
+and computer vision have propelled the use of large video datasets
+with new applications including drone analytics [60, 62], citywide
+traffic analytics [2, 24], civil engineering [4, 23], and many oth-
+ers [22, 45, 49, 54, 55]. Although video database management sys-
+tems (VDBMSs) have recently re-emerged as an active research
+area to support these applications [3, 5, 17, 21, 34, 35, 42], existing
+systems fall short of supporting many use cases.
+Consider a traffic analytics application: A traffic engineer may
+want to understand road hazards involving car and motorcycle in-
+teractions (e.g., motorcycles swerving abruptly in front of turning
+cars). Although many computer vision models exist that detect com-
+mon objects (e.g., “cars” and “motorcycles”) [63] and relate objects
+spatially (e.g., “bottom of”, “left of”, “near”) [12], a specific classi-
+fier that identifies “a motorcycle swerving in front of a car, while
+the car is turning at an intersection” is unlikely to exist [65, 67].
+Worse, training one would require many hours of user effort in
+labeling for a single query. Given the relative rarity of most interest-
+ing events, finding sufficient positive instances further exacerbates
+these labeling requirements. In our example, there will be many in-
+stances of cars and motorcycles in intersections. Only rarely would
+a motorcycle swerve in front of a turning car.
+Assuming that we can run existing computer vision mod-
+els on videos to identify objects, extract attributes, and reason
+about their pairwise relationships, some recent video data man-
+agement systems support users by providing an interface to ex-
+press a declarative query as a composition of extracted informa-
+tion [7, 11, 14, 21, 42, 46, 65]. For example, a user might be able
+EQUI-VOCAL
+Query output: 
+matching video segments
+Ask 
+for more 
+labels 
+Provide a 
+few binary 
+labels 
+Videos
+User-defined functions
+(e.g., Fast R-CNN)
+1
+Event of 
+interest
+2
+5
+3
+4
+Figure 1: Given 1○ a video dataset, 2○ user-defined functions
+that extract semantic information from videos, and 3○ a few
+user-provided labels, EQUI-VOCAL synthesizes a query to
+find instances of an event of interest. It iteratively 4○ asks
+the user for more labels to reduce its uncertainty. Once syn-
+thesized, it 5○ executes the query to return matching events
+on unseen videos.
+to query for an event using a specification that searches for video
+clips containing car and motorcycle objects, and specifying their
+desired relationships (near then front_of, etc.). These systems
+expect users to possess a level of database expertise to be able to
+express such queries. Additionally, real-world events can be difficult
+to express declaratively—even for experts. For example, there are
+multiple ways to express our sample query. The best way depends
+on the data (e.g., in a given video, motorcycles may be swerving
+from outside the frame, as illustrated in Figure 2, discussed later).
+Other similar systems ask users to sketch their events [10, 14]; this
+is equally challenging for the same reason: an event of interest may
+deviate from the exact user-provided sketch.
+In this paper, we present EQUI-VOCAL (Figure 1), a system
+that addresses the above challenge by synthesizing declarative
+queries on behalf of users from a small number of labeled video
+segments. Put another way, EQUI-VOCAL Evolves Queries for
+Users Iteratively and is part of our larger VOCAL system [19]. In
+our example, a user provides as few as two positive and ten negative
+examples containing the event of interest (see Section 5). EQUI-
+VOCAL then synthesizes a declarative query and executes it on the
+remaining large pool of video data to identify other examples of the
+desired event. The source code of EQUI-VOCAL is available at [1].
+EQUI-VOCAL supports users’ search for complex events without
+requiring any specific domain knowledge, with limited effort, and
+without requiring precise declarative specifications or sketches.
+Similar to other systems, we use the insight that while most user
+queries are new and unseen, they are usually composed of known,
+previously seen atoms. Computer vision models for common atoms
+1
+arXiv:2301.00929v1  [cs.DB]  3 Jan 2023
+
+OFOalready exist such as objects (e.g., “car”, “backpack”) and their spa-
+tial and semantic relationships (e.g., “left of”, “holding”). Formally,
+such a composition of visual scenes is referred to as a scene graph
+in the computer vision community [31, 38] and was developed
+from its cognitive grounding in human perception [8, 40, 68]. Key
+to EQUI-VOCAL’s contribution is to encapsulate spatio-temporal
+scene graphs in its data model and use them to define a query
+language. A spatio-temporal scene graph conceptualizes the con-
+tents of a video as a sequence of graphs, each graph representing
+a single video frame. Each graph contains vertices, which repre-
+sent objects in the frame; edges represent the relationships be-
+tween those objects. Each object can possess a set of attributes
+that describe its properties (e.g., “red”, “leather”). EQUI-VOCAL ex-
+tracts relevant data from each video using user-provided functions:
+i.e., pre-existing detectors and classifiers; it synthesizes queries
+as a composition of extracted scene graph atoms. We show that
+EQUI-VOCAL’s data model and query language, both based on the
+relational model, can express a variety of compositional queries.
+Leveraging scene graphs, EQUI-VOCAL contributes a new query
+synthesis approach that finds user events with far fewer labeled
+examples than would be required to train a specialized machine
+learning model directly, and that works on noisy, video-scale data,
+and complex events. To support query synthesis in such environ-
+ment, EQUI-VOCAL solves two technical challenges: it reduces
+computational effort and user effort.
+EQUI-VOCAL reduces computational effort by limiting query
+search using scene graphs, by pruning search paths using beam
+search, and by avoiding expensive database operations. First, un-
+like prior query-by-example techniques that synthesize arbitrary
+SQL [20, 41, 50, 56, 58], EQUI-VOCAL reduces the search space by
+limiting the query search to sequences of scene graphs. Second,
+synthesizing queries over sequences of scene graphs can still be
+a computationally slow process to traverse the search space of
+possible queries. Existing query-by-example systems enumerate
+all possible queries; although pruning techniques like equivalence
+classes [57], over-approximation [58], and lifting projection opera-
+tors [56] can be used to reduce the search space, these mechanisms
+are not sufficient to make exhaustive exploration tractable. Instead,
+EQUI-VOCAL adopts a beam search strategy to explore the query
+space efficiently. Beam search limits exploration to a subset of the
+most promising branches at each step. Third, executing the many
+candidate queries on the user examples is prohibitively expensive.
+Existing systems [56, 58] evaluate candidate queries with many
+joins and thus do not scale well when the size of user examples be-
+comes large. EQUI-VOCAL carefully generates efficient queries that
+avoid expensive operations such as recursive joins. EQUI-VOCAL
+comes with a set of optimizations to generate efficient SQL state-
+ments and uses the PostgreSQL database engine to execute them.
+EQUI-VOCAL reduces user effort by using active learning and
+by being robust to noise. With active learning, EQUI-VOCAL re-
+duces the number of labeled examples needed: Instead of asking
+a user to provide all examples up front, EQUI-VOCAL iteratively
+requests labels of carefully selected additional examples to reduce
+the uncertainty in query synthesis. Noise can naturally creep into
+systems that interface with user labeling, machine learning models,
+and potentially ambiguous real-world events. Distinct from other
+existing systems [46, 56], EQUI-VOCAL searches for queries that
+Figure 2: Example frames of multiple, simultaneous car-
+motorcycle interactions (generated using [26]).
+𝑜!!
+𝑜!"
+𝑟!!
+car
+motorcycle
+left of 
+Figure 3: EQUI-VOCAL represents video content as a se-
+quence of region graphs in its data model. Each region graph
+models a single video frame (left), with nodes representing
+objects and edges representing relationships. A region graph
+(right) is a subset of the full scene graph (not shown).
+best match potentially noisy data and input. It also retains imper-
+fect query candidates at every iteration and uses regularization to
+prevent overfitting to noise or limited user input.
+In summary, EQUI-VOCAL makes the following contributions:
+• We introduce an expressive data model and a query language
+based on spatio-temporal scene graphs (Section 2).
+• We propose a new approach that efficiently synthesizes the
+user’s intended query from examples. We limit the query
+search space using scene graphs, prune search paths using
+beam search, leverage active learning to reduce user effort,
+and handle noisy data (Section 3).
+• We implement a set of optimizations to generate efficient SQL
+query statements and reduce computational effort during
+query synthesis (Section 4).
+• We evaluate our approach on the CLEVRER dataset [66] and
+show that it outperforms two baselines [46, 56]—in terms of
+F1 score, synthesis time, and robustness to data noise—and
+can flexibly synthesize complex queries that the baselines
+do not support (Section 5).
+Overall, EQUI-VOCAL is an important step toward making video
+database management systems more accessible to experts and non-
+experts alike, by easing the task of expressing queries over video
+data.
+2
+EQUI-VOCAL DATA MODEL
+This section describes EQUI-VOCAL’s data model and query lan-
+guage, which we briefly introduced in our vision paper [19], but
+develop in depth here. Section 3 shows how EQUI-VOCAL synthe-
+sizes queries from user input using this data model.
+For ease of presentation, we use a simplified, running example,
+where a traffic engineer seeks to find instances of “a car arriving
+from the left and passing a motorcycle at the intersection.” Figure 2
+shows two representative frames from a video that contains such
+an event. We show other example queries in Section 5.
+2
+
+EQUI-VOCAL: Synthesizing Queries for Compositional Video Events from Limited User Interactions
+2.1
+Scene Graphs as our data model
+EQUI-VOCAL represents a video 𝑉 as a set of short, non-
+overlapping video segments, 𝑣 ∈ 𝑉 (5-second segments in our pro-
+totype implementation). Each video segment is a sequence of 𝑁
+frames {𝑓1, . . . , 𝑓𝑁 }. The visual content of each frame is represented
+by a scene graph [38]: A scene graph 𝑔𝑖 = (o𝑖, r𝑖) contains the set
+of all objects o𝑖 in a frame, along with a set of all relationships r𝑖
+between those objects. Often a scene graph contains more informa-
+tion than is necessary to identify an event, and so the literature also
+defines a region graph 𝑔𝑖𝑗, which is a subgraph of 𝑔𝑖, i.e., 𝑔𝑖𝑗 ⊆ 𝑔𝑖.
+Figure 3 shows an example frame and region graph.
+We define an object in a frame as 𝑜 = (𝑣𝑖𝑑, 𝑓𝑖,𝑜𝑖𝑑,𝑐𝑖𝑑,𝑏𝑏𝑜𝑥),
+where 𝑓𝑖 is the sequence number of frame 𝑖 in video segment, 𝑣𝑖𝑑.
+𝑜𝑖𝑑 is a unique identifier of the object in the video segment, 𝑐𝑖𝑑 is
+the identifier for the class of the object (e.g., “car”, “motorcycle”),
+and 𝑏𝑏𝑜𝑥 is the bounding box containing the object in frame, 𝑓𝑖. A
+𝑏𝑏𝑜𝑥 is represented by its upper-left and bottom-right coordinates,
+i.e., 𝑏𝑏𝑜𝑥 = (𝑥1,𝑦1,𝑥2,𝑦2).
+Objects can have intra-frame relationships defined as 𝑟
+=
+(𝑣𝑖𝑑, 𝑓𝑖,𝑟𝑖𝑑,𝑜𝑖𝑑sub, 𝑝𝑖𝑑,𝑜𝑖𝑑tar), where 𝑟𝑖𝑑 is a unique identifier of
+the relationship in frame, 𝑓𝑖. Subject, 𝑜sub, is connected to target,
+𝑜tar, with the relationship class identifier, 𝑝𝑖𝑑 (e.g., “near” or “holds”).
+Both subject and target belong to frame 𝑓𝑖: i.e., 𝑜sub,𝑜tar ∈ o𝑖.
+Objects can have attributes 𝑎 = (𝑣𝑖𝑑, 𝑓𝑖,𝑜𝑖𝑑,𝑘, 𝑣), where 𝑘 is the
+name of the attribute, 𝑣 is the value of the attribute, 𝑣𝑖𝑑, 𝑓𝑖 and 𝑜𝑖𝑑
+identify the video segment, frame, and object. EQUI-VOCAL distin-
+guishes location and property attributes. The former change over
+time and are typically computed from the bounding box of an object
+(e.g., “location=bottom”). The latter capture intrinsic properties of
+objects and are immutable (e.g., “color=red”).
+Finally, an event 𝑒 is a temporally ordered sequence of region
+graphs 𝑒 = (𝑒𝑖𝑑, {𝑔1, . . . ,𝑔𝑘}). Region graphs in an event do not
+need to be contiguous or distinct.
+Example. Suppose that the two frames in Figure 2 are the 10th and
+15th frames of a video segment V1, and that the motorcycle and car
+are the 7th and 9th objects detected in V1. Then, for the left frame,
+we generate the region graph 𝑔1 = (o1, r1), where: o1 = {𝑜11,𝑜12}
+represents the car 𝑜11 = (V1, F10, OID9, car,𝑏𝑏𝑜𝑥1) and the motor-
+cycle 𝑜12 = (V1, F10, OID7, motorcycle,𝑏𝑏𝑜𝑥2) and r1 = {𝑟11} con-
+tains a relationship 𝑟11 = (V1, F10, RID1, OID9, leftOf, OID7). The
+car also has an attribute 𝑎11 = (V1, F10, OID9, location, bottom).
+We can define the region graph 𝑔2 for the right frame similarly.
+The only difference will be the relationship between the object will
+indicate that the car is now rightOf the motorcycle. Finally, the
+event 𝑒 = (EID1, {𝑔1,𝑔2}) represents a car arriving from the left
+and passing a motorcycle at the intersection.
+The relational schema in Table 1 captures the above data model.
+The benefit of using a relational schema is that we can execute re-
+lational queries to specify region graphs and find events of interest,
+which is flexible and follows the well-understood semantics of the
+relational model. For each video (or collection of related videos),
+EQUI-VOCAL creates a view with this schema. In Section 4 we
+describe when and how relations in the view are materialized.
+To populate each relation, EQUI-VOCAL uses available user-
+defined functions (Figure 1). User-defined functions can be provided
+by the user or be available publicly in the form of existing machine
+Objects(vid, fid, oid, cid, 𝑥1, 𝑦1, 𝑥2, 𝑦2)
+Relationships(vid, fid, rid, oid1, pid, oid2)
+Attributes(vid, fid, oid, key, value)
+Table 1: Relational schema representation of data model.
+learning models, such as object detectors. Various user-defined func-
+tions can be declared in EQUI-VOCAL: (i) an object detector [51]
+that takes a video frame as input and outputs the set of objects
+and their bounding boxes, (ii) an object tracking algorithm [63]
+that takes objects in consecutive frames as input and, for each pair
+of objects, determines if they are the same, and (iii) a set of pre-
+trained models (e.g., [12]) or rules that can take two objects in the
+same frame as input and determine their relationship (e.g., “near”,
+“behind”, “riding”, “holding”) or that can take one object as input
+and determine its attributes (e.g., “location=bottom”, “color=red”).
+In our experiments, EQUI-VOCAL uses a general-purpose object
+detector [27] to locate objects and intrinsic attributes of objects to
+generate trajectories across frames [66].
+2.2
+Scene graphs as our query language
+EQUI-VOCAL could execute arbitrary relational queries on the
+view defined above. However, this would form an intractable search
+space, making query synthesis unusably slow for most real-world
+applications. Instead, we define a query language that is more re-
+strictive, affording a smaller search space and therefore, faster syn-
+thesis. We constrain queries to (i) a temporally ordered sequence
+of region graphs, (ii) a set of predicates, (iii) a set of duration con-
+straints, (iv) a window specification, and (v) to output video segment
+identifiers. Using Datalog and with some abuse of notation, a query
+in EQUI-VOCAL can be expressed as:
+𝑞(𝑣𝑖𝑑) :- 𝑔1, . . . ,𝑔𝑘, p, d,𝑤, where:
+• A temporally ordered sequence of region graphs 𝑔1, . . . ,𝑔𝑘
+specifies that a matching event consists of 𝑔1, followed by 𝑔2,
+followed by 𝑔3, etc. Each 𝑔𝑖 is specified with a set of atoms:
+Objects, Relationships, and Attributes joined on a shared
+𝑣𝑖𝑑 and 𝑓 𝑖𝑑. Moreover, each 𝑔𝑖 can persist for multiple frames
+and there can be other frames between 𝑔𝑖 and 𝑔𝑖+1.
+• A set of predicates p can be applied to objects, relationships,
+and attributes. In our example, predicates would specify that
+the query is looking for “car” and “motorcycle” objects, that
+the car needs to be “left of” then “right of” the motorcycle,
+and that the car should be at the “bottom” of the frame.
+• A set of duration constraints d can be applied to region graphs
+and define the minimum number of contiguous frames that
+a region graph 𝑔𝑖 should be valid before transitioning to the
+next region graph 𝑔𝑖+1.
+• A window specification 𝑤 is the maximum number of frames
+that can separate 𝑔1 from 𝑔𝑘.
+For example, following the above restricted template, the event
+in Section 2.1 can be expressed with the following Datalog rules:1
+g1(vid, fid, oid1, oid2) :-
+Objects(vid, fid, oid1, 'car', _, _, _, _),
+Objects(vid, fid, oid2, 'motorcycle', _, _, _, _),
+Relationships(vid, fid, _, oid1, 'leftOf', oid2),
+Attributes(vid, fid, oid1, 'location', 'bottom'), oid1 != oid2.
+1In this and the following examples we use English words instead of integers for 𝑐𝑖𝑑
+and 𝑝𝑖𝑑 values to make the examples more readable
+3
+
+Object detection
+Object tracking
+Relationship
+Attribute
+Conjunction
+Sequencing
+Iteration
+Window specification
+Query by example
+SVQ++ [11]
+✓
+✓
+Chen et al. [16]
+✓
+✓
+✓
+✓
+✓
+Caesar [42]
+✓
+✓
+✓
+✓
+✓
+STAR Retrieval [14]
+✓
+✓
+✓
+✓
+✓
+✓
+VidCEP [65]
+✓
+✓
+✓
+✓
+✓
+✓
+✓
+CVQL [39]
+✓
+✓
+✓
+✓
+✓
+Quivr [46]
+✓
+✓
+✓
+✓
+✓
+✓
+✓
+Rekall [21]
+✓
+✓
+✓
+✓
+✓
+✓
+✓
+✓
+EQUI-VOCAL
+✓
+✓
+✓
+✓
+✓
+✓
+✓
+✓
+✓
+Table 2: Comparison between compositional video analytics
+systems.
+The above rule will find frames in video segments that contain a
+car and a motorcycle, such that the car is on the left of the motorcy-
+cle, and the car is at the bottom of the frame, where the intersection
+is located. Next, the event will likely consist of a sequence of such
+frames, which can be captured with the following recursive rules:
+g1_star(vid, fid, fid, oid1, oid2) :- g1(vid, fid, oid1, oid2).
+g1_star(vid, fid_start, fid_end, oid1, oid2) :-
+g1_star(vid, fid_start, fid, oid1, oid2),
+g1(vid, fid_end, oid1, oid2), fid_end = fid + 1.
+We could use equivalent rules to define g2 and g2_star. Finally,
+the query that returns matching video segments takes the form:
+q(vid) :- g1_star(vid, fid11, fid12, oid1, oid2),
+g2_star(vid, fid21, fid22, oid1, oid2),
+fid21 > fid12, fid22 - fid11 < 1800
+The predicate 𝑓 𝑖𝑑21 > 𝑓 𝑖𝑑12 indicates that the second sequence
+of region graphs should follow the first one. The predicate allows
+for a gap between sequences, which may arise, for example, if
+something obstructs the vehicles from the camera’s view. Finally,
+𝑓 𝑖𝑑22 − 𝑓 𝑖𝑑11 < 1800 puts a time constraint on the event (a 30-
+second time-window, assuming 60 frames per second).
+To summarize, EQUI-VOCAL’s query language is a subset of
+Datalog with recursion, expressed on a specific schema. In Section 4,
+we explain how we avoid executing expensive recursive queries.
+2.3
+Expressiveness of our data model
+We compare the expressiveness of EQUI-VOCAL’s data model
+against other compositional video analytics systems in Table 2.
+Among them, SVQ++ [11] only supports spatial relationships be-
+tween two objects, Chen et al. [16] only supports temporal queries
+that count co-occurring objects. Caesar [42], STAR Retrieval [14],
+and VidCEP [65] lack the important feature of iteration that allows a
+region graph to persist for multiple frames and thus cannot support
+duration constraints. CVQL [39] does not track objects and all its
+predicates are defined at the object class level. Quivr [46] has similar
+expressiveness to us, but is limited to trajectory queries. Rekall [21]
+introduces a flexible Python library for video event specification,
+but requires that users manually write and refine queries.
+Our data model currently does not support tertiary relationships
+between three objects, such as “a person hitting a ball with a bat”.
+The closest approximation we have is “a person holding a bat” and
+“a bat hitting a ball”. That being said, it is possible to extend our data
+model to support tertiary relationships. Finally, we do not support
+disjunction or negation. We plan to support these in the future.
+3
+QUERY SYNTHESIS
+With our data model and query language defined, we now for-
+mally present the query synthesis problem statement, then describe
+all the components of our proposed solution.
+3.1
+Query synthesis problem statement
+Following the aforementioned data model, a user would like to
+execute a query 𝑞𝑢 on video database 𝐷 that returns a set of video
+segment identifiers, 𝑉𝑜𝑢𝑡𝑝𝑢𝑡 = 𝑞𝑢 (𝐷). Given the user’s intended
+query, 𝑞𝑢, each video segment 𝑣𝑖 can be seen as having a ground-
+truth label 𝑦𝑖 ∈ {0, 1} indicating whether it matches 𝑞𝑢. Initially,
+both the ground-truth labels and 𝑞𝑢 are unknown because the user
+is unable to specify 𝑞𝑢 and can only label video segments as positive
+or negative instances. The goal of EQUI-VOCAL is to synthesize
+a target query, 𝑞𝑡 ∈ 𝑄, that is the best approximation of 𝑞𝑢 in its
+search space. When executed over database 𝐷, 𝑞𝑡 should yield the
+best measure performance (e.g., F1 score):
+𝑞𝑡 = arg max
+𝑞∈𝑄 measure(𝑞(𝐷),𝑞𝑢 (𝐷))
+EQUI-VOCAL can request a label from the user 𝑂: ˆ𝑦𝑖 = 𝑂 (𝑣𝑖).
+Since user labels may be noisy, it is possible that ˆ𝑦𝑖 ≠ 𝑦𝑖. Given that
+enumerative search is intractable, EQUI-VOCAL uses a heuristic
+approach to find an approximately-best query ˆ𝑞𝑡 to the objective,
+while reducing both user effort and query synthesis time.
+3.2
+Query synthesis algorithm overview
+Algorithm 1 provides an overview of EQUI-VOCAL’s synthesis
+algorithm. It traverses the space of possible queries to synthesize
+a final target query that matches the user’s intent. The algorithm
+takes as input the set of unlabeled video segments, 𝑈 , and a small
+set of labeled segments, 𝐿. 𝐿 is provided by the user and should
+include both positive and negative examples of the desired event.
+The algorithm also takes as input a set of user-defined functions,
+𝑃, comprised of indicator functions for object classes (e.g., “car”),
+relationship values (e.g., “near”), and attribute key-value pairs (e.g.,
+“color=red”). Algorithm 1 enumerates 𝑃 when expanding its search
+space.
+The algorithm returns a set of target queries, 𝑄𝑡, which is the top-
+𝑘 synthesized queries ranked by a performance measure (F1 score
+in our prototype implementation) and is updated regularly through
+the search process. The learned queries can be applied to unseen
+videos to find video segments containing the matching event. By
+default, EQUI-VOCAL returns the top-𝑘 queries but then executes
+the best one over unseen videos. The user can also randomly sample
+one query from 𝑄𝑡, execute multiple queries from 𝑄𝑡 and aggregate
+the results, or manually examine 𝑄𝑡 to pick the intended query.
+4
+
+EQUI-VOCAL: Synthesizing Queries for Compositional Video Events from Limited User Interactions
+Algorithm 1: Query synthesis algorithm that returns top-𝑘
+synthesized queries matching user input.
+Input
+:𝑈 - set of unlabeled video segments
+𝐿 - set of labeled video segments
+𝑃 - set of user-defined functions
+𝑏, 𝑏𝑤, 𝑘 - hyperparameters
+Output:𝑄𝑡 - set of top-𝑘 target queries
+1 𝑆 ← {𝑞∅}
+2 while |𝑆| > 0 do
+3
+𝑆′ ← {}
+4
+foreach 𝑞 ∈ 𝑆 do
+5
+𝑆′ ← 𝑆′ ∪ ExpandQuery(𝑞, 𝑃)
+6
+if |𝐿| < 𝑏 then
+7
+𝐿,𝑈 ← PickNextSegments(𝐿,𝑈,𝑆′)
+8
+𝑆 ← SampleQueries(𝑆′, 𝐿,𝑏𝑤)
+9
+𝑄𝑡 ← 𝑄𝑡 ∪ 𝑆′
+10
+𝑄𝑡 ← RetainTopQueries(𝑄𝑡, 𝐿,𝑘)
+11 return 𝑄𝑡
+To populate 𝑄𝑡 efficiently, EQUI-VOCAL synthesizes queries
+in a bottom-up fashion. It starts with an empty query, 𝑞∅, and
+incrementally adds predicates to it, up to a certain complexity (see
+the ExpandQuery method on line 5 and in Section 3.3). At each step,
+different predicates can be added to a query, expanding the search
+in multiple directions. To limit the size of the search space and
+ensure fast query synthesis, EQUI-VOCAL adopts a beam search-
+style strategy (see the SampleQueries method on line 8 and in
+Section 3.4).
+An important challenge for EQUI-VOCAL is that learning a
+query from a small number of user-provided examples is difficult.
+To address this challenge, EQUI-VOCAL uses active learning to
+effectively guide the query synthesis process and identify good
+target queries with limited initial and additional user effort. Specifi-
+cally, method PickNextSegments on line 7 and in Section 3.5 uses
+active learning to select additional video segments for the user to
+label in order to effectively differentiate between multiple candidate
+subqueries, and expand the search in the most promising directions.
+At the end of each iteration, method RetainTopQueries (line
+10 and Section 3.6) maintains a list of top-𝑘 queries seen so far,
+which is larger than the number of queries selected for additional
+expansion, in case a query seen earlier in the search ends up with
+the best score on the final labeled set of segments, or, as mentioned
+above, to give users options if they would like to try alternative,
+high-performing queries.
+The algorithm has several hyperparameters, which include a
+labeling budget 𝑏 (i.e., the maximum number of labels that the user
+is willing to provide), the number of candidate queries to retain
+during exploration 𝑏𝑤 (i.e., the beam width), and the number of
+queries in the final answer 𝑘. Figure 4 illustrates the algorithm
+using the running example.
+3.3
+Query expansion
+Existing query-by-example systems use sketch-based query synthe-
+sis approaches [46, 56, 58] to enumerate candidate queries. A sketch
+query is a query with unspecified parts in the form of holes and
+these approaches enumerate the search space by first generating
+high-level sketch queries and then filling them with low-level de-
+tails. However, enumerative search is slow and memory-intensive.
+Inspired by execution-guided synthesis approaches [13, 30], which
+treat a program as a sequence of manipulations and use the results
+of partial programs to guide the search, EQUI-VOCAL explores
+the search space based on the results of executing intermediate
+queries on the examples. Instead of synthesizing sketch queries
+with uninstantiated holes that cannot be executed directly, or di-
+rectly applying techniques from [13, 30], which require a large
+amount of data to train a neural synthesizer, EQUI-VOCAL expands
+queries by adding instantiated and executable constraints, execut-
+ing partial queries to assess the promise of each explored path, and
+iteratively refining a query towards the target query.
+In our synthesis algorithm, method ExpandQuery(𝑞, 𝑃) takes
+as input a query 𝑞 to expand, and a set of user-defined functions, 𝑃
+to construct more complex queries. The function returns a set of
+expanded queries as illustrated in Figure 4(a).
+Our query synthesis approach uses a compact query notation,
+which can be seen as a DSL. The DSL captures the logical structure
+of the queries to synthesize and key query parameters, but omits
+the details of the full, underlying SQL (or Datalog). This approach
+is important for several reasons: First, since we do not generate
+arbitrary SQL, but rather queries that conform to the structure
+presented in Section 2.2, the DSL captures that structure precisely,
+simplifying the search space and guiding synthesis toward the
+correctly-structured queries. Second, this approach helps to de-
+couple the logical query specification from the details of the SQL
+queries that are ultimately executed. As we present in Section 4,
+critical optimizations are necesssary during the translation from
+our DSL to SQL to achieve efficient query execution.
+In our DSL, we use a variable 𝑜 to represent an object in a query.
+Different variables represent objects with different 𝑜𝑖𝑑’s. All pred-
+icates of a region graph are connected by commas and are repre-
+sented with shorthand notations that specify only their key-value
+pairs (for property attributes, e.g., Color(𝑜1, ‘cyan’)), value (for
+location attributes, e.g., Bottom(𝑜1)), or class (for objects and re-
+lationships, e.g., Car(𝑜1)). Then, region graphs are connected in
+sequence with semicolons. For example, the query for the event
+from Section 2.1 can be represented as:
+𝑞 =(Car(𝑜1), Motorcycle(𝑜2), LeftOf(𝑜1,𝑜2), Bottom(𝑜1));
+(Car(𝑜1), Motorcycle(𝑜2), RightOf(𝑜1,𝑜2), Bottom(𝑜1))
+We further use notation Duration(𝑔,𝑑) to require that the region
+graph 𝑔 exist in at least 𝑑 consecutive frames.
+During query synthesis, EQUI-VOCAL expands queries written
+in our DSL. ExpandQuery can take any of the following three
+actions:
+• Graph construction (GC): Add a predicate to an existing
+region graph.
+• Sequence construction (SC): Insert a new region graph con-
+sisting of one predicate into any position of the existing
+sequence of region graphs.
+• Duration refinement (DR): Increment the duration constraint
+of one existing region graph in the sequence.
+5
+
+0.6
+0.4
+0.4
+0.3
+0.2
+Pick
+Label: ✅ or ❌
+0.6
+0.3
+0.5
+0.4
+0.1
+Top-k 
+queries:
+(a) Query expansion
+(b) Example 
+selection
+Top-k 
+queries:
+(c) Query 
+sampling
+(d) Top-k 
+queries
+(a) Query expansion
+(a) Query expansion
+(d) Top-k 
+queries
+Iteration: 1
+Iteration: 2
+Iteration: 8
+𝑃 = Car, Motorcycle, LeftOf, RightOf, Bottom , 𝑏𝑤 = 2, 𝑘 = 1
+𝑞∅
+𝑞" 	= LeftOf 𝑜", 𝑜#
+𝑞# = RightOf(𝑜", 𝑜#)
+𝑞$ = Bottom(𝑜")
+𝑞% = Car(𝑜")
+𝑞& = Motorcycle(𝑜")
+𝑄' = 𝑞"
+𝒒𝟏
+𝑞#
+𝒒𝟑
+𝑞%
+𝑞&
+𝑞"
+𝑞$
+𝑞* = (LeftOf 𝑜", 𝑜# , Bottom 𝑜" )
+𝑞+ = LeftOf 𝑜", 𝑜# ; 	Bottom 𝑜"
+𝑞, = Duration LeftOf 𝑜", 𝑜# , 5
+𝑞! = (LeftOf	 𝑜", 𝑜# , Bottom 𝑜" ,	
+Car 𝑜" , Motorcycle 𝑜# );
+						((RightOf 𝑜", 𝑜# , Bottom 𝑜" ,
+Car 𝑜" , Motorcycle 𝑜# )
+𝑄' = 𝑞-
+Figure 4: Running example of the query synthesis algorithm. The algorithm starts with 𝑞∅ and synthesizes queries by iter-
+atively (a) expanding queries, (b) selecting new video segments for the user to label, (c) sampling a small set of queries for
+further expansion, and (d) updating a list of top-𝑘 queries after each iteration. The algorithm returns 𝑄𝑡 = {𝑞𝑖}.
+∅
+𝐴
+𝐵
+𝐶
+𝐴; 𝐵
+𝐵; 𝐶
+𝐴; 𝐵; 𝐶
+𝐵; 𝐶; 𝐴
+0.2
+0.5
+0.6
+0.6
+0.7
+0.5
+0.9
+⋯
+⋯
+(a) Restrictive expansion.
+∅
+𝐴
+𝐵
+𝐶
+𝐴; 𝐵
+𝐵; 𝐶
+𝐴; 𝐵; 𝐶
+𝐵; 𝐶; 𝐴
+0.2
+0.5
+0.6
+0.6
+0.7
+0.5
+0.9
+⋯
+⋯
+(b) Relaxed expansion.
+Figure 5: A restrictive rule constrains each query to have one
+construction path, while a relaxed rule allows for multiple
+paths. Though a relaxed rule results in a larger search space,
+it is more likely to find queries with high performance than
+a restrictive rule.
+As shown in Figure 4(a), EQUI-VOCAL starts with an empty query
+𝑞0 = 𝑞∅. In iteration 1, EQUI-VOCAL takes action SC to expand
+𝑞0. This results in 𝑞1 to 𝑞5, each consisting of a single region
+graph with one predicate drawn from 𝑃. In iteration 2 of Fig-
+ure 4(a), EQUI-VOCAL first expands 𝑞1 = LeftOf(𝑜1,𝑜2). Per-
+forming action GC leads to 𝑞6 = (LeftOf(𝑜1,𝑜2), Bottom(𝑜1));
+SC leads to 𝑞7 = LeftOf(𝑜1,𝑜2); Bottom(𝑜1); and DR leads to
+𝑞8 = Duration(LeftOf(𝑜1,𝑜2), 5), assuming the granularity of
+Duration is 5 frames.
+Deciding which actions to take to expand queries is an important
+design decision in EQUI-VOCAL. Imagine the search space as a di-
+rected acyclic graph (DAG), where each node represents a possible
+query in the search space, the root node represents the empty query
+𝑞∅, and the query of each child node is constructed by applying
+one action to its parent node. We consider two extreme expansion
+rules: A restrictive rule (Figure 5a) only allows each query to have
+one construction path in the DAG, while a relaxed rule allows each
+query to be generated through all possible permutations of actions
+with multiple construction paths (Figure 5b). Our early experiments
+showed that the restrictive rule led to poor performance: In some
+cases, queries with high performance were not found because their
+ancestor queries (i.e., intermediate parent queries) were not good
+and were getting pruned. Since different predicates and their com-
+binations have different selectivity and contribute differently to the
+final query performance, the relaxed rule allows the algorithm to
+focus on synthesizing the more dominant part of the query during
+early iterations, avoiding all paths to the target query being acci-
+dentally pruned. The cost of a relaxed rule is a larger search space
+and slower query synthesis. In the example of Figure 5, queries are
+constructed following the paths highlighted in blue, assuming only
+the best query is expanded at each step. The restrictive rule selects
+the query 𝐵;𝐶;𝐴 with a score of 0.5 rather than 𝐴; 𝐵;𝐶 with a score
+of 0.9, because 𝐴 has a lower F1 score than 𝐶 and 𝐵. On the other
+hand, using the relaxed rule selects 𝐴; 𝐵;𝐶 by first constructing 𝐶,
+then adding 𝐵 and 𝐴 to the sequence. EQUI-VOCAL follows the
+relaxed rule: GC can add any predicates that are not already in
+the region graph (predicates of the same user-defined function but
+different variables are considered different); SC can insert a new
+region graph before and after any existing region graph; DR can
+increment the duration constraint of any existing region graph.
+3.4
+Beam search
+When traversing the search space, we can either design our al-
+gorithm to only expand the top query or to expand all queries.
+The former is quick to compute but may not find the target query
+at the end, while the latter is optimal but comes with a prohibi-
+tive computational cost. EQUI-VOCAL uses beam search to bal-
+ance query performance and synthesis efficiency. Beam search is
+used in NLP [43] where problems commonly have large search
+spaces. Beam search reduces runtime by limiting the number of
+explored branches at each iteration. However, the search out-
+puts are not guaranteed to be optimal. Search quality depends on
+how branches are expanded, scored, and pruned. In our approach,
+method SampleQueries(𝑆′, 𝐿,𝑏𝑤) retains the top 𝑏𝑤 queries to
+expand by evaluating the F1 score of the 𝑆′ candidate queries on
+the set of labeled video segments 𝐿 (see Figure 4(c)).
+3.5
+Active learning
+One challenge with asking the user for only a handful of examples
+of the intended event is that we risk overfitting, but asking the
+user to find a larger number of initial examples is difficult. To
+address this challenge, our approach is to use active learning during
+query synthesis. At every iteration, EQUI-VOCAL asks the user
+for a handful of additional labels to differentiate between the 𝑏𝑤
+queries retained from the previous iteration. We choose which
+video segments to ask for labels using a disagreement-based active
+learning algorithm [37]. Intuitively, this method picks the video
+6
+
+VEQUI-VOCAL: Synthesizing Queries for Compositional Video Events from Limited User Interactions
+segments where the retained queries disagree the most. Using active
+learning reduces the number of data points a user has to provide;
+additionally, we posit that it is easier for users to label system-
+selected examples than to come up with their own.
+During each call to PickNextSegments(𝐿,𝑈,𝑆′), EQUI-VOCAL
+computes a score for a sample of unlabeled video segments 𝑈 over
+the candidate queries 𝑆′ and then picks the video segments with
+the largest disagreement. The score of each sampled, unlabeled
+video segment is computed as the weighted disagreement between
+the candidate queries. The weight of each candidate query is set
+to its regularized performance over the labeled set 𝐿 (Section 3.7).
+While the algorithm in [37] is designed for the setting where la-
+beling candidates are streamed, EQUI-VOCAL instead maintains a
+pool of candidates. Therefore, for each call to PickNextSegments,
+EQUI-VOCAL computes the score for a sample of unlabeled video
+segments over a sample of candidate queries and then picks the
+best one. The function updates 𝐿 and 𝑈 given the new user labels.
+As shown in Figure 4(a), EQUI-VOCAL generates five candidate
+queries 𝑞1 to 𝑞5 in iteration 1. Among them, 𝑞1 has the highest score
+0.6, while 𝑞2 and 𝑞3 give the same second highest score 0.4. EQUI-
+VOCAL requests a new video segment to be labeled (Figure 4(b)),
+which distinguishes 𝑞3 from 𝑞2 (Figure 4(c)).
+Before query synthesis begins, users can set the labeling bud-
+get as a hyperparameter. EQUI-VOCAL precomputes the number
+of iterations of query expansion it will perform and uniformly di-
+vides the labeling budget amongst the iterations. At every iteration,
+EQUI-VOCAL asks for new labels only if it has a labeling budget
+left to do so; otherwise, it proceeds to synthesize queries without
+requesting new labels. EQUI-VOCAL precomputes the number of
+iterations by relying on the hyperparameters (Section 4) that bound
+the search process and the query expansion rules (Section 3.3),
+which increments the complexity of synthesized queries by one in
+each iteration.
+3.6
+Retaining top queries
+At the end of each iteration, EQUI-VOCAL updates its list of
+candidate final queries in order to retain 𝑘 best performing
+ones, as measured by their F1 scores on the labeled dataset.
+RetainTopQueries(𝑄𝑡, 𝐿,𝑘) evaluates all candidate queries 𝑄𝑡 on
+labeled set 𝐿 and returns the top 𝑘 (Figure 4(d)).
+In Figure 4(d), at the end of iteration 1, EQUI-VOCAL stores
+𝑞1 in 𝑄𝑡 because 𝑘=1, and 𝑞1 has the highest score. When the
+algorithm terminates, EQUI-VOCAL returns 𝑞𝑖 as the final query.
+In our evaluation, we set the default value of 𝑘 to 100 (and to 1000
+for a more complex dataset), which ensures both low overhead and
+high-performing final queries.
+3.7
+Regularization
+EQUI-VOCAL retains top queries using their F1 scores on the
+smaller 𝐿 labeled set. This can lead to overfitting since there are
+likely too few training examples to accurately evaluate each candi-
+date query. To prevent overfitting, we regularize the score of each
+candidate query by adding the term:𝑠𝑐𝑜𝑟𝑒reg(𝑞) = 𝑠𝑐𝑜𝑟𝑒(𝑞)−𝜆·𝑅(𝑞),
+where 𝜆 controls the importance of the regularization term. 𝑅(𝑞)
+𝑓!
+𝑓"
+𝑓#
+𝑓$
+𝑓%
+𝑓&
+𝑓'
+𝑣!
+⋯
+✅
+𝑓!
+∗
+𝑓"
+∗
+𝑓#
+∗
+𝑓$
+∗
+𝑓%
+∗
+𝑓&
+∗
+𝑣"
+❌
+𝑔!
+⋯
+𝑔"
+⋯
+𝑔#
+⋯
+𝑔!! 𝑔!" 𝑔!# 𝑔!$ 𝑔!% 𝑔!& 𝑔!'
+𝑔"! 𝑔"" 𝑔"# 𝑔"$ 𝑔"% 𝑔"& 𝑔"'
+𝑔#! 𝑔#" 𝑔## 𝑔#$ 𝑔#% 𝑔#& 𝑔#'
+𝑔!!
+∗
+𝑔!"
+∗
+𝑔!#
+∗
+𝑔!$
+∗
+𝑔!%
+∗
+𝑔!&
+∗
+𝑔"!
+∗
+𝑔""
+∗
+𝑔"#
+∗
+𝑔"$
+∗
+𝑔"%
+∗
+𝑔"&
+∗
+𝑔#!
+∗
+𝑔#"
+∗
+𝑔##
+∗
+𝑔#$
+∗
+𝑔#%
+∗
+𝑔#&
+∗
+Figure 6: Query execution example. EQUI-VOCAL reduces
+intermediate result sizes by computing the earliest match-
+ing sequences for each region graph specification.
+represents the complexity of the query 𝑞.
+𝑅(𝑞) =
+𝑘
+∑︁
+𝑖=1
+�𝛼1𝑛𝑝𝑖 + 𝛼2𝑛𝑑𝑖 + 𝛼3𝑛𝑑𝑖𝑛𝑝𝑖
+�
+Here, 𝑘 is the number of region graphs in 𝑞, 𝑛𝑝𝑖 is the number of
+predicates in the 𝑖th region graph 𝑔𝑖, and 𝑛𝑑𝑖 denotes the scale of
+the duration constraint of 𝑔𝑖. 𝛼1, 𝛼2, and 𝛼3 are hyperparameters
+that control the importance of each term.
+4
+QUERY EXECUTION
+As discussed in Section 3.3, we convert queries in our DSL into
+SQL and use a relational engine (PostgreSQL in our prototype) to
+execute them. In this section, we discuss important optimizations
+that we apply during this translation process.
+Query translation algorithm. In Section 2.2, we show that
+queries over videos are naturally recursive. Such queries, however,
+are slow to execute. To avoid recursion, we leverage the observation
+that those queries express the idea of iteratively matching a region
+graph in a contiguous sequence of frames, rather than arbitrary re-
+cursion. Therefore, we can express them using window functions
+instead of recursion.
+As a second optimization, we note that a query 𝑞 in our DSL
+(Section 3.3) takes as input a set of video segments 𝑉 and returns
+all segments in 𝑉 containing at least one event that matches 𝑞.
+Therefore, instead of finding all satisfying events for each video
+segment, the query execution needs to find only one. We leverage
+this observation to reduce intermediate result sizes by producing
+SQL that efficiently computes the earliest matching sequences for
+each region graph specification. This optimization applies to queries
+without window specifications and whose duration constraints are
+of the form “>” or “>=”, which are the only types of queries that
+EQUI-VOCAL currently generates.
+As a concrete example, consider Figure 6. Assume we want to
+execute a query 𝑞 = Duration(𝑔1, 2); Duration(𝑔2, 2);𝑔3, where
+𝑔1 = Far(𝑜1,𝑜2), 𝑔2 = Near(𝑜1,𝑜2) and 𝑔3 = Behind(𝑜1,𝑜2). Also,
+we only consider one permutation of objects for each video segment
+in this example for simplicity. The event that 𝑞 looks for consists
+of a contiguous sequence of 𝑔1 for at least 𝑑1 = 2 frames, followed
+by a contiguous sequence of 𝑔2 for at least 𝑑2 = 2 frames, etc.
+Consider video segment 𝑣1. For the first region graph 𝑔1, we
+only need to extract the earliest matching segment of 𝑔1, which is
+𝑒11 = {𝑔12,𝑔13}. This is because, for every satisfying event for the
+query, we can replace its matching segment of 𝑔1 with the earliest
+one and the resulting event will still be a match to the query. 𝑒2 =
+{𝑔13,𝑔14,𝑔25,𝑔26,𝑔37} is a matching event for 𝑞. By replacing 𝑒12 =
+7
+
+VCREATE VIEW g1_windowed AS (
+SELECT vid, fid,
+oid1, oid2,
+lead(fid, d1 - 1, 0) OVER (
+PARTITION BY vid, oid1,
+oid2 ORDER BY fid
+) as fid_offest
+FROM g1
+);
+(a) Q1: window
+CREATE VIEW g1_iterated AS (
+SELECT vid,
+min(fid_offest) AS fid,
+oid1, oid2
+FROM g1_windowed
+WHERE fid_offest
+= fid + (d1 - 1)
+GROUP BY vid, oid1, oid2
+);
+(b) Q2: iterate
+CREATE VIEW g2_filtered AS (
+SELECT t1.vid, t2.fid,
+t1.oid1, t1.oid2
+FROM g1_iterated t1, g2 t2
+WHERE t1.vid = t2.vid
+AND t1.oid1 = t2.oid1
+AND t1.oid2 = t2.oid2
+AND t1.fid < t2.fid
+);
+(c) Q3: filter
+Figure 7: SQL snippets. EQUI-VOCAL generates efficient SQL
+queries by avoiding recursions and pruning intermediate re-
+sults.
+{𝑔13,𝑔14} with 𝑒11 = {𝑔12,𝑔13}, we get 𝑒1 = {𝑔12,𝑔13,𝑔25,𝑔26,𝑔37},
+which is still a matching event for 𝑞. For subsequent region graph
+𝑔2, we first use the earliest matching segment from 𝑔1 to retain
+matching frames of 𝑔2 that are temporally after the end frame of
+the segment, which are 𝑔25 and 𝑔26. Then, we can use the same
+method as before to find the earliest matching segment from 𝑔2,
+which is 𝑒21 = {𝑔25,𝑔26}. The same procedure can be applied to 𝑔3,
+giving us 𝑒31 = {𝑔37}. Thus, the earliest matching event for 𝑞 in 𝑣1
+is 𝑒1. Video segment 𝑣2 does not contain any matching event of 𝑞.
+The above algorithm can be expressed using SQL queries, as
+shown in Figure 7. Q1 partitions the data on video ID and each
+unique permutation of groups of objects so that each group of
+objects is analyzed separately, and sorts the data on frame ID. Next,
+for each row, it gets the current frame ID as well as the frame ID that
+is 𝑑1 frames after the current frame ID, or 0 if no such frame exists.
+The result of Q1 gives a table of intervals. Q2 then finds the earliest
+matching segment for each combination of video and groups of
+objects. Together, Q1 and Q2 find the earliest matching segment for
+Duration(𝑔1, 2) without using recursive joins. Q3 uses the earliest
+matching segment from 𝑔1 to determine where to look for matching
+segments in 𝑔2. The example SQL snippets only contain two objects
+oid1 and oid2. In general, Q1 and Q2 partition and group the
+data by all objects in the region graph 𝑔𝑖, and Q3 specifies all the
+constraints on objects across two region graphs in the where clause.
+We demonstrate the impact of both optimizations in Section 5.3.
+Caching. We implement an application-level cache to reuse sub-
+query results. EQUI-VOCAL generates SQL queries as described
+above to find matching region graphs in the sequence specified
+by the DSL query one by one. We cache the intermediate results
+of all prefixes on the set of video segments to allow other queries
+sharing the same sub-queries to reuse the results. For example,
+when executing the query 𝑞 = 𝑔1;𝑔2;𝑔3 on video segments 𝑣1 and
+𝑣2, we cache the results of 𝑞1 = 𝑔1, 𝑞2 = 𝑔1;𝑔2, and 𝑞3 = 𝑔1;𝑔2;𝑔3
+on 𝑣1 and 𝑣2. Later on, if a query 𝑞′ = 𝑔1;𝑔2;𝑔4 is executed on 𝑣1
+and 𝑣3, EQUI-VOCAL will reuse the cached results of 𝑞2 = 𝑔1;𝑔2
+on 𝑣1 to improve performance.
+Bounding the search space. EQUI-VOCAL synthesizes queries
+up to a certain maximum size to ensure that the synthesis algo-
+rithm eventually terminates. Assume the number of user-defined
+functions for object classes and attributes is 𝑚1 and the number
+of user-defined functions for relationships is 𝑚2. We bound the
+query search space using a set of hyperparameters: the maximum
+number of region graphs (𝑛𝑔) in a query, the maximum number
+of predicates (𝑛𝑝) in the query, the number of possible values a
+duration constraint can take (𝑛𝑑), and the number of unique ob-
+jects allowed in a query (𝑛𝑣). The size of the search space is then
+|𝑄| = 𝑂((𝑛𝑔(𝑛𝑣𝑚1 + 𝑛2𝑣𝑚2))𝑛𝑝 · 𝑛𝑛𝑔
+𝑑 ).
+Predicate evaluation. The current implementation of EQUI-
+VOCAL populates the Objects relation and the property attributes
+in the Attributes relation before query synthesis (see Table 1),
+by executing ML models on all video frames. The Relationships
+relation and the location attributes in the Attributes relation are
+computed lazily during query execution since they do not require
+ML models in our prototype, and are thus inexpensive to compute.
+In general, optimizing what information is computed before query
+synthesis and what is computed lazily is not the focus of this paper.
+To simplify SQL query generation, we implement all predicates
+(except for join predicates) in EQUI-VOCAL as user-defined func-
+tions that take attributes and bounding box coordinates as input
+and return boolean values. In our prototype, we use PostgreSQL
+and implement user-defined functions for location and relation-
+ship predicates that operate on bounding boxes, and for property
+predicates that operate on key-value pairs.
+While EQUI-VOCAL provides the above user-defined functions,
+the user can also provide additional user-defined functions incre-
+mentally as they use the system, and they can be shared and reused
+across queries and users [61]. In practice, EQUI-VOCAL has to limit
+the number of user-defined functions to a reasonable amount to
+ensure the efficiency of query synthesis.
+5
+EVALUATION
+We conduct an experimental evaluation of EQUI-VOCAL. First,
+we show that, compared to existing systems, EQUI-VOCAL reduces
+the query synthesis time by 1-2 orders of magnitude, achieves
+comparable or better F1 scores under the same labeling budget,
+is more robust to noisy data, and explores and executes queries
+more efficiently. Second, we show that EQUI-VOCAL is capable of
+synthesizing more complex, flexible queries over arbitrary scene
+graphs, which existing systems cannot handle. Finally, we conduct
+an ablation study by varying the various design choices outlined in
+the previous sections.
+Baselines. To the best of our knowledge, there are no existing
+systems that can synthesize queries over scene graphs. The most
+similar system to EQUI-VOCAL is Quivr [46], which synthesizes
+queries over trajectories. Since object trajectories can be repre-
+sented using scene graphs, we compare our system against Quivr.
+We also compare against PATSQL [56], which is a state-of-the-art
+query-by-example system for relational data.
+Predicates. We use 13 predicates in experiments: six relationship
+predicates (Near, Far, LeftOf, RightOf, FrontOf, and Behind),
+four location predicates (Left, Right, Top, Bottom), and three prop-
+erty predicates (Color, Material, and Shape). We consider eight
+colors, two materials, and three shapes. Property predicates are
+only used for the scene graphs dataset. For duration constraints, we
+consider three possible values: duration(𝑔) ≥ 5, 10, or 15 frames.
+Section 4 discussed how EQUI-VOCAL evaluates these predicates.
+Data. We evaluate our system on the CLEVRER dataset [66], which
+comprises synthetic videos of moving objects. Each video is five
+seconds long with a resolution of 480 × 320. This dataset includes
+8
+
+EQUI-VOCAL: Synthesizing Queries for Compositional Video Events from Limited User Interactions
+ID
+Query
+Description
+Pos %
+TQ1
+(Near(𝑜1,𝑜2), Bottom(𝑜1))
+𝑜1 is close to 𝑜2 while 𝑜1 is at the bottom.
+7.0%
+TQ2
+Far(𝑜1,𝑜2); Near(𝑜1,𝑜2); Far(𝑜1,𝑜2)
+Two objects move from far to close, then to far again.
+7.5%
+TQ3
+Far(𝑜1,𝑜2); (Near(𝑜1,𝑜2), Behind(𝑜1,𝑜2))
+𝑜1 and 𝑜2 are far away, then they move close and 𝑜1 is behind 𝑜2.
+12%
+TQ4
+Far(𝑜1,𝑜2); (Near(𝑜1,𝑜2), Behind(𝑜1,𝑜2), Left(𝑜1))
+𝑜1 and 𝑜2 are far apart, then they move close and 𝑜1 is behind 𝑜2
+and 𝑜1 is on the left.
+6.5%
+TQ5
+(FrontOf(𝑜1,𝑜2), Top(𝑜1))
+𝑜1 is in front of 𝑜2 while 𝑜1 is at the top.
+45%
+TQ6
+Near(𝑜1,𝑜2); Far(𝑜1,𝑜2)
+Two objects move from close to far apart.
+10%
+TQ7
+(Near(𝑜1,𝑜2), Left(𝑜1), Behind(𝑜1,𝑜2))
+𝑜1 is close to and behind 𝑜2 while 𝑜1 is on the left.
+8.5%
+TQ8
+(Far(𝑜1,𝑜2), Bottom(𝑜1)); Near(𝑜1,𝑜2)
+𝑜1 at the bottom is far from 𝑜2, then they move close.
+6.9%
+TQ9
+(Far(𝑜1,𝑜2), Left(𝑜1)); (Near(𝑜1,𝑜2), Left(𝑜1))
+𝑜1 and 𝑜2 move from far to close while 𝑜1 is on the left.
+10%
+TQ1D
+Duration(Far(𝑜1,𝑜2), 5); Near(𝑜1,𝑜2); Far(𝑜1,𝑜2)
+Two objects are far apart for at least 5 frames, then they move
+close, then they are far again.
+5.6%
+TQ2D
+Duration(LeftOf(𝑜1,𝑜2), 5); (Near(𝑜1,𝑜2), Top(𝑜1));
+Duration(RightOf(𝑜1,𝑜2), 5)
+𝑜1 is on the left of 𝑜2 for at least 5 frames, then they move close
+to each other and 𝑜1 is at the top, then 𝑜1 is on the right of 𝑜2 for
+at least 5 frames.
+6%
+TQ3D
+Duration((Frontof(𝑜1,𝑜2), Left(𝑜1)), 15);
+Duration((Left(𝑜1), RightOf(𝑜1,𝑜2), Top(𝑜1)), 5)
+𝑜1 is in front of 𝑜2 while 𝑜1 is on the left for at least 15 frames,
+then 𝑜1 moves to the right of 𝑜2 while 𝑜1 is at the top left for at
+least 5 frames.
+8.3%
+Table 3: Example queries written for the trajectories dataset. TQ1D-TQ3D include duration constraints. “Pos %” is the percent-
+age of positive examples in the dataset.
+ID
+Query
+Description
+Pos %
+SQ1
+Duration((Far(𝑜1,𝑜2), Right(𝑜2)), 15);
+(FrontOf(𝑜1,𝑜2), Near(𝑜1,𝑜2));
+Duration((Far(𝑜1,𝑜2), Right(𝑜2)), 5)
+𝑜1 and 𝑜2 are far apart while 𝑜2 is on the right for at least 15
+frames, then they move close and 𝑜1 is in front of 𝑜2, then
+they are far apart while 𝑜2 is on the right for at least 5 frames.
+6.5%
+SQ2
+(Far(𝑜1,𝑜2), Shape(𝑜1, ‘sphere’), Color(𝑜3, ‘cyan’));
+(Near(𝑜1,𝑜2), FrontOf(𝑜1,𝑜2)); (Far(𝑜1,𝑜2), Top(𝑜2))
+A sphere 𝑜1 is far apart 𝑜2 while 𝑜3 in cyan is also in the
+frame, then 𝑜1 and 𝑜2 move close and 𝑜1 is in front of 𝑜2,
+then 𝑜1 and 𝑜2 are far apart with 𝑜2 at the top of the frame.
+6.8%
+SQ3
+Duration((LeftOf(𝑜1,𝑜2), Shape(𝑜2, ‘sphere’)), 15);
+(FrontOf(𝑜1,𝑜3), Near(𝑜1,𝑜3));
+Duration((Behind(𝑜1,𝑜3), Far(𝑜1,𝑜3)), 5)
+𝑜1 is on the left of a sphere 𝑜2 for at least 15 frames, then 𝑜1
+is in front of and close to 𝑜3, then 𝑜1 is behind and far away
+from 𝑜3 for at least 5 frames.
+5.8%
+Table 4: Example queries written for the scene graphs dataset. “Pos %” is the percentage of positive examples in the dataset.
+a variety of geometric shapes interacting in space and time, and
+comes with ground truth data, facilitating testing queries with
+varying complexities. We create two benchmarks from this data:
+trajectories and scene graphs datasets.
+Trajectories dataset. We create a first dataset to test queries
+over trajectories, which baseline systems support. We extract
+10,080 pairs of object trajectories that overlap in time from 500,
+5-second video segments (128 frames each). Each trajectory pair is
+essentially a temporally ordered sequence of bounding box pairs
+𝑏 = (𝑥𝑎1,𝑦𝑎1,𝑥𝑎2,𝑦𝑎2,𝑥𝑏1,𝑦𝑏1,𝑥𝑏2,𝑦𝑏2) representing two objects
+in a video segment. We manually generate a set of queries with
+varying complexities (Table 3); TQ1D-TQ3D include duration con-
+straints. To generate ground truth labels, we run each target query
+on the dataset. We sample 500 trajectory pairs as training data (i.e.,
+data used during query synthesis) and use the rest as test data (i.e.,
+to measure the quality of synthesized queries).
+Scene graphs dataset. This more complex dataset contains scene
+graphs extracted from the CLEVRER dataset, which baseline sys-
+tems do not support. We extract scene graphs from 10,000, 5-second
+video segments (128 frames each). For every frame of a video seg-
+ment, we store the object track ID (𝑜𝑖𝑑), bounding box coordinates,
+and object attributes (shape, color, and material) of every object in
+the frame (the Objects and Attributes relations in Table 1). As
+for trajectories, we sample 500 video segments as training data and
+use the rest as test data.
+We manually write three example queries shown in Table 4 to il-
+lustrate the kinds of more complex queries that can be asked on the
+scene graphs dataset. In our experiments, we automatically generate
+three classes of queries with different complexities: easy, medium,
+and hard. Each generated query contains exactly three variables
+(i.e., three distinct objects). Easy queries have exactly three predi-
+cates on relationships and locations, one property predicate, one
+region graph, and no duration constraints; medium queries have
+exactly five predicates on relationships and locations, two property
+9
+
+Method
+Simplified
+Normal
+TQ1
+TQ2
+TQ3
+TQ4
+TQ5
+TQ6
+TQ7
+TQ8
+TQ9
+TQ1
+TQ2
+TQ3
+TQ4
+TQ1D
+TQ2D
+TQ3D
+PATSQL
+1.54
+—
+538
+—
+4.00
+1.93
+552
+—
+—
+NA
+NA
+NA
+NA
+NA
+NA
+NA
+Quivr
+12.6
+6.11
+31.9
+94.8
+14.5
+5.18
+52.7
+21.3
+—
+8267
+6068
+8296
+—
+—
+—
+—
+Ours
+3.57
+1.85
+17.1
+46.6
+4.36
+1.66
+18.5
+22.7
+20.6
+75.6
+125
+106
+185
+223
+166
+183
+Table 5: Query synthesis time (median, in seconds) for each method to achieve at least 0.9 F1 scores. EQUI-VOCAL successfully
+learns all queries with at least 0.9 F1 scores and is faster than (or at least comparable to) the two baselines. (NA: not applicable,
+—: failed due to insufficient F1 score or timeout)
+predicates, three region graphs, and no duration constraints; hard
+queries have the same complexity as medium queries but also in-
+clude duration constraints with three possible values (i.e., 5, 10, and
+15 frames). Each class contains 40 queries.
+Metrics. We report the F1 score and query synthesis time. We
+synthesize queries using the training set and report the F1 score
+of the query (or the median F1 score if there are multiple queries
+with the same best score on the training set) over the test set. Each
+experiment is run 20 times for the trajectories dataset and five times
+for the scene graphs dataset, and we report the median F1 score
+and median query synthesis time over these runs.
+Implementation details. We implement our prototype in Python
+with PostgreSQL as the backend. We conduct all experiments on
+a computing cluster. When measuring runtimes, we request one
+node with Intel Xeon Gold 6230R CPUs at 2.10GHz and 100GB of
+RAM. Unless otherwise specified, we configure EQUI-VOCAL as
+follows. For the trajectories dataset, we search for queries with
+up to 5 predicates across up to 3 region graphs. We use beam
+width 𝑏𝑤 = 10, 𝜆 = 0.01 for regularization (with 𝛼1 = 1, 𝛼2 = 1,
+and 𝛼3 = 0.1, see Section 3.7), and we set 𝑘 = 100. For the scene
+graphs dataset, we search for queries with up to 7 predicates, 3
+region graphs, and 3 objects. Since the dataset is more complex
+and challenging, we use a smaller 𝜆 (because the query complexity
+term, 𝑅(𝑞) has a greater absolute value) and a greater 𝑘. We use
+beam width 𝑏𝑤 = 10, 𝜆 = 0.001 for regularization (with 𝛼1 = 1,
+𝛼2 = 1, and 𝛼3 = 0.1), and we set 𝑘 = 1000.
+We consider two variants of Quivr. As per the original paper, we
+limit the number of atomic predicates in Quivr’s queries to 5 and
+the depth of the nested constructs to 3, which leads to a similar
+search space as EQUI-VOCAL. When considering queries without
+duration constraints (TQ1-TQ9), we omit Kleene star operators from
+its search space. Otherwise (TQ1D-TQ3D), we include Kleene star
+in the query expansion and add one more predicate MinLength𝜃,
+which checks whether the duration of the input is at least 𝜃 frames.
+Quivr returns all queries that match the examples, so we select the
+queries with the simplest structure (which is determined by the
+number of atomic predicates, and, if the former is the same, by the
+depth) and report their median F1 score.
+Because PATSQL requires that all user-provided input tables be
+used in the solution query, when comparing against PATSQL, we
+restrict all systems to only the candidate predicates that appear
+in the target query. Because PATSQL cannot handle large tables
+efficiently, we also downsample each trajectory by 75% (we keep
+one frame out of four) to reduce the size of the input tables. We
+refer to this configuration as simplified tasks.
+5.1
+Results against baselines on trajectories
+We evaluate EQUI-VOCAL against the two baselines on the trajecto-
+ries dataset and the set of queries in Table 3. For TQ1-TQ9, we omit
+duration constraints from the search space. We run each method as
+follows. For each target query, we randomly select 2 positive and 10
+negative examples from the training set and use them as the input
+to the method. Each method then asks for 𝑏 additional examples
+during the search. For PATSQL, since it is not interactive, we simply
+sample 𝑏 more examples randomly from the remaining dataset and
+provide the 12 + 𝑏 examples to the system at the beginning. For
+Quivr and EQUI-VOCAL, we input the 12 initial examples, and each
+system actively requests more examples during the search process.
+EQUI-VOCAL is faster than baselines and can find high-
+performant queries even for complex queries. Table 5 shows
+the query synthesis time of each query to achieve at least a 0.9 F1
+score. For simplified tasks, EQUI-VOCAL outperforms baselines
+on 6 queries. PATSQL performs the best or close to the best when
+queries are simple (TQ1, 5, and 6) because these queries include
+only two predicates and PATSQL terminates once it finds one so-
+lution query. Quivr is slower than EQUI-VOCAL but comparable
+to it except for TQ9, in which case Quivr fails due to an insuffi-
+cient F1 score of the synthesized queries. Under the normal setting,
+EQUI-VOCAL is significantly more efficient than Quivr and can find
+high-performance queries in hundreds of seconds even for complex
+queries with duration constraints, while Quivr cannot synthesize
+TQ4 and TQ1D-TQ3D within 4 hours due to the large number of
+queries that need to be explored in the enumerative search. For
+TQ5-TQ9 on normal tasks (not shown in the table due to space con-
+straints), EQUI-VOCAL takes 50.4, 61.1, 106, 110, and 107 seconds
+to synthesize queries with at least 0.9 F1 scores, while Quivr takes
+7798 seconds for TQ5, 7366 seconds for TQ6, and cannot synthesize
+TQ7-TQ9 within 4 hours.
+On simplified tasks, EQUI-VOCAL and QUIVR find
+queries with similar F1 scores and outperform PATSQL for
+the same labeling budget. Figure 8 shows the F1 score of each
+system when varying the user labeling budget on the simplified
+tasks (which all systems can perform). We assign an F1 score of 0 if
+a system fails to find a query in 1 hour. Across all labeling budgets
+tested, as above, PATSQL performs well when the target query is
+simplest (TQ1, 5, and 6). For other queries, PATSQL either fails
+to find a solution query within 1 hour or the solution query has
+low performance (below a 0.9 F1 score). When the budget is 30, we
+observe a decrease in F1 scores for TQ3, 7, 8, and 9, because PATSQL
+has a less constrained search space than EQUI-VOCAL and does not
+scale well when the size of input and output tables increases as more
+examples are provided. Quivr performs better than EQUI-VOCAL
+10
+
+EQUI-VOCAL: Synthesizing Queries for Compositional Video Events from Limited User Interactions
+TQ1 TQ2 TQ3 TQ4 TQ5 TQ6 TQ7 TQ8 TQ9
+Query ID
+0.6
+0.8
+1.0
+F1 score
+Labeling budget: 12
+TQ1 TQ2 TQ3 TQ4 TQ5 TQ6 TQ7 TQ8 TQ9
+Query ID
+Labeling budget: 20
+TQ1 TQ2 TQ3 TQ4 TQ5 TQ6 TQ7 TQ8 TQ9
+Query ID
+Labeling budget: 30
+PATSQL
+Quivr
+EQUI-VOCAL
+Figure 8: F1 score for queries without duration constraints on simplified tasks, under different user labeling budgets. PATSQL
+performs well only when the target query is simple. EQUI-VOCAL performs worse than Quivr when no additional examples
+are requested, but catches up and outperforms Quivr with a larger labeling budget.
+FN rate
+Method
+Quivr
+Ours
+Mean (Range)
+Mean
+0.1
+68% (20%-100%)
+100%
+0.2
+54% (35%-75%)
+100%
+0.3
+47% (5%-75%)
+100%
+0.4
+34% (5%-55%)
+100%
+0.5
+17% (0%-50%)
+100%
+Table 6: Probability that a system returns at least one query
+on noisy data. (false positive rate is 0.1 of false negative rate)
+0.0
+0.5
+1.0
+F1 score
+Label noise: FN rate=0.1, FP rate=0.01
+TQ1
+TQ2
+TQ3
+TQ4
+TQ5
+TQ6
+TQ7
+TQ8
+TQ9
+Query ID
+0.0
+0.5
+1.0
+F1 score
+Label noise: FN rate=0.3, FP rate=0.03
+Quivr
+EQUI-VOCAL
+Figure 9: Impact of data noise, with a labeling budget of
+20. EQUI-VOCAL constantly outperforms Quivr and is thus
+more robust to data noise.
+when no additional examples are requested (budget=12) because it
+enumerates the entire search space and finds all consistent queries
+using the initial examples, while EQUI-VOCAL prunes candidate
+queries at every iteration to ensure efficient query synthesis. With
+a larger labeling budget, EQUI-VOCAL can request more labels
+during the search process to select better paths to explore, thus it
+catches up with Quivr and outperforms it when the budget is 30.
+EQUI-VOCAL is more robust to noisy data and produces
+higher quality queries than Quivr. We compare the perfor-
+mance of Quivr and EQUI-VOCAL when the data is noisy. We
+do not compare against PATSQL since its performance on video
+queries is already low even with perfect data. We inject noise into
+the original dataset by randomly flipping a fraction of the labels.
+We vary the false negative rate from 0.1 to 0.5 and set the false
+positive rate to 0.1 the false negative one. Table 6 shows the per-
+centage of runs when the system returned any queries for different
+noise rates. We evaluate the systems over 9 queries (TQ1-TQ9) in
+the simplified setting. For Quivr, we also report the range of the
+ID
+# queries explored
+Quivr
+Ours
+TQ1
+203018
+1123
+TQ2
+186333
+1071
+TQ3
+214536
+1101
+TQ5
+179155
+1118
+TQ6
+213419
+1077
+ID
+# predictions/s
+Quivr
+Ours
+TQ1
+174
+574
+TQ2
+176
+666
+TQ3
+170
+664
+TQ5
+151
+480
+TQ6
+138
+547
+Table 7: Number of queries explored and number of pre-
+dictions per second to achieve at least 0.9 F1 scores. EQUI-
+VOCAL reduces computational effort by exploring fewer
+queries and by executing queries faster.
+success rate besides the mean value. When the false negative rate
+is 0.1, Quivr has a 68% success rate, and this goes down to only
+17% when the false negative rate is 0.5. In contrast, EQUI-VOCAL
+always returns 𝑘 queries. To demonstrate that EQUI-VOCAL also
+produces higher quality queries, Figure 9 shows the F1 score of
+EQUI-VOCAL and Quivr with a labeling budget of 20, under two
+different noise rates. When Quivr fails to return any queries, we
+assign an F1 score of 0. EQUI-VOCAL performs better than Quivr
+for different queries and different noise rates.
+EQUI-VOCAL is more computationally efficient than
+Quivr by exploring fewer queries and executing every query
+faster. Table 7 measures the number of queries explored and the
+number of predictions per second to achieve at least a 0.9 F1 score.
+We define a prediction as evaluating a query on a video segment,
+so the number of predictions per second reflects the efficiency of
+query execution. We evaluate over TQ1-TQ9 in the normal setting
+and report the numbers for queries that Quivr can synthesize. By
+limiting the query search to valid sequences of region graphs and
+by using the beam search strategy, the number of queries explored
+by EQUI-VOCAL is as small as 0.50% of Quivr. EQUI-VOCAL also
+executes queries faster than Quivr by up to 4.0× in terms of the
+number of predictions per second. These two factors together make
+EQUI-VOCAL more efficient than Quivr.
+5.2
+Results on scene graphs
+Next, we evaluate EQUI-VOCAL on the scene graphs dataset. Nei-
+ther PATSQL nor Quivr support such flexible queries at video scale.
+Because the dataset is more complex, EQUI-VOCAL requires that
+the user provide a slightly larger (although still quite small) number
+of initial examples to avoid overfitting. We randomly select 15 posi-
+tive and 15 negative examples from the training dataset as the initial
+11
+
+30
+50
+100
+Budget
+0.0
+0.5
+1.0
+F1 score
+Easy
+30
+50
+100
+Budget
+Medium
+30
+50
+100
+Budget
+Hard
+Figure 10: Scene graph queries. EQUI-VOCAL can synthesize
+high-quality queries within a reasonable budget, even for
+complex queries.
+1
+2
+4
+8
+# CPUs
+20
+40
+Time (min)
+Figure 11: Vary # CPUs. EQUI-VOCAL can be easily paral-
+lelized to reduce synthesis time.
+examples. Later in Section 5.3, we discuss how varying the initial
+number of examples affects the performance of EQUI-VOCAL.
+Figure 10 shows the F1 score of EQUI-VOCAL under different
+user labeling budgets. With a larger budget, EQUI-VOCAL can learn
+queries with higher F1 scores, and achieves a median F1 score of
+1.0, 0.91, and 0.67 for easy, medium, and hard queries when the
+budget is 100. Results for a budget of 50 are nearly identical, which
+shows that EQUI-VOCAL can synthesize high-quality queries
+within a reasonable budget, even for complex queries.
+EQUI-VOCAL does not learn good queries in all cases. We note
+that some queries are easier to synthesize than others, which results
+in the high variance of the F1 score in Figure 10. In particular, EQUI-
+VOCAL assumes that the ancestor queries of the target query are
+informative and leverages the performance of those queries to guide
+the search. However, if the ancestor queries cannot be distinguished
+from other queries in the same iteration (because they all have the
+same, typically low, F1 score), EQUI-VOCAL struggles with learning
+good queries. One direction of future work could explore other types
+of more fine-grained user feedback besides binary labels to help
+EQUI-VOCAL learn intermediate queries [53], while still limiting
+user and computational efforts.
+EQUI-VOCAL’s query synthesis time on the scene graphs dataset
+is greater than on the trajectories dataset. It increases from tens
+of seconds (Table 5) to minutes. However, EQUI-VOCAL can
+easily be parallelized to reduce synthesis time, as executing
+queries in PostgreSQL is embarrassingly parallel. Figure 11
+shows the query synthesis time for the hard, scene-graphs queries
+when varying the number of CPUs and for a labeling budget of 50.
+Query synthesis time decreases significantly by using multiple cores
+and reduces to 13 minutes with 8 CPUs. EQUI-VOCAL achieves
+1.76× speedup with 2 CPUs and 3.03× speedup with 8 CPUs. Note
+that this is the largest labeling budget for our experiments. With
+a lower budget, EQUI-VOCAL is even faster. The query synthesis
+time can be further reduced by using more CPUs, further decreasing
+the labeling budget, decreasing 𝑏𝑤, or decreasing 𝑘.
+10
+20
+30
+40
+50
+# Initial examples
+0.0
+0.5
+1.0
+F1 score
+Easy
+10
+20
+30
+40
+50
+# Initial examples
+Medium
+10
+20
+30
+40
+50
+# Initial examples
+Hard
+Figure 12: Providing more initial examples helps EQUI-
+VOCAL learn better queries and avoid overfitting.
+10
+20
+30
+40
+50
+# Initial examples
+4
+7
+10
+13
+16
+Query complexity
+1.00
+1.00
+1.00
+1.00
+1.00
+0.41
+0.72
+0.90
+0.93
+0.95
+0.41
+0.54
+0.69
+0.89
+0.84
+0.36
+0.52
+0.65
+0.77
+0.83
+0.21
+0.42
+0.60
+0.60
+0.64
+F1 score
+0.4
+0.6
+0.8
+Figure 13: F1 score for various numbers of initial exam-
+ples and queries with different complexity. EQUI-VOCAL ob-
+tains higher F1 scores with more initial examples and when
+the target queries are simpler.
+5.3
+Ablation study
+We study the impact of hyperparameters and the sensitivity of
+EQUI-VOCAL to hyperparameter tuning. Unless otherwise speci-
+fied, we evaluate EQUI-VOCAL on the scene graphs dataset and use
+the default values for hyperparameters that we are not studying.
+EQUI-VOCAL synthesizes useful queries with a small
+number of initial examples, but providing more initial ex-
+amples improves performance. Figure 12 shows EQUI-VOCAL’s
+F1 score for different numbers of initial examples, from five positives
+and five negatives (ten in total), to 25 positives and 25 negatives
+(50 in total), while the total labeling budget remains 100. When
+the number of initial examples is smallest, EQUI-VOCAL can incor-
+rectly retain queries that overfit the examples in early iterations,
+leading to final results with low F1 scores. Results, however, quickly
+improve, and with as few as 30 initial examples, median F1 scores
+are high at 0.998, 0.911, and 0.670 for the easy, medium, and hard
+queries respectively. Figure 13 shows the F1 score of EQUI-VOCAL
+under different numbers of initial examples and queries with dif-
+ferent complexity, while the total labeling budget remains 100. As
+expected, EQUI-VOCAL obtains higher F1 scores with more initial
+examples and when the target queries are simpler. The heatmap
+aggregates the results of all easy, medium, and hard queries, and
+we use 𝛼1 = 1, 𝛼2 = 0.9, 𝛼3 = 0.2 when computing the query com-
+plexity for better visualization. Because manually finding initial
+examples requires more user effort than labeling video segments
+selected by EQUI-VOCAL, our approach is to balance user effort
+and system’s performance, using 30 initial examples in all other
+experiments on the scene graphs dataset. Interestingly, since EQUI-
+VOCAL can produce good queries in many cases even with a few
+examples, users always have the option to first start with a small
+set and restart with more examples if needed.
+12
+
+EQUI-VOCAL: Synthesizing Queries for Compositional Video Events from Limited User Interactions
+TQ1
+TQ2
+TQ3
+TQ4
+TQ5
+TQ6
+TQ7
+TQ8
+TQ9
+TQ1D
+TQ2D
+TQ3D
+0.0
+0.5
+1.0
+F1 score
+Trajectories (budget: 20)
+Easy
+Medium
+Hard
+0.0
+0.5
+1.0
+F1 score
+Scene graphs (budget: 50)
+=0.0
+=0.001
+=0.01
+Figure 14: Regularization (𝜆 = 0.001 and 𝜆 = 0.01) helps EQUI-VOCAL to learn better queries in many cases and does not harm
+the performance in other cases.
+10
+20
+30
+40
+# Initial examples
+0.0
+0.5
+1.0
+F1 score
+Easy
+10
+20
+30
+40
+# Initial examples
+Medium
+10
+20
+30
+40
+# Initial examples
+Hard
+Active learning
+Random sampling
+Figure 15: Active learning helps EQUI-VOCAL learn better
+queries compared to random sampling, and the improve-
+ment is more significant when the number of initial exam-
+ples is larger.
+Regularization helps EQUI-VOCAL learn better queries in
+many cases and does not harm performance in other cases.
+Figure 14 shows the F1 score of EQUI-VOCAL on both the trajec-
+tories dataset and the scene graphs dataset, under different values
+of 𝜆. For the trajectory pairs dataset, we set the labeling budget
+to 20 and the initial number of examples to 12 (i.e., two positive
+and ten negative examples); for the scene graphs dataset, we set
+the labeling budget to 50 and the initial number of examples to
+30. For the trajectories dataset, we see an improvement for many
+queries with regularization, especially when the queries are simple
+(e.g., with fewer predicates). With 𝜆 = 0.01, we see a median im-
+provement between 44.6% and −3.44% for the queries from Table 3.
+TQ4, TQ9, TQ2D, and TQ3D are more complex than other queries
+and have four predicates, so applying regularization is unlikely
+to improve the performance. For the more complex scene graphs
+dataset, regularization does not improve F1 scores but also does
+not compromise system performance. With 𝜆 = 0.001, the median
+change in F1 score stays between 2.84% and −7.69%. For this reason,
+EQUI-VOCAL defaults to using a small regularization factor.
+Active learning helps EQUI-VOCAL learn better queries
+quicker than randomly sampling video segments to label.
+Figure 15 shows EQUI-VOCAL’s F1 score when either using ac-
+tive learning or randomly selecting additional examples during
+query synthesis. We set the labeling budget to 50, and we vary the
+number of initial examples. The improvement of active learning
+over random sampling is more significant when the number of ini-
+tial examples is larger since fewer labels are requested per iteration
+and the selection of informative examples becomes more important.
+1
+5
+10
+15
+20
+bw
+0.0
+0.5
+1.0
+F1 score
+Easy
+1
+5
+10
+15
+20
+bw
+Medium
+1
+5
+10
+15
+20
+bw
+Hard
+Figure 16: Increasing𝑏𝑤 improves the performance of EQUI-
+VOCAL.
+1
+10
+100 1000
+k
+0.0
+0.5
+1.0
+F1 score
+Easy
+1
+10
+100 1000
+k
+Medium
+1
+10
+100 1000
+k
+Hard
+Figure 17: Increasing 𝑘 slightly improves the performance
+of EQUI-VOCAL.
+Increasing the beam width improves performance, but
+also increases the synthesis time. Figure 16 shows the F1 score
+of EQUI-VOCAL under different values of 𝑏𝑤. Increasing 𝑏𝑤 from
+1 to 10 increases median F1 scores on easy queries by 3.10%, on
+medium queries by 23.8%, and on hard queries by 34.3%, but in-
+creasing from 10 to 20 only further improves median scores by up
+to 3.07%. At the same time, increasing 𝑏𝑤 from 10 to 15 increases
+the median synthesis time on hard queries by 27.8%. EQUI-VOCAL
+defaults to 𝑏𝑤 = 10 to strike a balance between query performance
+and synthesis time.
+increasing 𝑘 slightly improves performance. We evaluate
+the F1 score of EQUI-VOCAL under different values of 𝑘, as shown
+in Figure 17. For medium and hard scene graph queries, increas-
+ing 𝑘 from 1 to 1000 slightly improves the median F1 score by
+2.36% and 9.12% respectively. Since the target queries are complex,
+they are not explored until later iterations of the algorithm, when
+most labeling budget has been used. Labels requested afterward
+have little impact on the ranking of the queries in the list, so the
+performance improvement from increasing 𝑘 is also limited. Easy
+queries already have near-perfect F1 scores, so increasing 𝑘 does
+not improve performance.
+13
+
+SQ1
+SQ2
+SQ3
+0
+50
+100
+Time (s)
+EQUI-VOCAL
+No-Pruning
+No-Recursion
+ -Avoidance
+(a) Single query execution time with
+different SQL optimizations.
+Caching
+No caching
+0
+20
+40
+Time (min)
+(b) Synthesis time with
+and without caching.
+Figure 18: Impact of query execution optimizations. EQUI-
+VOCAL reduces query synthesis time by (a) generating effi-
+cient SQL queries that prune intermediate results and avoid
+recursion, and (b) using a caching mechanism.
+EQUI-VOCAL’s query execution optimizations signifi-
+cantly reduce query synthesis times. Figure 18a shows the av-
+erage query execution time for variants of our query translation
+algorithm on scene graph queries. We generate SQL queries using
+our complete query translation algorithm, and two variants that
+omit optimizations of pruning intermediate results and avoiding
+recursion, respectively, and then execute them over all 10,000 video
+segments from the scene graphs dataset in PostgreSQL. We evaluate
+EQUI-VOCAL on SQ1-SQ3, and each query is executed 20 times.
+As the figure shows, the query execution time increases dramat-
+ically without either optimization. Pruning intermediate results
+boosts performance by up to 1.66×. Recursion avoidance is impor-
+tant when the query contains duration constraints (SQ1 and SQ3),
+providing up to a 4.14× speedup.
+Figure 18b shows the impact of our caching mechanism. Caching
+query results leads to a 1.97× speedup. We experimented on hard
+queries using four CPUs and a labeling budget of 50.
+6
+RELATED WORK
+Video analytics systems. Many recent VDBMSs have been pro-
+posed and focus on a wide range of data management challenges,
+including fast inference over videos [3, 5, 34, 44, 47], storage op-
+timization [18, 25, 64], efficient dataflow processing [49], prepro-
+cessing and indexing [6, 28, 36], exploration and organization [19],
+privacy [9], and tuning configurations [29, 32, 52]. Those techniques
+are orthogonal to our work.
+Compositional query processing over videos. Several sys-
+tems have explored compositional queries over videos. However,
+they either require that users express compositional queries ex-
+plicitly [14, 21, 42], or train customized models for such events [7,
+11, 17]. Instead, EQUI-VOCAL uses a query-by-example approach
+to minimize user effort and learn and refine a query specification
+iteratively from user feedback.
+As detailed in Section 2.3, TQVS [15] and Chen et al. [16] limit
+the types of compositional queries to temporal queries that count
+co-occurring objects. Vaas [7] enables users to execute a complex
+query by combining the output of simpler queries, but does not
+introduce any algebra to reason about spatiotemporal events, and
+SVQ++ [11] only supports spatial queries. In contrast, EQUI-VOCAL
+introduces a more general data model that supports compositional
+queries involving relationships between objects in space and time.
+Accelerating query execution over videos. Prior work on re-
+cent VDBMSs focuses on accelerating queries by filtering frames
+using cheaper predicates [44], optimizing the sampling rate [5, 47],
+and building specialized models [3, 34]. EQUI-VOCAL currently
+extracts all objects in a video before the search process, but could
+leverage those existing techniques to avoid running expensive ob-
+ject detection and tracking algorithms on all frames.
+Query by example. Many query-by-example systems have
+been proposed to support users in expressing SQL queries over
+relational data [20, 41, 50, 56, 58]. EQUI-VOCAL focuses on learn-
+ing queries for video events, which can be expressed using a form
+that is more constrained than general relational queries, but also re-
+quires greater scalability and tolerance to noise. The most relevant
+work to our system is Quivr [46], which also targets video queries.
+However, Quivr only operates over object trajectories instead of
+entire scene graphs, and assumes noiseless inputs. SQuID [20] syn-
+thesizes queries based on semantic similarity but does not support
+self-joins and non-equi joins, which are necessary to find our video
+events, and S4 [50] ranks queries based on the degree of input
+containment but is limited to project-join queries.
+Program synthesis. Program synthesis has been used in recent
+research for a wide range of tasks. Huang et al. [30] generates re-
+ferring relational programs to uniquely identify an object in an
+image in terms of its attributes and relations with other objects.
+Falx [59] supports users in authoring visualizations from examples.
+Wrangler [33] learns relational data transformation through user
+interactions. BUSTLE [48] synthesizes programs for string process-
+ing. Unlike these systems, EQUI-VOCAL synthesizes queries in the
+domain of videos, which is significantly different in terms of the
+spatio-temporal complexity and the prevalence of noise.
+Active learning. EQUI-VOCAL has an active learning compo-
+nent to select the most promising queries to expand, instead of
+training a better classifier. Model Picker [37] considers a set of pre-
+trained classifiers and proposes an active learning approach that
+selects informative examples to distinguish the best model in an
+online setting. EQUI-VOCAL adapts this approach to distinguish
+the most promising queries to explore at each iteration. Quivr [46]
+also prunes candidate queries using active learning, but only af-
+ter enumerating all candidates. EQUI-VOCAL instead integrates
+active learning and labeling into the search process to expand only
+the most promising queries in each iteration. This ensures both
+synthesis efficiency and query performance.
+7
+CONCLUSION
+In this paper, we presented EQUI-VOCAL, a new system that synthe-
+sizes compositional queries from examples. EQUI-VOCAL models
+compositional events as spatio-temporal scene graphs, explores the
+query search space using results of executing intermediate queries
+and beam search, leverages active learning to reduce user effort,
+and generates efficient SQL queries to reduce computational effort.
+ACKNOWLEDGMENTS
+This work was supported in part by the NSF through awards CCF-
+1703051 and IIS-2211133 as well as a grant from CISCO.
+14
+
+EQUI-VOCAL: Synthesizing Queries for Compositional Video Events from Limited User Interactions
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf,len=1336
+page_content='EQUI-VOCAL: Synthesizing Queries for Compositional Video  Events from Limited User Interactions  Technical Report  Enhao Zhang1, Maureen Daum1, Dong He1,  Magdalena Balazinska1, Brandon Haynes2, and Ranjay Krishna1  1University of Washington,  {enhaoz, mdaum, donghe, magda, ranjay}@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='edu  2Microsoft Gray Systems Lab,  brandon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='haynes@microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='com  ABSTRACT  We introduce EQUI-VOCAL: a new system that automatically syn-  thesizes queries over videos from limited user interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The  user only provides a handful of positive and negative examples  of what they are looking for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL utilizes these initial  examples and additional ones collected through active learning to  efficiently synthesize complex user queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Our approach enables  users to find events without database expertise, with limited label-  ing effort, and without declarative specifications or sketches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Core  to EQUI-VOCAL’s design is the use of spatio-temporal scene graphs  in its data model and query language and a novel query synthesis  approach that works on large and noisy video data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Our system  outperforms two baseline systems—in terms of F1 score, synthesis  time, and robustness to noise—and can flexibly synthesize complex  queries that the baselines do not support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  1  INTRODUCTION  Video data is increasingly becoming a prized commodity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Inexpen-  sive large-scale video storage and advances in machine learning  and computer vision have propelled the use of large video datasets  with new applications including drone analytics [60, 62], citywide  traffic analytics [2, 24], civil engineering [4, 23], and many oth-  ers [22, 45, 49, 54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Although video database management sys-  tems (VDBMSs) have recently re-emerged as an active research  area to support these applications [3, 5, 17, 21, 34, 35, 42], existing  systems fall short of supporting many use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Consider a traffic analytics application: A traffic engineer may  want to understand road hazards involving car and motorcycle in-  teractions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', motorcycles swerving abruptly in front of turning  cars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Although many computer vision models exist that detect com-  mon objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', “cars” and “motorcycles”) [63] and relate objects  spatially (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', “bottom of”, “left of”, “near”) [12], a specific classi-  fier that identifies “a motorcycle swerving in front of a car, while  the car is turning at an intersection” is unlikely to exist [65, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Worse, training one would require many hours of user effort in  labeling for a single query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Given the relative rarity of most interest-  ing events, finding sufficient positive instances further exacerbates  these labeling requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In our example, there will be many in-  stances of cars and motorcycles in intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Only rarely would  a motorcycle swerve in front of a turning car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Assuming that we can run existing computer vision mod-  els on videos to identify objects, extract attributes, and reason  about their pairwise relationships, some recent video data man-  agement systems support users by providing an interface to ex-  press a declarative query as a composition of extracted informa-  tion [7, 11, 14, 21, 42, 46, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For example, a user might be able  EQUI-VOCAL  Query output:  matching video segments  Ask  for more  labels  Provide a  few binary  labels  Videos  User-defined functions  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', Fast R-CNN)  1  Event of  interest  2  5  3  4  Figure 1: Given 1○ a video dataset, 2○ user-defined functions  that extract semantic information from videos, and 3○ a few  user-provided labels, EQUI-VOCAL synthesizes a query to  find instances of an event of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' It iteratively 4○ asks  the user for more labels to reduce its uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Once syn-  thesized, it 5○ executes the query to return matching events  on unseen videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  to query for an event using a specification that searches for video  clips containing car and motorcycle objects, and specifying their  desired relationships (near then front_of, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' These systems  expect users to possess a level of database expertise to be able to  express such queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Additionally, real-world events can be difficult  to express declaratively—even for experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For example, there are  multiple ways to express our sample query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The best way depends  on the data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', in a given video, motorcycles may be swerving  from outside the frame, as illustrated in Figure 2, discussed later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Other similar systems ask users to sketch their events [10, 14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' this  is equally challenging for the same reason: an event of interest may  deviate from the exact user-provided sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  In this paper, we present EQUI-VOCAL (Figure 1), a system  that addresses the above challenge by synthesizing declarative  queries on behalf of users from a small number of labeled video  segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Put another way, EQUI-VOCAL Evolves Queries for  Users Iteratively and is part of our larger VOCAL system [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In  our example, a user provides as few as two positive and ten negative  examples containing the event of interest (see Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-  VOCAL then synthesizes a declarative query and executes it on the  remaining large pool of video data to identify other examples of the  desired event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The source code of EQUI-VOCAL is available at [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  EQUI-VOCAL supports users’ search for complex events without  requiring any specific domain knowledge, with limited effort, and  without requiring precise declarative specifications or sketches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Similar to other systems, we use the insight that while most user  queries are new and unseen, they are usually composed of known,  previously seen atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Computer vision models for common atoms  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='00929v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='DB]  3 Jan 2023  OFOalready exist such as objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', “car”, “backpack”) and their spa-  tial and semantic relationships (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', “left of”, “holding”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Formally,  such a composition of visual scenes is referred to as a scene graph  in the computer vision community [31, 38] and was developed  from its cognitive grounding in human perception [8, 40, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Key  to EQUI-VOCAL’s contribution is to encapsulate spatio-temporal  scene graphs in its data model and use them to define a query  language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' A spatio-temporal scene graph conceptualizes the con-  tents of a video as a sequence of graphs, each graph representing  a single video frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Each graph contains vertices, which repre-  sent objects in the frame;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' edges represent the relationships be-  tween those objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Each object can possess a set of attributes  that describe its properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', “red”, “leather”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL ex-  tracts relevant data from each video using user-provided functions:  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', pre-existing detectors and classifiers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' it synthesizes queries  as a composition of extracted scene graph atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We show that  EQUI-VOCAL’s data model and query language, both based on the  relational model, can express a variety of compositional queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Leveraging scene graphs, EQUI-VOCAL contributes a new query  synthesis approach that finds user events with far fewer labeled  examples than would be required to train a specialized machine  learning model directly, and that works on noisy, video-scale data,  and complex events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' To support query synthesis in such environ-  ment, EQUI-VOCAL solves two technical challenges: it reduces  computational effort and user effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  EQUI-VOCAL reduces computational effort by limiting query  search using scene graphs, by pruning search paths using beam  search, and by avoiding expensive database operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' First, un-  like prior query-by-example techniques that synthesize arbitrary  SQL [20, 41, 50, 56, 58], EQUI-VOCAL reduces the search space by  limiting the query search to sequences of scene graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Second,  synthesizing queries over sequences of scene graphs can still be  a computationally slow process to traverse the search space of  possible queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Existing query-by-example systems enumerate  all possible queries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' although pruning techniques like equivalence  classes [57], over-approximation [58], and lifting projection opera-  tors [56] can be used to reduce the search space, these mechanisms  are not sufficient to make exhaustive exploration tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Instead,  EQUI-VOCAL adopts a beam search strategy to explore the query  space efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Beam search limits exploration to a subset of the  most promising branches at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Third, executing the many  candidate queries on the user examples is prohibitively expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Existing systems [56, 58] evaluate candidate queries with many  joins and thus do not scale well when the size of user examples be-  comes large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL carefully generates efficient queries that  avoid expensive operations such as recursive joins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL  comes with a set of optimizations to generate efficient SQL state-  ments and uses the PostgreSQL database engine to execute them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  EQUI-VOCAL reduces user effort by using active learning and  by being robust to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' With active learning, EQUI-VOCAL re-  duces the number of labeled examples needed: Instead of asking  a user to provide all examples up front, EQUI-VOCAL iteratively  requests labels of carefully selected additional examples to reduce  the uncertainty in query synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Noise can naturally creep into  systems that interface with user labeling, machine learning models,  and potentially ambiguous real-world events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Distinct from other  existing systems [46, 56], EQUI-VOCAL searches for queries that  Figure 2: Example frames of multiple, simultaneous car-  motorcycle interactions (generated using [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  𝑜!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  𝑜!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='"  𝑟!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  car  motorcycle  left of  Figure 3: EQUI-VOCAL represents video content as a se-  quence of region graphs in its data model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Each region graph  models a single video frame (left), with nodes representing  objects and edges representing relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' A region graph  (right) is a subset of the full scene graph (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  best match potentially noisy data and input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' It also retains imper-  fect query candidates at every iteration and uses regularization to  prevent overfitting to noise or limited user input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  In summary, EQUI-VOCAL makes the following contributions:  We introduce an expressive data model and a query language  based on spatio-temporal scene graphs (Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  We propose a new approach that efficiently synthesizes the  user’s intended query from examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We limit the query  search space using scene graphs, prune search paths using  beam search, leverage active learning to reduce user effort,  and handle noisy data (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  We implement a set of optimizations to generate efficient SQL  query statements and reduce computational effort during  query synthesis (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  We evaluate our approach on the CLEVRER dataset [66] and  show that it outperforms two baselines [46, 56]—in terms of  F1 score, synthesis time, and robustness to data noise—and  can flexibly synthesize complex queries that the baselines  do not support (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Overall, EQUI-VOCAL is an important step toward making video  database management systems more accessible to experts and non-  experts alike, by easing the task of expressing queries over video  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  2  EQUI-VOCAL DATA MODEL  This section describes EQUI-VOCAL’s data model and query lan-  guage, which we briefly introduced in our vision paper [19], but  develop in depth here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Section 3 shows how EQUI-VOCAL synthe-  sizes queries from user input using this data model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  For ease of presentation, we use a simplified, running example,  where a traffic engineer seeks to find instances of “a car arriving  from the left and passing a motorcycle at the intersection.” Figure 2  shows two representative frames from a video that contains such  an event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We show other example queries in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  2  EQUI-VOCAL: Synthesizing Queries for Compositional Video Events from Limited User Interactions  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='1  Scene Graphs as our data model  EQUI-VOCAL represents a video 𝑉 as a set of short, non-  overlapping video segments, 𝑣 ∈ 𝑉 (5-second segments in our pro-  totype implementation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Each video segment is a sequence of 𝑁  frames {𝑓1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' , 𝑓𝑁 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The visual content of each frame is represented  by a scene graph [38]: A scene graph 𝑔𝑖 = (o𝑖, r𝑖) contains the set  of all objects o𝑖 in a frame, along with a set of all relationships r𝑖  between those objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Often a scene graph contains more informa-  tion than is necessary to identify an event, and so the literature also  defines a region graph 𝑔𝑖𝑗, which is a subgraph of 𝑔𝑖, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', 𝑔𝑖𝑗 ⊆ 𝑔𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Figure 3 shows an example frame and region graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  We define an object in a frame as 𝑜 = (𝑣𝑖𝑑, 𝑓𝑖,𝑜𝑖𝑑,𝑐𝑖𝑑,𝑏𝑏𝑜𝑥),  where 𝑓𝑖 is the sequence number of frame 𝑖 in video segment, 𝑣𝑖𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  𝑜𝑖𝑑 is a unique identifier of the object in the video segment, 𝑐𝑖𝑑 is  the identifier for the class of the object (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', “car”, “motorcycle”),  and 𝑏𝑏𝑜𝑥 is the bounding box containing the object in frame, 𝑓𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' A  𝑏𝑏𝑜𝑥 is represented by its upper-left and bottom-right coordinates,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', 𝑏𝑏𝑜𝑥 = (𝑥1,𝑦1,𝑥2,𝑦2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Objects can have intra-frame relationships defined as 𝑟  =  (𝑣𝑖𝑑, 𝑓𝑖,𝑟𝑖𝑑,𝑜𝑖𝑑sub, 𝑝𝑖𝑑,𝑜𝑖𝑑tar), where 𝑟𝑖𝑑 is a unique identifier of  the relationship in frame, 𝑓𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Subject, 𝑜sub, is connected to target,  𝑜tar, with the relationship class identifier, 𝑝𝑖𝑑 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', “near” or “holds”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Both subject and target belong to frame 𝑓𝑖: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', 𝑜sub,𝑜tar ∈ o𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Objects can have attributes 𝑎 = (𝑣𝑖𝑑, 𝑓𝑖,𝑜𝑖𝑑,𝑘, 𝑣), where 𝑘 is the  name of the attribute, 𝑣 is the value of the attribute, 𝑣𝑖𝑑, 𝑓𝑖 and 𝑜𝑖𝑑  identify the video segment, frame, and object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL distin-  guishes location and property attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The former change over  time and are typically computed from the bounding box of an object  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', “location=bottom”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The latter capture intrinsic properties of  objects and are immutable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', “color=red”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Finally, an event 𝑒 is a temporally ordered sequence of region  graphs 𝑒 = (𝑒𝑖𝑑, {𝑔1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' ,𝑔𝑘}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Region graphs in an event do not  need to be contiguous or distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Suppose that the two frames in Figure 2 are the 10th and  15th frames of a video segment V1, and that the motorcycle and car  are the 7th and 9th objects detected in V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Then, for the left frame,  we generate the region graph 𝑔1 = (o1, r1), where: o1 = {𝑜11,𝑜12}  represents the car 𝑜11 = (V1, F10, OID9, car,𝑏𝑏𝑜𝑥1) and the motor-  cycle 𝑜12 = (V1, F10, OID7, motorcycle,𝑏𝑏𝑜𝑥2) and r1 = {𝑟11} con-  tains a relationship 𝑟11 = (V1, F10, RID1, OID9, leftOf, OID7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The  car also has an attribute 𝑎11 = (V1, F10, OID9, location, bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  We can define the region graph 𝑔2 for the right frame similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  The only difference will be the relationship between the object will  indicate that the car is now rightOf the motorcycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Finally, the  event 𝑒 = (EID1, {𝑔1,𝑔2}) represents a car arriving from the left  and passing a motorcycle at the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  The relational schema in Table 1 captures the above data model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  The benefit of using a relational schema is that we can execute re-  lational queries to specify region graphs and find events of interest,  which is flexible and follows the well-understood semantics of the  relational model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For each video (or collection of related videos),  EQUI-VOCAL creates a view with this schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In Section 4 we  describe when and how relations in the view are materialized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  To populate each relation, EQUI-VOCAL uses available user-  defined functions (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' User-defined functions can be provided  by the user or be available publicly in the form of existing machine  Objects(vid, fid, oid, cid, 𝑥1, 𝑦1, 𝑥2, 𝑦2)  Relationships(vid, fid, rid, oid1, pid, oid2)  Attributes(vid, fid, oid, key, value)  Table 1: Relational schema representation of data model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  learning models, such as object detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Various user-defined func-  tions can be declared in EQUI-VOCAL: (i) an object detector [51]  that takes a video frame as input and outputs the set of objects  and their bounding boxes, (ii) an object tracking algorithm [63]  that takes objects in consecutive frames as input and, for each pair  of objects, determines if they are the same, and (iii) a set of pre-  trained models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', [12]) or rules that can take two objects in the  same frame as input and determine their relationship (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', “near”,  “behind”, “riding”, “holding”) or that can take one object as input  and determine its attributes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', “location=bottom”, “color=red”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  In our experiments, EQUI-VOCAL uses a general-purpose object  detector [27] to locate objects and intrinsic attributes of objects to  generate trajectories across frames [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='2  Scene graphs as our query language  EQUI-VOCAL could execute arbitrary relational queries on the  view defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' However, this would form an intractable search  space, making query synthesis unusably slow for most real-world  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Instead, we define a query language that is more re-  strictive, affording a smaller search space and therefore, faster syn-  thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We constrain queries to (i) a temporally ordered sequence  of region graphs, (ii) a set of predicates, (iii) a set of duration con-  straints, (iv) a window specification, and (v) to output video segment  identifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Using Datalog and with some abuse of notation, a query  in EQUI-VOCAL can be expressed as:  𝑞(𝑣𝑖𝑑) :- 𝑔1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' ,𝑔𝑘, p, d,𝑤, where:  A temporally ordered sequence of region graphs 𝑔1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' ,𝑔𝑘  specifies that a matching event consists of 𝑔1, followed by 𝑔2,  followed by 𝑔3, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Each 𝑔𝑖 is specified with a set of atoms:  Objects, Relationships, and Attributes joined on a shared  𝑣𝑖𝑑 and 𝑓 𝑖𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Moreover, each 𝑔𝑖 can persist for multiple frames  and there can be other frames between 𝑔𝑖 and 𝑔𝑖+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  A set of predicates p can be applied to objects, relationships,  and attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In our example, predicates would specify that  the query is looking for “car” and “motorcycle” objects, that  the car needs to be “left of” then “right of” the motorcycle,  and that the car should be at the “bottom” of the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  A set of duration constraints d can be applied to region graphs  and define the minimum number of contiguous frames that  a region graph 𝑔𝑖 should be valid before transitioning to the  next region graph 𝑔𝑖+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  A window specification 𝑤 is the maximum number of frames  that can separate 𝑔1 from 𝑔𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  For example, following the above restricted template, the event  in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content="1 can be expressed with the following Datalog rules:1  g1(vid, fid, oid1, oid2) :-  Objects(vid, fid, oid1, 'car', _, _, _, _),  Objects(vid, fid, oid2, 'motorcycle', _, _, _, _),  Relationships(vid, fid, _, oid1, 'leftOf', oid2),  Attributes(vid, fid, oid1, 'location', 'bottom'), oid1 !" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='= oid2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  1In this and the following examples we use English words instead of integers for 𝑐𝑖𝑑  and 𝑝𝑖𝑑 values to make the examples more readable  3  Object detection  Object tracking  Relationship  Attribute  Conjunction  Sequencing  Iteration  Window specification  Query by example  SVQ++ [11]  ✓  ✓  Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' [16]  ✓  ✓  ✓  ✓  ✓  Caesar [42]  ✓  ✓  ✓  ✓  ✓  STAR Retrieval [14]  ✓  ✓  ✓  ✓  ✓  ✓  VidCEP [65]  ✓  ✓  ✓  ✓  ✓  ✓  ✓  CVQL [39]  ✓  ✓  ✓  ✓  ✓  Quivr [46]  ✓  ✓  ✓  ✓  ✓  ✓  ✓  Rekall [21]  ✓  ✓  ✓  ✓  ✓  ✓  ✓  ✓  EQUI-VOCAL  ✓  ✓  ✓  ✓  ✓  ✓  ✓  ✓  ✓  Table 2: Comparison between compositional video analytics  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  The above rule will find frames in video segments that contain a  car and a motorcycle, such that the car is on the left of the motorcy-  cle, and the car is at the bottom of the frame, where the intersection  is located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Next, the event will likely consist of a sequence of such  frames, which can be captured with the following recursive rules:  g1_star(vid, fid, fid, oid1, oid2) :- g1(vid, fid, oid1, oid2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  g1_star(vid, fid_start, fid_end, oid1, oid2) :-  g1_star(vid, fid_start, fid, oid1, oid2),  g1(vid, fid_end, oid1, oid2), fid_end = fid + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  We could use equivalent rules to define g2 and g2_star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Finally,  the query that returns matching video segments takes the form:  q(vid) :- g1_star(vid, fid11, fid12, oid1, oid2),  g2_star(vid, fid21, fid22, oid1, oid2),  fid21 > fid12, fid22 - fid11 < 1800  The predicate 𝑓 𝑖𝑑21 > 𝑓 𝑖𝑑12 indicates that the second sequence  of region graphs should follow the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The predicate allows  for a gap between sequences, which may arise, for example, if  something obstructs the vehicles from the camera’s view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Finally,  𝑓 𝑖𝑑22 − 𝑓 𝑖𝑑11 < 1800 puts a time constraint on the event (a 30-  second time-window, assuming 60 frames per second).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  To summarize, EQUI-VOCAL’s query language is a subset of  Datalog with recursion, expressed on a specific schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In Section 4,  we explain how we avoid executing expensive recursive queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='3  Expressiveness of our data model  We compare the expressiveness of EQUI-VOCAL’s data model  against other compositional video analytics systems in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Among them, SVQ++ [11] only supports spatial relationships be-  tween two objects, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' [16] only supports temporal queries  that count co-occurring objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Caesar [42], STAR Retrieval [14],  and VidCEP [65] lack the important feature of iteration that allows a  region graph to persist for multiple frames and thus cannot support  duration constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' CVQL [39] does not track objects and all its  predicates are defined at the object class level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Quivr [46] has similar  expressiveness to us, but is limited to trajectory queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Rekall [21]  introduces a flexible Python library for video event specification,  but requires that users manually write and refine queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Our data model currently does not support tertiary relationships  between three objects, such as “a person hitting a ball with a bat”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  The closest approximation we have is “a person holding a bat” and  “a bat hitting a ball”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' That being said, it is possible to extend our data  model to support tertiary relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Finally, we do not support  disjunction or negation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We plan to support these in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  3  QUERY SYNTHESIS  With our data model and query language defined, we now for-  mally present the query synthesis problem statement, then describe  all the components of our proposed solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='1  Query synthesis problem statement  Following the aforementioned data model, a user would like to  execute a query 𝑞𝑢 on video database 𝐷 that returns a set of video  segment identifiers, 𝑉𝑜𝑢𝑡𝑝𝑢𝑡 = 𝑞𝑢 (𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Given the user’s intended  query, 𝑞𝑢, each video segment 𝑣𝑖 can be seen as having a ground-  truth label 𝑦𝑖 ∈ {0, 1} indicating whether it matches 𝑞𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Initially,  both the ground-truth labels and 𝑞𝑢 are unknown because the user  is unable to specify 𝑞𝑢 and can only label video segments as positive  or negative instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The goal of EQUI-VOCAL is to synthesize  a target query, 𝑞𝑡 ∈ 𝑄, that is the best approximation of 𝑞𝑢 in its  search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' When executed over database 𝐷, 𝑞𝑡 should yield the  best measure performance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', F1 score):  𝑞𝑡 = arg max  𝑞∈𝑄 measure(𝑞(𝐷),𝑞𝑢 (𝐷))  EQUI-VOCAL can request a label from the user 𝑂: ˆ𝑦𝑖 = 𝑂 (𝑣𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Since user labels may be noisy, it is possible that ˆ𝑦𝑖 ≠ 𝑦𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Given that  enumerative search is intractable, EQUI-VOCAL uses a heuristic  approach to find an approximately-best query ˆ𝑞𝑡 to the objective,  while reducing both user effort and query synthesis time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='2  Query synthesis algorithm overview  Algorithm 1 provides an overview of EQUI-VOCAL’s synthesis  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' It traverses the space of possible queries to synthesize  a final target query that matches the user’s intent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The algorithm  takes as input the set of unlabeled video segments, 𝑈 , and a small  set of labeled segments, 𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝐿 is provided by the user and should  include both positive and negative examples of the desired event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  The algorithm also takes as input a set of user-defined functions,  𝑃, comprised of indicator functions for object classes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', “car”),  relationship values (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', “near”), and attribute key-value pairs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=',  “color=red”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Algorithm 1 enumerates 𝑃 when expanding its search  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  The algorithm returns a set of target queries, 𝑄𝑡, which is the top-  𝑘 synthesized queries ranked by a performance measure (F1 score  in our prototype implementation) and is updated regularly through  the search process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The learned queries can be applied to unseen  videos to find video segments containing the matching event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' By  default, EQUI-VOCAL returns the top-𝑘 queries but then executes  the best one over unseen videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The user can also randomly sample  one query from 𝑄𝑡, execute multiple queries from 𝑄𝑡 and aggregate  the results, or manually examine 𝑄𝑡 to pick the intended query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  4  EQUI-VOCAL: Synthesizing Queries for Compositional Video Events from Limited User Interactions  Algorithm 1: Query synthesis algorithm that returns top-𝑘  synthesized queries matching user input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Input  :𝑈 - set of unlabeled video segments  𝐿 - set of labeled video segments  𝑃 - set of user-defined functions  𝑏,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝑏𝑤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝑘 - hyperparameters  Output:𝑄𝑡 - set of top-𝑘 target queries  1 𝑆 ← {𝑞∅}  2 while |𝑆| > 0 do  3  𝑆′ ← {}  4  foreach 𝑞 ∈ 𝑆 do  5  𝑆′ ← 𝑆′ ∪ ExpandQuery(𝑞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝑃)  6  if |𝐿| < 𝑏 then  7  𝐿,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='𝑈 ← PickNextSegments(𝐿,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='𝑈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='𝑆′)  8  𝑆 ← SampleQueries(𝑆′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝐿,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='𝑏𝑤)  9  𝑄𝑡 ← 𝑄𝑡 ∪ 𝑆′  10  𝑄𝑡 ← RetainTopQueries(𝑄𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝐿,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='𝑘)  11 return 𝑄𝑡  To populate 𝑄𝑡 efficiently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL synthesizes queries  in a bottom-up fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' It starts with an empty query, 𝑞∅, and  incrementally adds predicates to it, up to a certain complexity (see  the ExpandQuery method on line 5 and in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' At each step,  different predicates can be added to a query, expanding the search  in multiple directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' To limit the size of the search space and  ensure fast query synthesis, EQUI-VOCAL adopts a beam search-  style strategy (see the SampleQueries method on line 8 and in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  An important challenge for EQUI-VOCAL is that learning a  query from a small number of user-provided examples is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  To address this challenge, EQUI-VOCAL uses active learning to  effectively guide the query synthesis process and identify good  target queries with limited initial and additional user effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Specifi-  cally, method PickNextSegments on line 7 and in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5 uses  active learning to select additional video segments for the user to  label in order to effectively differentiate between multiple candidate  subqueries, and expand the search in the most promising directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  At the end of each iteration, method RetainTopQueries (line  10 and Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='6) maintains a list of top-𝑘 queries seen so far,  which is larger than the number of queries selected for additional  expansion, in case a query seen earlier in the search ends up with  the best score on the final labeled set of segments, or, as mentioned  above, to give users options if they would like to try alternative,  high-performing queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  The algorithm has several hyperparameters, which include a  labeling budget 𝑏 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', the maximum number of labels that the user  is willing to provide), the number of candidate queries to retain  during exploration 𝑏𝑤 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', the beam width), and the number of  queries in the final answer 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Figure 4 illustrates the algorithm  using the running example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='3  Query expansion  Existing query-by-example systems use sketch-based query synthe-  sis approaches [46, 56, 58] to enumerate candidate queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' A sketch  query is a query with unspecified parts in the form of holes and  these approaches enumerate the search space by first generating  high-level sketch queries and then filling them with low-level de-  tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' However, enumerative search is slow and memory-intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Inspired by execution-guided synthesis approaches [13, 30], which  treat a program as a sequence of manipulations and use the results  of partial programs to guide the search, EQUI-VOCAL explores  the search space based on the results of executing intermediate  queries on the examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Instead of synthesizing sketch queries  with uninstantiated holes that cannot be executed directly, or di-  rectly applying techniques from [13, 30], which require a large  amount of data to train a neural synthesizer, EQUI-VOCAL expands  queries by adding instantiated and executable constraints, execut-  ing partial queries to assess the promise of each explored path, and  iteratively refining a query towards the target query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  In our synthesis algorithm, method ExpandQuery(𝑞, 𝑃) takes  as input a query 𝑞 to expand, and a set of user-defined functions, 𝑃  to construct more complex queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The function returns a set of  expanded queries as illustrated in Figure 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Our query synthesis approach uses a compact query notation,  which can be seen as a DSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The DSL captures the logical structure  of the queries to synthesize and key query parameters, but omits  the details of the full, underlying SQL (or Datalog).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' This approach  is important for several reasons: First, since we do not generate  arbitrary SQL, but rather queries that conform to the structure  presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='2, the DSL captures that structure precisely,  simplifying the search space and guiding synthesis toward the  correctly-structured queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Second, this approach helps to de-  couple the logical query specification from the details of the SQL  queries that are ultimately executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' As we present in Section 4,  critical optimizations are necesssary during the translation from  our DSL to SQL to achieve efficient query execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  In our DSL, we use a variable 𝑜 to represent an object in a query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Different variables represent objects with different 𝑜𝑖𝑑’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' All pred-  icates of a region graph are connected by commas and are repre-  sented with shorthand notations that specify only their key-value  pairs (for property attributes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', Color(𝑜1, ‘cyan’)), value (for  location attributes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', Bottom(𝑜1)), or class (for objects and re-  lationships, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', Car(𝑜1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Then, region graphs are connected in  sequence with semicolons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For example, the query for the event  from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='1 can be represented as:  𝑞 =(Car(𝑜1), Motorcycle(𝑜2), LeftOf(𝑜1,𝑜2), Bottom(𝑜1));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  (Car(𝑜1), Motorcycle(𝑜2), RightOf(𝑜1,𝑜2), Bottom(𝑜1))  We further use notation Duration(𝑔,𝑑) to require that the region  graph 𝑔 exist in at least 𝑑 consecutive frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  During query synthesis, EQUI-VOCAL expands queries written  in our DSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' ExpandQuery can take any of the following three  actions:  Graph construction (GC): Add a predicate to an existing  region graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Sequence construction (SC): Insert a new region graph con-  sisting of one predicate into any position of the existing  sequence of region graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Duration refinement (DR): Increment the duration constraint  of one existing region graph in the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='2  Pick  Label: ✅ or ❌  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='1  Top-k  queries:  (a) Query expansion  (b) Example  selection  Top-k  queries:  (c) Query  sampling  (d) Top-k  queries  (a) Query expansion  (a) Query expansion  (d) Top-k  queries  Iteration: 1  Iteration: 2  Iteration: 8  𝑃 = Car, Motorcycle, LeftOf, RightOf, Bottom , 𝑏𝑤 = 2, 𝑘 = 1  𝑞∅  𝑞"  = LeftOf 𝑜", 𝑜#  𝑞# = RightOf(𝑜", 𝑜#)  𝑞$ = Bottom(𝑜")  𝑞% = Car(𝑜")  𝑞& = Motorcycle(𝑜")  𝑄\' = 𝑞"  𝒒𝟏  𝑞#  𝒒𝟑  𝑞%  𝑞&  𝑞"  𝑞$  𝑞* = (LeftOf 𝑜", 𝑜# , Bottom 𝑜" )  𝑞+ = LeftOf 𝑜", 𝑜# ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Bottom 𝑜"  𝑞, = Duration LeftOf 𝑜", 𝑜# , 5  𝑞!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' = (LeftOf  𝑜", 𝑜# , Bottom 𝑜" ,  Car 𝑜" , Motorcycle 𝑜# );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  ((RightOf 𝑜", 𝑜# , Bottom 𝑜" ,  Car 𝑜" , Motorcycle 𝑜# )  𝑄\' = 𝑞-  Figure 4: Running example of the query synthesis algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The algorithm starts with 𝑞∅ and synthesizes queries by iter-  atively (a) expanding queries, (b) selecting new video segments for the user to label, (c) sampling a small set of queries for  further expansion, and (d) updating a list of top-𝑘 queries after each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The algorithm returns 𝑄𝑡 = {𝑞𝑖}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  ∅  𝐴  𝐵  𝐶  𝐴;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝐵  𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝐶  𝐴;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝐶  𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝐶;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝐴  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='9  ⋯  ⋯  (a) Restrictive expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  ∅  𝐴  𝐵  𝐶  𝐴;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝐵  𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝐶  𝐴;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝐶  𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝐶;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝐴  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='9  ⋯  ⋯  (b) Relaxed expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Figure 5: A restrictive rule constrains each query to have one  construction path, while a relaxed rule allows for multiple  paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Though a relaxed rule results in a larger search space,  it is more likely to find queries with high performance than  a restrictive rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  As shown in Figure 4(a), EQUI-VOCAL starts with an empty query  𝑞0 = 𝑞∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In iteration 1, EQUI-VOCAL takes action SC to expand  𝑞0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' This results in 𝑞1 to 𝑞5, each consisting of a single region  graph with one predicate drawn from 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In iteration 2 of Fig-  ure 4(a), EQUI-VOCAL first expands 𝑞1 = LeftOf(𝑜1,𝑜2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Per-  forming action GC leads to 𝑞6 = (LeftOf(𝑜1,𝑜2), Bottom(𝑜1));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  SC leads to 𝑞7 = LeftOf(𝑜1,𝑜2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Bottom(𝑜1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' and DR leads to  𝑞8 = Duration(LeftOf(𝑜1,𝑜2), 5), assuming the granularity of  Duration is 5 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Deciding which actions to take to expand queries is an important  design decision in EQUI-VOCAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Imagine the search space as a di-  rected acyclic graph (DAG), where each node represents a possible  query in the search space, the root node represents the empty query  𝑞∅, and the query of each child node is constructed by applying  one action to its parent node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We consider two extreme expansion  rules: A restrictive rule (Figure 5a) only allows each query to have  one construction path in the DAG, while a relaxed rule allows each  query to be generated through all possible permutations of actions  with multiple construction paths (Figure 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Our early experiments  showed that the restrictive rule led to poor performance: In some  cases, queries with high performance were not found because their  ancestor queries (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', intermediate parent queries) were not good  and were getting pruned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Since different predicates and their com-  binations have different selectivity and contribute differently to the  final query performance, the relaxed rule allows the algorithm to  focus on synthesizing the more dominant part of the query during  early iterations, avoiding all paths to the target query being acci-  dentally pruned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The cost of a relaxed rule is a larger search space  and slower query synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In the example of Figure 5, queries are  constructed following the paths highlighted in blue, assuming only  the best query is expanded at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The restrictive rule selects  the query 𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='𝐶;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='𝐴 with a score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5 rather than 𝐴;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='𝐶 with a score  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='9, because 𝐴 has a lower F1 score than 𝐶 and 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' On the other  hand, using the relaxed rule selects 𝐴;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='𝐶 by first constructing 𝐶,  then adding 𝐵 and 𝐴 to the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL follows the  relaxed rule: GC can add any predicates that are not already in  the region graph (predicates of the same user-defined function but  different variables are considered different);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' SC can insert a new  region graph before and after any existing region graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' DR can  increment the duration constraint of any existing region graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='4  Beam search  When traversing the search space, we can either design our al-  gorithm to only expand the top query or to expand all queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  The former is quick to compute but may not find the target query  at the end, while the latter is optimal but comes with a prohibi-  tive computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL uses beam search to bal-  ance query performance and synthesis efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Beam search is  used in NLP [43] where problems commonly have large search  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Beam search reduces runtime by limiting the number of  explored branches at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' However, the search out-  puts are not guaranteed to be optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Search quality depends on  how branches are expanded, scored, and pruned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In our approach,  method SampleQueries(𝑆′, 𝐿,𝑏𝑤) retains the top 𝑏𝑤 queries to  expand by evaluating the F1 score of the 𝑆′ candidate queries on  the set of labeled video segments 𝐿 (see Figure 4(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5  Active learning  One challenge with asking the user for only a handful of examples  of the intended event is that we risk overfitting, but asking the  user to find a larger number of initial examples is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' To  address this challenge, our approach is to use active learning during  query synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' At every iteration, EQUI-VOCAL asks the user  for a handful of additional labels to differentiate between the 𝑏𝑤  queries retained from the previous iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We choose which  video segments to ask for labels using a disagreement-based active  learning algorithm [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Intuitively, this method picks the video  6  VEQUI-VOCAL: Synthesizing Queries for Compositional Video Events from Limited User Interactions  segments where the retained queries disagree the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Using active  learning reduces the number of data points a user has to provide;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  additionally, we posit that it is easier for users to label system-  selected examples than to come up with their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  During each call to PickNextSegments(𝐿,𝑈,𝑆′), EQUI-VOCAL  computes a score for a sample of unlabeled video segments 𝑈 over  the candidate queries 𝑆′ and then picks the video segments with  the largest disagreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The score of each sampled, unlabeled  video segment is computed as the weighted disagreement between  the candidate queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The weight of each candidate query is set  to its regularized performance over the labeled set 𝐿 (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  While the algorithm in [37] is designed for the setting where la-  beling candidates are streamed, EQUI-VOCAL instead maintains a  pool of candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Therefore, for each call to PickNextSegments,  EQUI-VOCAL computes the score for a sample of unlabeled video  segments over a sample of candidate queries and then picks the  best one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The function updates 𝐿 and 𝑈 given the new user labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  As shown in Figure 4(a), EQUI-VOCAL generates five candidate  queries 𝑞1 to 𝑞5 in iteration 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Among them, 𝑞1 has the highest score  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='6, while 𝑞2 and 𝑞3 give the same second highest score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-  VOCAL requests a new video segment to be labeled (Figure 4(b)),  which distinguishes 𝑞3 from 𝑞2 (Figure 4(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Before query synthesis begins, users can set the labeling bud-  get as a hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL precomputes the number  of iterations of query expansion it will perform and uniformly di-  vides the labeling budget amongst the iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' At every iteration,  EQUI-VOCAL asks for new labels only if it has a labeling budget  left to do so;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' otherwise, it proceeds to synthesize queries without  requesting new labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL precomputes the number of  iterations by relying on the hyperparameters (Section 4) that bound  the search process and the query expansion rules (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='3),  which increments the complexity of synthesized queries by one in  each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='6  Retaining top queries  At the end of each iteration, EQUI-VOCAL updates its list of  candidate final queries in order to retain 𝑘 best performing  ones, as measured by their F1 scores on the labeled dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  RetainTopQueries(𝑄𝑡, 𝐿,𝑘) evaluates all candidate queries 𝑄𝑡 on  labeled set 𝐿 and returns the top 𝑘 (Figure 4(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  In Figure 4(d), at the end of iteration 1, EQUI-VOCAL stores  𝑞1 in 𝑄𝑡 because 𝑘=1, and 𝑞1 has the highest score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' When the  algorithm terminates, EQUI-VOCAL returns 𝑞𝑖 as the final query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  In our evaluation, we set the default value of 𝑘 to 100 (and to 1000  for a more complex dataset), which ensures both low overhead and  high-performing final queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='7  Regularization  EQUI-VOCAL retains top queries using their F1 scores on the  smaller 𝐿 labeled set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' This can lead to overfitting since there are  likely too few training examples to accurately evaluate each candi-  date query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' To prevent overfitting, we regularize the score of each  candidate query by adding the term:𝑠𝑐𝑜𝑟𝑒reg(𝑞) = 𝑠𝑐𝑜𝑟𝑒(𝑞)−𝜆·𝑅(𝑞),  where 𝜆 controls the importance of the regularization term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝑅(𝑞)  𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  𝑓"  𝑓#  𝑓$  𝑓%  𝑓&  𝑓\'  𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  ⋯  ✅  𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  ∗  𝑓"  ∗  𝑓#  ∗  𝑓$  ∗  𝑓%  ∗  𝑓&  ∗  𝑣"  ❌  𝑔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  ⋯  𝑔"  ⋯  𝑔#  ⋯  𝑔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝑔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='" 𝑔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='# 𝑔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='$ 𝑔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='% 𝑔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='& 𝑔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='\'  𝑔"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝑔"" 𝑔"# 𝑔"$ 𝑔"% 𝑔"& 𝑔"\'  𝑔#!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝑔#" 𝑔## 𝑔#$ 𝑔#% 𝑔#& 𝑔#\'  𝑔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  ∗  𝑔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='"  ∗  𝑔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='#  ∗  𝑔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='$  ∗  𝑔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='%  ∗  𝑔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='&  ∗  𝑔"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  ∗  𝑔""  ∗  𝑔"#  ∗  𝑔"$  ∗  𝑔"%  ∗  𝑔"&  ∗  𝑔#!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  ∗  𝑔#"  ∗  𝑔##  ∗  𝑔#$  ∗  𝑔#%  ∗  𝑔#&  ∗  Figure 6: Query execution example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL reduces  intermediate result sizes by computing the earliest match-  ing sequences for each region graph specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  represents the complexity of the query 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  𝑅(𝑞) =  𝑘  ∑︁  𝑖=1  �𝛼1𝑛𝑝𝑖 + 𝛼2𝑛𝑑𝑖 + 𝛼3𝑛𝑑𝑖𝑛𝑝𝑖  �  Here, 𝑘 is the number of region graphs in 𝑞, 𝑛𝑝𝑖 is the number of  predicates in the 𝑖th region graph 𝑔𝑖, and 𝑛𝑑𝑖 denotes the scale of  the duration constraint of 𝑔𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝛼1, 𝛼2, and 𝛼3 are hyperparameters  that control the importance of each term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  4  QUERY EXECUTION  As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='3, we convert queries in our DSL into  SQL and use a relational engine (PostgreSQL in our prototype) to  execute them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In this section, we discuss important optimizations  that we apply during this translation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Query translation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='2, we show that  queries over videos are naturally recursive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Such queries, however,  are slow to execute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' To avoid recursion, we leverage the observation  that those queries express the idea of iteratively matching a region  graph in a contiguous sequence of frames, rather than arbitrary re-  cursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Therefore, we can express them using window functions  instead of recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  As a second optimization, we note that a query 𝑞 in our DSL  (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='3) takes as input a set of video segments 𝑉 and returns  all segments in 𝑉 containing at least one event that matches 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Therefore, instead of finding all satisfying events for each video  segment, the query execution needs to find only one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We leverage  this observation to reduce intermediate result sizes by producing  SQL that efficiently computes the earliest matching sequences for  each region graph specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' This optimization applies to queries  without window specifications and whose duration constraints are  of the form “>” or “>=”, which are the only types of queries that  EQUI-VOCAL currently generates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  As a concrete example, consider Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Assume we want to  execute a query 𝑞 = Duration(𝑔1, 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Duration(𝑔2, 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='𝑔3, where  𝑔1 = Far(𝑜1,𝑜2), 𝑔2 = Near(𝑜1,𝑜2) and 𝑔3 = Behind(𝑜1,𝑜2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Also,  we only consider one permutation of objects for each video segment  in this example for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The event that 𝑞 looks for consists  of a contiguous sequence of 𝑔1 for at least 𝑑1 = 2 frames, followed  by a contiguous sequence of 𝑔2 for at least 𝑑2 = 2 frames, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Consider video segment 𝑣1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For the first region graph 𝑔1, we  only need to extract the earliest matching segment of 𝑔1, which is  𝑒11 = {𝑔12,𝑔13}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' This is because, for every satisfying event for the  query, we can replace its matching segment of 𝑔1 with the earliest  one and the resulting event will still be a match to the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 𝑒2 =  {𝑔13,𝑔14,𝑔25,𝑔26,𝑔37} is a matching event for 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' By replacing 𝑒12 =  7  VCREATE VIEW g1_windowed AS (  SELECT vid, fid,  oid1, oid2,  lead(fid, d1 - 1, 0) OVER (  PARTITION BY vid, oid1,  oid2 ORDER BY fid  ) as fid_offest  FROM g1  );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  (a) Q1: window  CREATE VIEW g1_iterated AS (  SELECT vid,  min(fid_offest) AS fid,  oid1, oid2  FROM g1_windowed  WHERE fid_offest  = fid + (d1 - 1)  GROUP BY vid, oid1, oid2  );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  (b) Q2: iterate  CREATE VIEW g2_filtered AS (  SELECT t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='vid, t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='fid,  t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='oid1, t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='oid2  FROM g1_iterated t1, g2 t2  WHERE t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='vid = t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='vid  AND t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='oid1 = t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='oid1  AND t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='oid2 = t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='oid2  AND t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='fid < t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='fid  );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  (c) Q3: filter  Figure 7: SQL snippets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL generates efficient SQL  queries by avoiding recursions and pruning intermediate re-  sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  {𝑔13,𝑔14} with 𝑒11 = {𝑔12,𝑔13}, we get 𝑒1 = {𝑔12,𝑔13,𝑔25,𝑔26,𝑔37},  which is still a matching event for 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For subsequent region graph  𝑔2, we first use the earliest matching segment from 𝑔1 to retain  matching frames of 𝑔2 that are temporally after the end frame of  the segment, which are 𝑔25 and 𝑔26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Then, we can use the same  method as before to find the earliest matching segment from 𝑔2,  which is 𝑒21 = {𝑔25,𝑔26}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The same procedure can be applied to 𝑔3,  giving us 𝑒31 = {𝑔37}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Thus, the earliest matching event for 𝑞 in 𝑣1  is 𝑒1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Video segment 𝑣2 does not contain any matching event of 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  The above algorithm can be expressed using SQL queries, as  shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Q1 partitions the data on video ID and each  unique permutation of groups of objects so that each group of  objects is analyzed separately, and sorts the data on frame ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Next,  for each row, it gets the current frame ID as well as the frame ID that  is 𝑑1 frames after the current frame ID, or 0 if no such frame exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  The result of Q1 gives a table of intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Q2 then finds the earliest  matching segment for each combination of video and groups of  objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Together, Q1 and Q2 find the earliest matching segment for  Duration(𝑔1, 2) without using recursive joins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Q3 uses the earliest  matching segment from 𝑔1 to determine where to look for matching  segments in 𝑔2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The example SQL snippets only contain two objects  oid1 and oid2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In general, Q1 and Q2 partition and group the  data by all objects in the region graph 𝑔𝑖, and Q3 specifies all the  constraints on objects across two region graphs in the where clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  We demonstrate the impact of both optimizations in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Caching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We implement an application-level cache to reuse sub-  query results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL generates SQL queries as described  above to find matching region graphs in the sequence specified  by the DSL query one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We cache the intermediate results  of all prefixes on the set of video segments to allow other queries  sharing the same sub-queries to reuse the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For example,  when executing the query 𝑞 = 𝑔1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='𝑔2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='𝑔3 on video segments 𝑣1 and  𝑣2, we cache the results of 𝑞1 = 𝑔1, 𝑞2 = 𝑔1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='𝑔2, and 𝑞3 = 𝑔1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='𝑔2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='𝑔3  on 𝑣1 and 𝑣2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Later on, if a query 𝑞′ = 𝑔1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='𝑔2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='𝑔4 is executed on 𝑣1  and 𝑣3, EQUI-VOCAL will reuse the cached results of 𝑞2 = 𝑔1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='𝑔2  on 𝑣1 to improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Bounding the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL synthesizes queries  up to a certain maximum size to ensure that the synthesis algo-  rithm eventually terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Assume the number of user-defined  functions for object classes and attributes is 𝑚1 and the number  of user-defined functions for relationships is 𝑚2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We bound the  query search space using a set of hyperparameters: the maximum  number of region graphs (𝑛𝑔) in a query, the maximum number  of predicates (𝑛𝑝) in the query, the number of possible values a  duration constraint can take (𝑛𝑑), and the number of unique ob-  jects allowed in a query (𝑛𝑣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The size of the search space is then  |𝑄| = 𝑂((𝑛𝑔(𝑛𝑣𝑚1 + 𝑛2𝑣𝑚2))𝑛𝑝 · 𝑛𝑛𝑔  𝑑 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Predicate evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The current implementation of EQUI-  VOCAL populates the Objects relation and the property attributes  in the Attributes relation before query synthesis (see Table 1),  by executing ML models on all video frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The Relationships  relation and the location attributes in the Attributes relation are  computed lazily during query execution since they do not require  ML models in our prototype, and are thus inexpensive to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  In general, optimizing what information is computed before query  synthesis and what is computed lazily is not the focus of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  To simplify SQL query generation, we implement all predicates  (except for join predicates) in EQUI-VOCAL as user-defined func-  tions that take attributes and bounding box coordinates as input  and return boolean values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In our prototype, we use PostgreSQL  and implement user-defined functions for location and relation-  ship predicates that operate on bounding boxes, and for property  predicates that operate on key-value pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  While EQUI-VOCAL provides the above user-defined functions,  the user can also provide additional user-defined functions incre-  mentally as they use the system, and they can be shared and reused  across queries and users [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In practice, EQUI-VOCAL has to limit  the number of user-defined functions to a reasonable amount to  ensure the efficiency of query synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  5  EVALUATION  We conduct an experimental evaluation of EQUI-VOCAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' First,  we show that, compared to existing systems, EQUI-VOCAL reduces  the query synthesis time by 1-2 orders of magnitude, achieves  comparable or better F1 scores under the same labeling budget,  is more robust to noisy data, and explores and executes queries  more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Second, we show that EQUI-VOCAL is capable of  synthesizing more complex, flexible queries over arbitrary scene  graphs, which existing systems cannot handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Finally, we conduct  an ablation study by varying the various design choices outlined in  the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' To the best of our knowledge, there are no existing  systems that can synthesize queries over scene graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The most  similar system to EQUI-VOCAL is Quivr [46], which synthesizes  queries over trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Since object trajectories can be repre-  sented using scene graphs, we compare our system against Quivr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  We also compare against PATSQL [56], which is a state-of-the-art  query-by-example system for relational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We use 13 predicates in experiments: six relationship  predicates (Near, Far, LeftOf, RightOf, FrontOf, and Behind),  four location predicates (Left, Right, Top, Bottom), and three prop-  erty predicates (Color, Material, and Shape).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We consider eight  colors, two materials, and three shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Property predicates are  only used for the scene graphs dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For duration constraints, we  consider three possible values: duration(𝑔) ≥ 5, 10, or 15 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Section 4 discussed how EQUI-VOCAL evaluates these predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We evaluate our system on the CLEVRER dataset [66], which  comprises synthetic videos of moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Each video is five  seconds long with a resolution of 480 × 320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' This dataset includes  8  EQUI-VOCAL: Synthesizing Queries for Compositional Video Events from Limited User Interactions  ID  Query  Description  Pos %  TQ1  (Near(𝑜1,𝑜2), Bottom(𝑜1))  𝑜1 is close to 𝑜2 while 𝑜1 is at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='0%  TQ2  Far(𝑜1,𝑜2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Near(𝑜1,𝑜2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Far(𝑜1,𝑜2)  Two objects move from far to close, then to far again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5%  TQ3  Far(𝑜1,𝑜2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' (Near(𝑜1,𝑜2), Behind(𝑜1,𝑜2))  𝑜1 and 𝑜2 are far away, then they move close and 𝑜1 is behind 𝑜2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  12%  TQ4  Far(𝑜1,𝑜2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' (Near(𝑜1,𝑜2), Behind(𝑜1,𝑜2), Left(𝑜1))  𝑜1 and 𝑜2 are far apart, then they move close and 𝑜1 is behind 𝑜2  and 𝑜1 is on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5%  TQ5  (FrontOf(𝑜1,𝑜2), Top(𝑜1))  𝑜1 is in front of 𝑜2 while 𝑜1 is at the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  45%  TQ6  Near(𝑜1,𝑜2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Far(𝑜1,𝑜2)  Two objects move from close to far apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  10%  TQ7  (Near(𝑜1,𝑜2), Left(𝑜1), Behind(𝑜1,𝑜2))  𝑜1 is close to and behind 𝑜2 while 𝑜1 is on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5%  TQ8  (Far(𝑜1,𝑜2), Bottom(𝑜1));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Near(𝑜1,𝑜2)  𝑜1 at the bottom is far from 𝑜2, then they move close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='9%  TQ9  (Far(𝑜1,𝑜2), Left(𝑜1));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' (Near(𝑜1,𝑜2), Left(𝑜1))  𝑜1 and 𝑜2 move from far to close while 𝑜1 is on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  10%  TQ1D  Duration(Far(𝑜1,𝑜2), 5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Near(𝑜1,𝑜2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Far(𝑜1,𝑜2)  Two objects are far apart for at least 5 frames, then they move  close, then they are far again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='6%  TQ2D  Duration(LeftOf(𝑜1,𝑜2), 5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' (Near(𝑜1,𝑜2), Top(𝑜1));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Duration(RightOf(𝑜1,𝑜2), 5)  𝑜1 is on the left of 𝑜2 for at least 5 frames, then they move close  to each other and 𝑜1 is at the top, then 𝑜1 is on the right of 𝑜2 for  at least 5 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  6%  TQ3D  Duration((Frontof(𝑜1,𝑜2), Left(𝑜1)), 15);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Duration((Left(𝑜1), RightOf(𝑜1,𝑜2), Top(𝑜1)), 5)  𝑜1 is in front of 𝑜2 while 𝑜1 is on the left for at least 15 frames,  then 𝑜1 moves to the right of 𝑜2 while 𝑜1 is at the top left for at  least 5 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='3%  Table 3: Example queries written for the trajectories dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' TQ1D-TQ3D include duration constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' “Pos %” is the percent-  age of positive examples in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  ID  Query  Description  Pos %  SQ1  Duration((Far(𝑜1,𝑜2), Right(𝑜2)), 15);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  (FrontOf(𝑜1,𝑜2), Near(𝑜1,𝑜2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Duration((Far(𝑜1,𝑜2), Right(𝑜2)), 5)  𝑜1 and 𝑜2 are far apart while 𝑜2 is on the right for at least 15  frames, then they move close and 𝑜1 is in front of 𝑜2, then  they are far apart while 𝑜2 is on the right for at least 5 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5%  SQ2  (Far(𝑜1,𝑜2), Shape(𝑜1, ‘sphere’), Color(𝑜3, ‘cyan’));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  (Near(𝑜1,𝑜2), FrontOf(𝑜1,𝑜2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' (Far(𝑜1,𝑜2), Top(𝑜2))  A sphere 𝑜1 is far apart 𝑜2 while 𝑜3 in cyan is also in the  frame, then 𝑜1 and 𝑜2 move close and 𝑜1 is in front of 𝑜2,  then 𝑜1 and 𝑜2 are far apart with 𝑜2 at the top of the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='8%  SQ3  Duration((LeftOf(𝑜1,𝑜2), Shape(𝑜2, ‘sphere’)), 15);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  (FrontOf(𝑜1,𝑜3), Near(𝑜1,𝑜3));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Duration((Behind(𝑜1,𝑜3), Far(𝑜1,𝑜3)), 5)  𝑜1 is on the left of a sphere 𝑜2 for at least 15 frames, then 𝑜1  is in front of and close to 𝑜3, then 𝑜1 is behind and far away  from 𝑜3 for at least 5 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='8%  Table 4: Example queries written for the scene graphs dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' “Pos %” is the percentage of positive examples in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  a variety of geometric shapes interacting in space and time, and  comes with ground truth data, facilitating testing queries with  varying complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We create two benchmarks from this data:  trajectories and scene graphs datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Trajectories dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We create a first dataset to test queries  over trajectories, which baseline systems support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We extract  10,080 pairs of object trajectories that overlap in time from 500,  5-second video segments (128 frames each).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Each trajectory pair is  essentially a temporally ordered sequence of bounding box pairs  𝑏 = (𝑥𝑎1,𝑦𝑎1,𝑥𝑎2,𝑦𝑎2,𝑥𝑏1,𝑦𝑏1,𝑥𝑏2,𝑦𝑏2) representing two objects  in a video segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We manually generate a set of queries with  varying complexities (Table 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' TQ1D-TQ3D include duration con-  straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' To generate ground truth labels, we run each target query  on the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We sample 500 trajectory pairs as training data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=',  data used during query synthesis) and use the rest as test data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=',  to measure the quality of synthesized queries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Scene graphs dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' This more complex dataset contains scene  graphs extracted from the CLEVRER dataset, which baseline sys-  tems do not support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We extract scene graphs from 10,000, 5-second  video segments (128 frames each).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For every frame of a video seg-  ment, we store the object track ID (𝑜𝑖𝑑), bounding box coordinates,  and object attributes (shape, color, and material) of every object in  the frame (the Objects and Attributes relations in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' As  for trajectories, we sample 500 video segments as training data and  use the rest as test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  We manually write three example queries shown in Table 4 to il-  lustrate the kinds of more complex queries that can be asked on the  scene graphs dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In our experiments, we automatically generate  three classes of queries with different complexities: easy, medium,  and hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Each generated query contains exactly three variables  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', three distinct objects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Easy queries have exactly three predi-  cates on relationships and locations, one property predicate, one  region graph, and no duration constraints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' medium queries have  exactly five predicates on relationships and locations, two property  9  Method  Simplified  Normal  TQ1  TQ2  TQ3  TQ4  TQ5  TQ6  TQ7  TQ8  TQ9  TQ1  TQ2  TQ3  TQ4  TQ1D  TQ2D  TQ3D  PATSQL  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='54  —  538  —  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='93  552  —  —  NA  NA  NA  NA  NA  NA  NA  Quivr  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='6  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='11  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='9  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='8  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='18  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='7  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='3  —  8267  6068  8296  —  —  —  —  Ours  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='57  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='85  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='1  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='36  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='66  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='7  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='6  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='6  125  106  185  223  166  183  Table 5: Query synthesis time (median, in seconds) for each method to achieve at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='9 F1 scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL successfully  learns all queries with at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='9 F1 scores and is faster than (or at least comparable to) the two baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' (NA: not applicable,  —: failed due to insufficient F1 score or timeout)  predicates, three region graphs, and no duration constraints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' hard  queries have the same complexity as medium queries but also in-  clude duration constraints with three possible values (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', 5, 10, and  15 frames).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Each class contains 40 queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We report the F1 score and query synthesis time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We  synthesize queries using the training set and report the F1 score  of the query (or the median F1 score if there are multiple queries  with the same best score on the training set) over the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Each  experiment is run 20 times for the trajectories dataset and five times  for the scene graphs dataset, and we report the median F1 score  and median query synthesis time over these runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We implement our prototype in Python  with PostgreSQL as the backend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We conduct all experiments on  a computing cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' When measuring runtimes, we request one  node with Intel Xeon Gold 6230R CPUs at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='10GHz and 100GB of  RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Unless otherwise specified, we configure EQUI-VOCAL as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For the trajectories dataset, we search for queries with  up to 5 predicates across up to 3 region graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We use beam  width 𝑏𝑤 = 10, 𝜆 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='01 for regularization (with 𝛼1 = 1, 𝛼2 = 1,  and 𝛼3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='1, see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='7), and we set 𝑘 = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For the scene  graphs dataset, we search for queries with up to 7 predicates, 3  region graphs, and 3 objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Since the dataset is more complex  and challenging, we use a smaller 𝜆 (because the query complexity  term, 𝑅(𝑞) has a greater absolute value) and a greater 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We use  beam width 𝑏𝑤 = 10, 𝜆 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='001 for regularization (with 𝛼1 = 1,  𝛼2 = 1, and 𝛼3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='1), and we set 𝑘 = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  We consider two variants of Quivr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' As per the original paper, we  limit the number of atomic predicates in Quivr’s queries to 5 and  the depth of the nested constructs to 3, which leads to a similar  search space as EQUI-VOCAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' When considering queries without  duration constraints (TQ1-TQ9), we omit Kleene star operators from  its search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Otherwise (TQ1D-TQ3D), we include Kleene star  in the query expansion and add one more predicate MinLength𝜃,  which checks whether the duration of the input is at least 𝜃 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Quivr returns all queries that match the examples, so we select the  queries with the simplest structure (which is determined by the  number of atomic predicates, and, if the former is the same, by the  depth) and report their median F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Because PATSQL requires that all user-provided input tables be  used in the solution query, when comparing against PATSQL, we  restrict all systems to only the candidate predicates that appear  in the target query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Because PATSQL cannot handle large tables  efficiently, we also downsample each trajectory by 75% (we keep  one frame out of four) to reduce the size of the input tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We  refer to this configuration as simplified tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='1  Results against baselines on trajectories  We evaluate EQUI-VOCAL against the two baselines on the trajecto-  ries dataset and the set of queries in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For TQ1-TQ9, we omit  duration constraints from the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We run each method as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For each target query, we randomly select 2 positive and 10  negative examples from the training set and use them as the input  to the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Each method then asks for 𝑏 additional examples  during the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For PATSQL, since it is not interactive, we simply  sample 𝑏 more examples randomly from the remaining dataset and  provide the 12 + 𝑏 examples to the system at the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For  Quivr and EQUI-VOCAL, we input the 12 initial examples, and each  system actively requests more examples during the search process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  EQUI-VOCAL is faster than baselines and can find high-  performant queries even for complex queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Table 5 shows  the query synthesis time of each query to achieve at least a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='9 F1  score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For simplified tasks, EQUI-VOCAL outperforms baselines  on 6 queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' PATSQL performs the best or close to the best when  queries are simple (TQ1, 5, and 6) because these queries include  only two predicates and PATSQL terminates once it finds one so-  lution query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Quivr is slower than EQUI-VOCAL but comparable  to it except for TQ9, in which case Quivr fails due to an insuffi-  cient F1 score of the synthesized queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Under the normal setting,  EQUI-VOCAL is significantly more efficient than Quivr and can find  high-performance queries in hundreds of seconds even for complex  queries with duration constraints, while Quivr cannot synthesize  TQ4 and TQ1D-TQ3D within 4 hours due to the large number of  queries that need to be explored in the enumerative search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For  TQ5-TQ9 on normal tasks (not shown in the table due to space con-  straints), EQUI-VOCAL takes 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='4, 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='1, 106, 110, and 107 seconds  to synthesize queries with at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='9 F1 scores, while Quivr takes  7798 seconds for TQ5, 7366 seconds for TQ6, and cannot synthesize  TQ7-TQ9 within 4 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  On simplified tasks, EQUI-VOCAL and QUIVR find  queries with similar F1 scores and outperform PATSQL for  the same labeling budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Figure 8 shows the F1 score of each  system when varying the user labeling budget on the simplified  tasks (which all systems can perform).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We assign an F1 score of 0 if  a system fails to find a query in 1 hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Across all labeling budgets  tested, as above, PATSQL performs well when the target query is  simplest (TQ1, 5, and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For other queries, PATSQL either fails  to find a solution query within 1 hour or the solution query has  low performance (below a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='9 F1 score).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' When the budget is 30, we  observe a decrease in F1 scores for TQ3, 7, 8, and 9, because PATSQL  has a less constrained search space than EQUI-VOCAL and does not  scale well when the size of input and output tables increases as more  examples are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Quivr performs better than EQUI-VOCAL  10  EQUI-VOCAL: Synthesizing Queries for Compositional Video Events from Limited User Interactions  TQ1 TQ2 TQ3 TQ4 TQ5 TQ6 TQ7 TQ8 TQ9  Query ID  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='0  F1 score  Labeling budget: 12  TQ1 TQ2 TQ3 TQ4 TQ5 TQ6 TQ7 TQ8 TQ9  Query ID  Labeling budget: 20  TQ1 TQ2 TQ3 TQ4 TQ5 TQ6 TQ7 TQ8 TQ9  Query ID  Labeling budget: 30  PATSQL  Quivr  EQUI-VOCAL  Figure 8: F1 score for queries without duration constraints on simplified tasks, under different user labeling budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' PATSQL  performs well only when the target query is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL performs worse than Quivr when no additional examples  are requested, but catches up and outperforms Quivr with a larger labeling budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  FN rate  Method  Quivr  Ours  Mean (Range)  Mean  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='1  68% (20%-100%)  100%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='2  54% (35%-75%)  100%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='3  47% (5%-75%)  100%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='4  34% (5%-55%)  100%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5  17% (0%-50%)  100%  Table 6: Probability that a system returns at least one query  on noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' (false positive rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='1 of false negative rate)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='0  F1 score  Label noise: FN rate=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='1, FP rate=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='01  TQ1  TQ2  TQ3  TQ4  TQ5  TQ6  TQ7  TQ8  TQ9  Query ID  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='0  F1 score  Label noise: FN rate=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='3, FP rate=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='03  Quivr  EQUI-VOCAL  Figure 9: Impact of data noise, with a labeling budget of  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL constantly outperforms Quivr and is thus  more robust to data noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  when no additional examples are requested (budget=12) because it  enumerates the entire search space and finds all consistent queries  using the initial examples, while EQUI-VOCAL prunes candidate  queries at every iteration to ensure efficient query synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' With  a larger labeling budget, EQUI-VOCAL can request more labels  during the search process to select better paths to explore, thus it  catches up with Quivr and outperforms it when the budget is 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  EQUI-VOCAL is more robust to noisy data and produces  higher quality queries than Quivr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We compare the perfor-  mance of Quivr and EQUI-VOCAL when the data is noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We  do not compare against PATSQL since its performance on video  queries is already low even with perfect data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We inject noise into  the original dataset by randomly flipping a fraction of the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  We vary the false negative rate from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5 and set the false  positive rate to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='1 the false negative one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Table 6 shows the per-  centage of runs when the system returned any queries for different  noise rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We evaluate the systems over 9 queries (TQ1-TQ9) in  the simplified setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For Quivr, we also report the range of the  ID  # queries explored  Quivr  Ours  TQ1  203018  1123  TQ2  186333  1071  TQ3  214536  1101  TQ5  179155  1118  TQ6  213419  1077  ID  # predictions/s  Quivr  Ours  TQ1  174  574  TQ2  176  666  TQ3  170  664  TQ5  151  480  TQ6  138  547  Table 7: Number of queries explored and number of pre-  dictions per second to achieve at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='9 F1 scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-  VOCAL reduces computational effort by exploring fewer  queries and by executing queries faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  success rate besides the mean value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' When the false negative rate  is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='1, Quivr has a 68% success rate, and this goes down to only  17% when the false negative rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In contrast, EQUI-VOCAL  always returns 𝑘 queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' To demonstrate that EQUI-VOCAL also  produces higher quality queries, Figure 9 shows the F1 score of  EQUI-VOCAL and Quivr with a labeling budget of 20, under two  different noise rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' When Quivr fails to return any queries, we  assign an F1 score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL performs better than Quivr  for different queries and different noise rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  EQUI-VOCAL is more computationally efficient than  Quivr by exploring fewer queries and executing every query  faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Table 7 measures the number of queries explored and the  number of predictions per second to achieve at least a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='9 F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  We define a prediction as evaluating a query on a video segment,  so the number of predictions per second reflects the efficiency of  query execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We evaluate over TQ1-TQ9 in the normal setting  and report the numbers for queries that Quivr can synthesize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' By  limiting the query search to valid sequences of region graphs and  by using the beam search strategy, the number of queries explored  by EQUI-VOCAL is as small as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='50% of Quivr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL also  executes queries faster than Quivr by up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='0× in terms of the  number of predictions per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' These two factors together make  EQUI-VOCAL more efficient than Quivr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='2  Results on scene graphs  Next, we evaluate EQUI-VOCAL on the scene graphs dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Nei-  ther PATSQL nor Quivr support such flexible queries at video scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Because the dataset is more complex, EQUI-VOCAL requires that  the user provide a slightly larger (although still quite small) number  of initial examples to avoid overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We randomly select 15 posi-  tive and 15 negative examples from the training dataset as the initial  11  30  50  100  Budget  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='0  F1 score  Easy  30  50  100  Budget  Medium  30  50  100  Budget  Hard  Figure 10: Scene graph queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL can synthesize  high-quality queries within a reasonable budget, even for  complex queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  1  2  4  8  # CPUs  20  40  Time (min)  Figure 11: Vary # CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL can be easily paral-  lelized to reduce synthesis time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Later in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='3, we discuss how varying the initial  number of examples affects the performance of EQUI-VOCAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Figure 10 shows the F1 score of EQUI-VOCAL under different  user labeling budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' With a larger budget, EQUI-VOCAL can learn  queries with higher F1 scores, and achieves a median F1 score of  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='91, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='67 for easy, medium, and hard queries when the  budget is 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Results for a budget of 50 are nearly identical, which  shows that EQUI-VOCAL can synthesize high-quality queries  within a reasonable budget, even for complex queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  EQUI-VOCAL does not learn good queries in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We note  that some queries are easier to synthesize than others, which results  in the high variance of the F1 score in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In particular, EQUI-  VOCAL assumes that the ancestor queries of the target query are  informative and leverages the performance of those queries to guide  the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' However, if the ancestor queries cannot be distinguished  from other queries in the same iteration (because they all have the  same, typically low, F1 score), EQUI-VOCAL struggles with learning  good queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' One direction of future work could explore other types  of more fine-grained user feedback besides binary labels to help  EQUI-VOCAL learn intermediate queries [53], while still limiting  user and computational efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  EQUI-VOCAL’s query synthesis time on the scene graphs dataset  is greater than on the trajectories dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' It increases from tens  of seconds (Table 5) to minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' However, EQUI-VOCAL can  easily be parallelized to reduce synthesis time, as executing  queries in PostgreSQL is embarrassingly parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Figure 11  shows the query synthesis time for the hard, scene-graphs queries  when varying the number of CPUs and for a labeling budget of 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Query synthesis time decreases significantly by using multiple cores  and reduces to 13 minutes with 8 CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL achieves  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='76× speedup with 2 CPUs and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='03× speedup with 8 CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Note  that this is the largest labeling budget for our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' With  a lower budget, EQUI-VOCAL is even faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The query synthesis  time can be further reduced by using more CPUs, further decreasing  the labeling budget, decreasing 𝑏𝑤, or decreasing 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  10  20  30  40  50  # Initial examples  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='0  F1 score  Easy  10  20  30  40  50  # Initial examples  Medium  10  20  30  40  50  # Initial examples  Hard  Figure 12: Providing more initial examples helps EQUI-  VOCAL learn better queries and avoid overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  10  20  30  40  50  # Initial examples  4  7  10  13  16  Query complexity  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
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+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='93  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='54  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='69  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='89  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='84  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='36  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='83  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='42  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='64  F1 score  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='8  Figure 13: F1 score for various numbers of initial exam-  ples and queries with different complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL ob-  tains higher F1 scores with more initial examples and when  the target queries are simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='3  Ablation study  We study the impact of hyperparameters and the sensitivity of  EQUI-VOCAL to hyperparameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Unless otherwise speci-  fied, we evaluate EQUI-VOCAL on the scene graphs dataset and use  the default values for hyperparameters that we are not studying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  EQUI-VOCAL synthesizes useful queries with a small  number of initial examples, but providing more initial ex-  amples improves performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Figure 12 shows EQUI-VOCAL’s  F1 score for different numbers of initial examples, from five positives  and five negatives (ten in total), to 25 positives and 25 negatives  (50 in total), while the total labeling budget remains 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' When  the number of initial examples is smallest, EQUI-VOCAL can incor-  rectly retain queries that overfit the examples in early iterations,  leading to final results with low F1 scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Results, however, quickly  improve, and with as few as 30 initial examples, median F1 scores  are high at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='998, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='911, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='670 for the easy, medium, and hard  queries respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Figure 13 shows the F1 score of EQUI-VOCAL  under different numbers of initial examples and queries with dif-  ferent complexity, while the total labeling budget remains 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' As  expected, EQUI-VOCAL obtains higher F1 scores with more initial  examples and when the target queries are simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The heatmap  aggregates the results of all easy, medium, and hard queries, and  we use 𝛼1 = 1, 𝛼2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='9, 𝛼3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='2 when computing the query com-  plexity for better visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Because manually finding initial  examples requires more user effort than labeling video segments  selected by EQUI-VOCAL, our approach is to balance user effort  and system’s performance, using 30 initial examples in all other  experiments on the scene graphs dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Interestingly, since EQUI-  VOCAL can produce good queries in many cases even with a few  examples, users always have the option to first start with a small  set and restart with more examples if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  12  EQUI-VOCAL: Synthesizing Queries for Compositional Video Events from Limited User Interactions  TQ1  TQ2  TQ3  TQ4  TQ5  TQ6  TQ7  TQ8  TQ9  TQ1D  TQ2D  TQ3D  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='0  F1 score  Trajectories (budget: 20)  Easy  Medium  Hard  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='0  F1 score  Scene graphs (budget: 50)  =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='0  =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='001  =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='01  Figure 14: Regularization (𝜆 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='001 and 𝜆 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='01) helps EQUI-VOCAL to learn better queries in many cases and does not harm  the performance in other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  10  20  30  40  # Initial examples  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='0  F1 score  Easy  10  20  30  40  # Initial examples  Medium  10  20  30  40  # Initial examples  Hard  Active learning  Random sampling  Figure 15: Active learning helps EQUI-VOCAL learn better  queries compared to random sampling, and the improve-  ment is more significant when the number of initial exam-  ples is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Regularization helps EQUI-VOCAL learn better queries in  many cases and does not harm performance in other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Figure 14 shows the F1 score of EQUI-VOCAL on both the trajec-  tories dataset and the scene graphs dataset, under different values  of 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For the trajectory pairs dataset, we set the labeling budget  to 20 and the initial number of examples to 12 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', two positive  and ten negative examples);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' for the scene graphs dataset, we set  the labeling budget to 50 and the initial number of examples to  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For the trajectories dataset, we see an improvement for many  queries with regularization, especially when the queries are simple  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=', with fewer predicates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' With 𝜆 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='01, we see a median im-  provement between 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='6% and −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='44% for the queries from Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  TQ4, TQ9, TQ2D, and TQ3D are more complex than other queries  and have four predicates, so applying regularization is unlikely  to improve the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For the more complex scene graphs  dataset, regularization does not improve F1 scores but also does  not compromise system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' With 𝜆 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='001, the median  change in F1 score stays between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='84% and −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='69%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For this reason,  EQUI-VOCAL defaults to using a small regularization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Active learning helps EQUI-VOCAL learn better queries  quicker than randomly sampling video segments to label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Figure 15 shows EQUI-VOCAL’s F1 score when either using ac-  tive learning or randomly selecting additional examples during  query synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We set the labeling budget to 50, and we vary the  number of initial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The improvement of active learning  over random sampling is more significant when the number of ini-  tial examples is larger since fewer labels are requested per iteration  and the selection of informative examples becomes more important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  1  5  10  15  20  bw  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='0  F1 score  Easy  1  5  10  15  20  bw  Medium  1  5  10  15  20  bw  Hard  Figure 16: Increasing𝑏𝑤 improves the performance of EQUI-  VOCAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  1  10  100 1000  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='0  F1 score  Easy  1  10  100 1000  k  Medium  1  10  100 1000  k  Hard  Figure 17: Increasing 𝑘 slightly improves the performance  of EQUI-VOCAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Increasing the beam width improves performance, but  also increases the synthesis time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Figure 16 shows the F1 score  of EQUI-VOCAL under different values of 𝑏𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Increasing 𝑏𝑤 from  1 to 10 increases median F1 scores on easy queries by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='10%, on  medium queries by 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='8%, and on hard queries by 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='3%, but in-  creasing from 10 to 20 only further improves median scores by up  to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='07%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' At the same time, increasing 𝑏𝑤 from 10 to 15 increases  the median synthesis time on hard queries by 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL  defaults to 𝑏𝑤 = 10 to strike a balance between query performance  and synthesis time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  increasing 𝑘 slightly improves performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We evaluate  the F1 score of EQUI-VOCAL under different values of 𝑘, as shown  in Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' For medium and hard scene graph queries, increas-  ing 𝑘 from 1 to 1000 slightly improves the median F1 score by  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='36% and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='12% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Since the target queries are complex,  they are not explored until later iterations of the algorithm, when  most labeling budget has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Labels requested afterward  have little impact on the ranking of the queries in the list, so the  performance improvement from increasing 𝑘 is also limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Easy  queries already have near-perfect F1 scores, so increasing 𝑘 does  not improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  13  SQ1  SQ2  SQ3  0  50  100  Time (s)  EQUI-VOCAL  No-Pruning  No-Recursion  Avoidance  (a) Single query execution time with  different SQL optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Caching  No caching  0  20  40  Time (min)  (b) Synthesis time with  and without caching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Figure 18: Impact of query execution optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-  VOCAL reduces query synthesis time by (a) generating effi-  cient SQL queries that prune intermediate results and avoid  recursion, and (b) using a caching mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  EQUI-VOCAL’s query execution optimizations signifi-  cantly reduce query synthesis times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Figure 18a shows the av-  erage query execution time for variants of our query translation  algorithm on scene graph queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We generate SQL queries using  our complete query translation algorithm, and two variants that  omit optimizations of pruning intermediate results and avoiding  recursion, respectively, and then execute them over all 10,000 video  segments from the scene graphs dataset in PostgreSQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We evaluate  EQUI-VOCAL on SQ1-SQ3, and each query is executed 20 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  As the figure shows, the query execution time increases dramat-  ically without either optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Pruning intermediate results  boosts performance by up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='66×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Recursion avoidance is impor-  tant when the query contains duration constraints (SQ1 and SQ3),  providing up to a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='14× speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Figure 18b shows the impact of our caching mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Caching  query results leads to a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='97× speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' We experimented on hard  queries using four CPUs and a labeling budget of 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  6  RELATED WORK  Video analytics systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Many recent VDBMSs have been pro-  posed and focus on a wide range of data management challenges,  including fast inference over videos [3, 5, 34, 44, 47], storage op-  timization [18, 25, 64], efficient dataflow processing [49], prepro-  cessing and indexing [6, 28, 36], exploration and organization [19],  privacy [9], and tuning configurations [29, 32, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Those techniques  are orthogonal to our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Compositional query processing over videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Several sys-  tems have explored compositional queries over videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' However,  they either require that users express compositional queries ex-  plicitly [14, 21, 42], or train customized models for such events [7,  11, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Instead, EQUI-VOCAL uses a query-by-example approach  to minimize user effort and learn and refine a query specification  iteratively from user feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  As detailed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='3, TQVS [15] and Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' [16] limit  the types of compositional queries to temporal queries that count  co-occurring objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Vaas [7] enables users to execute a complex  query by combining the output of simpler queries, but does not  introduce any algebra to reason about spatiotemporal events, and  SVQ++ [11] only supports spatial queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In contrast, EQUI-VOCAL  introduces a more general data model that supports compositional  queries involving relationships between objects in space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Accelerating query execution over videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Prior work on re-  cent VDBMSs focuses on accelerating queries by filtering frames  using cheaper predicates [44], optimizing the sampling rate [5, 47],  and building specialized models [3, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL currently  extracts all objects in a video before the search process, but could  leverage those existing techniques to avoid running expensive ob-  ject detection and tracking algorithms on all frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Query by example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Many query-by-example systems have  been proposed to support users in expressing SQL queries over  relational data [20, 41, 50, 56, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL focuses on learn-  ing queries for video events, which can be expressed using a form  that is more constrained than general relational queries, but also re-  quires greater scalability and tolerance to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' The most relevant  work to our system is Quivr [46], which also targets video queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  However, Quivr only operates over object trajectories instead of  entire scene graphs, and assumes noiseless inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' SQuID [20] syn-  thesizes queries based on semantic similarity but does not support  self-joins and non-equi joins, which are necessary to find our video  events, and S4 [50] ranks queries based on the degree of input  containment but is limited to project-join queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Program synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Program synthesis has been used in recent  research for a wide range of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' [30] generates re-  ferring relational programs to uniquely identify an object in an  image in terms of its attributes and relations with other objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Falx [59] supports users in authoring visualizations from examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Wrangler [33] learns relational data transformation through user  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' BUSTLE [48] synthesizes programs for string process-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Unlike these systems, EQUI-VOCAL synthesizes queries in the  domain of videos, which is significantly different in terms of the  spatio-temporal complexity and the prevalence of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL has an active learning compo-  nent to select the most promising queries to expand, instead of  training a better classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Model Picker [37] considers a set of pre-  trained classifiers and proposes an active learning approach that  selects informative examples to distinguish the best model in an  online setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL adapts this approach to distinguish  the most promising queries to explore at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Quivr [46]  also prunes candidate queries using active learning, but only af-  ter enumerating all candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL instead integrates  active learning and labeling into the search process to expand only  the most promising queries in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' This ensures both  synthesis efficiency and query performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  7  CONCLUSION  In this paper, we presented EQUI-VOCAL, a new system that synthe-  sizes compositional queries from examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL models  compositional events as spatio-temporal scene graphs, explores the  query search space using results of executing intermediate queries  and beam search, leverages active learning to reduce user effort,  and generates efficient SQL queries to reduce computational effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This work was supported in part by the NSF through awards CCF-  1703051 and IIS-2211133 as well as a grant from CISCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  14  EQUI-VOCAL: Synthesizing Queries for Compositional Video Events from Limited User Interactions  REFERENCES  [1] 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' EQUI-VOCAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='com/uwdb/EQUI-VOCAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [2] Ganesh Ananthanarayanan, Paramvir Bahl, Peter Bodík, Krishna Chintalapudi,  Matthai Philipose, Lenin Ravindranath, and Sudipta Sinha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Real-Time Video  Analytics: The Killer App for Edge Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Computer 50, 10 (2017), 58–67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [3] Michael R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Anderson, Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Cafarella, Germán Ros, and Thomas F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Wenisch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Physical Representation-Based Predicate Optimization for a Visual Analyt-  ics Database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In ICDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 1466–1477.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
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+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Un-  manned Aerial Aircraft Systems for transportation engineering: Current practice  and future challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' IJTST 5, 3 (2016), 111–122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [5] Favyen Bastani, Songtao He, Arjun Balasingam, Karthik Gopalakrishnan, Mo-  hammad Alizadeh, Hari Balakrishnan, Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Cafarella, Tim Kraska, and  Sam Madden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' MIRIS: Fast Object Track Queries in Video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In SIGMOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  1907–1921.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [6] Favyen Bastani and Samuel Madden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' OTIF: Efficient Tracker Pre-processing  over Large Video Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In SIGMOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2091–2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [7] Favyen Bastani, Oscar R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Moll, and Samuel Madden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Vaas: Video Analytics  At Scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' PVLDB 13, 12 (2020), 2877–2880.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [8] Irving Biederman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Recognition-by-components: a theory of human image  understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Psychological review 94, 2 (1987), 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [9] Frank Cangialosi, Neil Agarwal, Venkat Arun, Srinivas Narayana, Anand D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Sarwate, and Ravi Netravali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Privid: Practical, Privacy-Preserving Video  Analytics Queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In NSDI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 209–228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [10] Shih-Fu Chang, William Chen, Horace J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Meng, Hari Sundaram, and Di Zhong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' VideoQ: An Automated Content Based Video Search System Using Visual  Cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In MM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 313–324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [11] Daren Chao, Nick Koudas, and Ioannis Xarchakos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' SVQ++: Querying for  Object Interactions in Video Streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In SIGMOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2769–2772.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [12] Vincent S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Chen, Paroma Varma, Ranjay Krishna, Michael S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Bernstein, Christo-  pher Ré, and Li Fei-Fei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Scene Graph Prediction With Limited Labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' ICCV  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [13] Xinyun Chen, Chang Liu, and Dawn Song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Execution-Guided Neural  Program Synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [14] Yueting Chen, Nick Koudas, Xiaohui Yu, and Ziqiang Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Spatial and  Temporal Constrained Ranked Retrieval over Videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' PVLDB 15, 11 (2022), 3226–  3239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [15] Yueting Chen, Xiaohui Yu, and Nick Koudas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' TQVS: Temporal Queries over  Video Streams in Action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In SIGMOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2737–2740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [16] Yueting Chen, Xiaohui Yu, Nick Koudas, and Ziqiang Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Evaluating  Temporal Queries Over Video Feeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In SIGMOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 287–299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [17] Pramod Chunduri, Jaeho Bang, Yao Lu, and Joy Arulraj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Zeus: Efficiently  Localizing Actions in Videos using Reinforcement Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In SIGMOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 545–  558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [18] Maureen Daum, Brandon Haynes, Dong He, Amrita Mazumdar, and Magdalena  Balazinska.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' TASM: A Tile-Based Storage Manager for Video Analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In  ICDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 1775–1786.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [19] Maureen Daum, Enhao Zhang, Dong He, Magdalena Balazinska, Brandon Haynes,  Ranjay Krishna, Apryle Craig, and Aaron Wirsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' VOCAL: Video Organi-  zation and Interactive Compositional AnaLytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In CIDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [20] Anna Fariha and Alexandra Meliou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Example-Driven Query Intent Dis-  covery: Abductive Reasoning using Semantic Similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' PVLDB 12, 11 (2019),  1262–1275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [21] Daniel Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Fu, Will Crichton, James Hong, Xinwei Yao, Haotian Zhang, Anh Truong,  Avanika Narayan, Maneesh Agrawala, Christopher Ré, and Kayvon Fatahalian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Rekall: Specifying Video Events using Compositions of Spatiotemporal  Labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' arXiv preprint arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='02993 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [22] Andreas Geiger, Philip Lenz, and Raquel Urtasun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Are we ready for au-  tonomous driving?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' the kitti vision benchmark suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In CVPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
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+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Towards Drone-sourced Live Video Analytics  for the Construction Industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In HotMobile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 3–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
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+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' A Reasoning Based Model for Anomaly Detection in the Smart City Domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  In IntelliSys (AISC), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
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+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' VSS: A Storage System for Video  Analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In SIGMOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 685–696.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [26] Brandon Haynes, Amrita Mazumdar, Magdalena Balazinska, Luis Ceze, and Alvin  Cheung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Visual Road: A Video Data Management Benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In SIGMOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  972–987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [27] Kaiming He, Georgia Gkioxari, Piotr Dollár, and Ross B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Girshick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Mask  R-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In ICCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2980–2988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [28] Wenjia He, Michael R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Anderson, Maxwell Strome, and Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Cafarella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  A Method for Optimizing Opaque Filter Queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In SIGMOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 1257–1272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [29] Wenjia He and Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Cafarella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Controlled Intentional Degradation in  Analytical Video Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In SIGMOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2105–2119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [30] Jiani Huang, Calvin Smith, Osbert Bastani, Rishabh Singh, Aws Albarghouthi,  and Mayur Naik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Generating Programmatic Referring Expressions via  Program Synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In ICML, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 4495–4506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [31] Jingwei Ji, Ranjay Krishna, Li Fei-Fei, and Juan Carlos Niebles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Action  genome: Actions as compositions of spatio-temporal scene graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In CVPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  10236–10247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
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+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Chameleon: scalable adaptation of video analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In SIGCOMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  253–266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [33] Sean Kandel, Andreas Paepcke, Joseph M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Hellerstein, and Jeffrey Heer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Wrangler: interactive visual specification of data transformation scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In CHI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  3363–3372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
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+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' BlazeIt: Optimizing Declara-  tive Aggregation and Limit Queries for Neural Network-Based Video Analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  PVLDB 13, 4 (2019), 533–546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
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+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  NoScope: Optimizing Deep CNN-Based Queries over Video Streams at Scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  PVLDB 10, 11 (2017), 1586–1597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [36] Daniel Kang, John Guibas, Peter D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Bailis, Tatsunori Hashimoto, and Matei  Zaharia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' TASTI: Semantic Indexes for Machine Learning-based Queries  over Unstructured Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In SIGMOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 1934–1947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [37] Mohammad Reza Karimi, Nezihe Merve Gürel, Bojan Karlas, Johannes Rausch, Ce  Zhang, and Andreas Krause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Online Active Model Selection for Pre-trained  Classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In AISTATS, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
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+page_content=' 307–315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
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+page_content=' Shamma, Michael S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
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+page_content=' Visual Genome: Connecting Language and Vision  Using Crowdsourced Dense Image Annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' IJCV 123, 1 (2017), 32–73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
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+page_content=' Kuo and Arbee L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
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+page_content=' Content-Based Query Processing for  Video Databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' TMM 2, 1 (2000), 1–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
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+page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Segmentation in the perception  and memory of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' TiCS 12, 2 (2008), 72–79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
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+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Query From Examples: An  Iterative, Data-Driven Approach to Query Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' PVLDB 8, 13 (2015),  2158–2169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Manjunath, Kevin S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Chan,  and Ramesh Govindan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Caesar: cross-camera complex activity recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  In SenSys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 232–244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
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+page_content=' Smith, and  Yejin Choi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' NeuroLogic A*esque Decoding: Constrained Text Generation  with Lookahead Heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In NAACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 780–799.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
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+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Accelerating Machine Learning Inference with Probabilistic Predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In SIG-  MOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 1493–1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
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+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Bird Cams Lab  Biological Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' https://ecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='edu/handle/1813/110264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [46] Stephen Mell, Favyen Bastani, Stephan Zdancewic, and Osbert Bastani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Synthesizing Video Trajectory Queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In AIPLANS Workshop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [47] Oscar Moll, Favyen Bastani, Sam Madden, Michael Stonebraker, Vijay N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Gade-  pally, and Tim Kraska.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' ExSample: Efficient Searches on Video Repositories  through Adaptive Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' ICDE (2020), 3065–3077.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [48] Augustus Odena, Kensen Shi, David Bieber, Rishabh Singh, Charles Sutton, and  Hanjun Dai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' BUSTLE: Bottom-Up Program Synthesis Through Learning-  Guided Exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [49] Alex Poms, Will Crichton, Pat Hanrahan, and Kayvon Fatahalian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Scanner:  Efficient Video Analysis at Scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 37, 4 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [50] Fotis Psallidas, Bolin Ding, Kaushik Chakrabarti, and Surajit Chaudhuri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' S4:  Top-k Spreadsheet-Style Search for Query Discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In SIGMOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2001–2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [51] Shaoqing Ren, Kaiming He, Ross B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Girshick, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Faster R-CNN:  Towards Real-Time Object Detection with Region Proposal Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  91–99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [52] Francisco Romero, Mark Zhao, Neeraja J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Yadwadkar, and Christos Kozyrakis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Llama: A Heterogeneous & Serverless Framework for Auto-Tuning Video  Analytics Pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In SoCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 1–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [53] Olga Russakovsky, Li-Jia Li, and Li Fei-Fei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Best of both worlds: Human-  machine collaboration for object annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In CVPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2121–2131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [54] Andrew W Senior, L Brown, Arun Hampapur, C-F Shu, Yun Zhai, Rogério Schmidt  Feris, Y-L Tian, Sergio Borger, and C Carlson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Video analytics for retail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In  AVSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 423–428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [55] Manuel Stein, Halldor Janetzko, Andreas Lamprecht, Thorsten Breitkreutz,  Philipp Zimmermann, Bastian Goldlücke, Tobias Schreck, Gennady L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Andrienko,  Michael Grossniklaus, and Daniel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Keim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Bring It to the Pitch: Combining  Video and Movement Data to Enhance Team Sport Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' TVCG 24, 1 (2018),  13–22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [56] Keita Takenouchi, Takashi Ishio, Joji Okada, and Yuji Sakata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' PATSQL:  Efficient Synthesis of SQL Queries from Example Tables with Quick Inference of  15  Projected Columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' PVLDB 14, 11 (2021), 1937–1949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [57] Abhishek Udupa, Arun Raghavan, Jyotirmoy V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Deshmukh, Sela Mador-Haim,  Milo M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Martin, and Rajeev Alur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' TRANSIT: specifying protocols with  concolic snippets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In PLDI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 287–296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [58] Chenglong Wang, Alvin Cheung, and Rastislav Bodík.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Synthesizing highly  expressive SQL queries from input-output examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' PLDI (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [59] Chenglong Wang, Yu Feng, Rastislav Bodík, Isil Dillig, Alvin Cheung, and Amy J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Ko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Falx: Synthesis-Powered Visualization Authoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In CHI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 106:1–106:15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [60] Junjue Wang, Ziqiang Feng, Zhuo Chen, Shilpa George, Mihir Bala, Padmanabhan  Pillai, Shao-Wen Yang, and Mahadev Satyanarayanan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Bandwidth-Efficient  Live Video Analytics for Drones Via Edge Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In SEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 159–173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [61] Sida I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Wang, Samuel Ginn, Percy Liang, and Christopher D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Manning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  Naturalizing a Programming Language via Interactive Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In ACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 929–  938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [62] Xiaoli Wang, Aakanksha Chowdhery, and Mung Chiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' SkyEyes: adaptive  video streaming from UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In HotWireless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [63] Nicolai Wojke and Alex Bewley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Deep Cosine Metric Learning for Person  Re-identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In WACV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 748–756.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [64] Tiantu Xu, Luis Materon Botelho, and Felix Xiaozhu Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' VStore: A Data  Store for Analytics on Large Videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In EuroSys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 16:1–16:17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [65] Piyush Yadav and Edward Curry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' VidCEP: Complex Event Processing  Framework to Detect Spatiotemporal Patterns in Video Streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In Big Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  2513–2522.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [66] Kexin Yi, Chuang Gan, Yunzhu Li, Pushmeet Kohli, Jiajun Wu, Antonio Tor-  ralba, and Joshua B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Tenenbaum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' CLEVRER: Collision Events for Video  Representation and Reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [67] Quanzeng You, Hailin Jin, Zhaowen Wang, Chen Fang, and Jiebo Luo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Image  Captioning with Semantic Attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' In CVPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 4651–4659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  [68] Jeffrey M Zacks, Barbara Tversky, and Gowri Iyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Perceiving, remembering,  and communicating structure in events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content=' Journal of experimental psychology:  General 130, 1 (2001), 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
+page_content='  16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfAfr_/content/2301.00929v1.pdf'}
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+arXiv:2301.08576v1  [math.AP]  20 Jan 2023
+STABILITY ESTIMATES FOR NONLOCAL BALANCE LAWS ARISING IN
+TRAFFIC MODELLING
+FELISIA ANGELA CHIARELLO AND HAROLD DEIVI CONTRERAS
+Abstract. This paper focuses on the proof of the stability of entropy weak solutions of a non-
+local balance law modelling vehicular traffic flow on a road with on- and off-ramps. The stabil-
+ity is obtained with respect to a kernel function in the source term. We get an estimate of the
+L1−dependence of the solution with respect to the initial datum, the on-ramp rate, the off-ramp
+rate and the mentioned kernel function. We also show numerical experiments in order to perform
+an optimization problem in traffic flow with on-ramps.
+1. Introduction
+1.1. Aim. In traffic flow modeling, nonlocal conservation laws are intended to describe the be-
+haviour of drivers that adapt their velocity with respect to what happens in front of them, see
+[1, 3, 4, 7, 8]; the classical LWR (Lighthill-Whitham [10] and Richards [13]) is not able to model
+this type of situation. For this reason, in order to generalise the LWR model in such a way to make
+it able to describe more realistic settings, in [2] a nonlocal balance law intended to model vehicular
+traffic flow on a main road wit on- and off-ramps is introduced and it is given by
+(1.1)
+ρt + (ρv(ρ ∗ ωη))x = Son(·, ·, ρ, ρ ∗ ωη,δ) − Soff(·, ·, ρ),
+x ∈ R,
+where Son and Soff represents the traffic flow entering and exiting through an on- and off-ramp
+respectively, and the convolution term in Son is defined as follows
+(ρ ∗ ωη,δ)(t, x) =
+� x+η+δ
+x−η+δ
+ρ(t, y)ωη,δ(y − x)dy,
+with η ∈ [0, 1] and δ ∈ [−η, η]. Here, the parameter η represents the length of the support of the
+kernel function ωη,δ, while δ is the point at which the maximum of the kernel is attained. This
+choice of the kernel function models the fact that drivers on the on-ramp can see what happens on
+backward and forward on the main road.
+It is well known that on-ramp merging has a great impact on traffic efficiency, if one concentrates on
+highway networks the reduction of the capacity is often due to on- and off-ramps, for this reason in
+this paper we are interested to study the dependence of solutions to (1.1) on the convolution kernel
+given in the source term Son. The strategies that we employ are inspired by the results obtained in
+[2, 5]. In particular, we adopt the results about existence and uniqueness to (1.1) presented in [2]
+and we propose the analytical study of the dependence of solutions to (1.1) on the kernel function
+in the source term.
+1.2. Related work. In literature, the LWR model has been extended to include on and off-ramps
+by means of source and sink terms, see [6, 9, 11, 12, 14, 15]. Starting from the model introduced in
+[11], in [2] the authors introduce the new nonlocal balance law (1.1), in which the velocity function
+Date: January 23, 2023.
+1
+
+2
+FELISIA ANGELA CHIARELLO AND HAROLD DEIVI CONTRERAS
+depends on a weighted mean of the downstream traffic density, in order to model the behaviour
+of drivers that adapt their velocity with respect to what happens in front of them; moreover the
+nonlocal source term is intended to model the fact that drivers on the on-ramp can see what
+happens on backward and forward on the main road. Regarding the aferomentioned model the
+existence and uniqueness of entropy weak solution has been proved, specifically, approximating
+the problem using an Upwind-type numerical scheme and providing compactness estimates for the
+sequence of approximate solutions that allows to get the convergence and thus, the existence of
+solution. Furthemore, by using Kruˇzkov’s doubling of variables technique the authors get an L1
+stability estimate depending on initial data, the on-ramp rate and the off-ramp rate. On the other
+hand, motivated by control and optimisation problems, in [5] the authors study a conservation law
+with nonlocal flux function given by
+∂tρ + ∂x (f(t, x, ρ)v(ρ ∗ ω)(t, x)) = 0.
+(1.2)
+The authors obtain an estimate of the dependence of the solution to (1.2) with respect to the kernel
+function, the velocity and the initial datum. The stability is obtained from an entropy condition
+through the doubling of variables technique.
+1.3. Outline of this paper. This work is organized as follows: In Section 2 we recall the mathe-
+matical model on which we will focus our study as well as the definitions of weak and entropy weak
+solution. In section 3 we present the main results of this work, i.e. the L1− Lipschitz continuous
+dependence of the solutions to (1.1) on to the initial datum, the on-ramp rate, the off-ramp rate
+and the kernel function in the source term. Finally, in Section 4 we present numerical examples in
+order to illustrate the behaviour of solutions when some parameters of the kernel function in the
+source term are varying.
+2. Mathematical model
+Let us consider the equation (1.1) with terms Son and Soff defined as
+Son(t, x, ρ, ρ ∗ ωη,δ)
+=
+1on(x)qon(t)
+�
+1 −
+ρ
+ρmax
+� �
+1 − ρ ∗ ωη,δ
+ρmax
+�
+,
+(2.1)
+Soff(t, x, ρ)
+=
+1off(x)qoff(t)
+ρ
+ρmax
+,
+(2.2)
+with ρmax = 1 for the sake of simplicity, and we also endow the nonlocal traffic reaction model (1.1)-
+(2.1)-(2.2) with an initial condition as follows
+ρ(x, 0) = ρ0(x) ∈
+�
+L1 ∩ BV
+�
+(R; [0, ρmax]).
+(2.3)
+Let us assume the following assumptions on the parameters of model (1.1).
+Assumptions 2.1. We assume
+(i) qramp
+on
+∈ L∞(R+; R+), qramp
+off
+∈ L∞(R+; R+).
+(ii) v ∈ C2([0, ρmax]; R+), v′(ρ) ≤ 0, ρ ∈ [0, ρmax].
+(iii) ωη ∈ C1([0, η]; R+) with ω′
+η(x) ≤ 0,
+� η
+0 ωη(x)dx = 1, ∀η > 0.
+(iv) ωη,δ ∈ (C1 ∩ L1)([δ − η, δ + η]; R+) with ω′(x)η,δ ≥ 0 for x ∈ [δ − η, 0], ω′(x)η,δ ≤ 0 for x ∈
+[0, δ + η], and
+� δ+η
+δ−η ωη,δ(x)dx = 1, ∀η > 0.
+
+STABILITY ESTIMATES FOR NONLOCAL BALANCE LAWS ARISING IN TRAFFIC MODELLING
+3
+We consider solutions in a weak sense as follows
+Definition 2.1. Let ρ0 ∈ (L1 ∩ BV)(R; [0, ρmax]). We say that ρ ∈ C([0, T]; L1(R; [0, ρmax])),
+with ρ(t, ·) ∈ BV(R; [0, ρmax]) for t ∈ [0, T], is a weak solution to (1.1)-(2.3) if for any ϕ ∈
+C1
+c([0, T[×R; R)
+� T
+0
+�
+R
+(ρϕt + ρV ϕx) dxdt +
+� T
+0
+�
+Ωon
+Sonϕdxdt
+−
+� T
+0
+�
+Ωoff
+Soffϕdxdt +
+�
+R
+ρ0(x)ϕ(0, x)dx = 0,
+where V (t, x) = v((ρ ∗ ω)(t, x)).
+Furthermore, we consider entropy weak solutions in Kruˇzkov sense, i.e.
+Definition 2.2. Let ρ0 ∈ (L1 ∩ BV)(R; [0, ρmax]). We say that ρ ∈ C([0, T]; L1(R; [0, ρmax])), with
+ρ(t, ·) ∈ BV(R; [0, ρmax]) for t ∈ [0, T], is a entropy weak solution to (1.1) with initial datum ρ0 if
+for any ϕ ∈ C1
+c([0, T[×R; R) and for all k ∈ R
+� T
+0
+�
+R
+(|ρ − k|ϕt + |ρ − k| V ϕx − sgn(ρ − k)kVxϕ) dxdt
++
+� T
+0
+�
+Ωon
+sgn(ρ − k)Sonϕdxdt −
+� T
+0
+�
+Ωoff
+sgn(ρ − k)Soffϕdxdt
++
+�
+R
+|ρ0 − k|ϕ(0, x)dx ≥ 0,
+where Ωon and Ωoff are the spatial positions of on-ramp and off-ramp on the main road, respec-
+tively.
+3. Main Result
+Before giving the main results of this work, we first recall the main Theorem in [2, Theorem 2.1].
+Theorem 3.1. Let ρ0 ∈
+�
+L1 ∩ BV
+�
+(R; [0, 1]). Let the Assumptions 2.1 hold. Then, for all T > 0,
+the problem (1.1) has a unique solution ρ ∈ C0 �
+[0, T]; L1(R; [0, 1])
+�
+in the sense of Definition 2.2.
+Moreover, the following estimates hold: for any t ∈ [0, T]
+∥ρ(t)∥L1(R) ≤ R1(t),
+0 ≤ ρ(t, x) ≤ 1,
+TV (ρ(t)) ≤ etH (TV (ρ0) + tQT ) ,
+where
+R1 = ∥ρ0∥L1(R) + ∥qramp
+on
+(·)∥L1([0,t]) − min
+x∈Ωon ∥qramp
+on
+(·)ρ(·, x)∥L1([0,t])
+− min
+x∈Ωoff
+��qramp
+off
+(·)ρ(·, x)
+��
+L1([0,t])
+(3.1)
+QT
+=
+2
+�
+∥qon∥L∞([0,T]) + ∥qoff∥L∞([0,T])
+�
+(3.2)
+H
+=
+2 ∥qon∥L∞([0,T]) + ∥qoff∥L∞([0,T]) + ωη(0)L
+(3.3)
+L
+=
+�
+∥v∥L∞([0,1]) + ∥v′∥L∞([0,1])
+�
+.
+(3.4)
+
+4
+FELISIA ANGELA CHIARELLO AND HAROLD DEIVI CONTRERAS
+The following theorem is the main result of this work and it states the L1- Lipschitz continuous
+dependence of solutions to (1.1) on to the initial datum, the on-ramp rate, the off-ramp rate and
+the kernel function.
+Theorem 3.2. Let ρ and ˜ρ be two solutions to problem (1.1) in the sense of Definition 2.2, with
+initial data ρ0, ˜ρ0 ∈ L1 ∩ BV (R; [0, 1]), with on-ramp rates qon, ˜qon, off-ramp rates qoff, ˜qoff and
+kernel functions ωη,δ, ˜ωη,δ, respectively. Assume v ∈ C2 ([0, 1], R+). Then, for a.e. t ∈ [0, T],
+∥ρ(t) − ˜ρ(t)∥L1(R)
+≤
+eCT
+�
+∥ρ0 − ˜ρ0∥L1(R) + L
+�
+∥qon − ˜qon∥L1([0,t]) + ∥qoff − ˜qoff∥L1([0,T])
+�
++r(T)∥ωη,δ − ˜ωη,δ∥L1(R)
+�
+,
+where r(T) depends on the L1−norms of initial conditions, expected inflow flow of the on-ramp and
+expected output flow of the off-ramp and C is defined as in [2, (3.24)].
+Proof. Since ρ and ˜ρ are entropy weak solutions to (1.1) then
+�
+ρt + (ρV (t, x))x = Son (t, x, qon, ρ, Ron) − Soff (t, x, qoff, ρ)
+ρ(0, x) = ρ0(x),
+
+
+
+˜ρt + (˜ρ ˜V (t, x))x = Son
+�
+t, x, ˜qon, ˜ρ, ˜Ron
+�
+− Soff (t, x, ˜qoff, ρ)
+˜ρ(0, x) = ˜ρ0(x),
+in a distributional sense, where V (t, x) = v((ρ ∗ ωη)(t, x)), Ron = (ρ ∗ ωη,δ)(t, x), ˜V (t, x) = v((˜ρ ∗
+ωη)(t, x)), ˜Ron = (˜ρ ∗ ˜ωη,δ)(t, x). Following the argument in [2] and using Kruˇzkov’s doubling of
+variables technique we get
+∥ρ(T, ·) − ˜ρ(T, ·)∥L1(R)
+≤
+∥ρ0 − ˜ρ0∥L1(R) +
+� T
+0
+�
+Ωon
+��� ˜Son
+��� dxdt +
+� T
+0
+�
+Ωoff
+��� ˜Soff
+��� dxdt
++
+� T
+0
+�
+R
+|V| |∂xρ(t, x)| dxdt +
+� T
+0
+�
+R
+|Vx| |ρ(t, x)| dxdt,
+(3.5)
+where
+˜Son
+=
+Son (t, x, qon, ρ, Ron) − Son
+�
+t, x, ˜qon, ˜ρ, ˜Ron
+�
+,
+˜Soff
+=
+Soff (t, x, qon, ρ) − Soff (t, x, ˜qon, ˜ρ) ,
+V
+=
+v(R) − v(P),
+Vx
+=
+∂xv(R) − ∂xv(P).
+Except for the second term, all terms appearing at the right-hand side of (3.5) are computed as in
+[2, Theorem 3.1],
+� T
+0
+�
+Ωoff
+|Soff| dxdt
+≤
+∥qoff∥L∞([0,T])
+� T
+0
+∥ρ(t, ·) − ˜ρ(t, ·)∥L1(R) dt + L ∥qoff − ˜qoff∥L1([0,T]) ,
+� T
+0
+�
+R
+|V| |∂xρ(t, x)| dxdt
+≤
+ωη(0)
+��v′��
+L∞([0,1]) sup
+t∈[0,T]
+TV (ρ(t, ·))
+� T
+0
+∥ρ(t, ·) − ˜ρ(t, ·)∥L1(R) dt,
+� T
+0
+�
+R
+|Vx| |ρ(t, x)| dxdt
+≤
+W
+� T
+0
+∥ρ(t, ·) − ˜ρ(t, ·)∥L1(R) dt.
+
+STABILITY ESTIMATES FOR NONLOCAL BALANCE LAWS ARISING IN TRAFFIC MODELLING
+5
+Regarding the second term in (3.5), we have
+� T
+0
+�
+Ωon
+��� ˜Son
+��� dxdt
+=
+� T
+0
+�
+Ωon
+���Son (t, x, qon, ρ, Ron) − Son
+�
+t, x, ˜qon, ˜ρ, ˜Ron
+���� dxdt
+≤
+� T
+0
+�
+Ωon
+���� ˜S1
+on
+��� +
+��� ˜S2
+on
+��� +
+��� ˜S3
+on
+���
+�
+dxdt,
+where
+˜S1
+on
+=
+Son (t, x, qon, ρ, Ron) − Son
+�
+t, x, qon, ρ, ˜Ron
+�
+,
+˜S2
+on
+=
+Son
+�
+t, x, qon, ρ, ˜Ron
+�
+− Son
+�
+t, x, qon, ˜ρ, ˜Ron
+�
+,
+˜S3
+on
+=
+Son
+�
+t, x, qon, ˜ρ, ˜Ron
+�
+− Son
+�
+t, x, ˜qon, ˜ρ, ˜Ron
+�
+.
+We bound ˜S2
+on and ˜S3
+on writing
+� T
+0
+�
+Ωon
+��� ˜S2
+on
+��� dxdt
+≤
+∥qon∥L∞([0,T])
+� T
+0
+∥ρ(t, ·) − ˜ρ(t, ·)∥L1(R) dt,
+� T
+0
+�
+Ωon
+��� ˜S3
+on
+��� dxdt
+≤
+� T
+0
+�
+Ωon
+|qon − ˜qon| dxdt
+≤
+L ∥qon − ˜qon∥L1([0,T]) .
+Next, we are going to bound ˜S1
+on term,
+��� ˜S1
+on
+���
+=
+���1onqon (1 − ρ)
+�
+(1 − Ron) −
+�
+1 − ˜Ron
+�����
+≤
+∥qon∥L∞([0,T])
+��� ˜Ron − Ron
+��� ,
+thus
+� T
+0
+�
+Ωon
+��� ˜S1
+on
+��� dxdt
+≤
+∥qon∥L∞([0,T])
+� T
+0
+�
+Ωon
+��� ˜Ron − Ron
+��� dxdt,
+and observe that
+�
+Ωon
+���Ron − ˜Ron
+��� dx
+=
+�
+Ωon
+|ωη,δ ∗ ρ − ˜ωη,δ ∗ ˜ρ| dx
+=
+�
+Ωon
+|ωη,δ ∗ ρ − ωη,δ ∗ ˜ρ + ωη,δ ∗ ˜ρ − ˜ωη,δ ∗ ˜ρ| dx
+≤
+�
+Ωon
+|ωη,δ ∗ ρ − ωη,δ ∗ ˜ρ| + |ωη,δ ∗ ˜ρ − ˜ωη,δ ∗ ˜ρ| dx
+=
+�
+Ωon
+|ωη,δ ∗ (ρ − ˜ρ)| +
+�
+Ωon
+|(ωη,δ − ˜ωη,δ) ∗ ˜ρ| dx
+≤
+∥ωη,δ∥L1(Ωon) ∥ρ − ˜ρ∥L1(Ωon) + ∥ωη,δ − ˜ωη,δ∥L1(Ωon) ∥˜ρ∥L1(Ωon)
+≤
+∥ωη,δ∥L1(R) ∥ρ − ˜ρ∥L1(R) + ∥ωη,δ − ˜ωη,δ∥L1(R) ∥˜ρ∥L1(R)
+≤
+∥ρ − ˜ρ∥L1(R) + ∥ωη,δ − ˜ωη,δ∥L1(R) R1(t),
+since
+�
+R ωη,δ(x)dx = 1. Here R1 is defined as in (3.1). Then,
+� T
+0
+�
+Ωon
+��� ˜S1
+on
+��� dxdt
+≤
+∥qon∥L∞([0,T])
+� T
+0
+∥ρ(t, ·) − ˜ρ(t, ·)∥L1(R) dt
++
+� T
+0
+∥ωη,δ − ˜ωη,δ∥L1(R) R1(t)dt
+
+6
+FELISIA ANGELA CHIARELLO AND HAROLD DEIVI CONTRERAS
+=
+∥qon∥L∞([0,T])
+� T
+0
+∥ρ(t, ·) − ˜ρ(t, ·)∥L1(R) dt
++ ∥ωη,δ − ˜ωη,δ∥L1(R)
+� T
+0
+R1(t)dt.
+Therefore, the inequality (3.5) is equivalent to following inequality
+∥ρ(T, ·) − ˜ρ(T, ·)∥L1(R)
+≤
+∥ρ0 − ˜ρ0∥L1(R) + L
+�
+∥qon − ˜qon∥L1([0,t]) + ∥qoff − ˜qoff∥L1([0,t])
+�
++C
+� T
+0
+∥ρ(t, ·) − ˜ρ(t, ·)∥L1(R) dt + ∥ωη,δ − ˜ωη,δ∥L1(R)
+� T
+0
+R1(t)dt
+=
+∥ρ0 − ˜ρ0∥L1(R) + L
+�
+∥qon − ˜qon∥L1([0,t]) + ∥qoff − ˜qoff∥L1([0,t])
+�
++C
+� T
+0
+∥ρ(t, ·) − ˜ρ(t, ·)∥L1(R) dt + r(T) ∥ωη,δ − ˜ωη,δ∥L1(R) .
+An application of Gronwall’s Lemma to the above inequality completes the proof.
+□
+4. Numerical examples
+In this Section we use [2, Algorithm 3.1] in order to compute approximate solutions to (1.1)-(2.3)
+with the source term (2.1). We simulate an optimization problem in traffic merging, which consists
+in finding the optimal value δ keeping fixed η in the kernel function ωη,δ. According to the meaning
+of that kernel function, this is equivalent to find where a drivers located on the on-ramp should
+look in order to avoid the congestion on the main road. To this end, we consider one on-ramp with
+length L = 0.1 located from x = 1.0 until x = 1.1, and we consider the following kernel functions
+ωη(x)
+:=
+2η − x
+η2
+χ[0,η](x),
+ωη,δ(x)
+:=
+1
+η6
+16
+5π
+�
+η2 − (x − δ)2�5/2 χ[−η+δ,η+δ](x),
+for convective and reactive terms, respectively, with η = 0.5 and δ ∈ [−η, η], for x ∈ [−1, 4] at
+simulated time T = 6 and velocity function given by v(ρ) = 1 − ρ. We also consider ∆x = 1/200, a
+constant initial condition ρ0(x) = 0.3, and the rate of the on-ramp given by qon(t) = 1.2. Following
+[5], as a metric of traffic congestion we consider the two following functionals
+J(T) =
+� T
+0
+d|∂xρ(t, ·)|dt,
+Ψ(T; a, b) =
+� T
+0
+� b
+a
+ϕ(ρ(t, x))dxdt,
+where
+ϕ(r) =
+
+
+
+
+
+0,
+r < 0.75,
+10r − 7.5,
+0.75 ≤ r ≤ 0.85,
+1,
+0.85 < r ≤ 1.
+The functional J defined measures the integral with respect to time of the spatial total variation
+of the traffic density while the functional Ψ measures the traffic congestion in the interval [a, b] =
+[−1, 4]. Figure 1 shows the values of the functionals J and Ψ when we vary the value of δ, keeping
+fixed η. We can observe that the minimum value of Ψ is given when δ = 0.1. In Figure 2 we
+compare the solutions of (1.1)-(2.3) for different values of δ, keeping fixed η. Namely, we consider
+
+STABILITY ESTIMATES FOR NONLOCAL BALANCE LAWS ARISING IN TRAFFIC MODELLING
+7
+δ ∈ {−0.5, 0.1, 0.5} and η = 0.5. We can see that the solution for δ = 0.1 produces a smaller
+queue and therefore less increase of the density on the main road when vehicles enter through the
+on-ramp than the other values considered for δ.
+-0.5
+-0.4
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.86
+0.88
+0.9
+0.92
+0.94
+0.96
+0.98
+-0.5
+-0.4
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.2
+0.25
+0.3
+0.35
+0.4
+0.45
+0.5
+0.55
+0.6
+Figure 1. Left: Functional J with η = 0.5 and δ ∈ [−η, η]. Right: Functional Ψ
+with η = 0.5 and δ ∈ [−η, η]
+-1
+-0.5
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Figure 2. Solution of (1.1)-(2.3) for η = 0.5 and varying δ ∈ {−0.5, 0.1, 0.5}
+5. Conclusions
+In this work we proved the L1−stability of entropy weak solutions of a scalar nonlocal balance
+law with nonlocal source term arising in traffic modelling with on- and off-ramps introduced in [2].
+We reached an estimate of the dependence of the solution with respect to the kernel function in
+the source term, the on-ramp rate, the off-ramp rate and the initial datum. The stability has
+been obtained from the entropy condition through the doubling of variables technique. Finally,
+following [5], we have shown numerical simulations illustrating the dependencies above for two cost
+functionals derived from traffic flow applications.
+Acknowledgments
+FAC is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Ap-
+plicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). HDC are supported
+by the INRIA Associated Team “Efficient numerical schemes for non-local transport phenomena”
+
+8
+FELISIA ANGELA CHIARELLO AND HAROLD DEIVI CONTRERAS
+(NOLOCO; 2018–2020). HDC was partially supported by the National Agency for Research and
+Development, ANID-Chile through Scholarship Program, Doctorado Becas Chile 2021, 21210826
+and by Anillo project ACT210030. The authors would like to thank Luis Miguel Villada for helpful
+discussions about this work.
+References
+[1] S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow
+modeling, Numerische Mathematik, 132 (2016), pp. 217–241.
+[2] F. A. Chiarello, H. D. Contreras, and L. M. Villada, Nonlocal reaction traffic flow model with on-off
+ramps, Networks and Heterogeneous Media, 17 (2022), p. 203.
+[3] F. A. Chiarello, J. Friedrich, P. Goatin, S. G¨ottlich, and O. Kolb, A non-local traffic flow model for
+1-to-1 junctions, European Journal of Applied Mathematics, 31 (2020), pp. 1029–1049.
+[4] F. A. Chiarello and P. Goatin, Non-local multi-class traffic flow models, Networks & Heterogeneous Media,
+14 (2019), p. 371.
+[5] F. A. Chiarello, P. Goatin, and E. Rossi, Stability estimates for non-local scalar conservation laws, Non-
+linear Analysis: Real World Applications, 45 (2019), pp. 668–687.
+[6] M. L. Delle Monache, J. Reilly, S. Samaranayake, W. Krichene, P. Goatin, and A. M. Bayen, A
+pde-ode model for a junction with ramp buffer, SIAM Journal on Applied Mathematics, 74 (2014), pp. 22–39.
+[7] J. Friedrich, S. G¨ottlich, and E. Rossi, Nonlocal approaches for multilane traffic models, Communications
+in Mathematical Sciences, 19 (2021), pp. 2291–2317.
+[8] J. Friedrich, O. Kolb, and S. G¨ottlich, A godunov type scheme for a class of lwr traffic flow models with
+non-local flux, Networks & Heterogeneous Media, 13 (2018), pp. 531–547.
+[9] D. Helbing, A. Hennecke, V. Shvetsov, and M. Treiber, Master: macroscopic traffic simulation based on
+a gas-kinetic, non-local traffic model, Transportation Research Part B: Methodological, 35 (2001), pp. 183–211.
+[10] M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads,
+Proc. Roy. Soc. London. Ser. A., 229 (1955), pp. 317–345.
+[11] G. Lipt´ak, M. Pereira, B. Kulcs´ar, M. Kov´acs, and G. Szederk´enyi, Traffic reaction model, arXiv
+preprint arXiv:2101.10190, (2021).
+[12] G. Liu, A. S. Lyrintzis, and P. G. Michalopoulos, Modelling of freeway merging and diverging flow
+dynamics, Applied mathematical modelling, 20 (1996), pp. 459–469.
+[13] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), pp. 42–51.
+[14] T. Tie-Qiao, H. Hai-Jun, and S. Hua-Yan, Effects of the number of on-ramps on the ring traffic flow, Chinese
+Physics B, 19 (2010), p. 050517.
+[15] T. Tie-Qiao, H. Hai-Jun, S. Wong, G. Zi-You, and Z. Ying, A new macro model for traffic flow on a
+highway with ramps and numerical tests, Communications in Theoretical Physics, 51 (2009), p. 71.
+(Felisia Angela Chiarello)
+Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica
+Via Vetoio, Ed. Coppito 1 67100 L’Aquila AQ, Italy
+Email address: felisiaangela.chiarello@univaq.it
+(Harold Deivi Contreras)
+GIMNAP-Departamento de Matem´aticas, Universidad del B´ıo-B´ıo, Concepci´on, Chile,
+CI2MA-Universidad de Concepci´on, Casilla 160-C, Concepci´on, Chile.
+Email address: harold.contreras1801@alumnos.ubiobio.cl
+
diff --git a/ptFAT4oBgHgl3EQfeh1g/content/tmp_files/load_file.txt b/ptFAT4oBgHgl3EQfeh1g/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,391 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf,len=390
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='08576v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='AP]  20 Jan 2023  STABILITY ESTIMATES FOR NONLOCAL BALANCE LAWS ARISING IN  TRAFFIC MODELLING  FELISIA ANGELA CHIARELLO AND HAROLD DEIVI CONTRERAS  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' This paper focuses on the proof of the stability of entropy weak solutions of a non-  local balance law modelling vehicular traffic flow on a road with on- and off-ramps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' The stabil-  ity is obtained with respect to a kernel function in the source term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' We get an estimate of the  L1−dependence of the solution with respect to the initial datum, the on-ramp rate, the off-ramp  rate and the mentioned kernel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' We also show numerical experiments in order to perform  an optimization problem in traffic flow with on-ramps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Introduction  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Aim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' In traffic flow modeling, nonlocal conservation laws are intended to describe the be-  haviour of drivers that adapt their velocity with respect to what happens in front of them, see  [1, 3, 4, 7, 8];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' the classical LWR (Lighthill-Whitham [10] and Richards [13]) is not able to model  this type of situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' For this reason, in order to generalise the LWR model in such a way to make  it able to describe more realistic settings, in [2] a nonlocal balance law intended to model vehicular  traffic flow on a main road wit on- and off-ramps is introduced and it is given by  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1)  ρt + (ρv(ρ ∗ ωη))x = Son(·, ·, ρ, ρ ∗ ωη,δ) − Soff(·, ·, ρ),  x ∈ R,  where Son and Soff represents the traffic flow entering and exiting through an on- and off-ramp  respectively, and the convolution term in Son is defined as follows  (ρ ∗ ωη,δ)(t, x) =  � x+η+δ  x−η+δ  ρ(t, y)ωη,δ(y − x)dy,  with η ∈ [0, 1] and δ ∈ [−η, η].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Here, the parameter η represents the length of the support of the  kernel function ωη,δ, while δ is the point at which the maximum of the kernel is attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' This  choice of the kernel function models the fact that drivers on the on-ramp can see what happens on  backward and forward on the main road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  It is well known that on-ramp merging has a great impact on traffic efficiency, if one concentrates on  highway networks the reduction of the capacity is often due to on- and off-ramps, for this reason in  this paper we are interested to study the dependence of solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1) on the convolution kernel  given in the source term Son.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' The strategies that we employ are inspired by the results obtained in  [2, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' In particular, we adopt the results about existence and uniqueness to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1) presented in [2]  and we propose the analytical study of the dependence of solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1) on the kernel function  in the source term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' In literature, the LWR model has been extended to include on and off-ramps  by means of source and sink terms, see [6, 9, 11, 12, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Starting from the model introduced in  [11], in [2] the authors introduce the new nonlocal balance law (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1), in which the velocity function  Date: January 23, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  1  2  FELISIA ANGELA CHIARELLO AND HAROLD DEIVI CONTRERAS  depends on a weighted mean of the downstream traffic density, in order to model the behaviour  of drivers that adapt their velocity with respect to what happens in front of them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' moreover the  nonlocal source term is intended to model the fact that drivers on the on-ramp can see what  happens on backward and forward on the main road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Regarding the aferomentioned model the  existence and uniqueness of entropy weak solution has been proved, specifically, approximating  the problem using an Upwind-type numerical scheme and providing compactness estimates for the  sequence of approximate solutions that allows to get the convergence and thus, the existence of  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Furthemore, by using Kruˇzkov’s doubling of variables technique the authors get an L1  stability estimate depending on initial data, the on-ramp rate and the off-ramp rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' On the other  hand, motivated by control and optimisation problems, in [5] the authors study a conservation law  with nonlocal flux function given by  ∂tρ + ∂x (f(t, x, ρ)v(ρ ∗ ω)(t, x)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='2)  The authors obtain an estimate of the dependence of the solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='2) with respect to the kernel  function, the velocity and the initial datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' The stability is obtained from an entropy condition  through the doubling of variables technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Outline of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' This work is organized as follows: In Section 2 we recall the mathe-  matical model on which we will focus our study as well as the definitions of weak and entropy weak  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' In section 3 we present the main results of this work, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' the L1− Lipschitz continuous  dependence of the solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1) on to the initial datum, the on-ramp rate, the off-ramp rate  and the kernel function in the source term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Finally, in Section 4 we present numerical examples in  order to illustrate the behaviour of solutions when some parameters of the kernel function in the  source term are varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Mathematical model  Let us consider the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1) with terms Son and Soff defined as  Son(t, x, ρ, ρ ∗ ωη,δ)  =  1on(x)qon(t)  �  1 −  ρ  ρmax  � �  1 − ρ ∗ ωη,δ  ρmax  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1)  Soff(t, x, ρ)  =  1off(x)qoff(t)  ρ  ρmax  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='2)  with ρmax = 1 for the sake of simplicity, and we also endow the nonlocal traffic reaction model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1)-  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='2) with an initial condition as follows  ρ(x, 0) = ρ0(x) ∈  �  L1 ∩ BV  �  (R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' [0, ρmax]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='3)  Let us assume the following assumptions on the parameters of model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  Assumptions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' We assume  (i) qramp  on  ∈ L∞(R+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' R+), qramp  off  ∈ L∞(R+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  (ii) v ∈ C2([0, ρmax];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' R+), v′(ρ) ≤ 0, ρ ∈ [0, ρmax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  (iii) ωη ∈ C1([0, η];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' R+) with ω′  η(x) ≤ 0,  � η  0 ωη(x)dx = 1, ∀η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  (iv) ωη,δ ∈ (C1 ∩ L1)([δ − η, δ + η];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' R+) with ω′(x)η,δ ≥ 0 for x ∈ [δ − η, 0], ω′(x)η,δ ≤ 0 for x ∈  [0, δ + η], and  � δ+η  δ−η ωη,δ(x)dx = 1, ∀η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  STABILITY ESTIMATES FOR NONLOCAL BALANCE LAWS ARISING IN TRAFFIC MODELLING  3  We consider solutions in a weak sense as follows  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Let ρ0 ∈ (L1 ∩ BV)(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' [0, ρmax]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' We say that ρ ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' L1(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' [0, ρmax])),  with ρ(t, ·) ∈ BV(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' [0, ρmax]) for t ∈ [0, T], is a weak solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='3) if for any ϕ ∈  C1  c([0, T[×R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' R)  � T  0  �  R  (ρϕt + ρV ϕx) dxdt +  � T  0  �  Ωon  Sonϕdxdt  −  � T  0  �  Ωoff  Soffϕdxdt +  �  R  ρ0(x)ϕ(0, x)dx = 0,  where V (t, x) = v((ρ ∗ ω)(t, x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  Furthermore, we consider entropy weak solutions in Kruˇzkov sense, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Let ρ0 ∈ (L1 ∩ BV)(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' [0, ρmax]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' We say that ρ ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' L1(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' [0, ρmax])), with  ρ(t, ·) ∈ BV(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' [0, ρmax]) for t ∈ [0, T], is a entropy weak solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1) with initial datum ρ0 if  for any ϕ ∈ C1  c([0, T[×R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' R) and for all k ∈ R  � T  0  �  R  (|ρ − k|ϕt + |ρ − k| V ϕx − sgn(ρ − k)kVxϕ) dxdt  +  � T  0  �  Ωon  sgn(ρ − k)Sonϕdxdt −  � T  0  �  Ωoff  sgn(ρ − k)Soffϕdxdt  +  �  R  |ρ0 − k|ϕ(0, x)dx ≥ 0,  where Ωon and Ωoff are the spatial positions of on-ramp and off-ramp on the main road, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Main Result  Before giving the main results of this work, we first recall the main Theorem in [2, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Let ρ0 ∈  �  L1 ∩ BV  �  (R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Let the Assumptions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Then, for all T > 0,  the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1) has a unique solution ρ ∈ C0 �  [0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' L1(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' [0, 1])  �  in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  Moreover, the following estimates hold: for any t ∈ [0, T]  ∥ρ(t)∥L1(R) ≤ R1(t),  0 ≤ ρ(t, x) ≤ 1,  TV (ρ(t)) ≤ etH (TV (ρ0) + tQT ) ,  where  R1 = ∥ρ0∥L1(R) + ∥qramp  on  (·)∥L1([0,t]) − min  x∈Ωon ∥qramp  on  (·)ρ(·, x)∥L1([0,t])  − min  x∈Ωoff  ��qramp  off  (·)ρ(·, x)  ��  L1([0,t])  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1)  QT  =  2  �  ∥qon∥L∞([0,T]) + ∥qoff∥L∞([0,T])  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='2)  H  =  2 ∥qon∥L∞([0,T]) + ∥qoff∥L∞([0,T]) + ωη(0)L  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='3)  L  =  �  ∥v∥L∞([0,1]) + ∥v′∥L∞([0,1])  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='4)  4  FELISIA ANGELA CHIARELLO AND HAROLD DEIVI CONTRERAS  The following theorem is the main result of this work and it states the L1- Lipschitz continuous  dependence of solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1) on to the initial datum, the on-ramp rate, the off-ramp rate and  the kernel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Let ρ and ˜ρ be two solutions to problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1) in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='2, with  initial data ρ0, ˜ρ0 ∈ L1 ∩ BV (R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' [0, 1]), with on-ramp rates qon, ˜qon, off-ramp rates qoff, ˜qoff and  kernel functions ωη,δ, ˜ωη,δ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Assume v ∈ C2 ([0, 1], R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Then, for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' t ∈ [0, T],  ∥ρ(t) − ˜ρ(t)∥L1(R)  ≤  eCT  �  ∥ρ0 − ˜ρ0∥L1(R) + L  �  ∥qon − ˜qon∥L1([0,t]) + ∥qoff − ˜qoff∥L1([0,T])  �  +r(T)∥ωη,δ − ˜ωη,δ∥L1(R)  �  ,  where r(T) depends on the L1−norms of initial conditions, expected inflow flow of the on-ramp and  expected output flow of the off-ramp and C is defined as in [2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='24)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Since ρ and ˜ρ are entropy weak solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1) then  �  ρt + (ρV (t, x))x = Son (t, x, qon, ρ, Ron) − Soff (t, x, qoff, ρ)  ρ(0, x) = ρ0(x),  \uf8f1  \uf8f2  \uf8f3  ˜ρt + (˜ρ ˜V (t, x))x = Son  �  t, x, ˜qon, ˜ρ, ˜Ron  �  − Soff (t, x, ˜qoff, ρ)  ˜ρ(0, x) = ˜ρ0(x),  in a distributional sense, where V (t, x) = v((ρ ∗ ωη)(t, x)), Ron = (ρ ∗ ωη,δ)(t, x), ˜V (t, x) = v((˜ρ ∗  ωη)(t, x)), ˜Ron = (˜ρ ∗ ˜ωη,δ)(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Following the argument in [2] and using Kruˇzkov’s doubling of  variables technique we get  ∥ρ(T, ·) − ˜ρ(T, ·)∥L1(R)  ≤  ∥ρ0 − ˜ρ0∥L1(R) +  � T  0  �  Ωon  ��� ˜Son  ��� dxdt +  � T  0  �  Ωoff  ��� ˜Soff  ��� dxdt  +  � T  0  �  R  |V| |∂xρ(t, x)| dxdt +  � T  0  �  R  |Vx| |ρ(t, x)| dxdt,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5)  where  ˜Son  =  Son (t, x, qon, ρ, Ron) − Son  �  t, x, ˜qon, ˜ρ, ˜Ron  �  ,  ˜Soff  =  Soff (t, x, qon, ρ) − Soff (t, x, ˜qon, ˜ρ) ,  V  =  v(R) − v(P),  Vx  =  ∂xv(R) − ∂xv(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  Except for the second term, all terms appearing at the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5) are computed as in  [2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1],  � T  0  �  Ωoff  |Soff| dxdt  ≤  ∥qoff∥L∞([0,T])  � T  0  ∥ρ(t, ·) − ˜ρ(t, ·)∥L1(R) dt + L ∥qoff − ˜qoff∥L1([0,T]) ,  � T  0  �  R  |V| |∂xρ(t, x)| dxdt  ≤  ωη(0)  ��v′��  L∞([0,1]) sup  t∈[0,T]  TV (ρ(t, ·))  � T  0  ∥ρ(t, ·) − ˜ρ(t, ·)∥L1(R) dt,  � T  0  �  R  |Vx| |ρ(t, x)| dxdt  ≤  W  � T  0  ∥ρ(t, ·) − ˜ρ(t, ·)∥L1(R) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  STABILITY ESTIMATES FOR NONLOCAL BALANCE LAWS ARISING IN TRAFFIC MODELLING  5  Regarding the second term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5), we have  � T  0  �  Ωon  ��� ˜Son  ��� dxdt  =  � T  0  �  Ωon  ���Son (t, x, qon, ρ, Ron) − Son  �  t, x, ˜qon, ˜ρ, ˜Ron  ���� dxdt  ≤  � T  0  �  Ωon  ���� ˜S1  on  ��� +  ��� ˜S2  on  ��� +  ��� ˜S3  on  ���  �  dxdt,  where  ˜S1  on  =  Son (t, x, qon, ρ, Ron) − Son  �  t, x, qon, ρ, ˜Ron  �  ,  ˜S2  on  =  Son  �  t, x, qon, ρ, ˜Ron  �  − Son  �  t, x, qon, ˜ρ, ˜Ron  �  ,  ˜S3  on  =  Son  �  t, x, qon, ˜ρ, ˜Ron  �  − Son  �  t, x, ˜qon, ˜ρ, ˜Ron  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  We bound ˜S2  on and ˜S3  on writing  � T  0  �  Ωon  ��� ˜S2  on  ��� dxdt  ≤  ∥qon∥L∞([0,T])  � T  0  ∥ρ(t, ·) − ˜ρ(t, ·)∥L1(R) dt,  � T  0  �  Ωon  ��� ˜S3  on  ��� dxdt  ≤  � T  0  �  Ωon  |qon − ˜qon| dxdt  ≤  L ∥qon − ˜qon∥L1([0,T]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' we are going to bound ˜S1  on term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  ��� ˜S1  on  ���  =  ���1onqon (1 − ρ)  �  (1 − Ron) −  �  1 − ˜Ron  �����  ≤  ∥qon∥L∞([0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='T])  ��� ˜Ron − Ron  ��� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  thus  � T  0  �  Ωon  ��� ˜S1  on  ��� dxdt  ≤  ∥qon∥L∞([0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='T])  � T  0  �  Ωon  ��� ˜Ron − Ron  ��� dxdt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  and observe that  �  Ωon  ���Ron − ˜Ron  ��� dx  =  �  Ωon  |ωη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='δ ∗ ρ − ˜ωη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='δ ∗ ˜ρ| dx  =  �  Ωon  |ωη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='δ ∗ ρ − ωη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='δ ∗ ˜ρ + ωη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='δ ∗ ˜ρ − ˜ωη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='δ ∗ ˜ρ| dx  ≤  �  Ωon  |ωη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='δ ∗ ρ − ωη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='δ ∗ ˜ρ| + |ωη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='δ ∗ ˜ρ − ˜ωη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='δ ∗ ˜ρ| dx  =  �  Ωon  |ωη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='δ ∗ (ρ − ˜ρ)| +  �  Ωon  |(ωη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='δ − ˜ωη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='δ) ∗ ˜ρ| dx  ≤  ∥ωη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='δ∥L1(Ωon) ∥ρ − ˜ρ∥L1(Ωon) + ∥ωη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='δ − ˜ωη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='δ∥L1(Ωon) ∥˜ρ∥L1(Ωon)  ≤  ∥ωη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='δ∥L1(R) ∥ρ − ˜ρ∥L1(R) + ∥ωη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='δ − ˜ωη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='δ∥L1(R) ∥˜ρ∥L1(R)  ≤  ∥ρ − ˜ρ∥L1(R) + ∥ωη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='δ − ˜ωη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='δ∥L1(R) R1(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  since  �  R ωη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='δ(x)dx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Here R1 is defined as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Then,  � T  0  �  Ωon  ��� ˜S1  on  ��� dxdt  ≤  ∥qon∥L∞([0,T])  � T  0  ∥ρ(t, ·) − ˜ρ(t, ·)∥L1(R) dt  +  � T  0  ∥ωη,δ − ˜ωη,δ∥L1(R) R1(t)dt  6  FELISIA ANGELA CHIARELLO AND HAROLD DEIVI CONTRERAS  =  ∥qon∥L∞([0,T])  � T  0  ∥ρ(t, ·) − ˜ρ(t, ·)∥L1(R) dt  + ∥ωη,δ − ˜ωη,δ∥L1(R)  � T  0  R1(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  Therefore, the inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5) is equivalent to following inequality  ∥ρ(T, ·) − ˜ρ(T, ·)∥L1(R)  ≤  ∥ρ0 − ˜ρ0∥L1(R) + L  �  ∥qon − ˜qon∥L1([0,t]) + ∥qoff − ˜qoff∥L1([0,t])  �  +C  � T  0  ∥ρ(t, ·) − ˜ρ(t, ·)∥L1(R) dt + ∥ωη,δ − ˜ωη,δ∥L1(R)  � T  0  R1(t)dt  =  ∥ρ0 − ˜ρ0∥L1(R) + L  �  ∥qon − ˜qon∥L1([0,t]) + ∥qoff − ˜qoff∥L1([0,t])  �  +C  � T  0  ∥ρ(t, ·) − ˜ρ(t, ·)∥L1(R) dt + r(T) ∥ωη,δ − ˜ωη,δ∥L1(R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  An application of Gronwall’s Lemma to the above inequality completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Numerical examples  In this Section we use [2, Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1] in order to compute approximate solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='3)  with the source term (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' We simulate an optimization problem in traffic merging, which consists  in finding the optimal value δ keeping fixed η in the kernel function ωη,δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' According to the meaning  of that kernel function, this is equivalent to find where a drivers located on the on-ramp should  look in order to avoid the congestion on the main road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' To this end, we consider one on-ramp with  length L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1 located from x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='0 until x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1, and we consider the following kernel functions  ωη(x)  :=  2η − x  η2  χ[0,η](x),  ωη,δ(x)  :=  1  η6  16  5π  �  η2 − (x − δ)2�5/2 χ[−η+δ,η+δ](x),  for convective and reactive terms, respectively, with η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5 and δ ∈ [−η, η], for x ∈ [−1, 4] at  simulated time T = 6 and velocity function given by v(ρ) = 1 − ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' We also consider ∆x = 1/200, a  constant initial condition ρ0(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='3, and the rate of the on-ramp given by qon(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Following  [5], as a metric of traffic congestion we consider the two following functionals  J(T) =  � T  0  d|∂xρ(t, ·)|dt,  Ψ(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' a, b) =  � T  0  � b  a  ϕ(ρ(t, x))dxdt,  where  ϕ(r) =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  0,  r < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='75,  10r − 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='75 ≤ r ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='85,  1,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='85 < r ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  The functional J defined measures the integral with respect to time of the spatial total variation  of the traffic density while the functional Ψ measures the traffic congestion in the interval [a, b] =  [−1, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Figure 1 shows the values of the functionals J and Ψ when we vary the value of δ, keeping  fixed η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' We can observe that the minimum value of Ψ is given when δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' In Figure 2 we  compare the solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='3) for different values of δ, keeping fixed η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Namely, we consider  STABILITY ESTIMATES FOR NONLOCAL BALANCE LAWS ARISING IN TRAFFIC MODELLING  7  δ ∈ {−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5} and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' We can see that the solution for δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1 produces a smaller  queue and therefore less increase of the density on the main road when vehicles enter through the  on-ramp than the other values considered for δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='86  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='6  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Left: Functional J with η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5 and δ ∈ [−η, η].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Right: Functional Ψ  with η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5 and δ ∈ [−η, η]  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5  4  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='9  1  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='3) for η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5 and varying δ ∈ {−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='5}  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Conclusions  In this work we proved the L1−stability of entropy weak solutions of a scalar nonlocal balance  law with nonlocal source term arising in traffic modelling with on- and off-ramps introduced in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  We reached an estimate of the dependence of the solution with respect to the kernel function in  the source term, the on-ramp rate, the off-ramp rate and the initial datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' The stability has  been obtained from the entropy condition through the doubling of variables technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Finally,  following [5], we have shown numerical simulations illustrating the dependencies above for two cost  functionals derived from traffic flow applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  Acknowledgments  FAC is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Ap-  plicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' HDC are supported  by the INRIA Associated Team “Efficient numerical schemes for non-local transport phenomena”  8  FELISIA ANGELA CHIARELLO AND HAROLD DEIVI CONTRERAS  (NOLOCO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' 2018–2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' HDC was partially supported by the National Agency for Research and  Development, ANID-Chile through Scholarship Program, Doctorado Becas Chile 2021, 21210826  and by Anillo project ACT210030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' The authors would like to thank Luis Miguel Villada for helpful  discussions about this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
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+page_content=' Tie-Qiao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Hai-Jun, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Hua-Yan, Effects of the number of on-ramps on the ring traffic flow, Chinese  Physics B, 19 (2010), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' 050517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  [15] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Tie-Qiao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Hai-Jun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Wong, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Zi-You, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Ying, A new macro model for traffic flow on a  highway with ramps and numerical tests, Communications in Theoretical Physics, 51 (2009), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  (Felisia Angela Chiarello)  Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica  Via Vetoio, Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content=' Coppito 1 67100 L’Aquila AQ, Italy  Email address: felisiaangela.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='chiarello@univaq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='it  (Harold Deivi Contreras)  GIMNAP-Departamento de Matem´aticas, Universidad del B´ıo-B´ıo, Concepci´on, Chile,  CI2MA-Universidad de Concepci´on, Casilla 160-C, Concepci´on, Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='  Email address: harold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
+page_content='contreras1801@alumnos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFAT4oBgHgl3EQfeh1g/content/2301.08576v1.pdf'}
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+Quarkonia production in ultra-peripheral PbPb collisions
+at LHCb∗
+Xiaolin Wang on behalf of the LHCb collaboration
+Guangdong Provincial Key Laboratory of Nuclear Science, Guangdong-Hong Kong
+Joint Laboratory of Quantum Matter, Institute of Quantum Matter, South China
+Normal University, Guangzhou, China
+Received January 3, 2023
+Measurements of coherent charmonium production cross sections to-
+gether with their ratio in ultra-peripheral PbPb collisions are studied at
+a nucleon-nucleon centre-of-mass energy of 5.02 TeV, the differential cross-
+sections are measured as a function of rapidity and transverse momentum,
+separately. The photo-production of J/ψ mesons at low transverse momen-
+tum is studied in peripheral PbPb collisions, which confirms coherent J/ψ
+production in hadronic collisions. These latest results significantly improve
+previous measurements and are compared with some theoretical predictions.
+1. Introduction
+The LHCb detector is a single-arm forward spectrometer covering the
+pseudorapidity range 2 < η < 5 [1,2], which has great performance with pre-
+cise vertex reconstruction, high particle momentum resolution, and excellent
+particle identification capability. Measurements of quarkonia production in
+(ultra-)peripheral heavy-ion collisions play an important role in studying
+the photon-nucleus interaction, the mechanism of vector meson production,
+and the partonic structure of nuclei. Meanwhile, coherent photo-production
+would provide an excellent laboratory to study the nuclear shadowing effects
+and the initial state of heavy ion collisions at small-x at the LHC [3].
+2. Charmonia production in ultra-peripheral PbPb collisions at
+LHCb [4]
+Ultra-peripheral collisions(UPCs) occur when the impact parameter is
+larger than the sum of the radii of two incoming nuclei [5]. The two ions
+∗ Presented at ”Diffraction and Low-x 2022”, Corigliano Calabro (Italy), September
+24-30, 2022
+(1)
+arXiv:2301.00222v1  [hep-ex]  31 Dec 2022
+
+2
+template
+printed on January 3, 2023
+interact via their cloud of semi-real photons, and photon-nuclear interactions
+dominate. In UPCs, J/ψ and ψ(2S) mesons are produced from the colorless
+reaction of a photon from one nucleus and a Pomeron from the other. If the
+photon interacts with a Pomeron emitted by the whole nucleus, there would
+be no nucleus break-up, and this is called coherent production, which is the
+process we are going to study.
+The charmonia candidates are reconstructed through the J/ψ → µ+µ−
+and ψ(2S) → µ+µ− decay channels using a PbPb data sample corresponding
+to an integrated luminosity of 228 ± 10 µb−1, which was collected by the
+LHCb experiment in 2018. According to the characteristics of UPCs, only
+events with low activity could be kept in the selection. After that, the signal
+extraction is done through two steps. A fit on the dimuon invariant mass
+spectrum is needed to estimate the non-resonant background yields within
+the J/ψ and ψ(2S) mass windows, then the fits to J/ψ and ψ(2S) transverse
+momentum distributions of selected candidates are performed to determine
+the coherent production events from the inclusive charmonia yields, as shown
+in Fig. 1.
+The measured differential cross sections as a function of y∗ and p∗
+T of
+coherent J/ψ and ψ(2S) are shown in Fig. 2, where the starred notation
+indicates that the observable is defined in the nucleus-nucleus centre-of-
+mass frame. The differential cross-section ratio of coherent ψ(2S) to J/ψ
+production is calculated as a function of rapidity for the first time and
+shown in Fig. 3. Compared with theoretical predictions, which are grouped
+as perturbative-QCD calculations [6,7] and colour-glass-condensate (CGC)
+models [8–15], it could be found that the measurements are in agreement
+with most of the prediction curves in general.
+3. Study of J/ψ photo-production in PbPb peripheral collisions
+at √sNN = 5 TeV [16]
+In peripheral collisions, the impact parameter is less than the sum of
+radii of the two colliding nuclei, so there is not only photo-nuclear interaction
+but also hadronic interaction. For hadronic productions, the gluon-gluon
+interaction produces a pair of charm quarks and lastly goes into the J/ψ
+meson. In this case, J/ψ meson will have transverse momentum typically
+between 1 and 2 GeV/c, and for coherent photo-production, J/ψ would have
+a very small transverse momentum, less than 300 MeV/c. Thanks to the
+high precision of the LHCb experiment, there are two visible clear peaks
+in the distribution of dimuon ln(p∗2
+T ), which could be used to discriminate
+photo-produced and hadronically produced J/ψ, as shown in Fig. 4.
+The differential photo-production yields of J/ψ versus the rapidity, the
+number of participants in the collision, and the double-differential J/ψ photo-
+
+template
+printed on January 3, 2023
+3
+3000
+3500
+4000
+]
+2c
+ [MeV/
+-
+µ
++
+µ
+m
+10
+2
+10
+3
+10
+4
+10
+5
+10
+)
+2c
+Candidates / (5 MeV/
+Data
+Fit
+ψ
+J/
+(2S)
+ψ
+Background
+LHCb
+ = 5.02 TeV
+NN
+s
+PbPb 
+ < 4.5
+y*
+2.0 < 
+15
+−
+10
+−
+5
+−
+0
+)]
+2c/
+2
+) [ln(GeV
+*2
+T
+p
+ln(
+0
+200
+400
+600
+800
+1000
+)] )
+2c/
+2
+Events / ( 0.125 [ln(GeV
+Data
+Fit
+ψ
+J/
+Coherent 
+ψ
+J/
+Incoherent 
+X
+ + 
+ψ
+J/
+ 
+→
+ 
+(2S)
+ψ
+Non-reson.
+LHCb
+ = 5.02 TeV
+NN
+s
+PbPb 
+ < 4.5
+y*
+2.0 < 
+J/ψ
+15
+−
+10
+−
+5
+−
+0
+)]
+2c/
+2
+) [ln(GeV
+*2
+T
+p
+ln(
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+)] )
+2c/
+2
+Events / ( 0.3 [ln(GeV
+Data
+Fit
+(2S)
+ψ
+Coherent 
+(2S)
+ψ
+Incoherent 
+Non-reson.
+LHCb
+ = 5.02 TeV
+NN
+s
+PbPb 
+ < 4.5
+y*
+2.0 < 
+ψ(2S)
+Fig. 1: Fit to the invariant mass distribution of dimuon candidates (up) and the
+ln(p∗2
+T ) distribution fit of dimuon candidates within the 2.0 < y∗ < 4.5 range for
+J/ψ candidates (left) and ψ(2S) candidates (right), where the starred notation
+indicates that the observable is defined in the nucleus-nucleus centre-of-mass frame.
+production yields versus transverse momentum as shown in Fig. 5. For the
+phenomenological models, one scenario does not consider the destructive
+effect due to the overlap between the two nuclei, whereas the other takes it
+into account. In general, the trends of J/ψ photo-production measurements
+and theoretical predictions are consistent. Meanwhile, the two theoretical
+curves do not show a significant difference because the collisions are peripheral
+and the nuclear overlapping effect is expected to increase in more central
+collisions. Currently, the measurement is limited to low values of Npart due
+to detector limitation, and the measurement of pT is consistent with the
+coherent J/ψ photo-production; the peak value of transverse momentum is
+around 60 MeV/c.
+
+4
+template
+printed on January 3, 2023
+0
+1
+2
+3
+4
+5
+y∗
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+4.5
+5.0
+5.5
+dσJ/ψ/dy∗ [mb]
+LHCb
+PbPb √sNN = 5.02 TeV
+Coherent J/ψ production
+Luminosity unc. : 4.4%
+0
+1
+2
+3
+4
+5
+y∗
+0.0
+0.5
+1.0
+1.5
+dσψ(2S)/dy∗ [mb]
+LHCb
+PbPb √sNN = 5.02 TeV
+Coherent ψ(2S) production
+Luminosity unc. : 4.4%
+data
+stat. unc.
+syst. unc.
+Guzey et al.
+LTA W
+LTA S
+EPS09
+Krelina et al.
+GBW+BT
+GBW+POW
+KST+BT
+GG-hs+BG
+M¨antysaari et al.
+No fluct. +BG
+Gon¸calves et al.
+bCGC+BG
+bCGC+GLC
+IP-SAT+BG
+IP-SAT+GLC
+0.00
+0.05
+0.10
+0.15
+0.20
+p∗
+T [GeV/c]
+0
+10
+20
+30
+40
+50
+60
+d2σJ/ψ/dp∗
+Tdy∗ [mb/(GeV/c)]
+LHCb
+PbPb √sNN = 5.02 TeV
+Coherent J/ψ production
+Luminosity unc. : 4.4%
+0.00
+0.05
+0.10
+0.15
+0.20
+p∗
+T [GeV/c]
+0.0
+1.5
+3.0
+4.5
+6.0
+d2σψ(2S)/dp∗
+Tdy∗ [mb/(GeV/c)]
+LHCb
+PbPb √sNN = 5.02 TeV
+Coherent ψ(2S) production
+2 < y∗ < 4.5
+Luminosity unc. : 4.4%
+data
+stat. unc.
+syst. unc.
+Guzey et al.
+LTA W
+LTA S
+EPPS16
+M¨antysaari et al.
+Is fluct. +BG
+No fluct. +BG
+Is fluct. +GLC
+No fluct. +GLC
+Fig. 2: Differential cross-section as a function y∗ (up) and p∗
+T (down) for coherent J/ψ
+(left) and ψ(2S) (right) production. Compared to theoretical predictions, which are
+grouped as perturbative-QCD calculations (red lines) and colour-glass-condensate
+models (blue lines).
+0
+1
+2
+3
+4
+5
+y∗
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+σψ(2S)/σJ/ψ
+LHCb
+PbPb √sNN = 5.02 TeV
+Coherent ψ(2S)/J/ψ production
+data
+stat. unc.
+syst. unc.
+Guzey et al.
+LTA W
+LTA S
+EPS09
+Krelina et al.
+GBW+BT
+GBW+POW
+KST+BT
+GG-hs+BG
+M¨antysaari et al.
+No fluct. +BG
+Gon¸calves et al.
+bCGC+BG
+bCGC+GLC
+IP-SAT+BG
+IP-SAT+GLC
+Fig. 3: Differential cross-section ratio of ψ(2S) to J/ψ as a function of y∗, compared
+to theoretical predictions, which are separated into perturbative QCD calculations
+(red lines) and colour-glass-condensate models (blue lines).
+
+template
+printed on January 3, 2023
+5
+3000
+3050
+3100
+3150
+3200
+]
+2c
+) [MeV/
+−
+µ
++
+µ
+(
+m
+0
+50
+100
+150
+200
+250
+300
+)
+2c
+Candidates /(5 MeV/
+Data
+Signal
+Background
+LHCb
+ = 5 TeV
+NN
+s
+PbPb 
+2.9
+±
+> = 10.6
+part
+N
+<
+2.0 < y < 4.5
+52
+±
+ = 1764
+Total
+ψ
+J/
+N
+5
+10
+15
+20
+])
+2c/
+2
+[MeV
+/
+2
+T
+p
+ln(
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+220
+Events/(0.40)
+LHCb
+ = 5 TeV
+NN
+s
+PbPb 
+2.9
+±
+> = 10.6
+part
+N
+<
+2.0 < y < 4.5
+19
+±
+ = 277
+ψ
+J/
+N
+Data
+ hadro-produced
+ψ
+J/
+ photo-produced
+ψ
+J/
+Fig. 4: Fit to the invariant mass spectrum of J/ψ candidates (left) and the ln(p2
+T)
+distribution fit for J/ψ candidates (right).
+2
+3
+4
+ψ
+J/
+y
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+3
+−
+10
+×
+y
+/d
+ψ
+J/
+dY
+LHCb
+ = 5 TeV
+NN
+s
+PbPb 
+µ
+µ
+ 
+→
+ 
+ψ
+Photo-produced J/
+9.2
+±
+> = 19.7
+part
+N
+<
+ < 300 MeV/c
+T
+ψ
+J/
+p
+Data
+No overlap effects
+Overlap effects
+10
+20
+30
+>
+part
+N
+<
+0.1
+0.2
+0.3
+0.4
+0.5
+3
+−
+10
+×
+y
+/d
+ψ
+J/
+dY
+LHCb
+ = 5 TeV
+NN
+s
+PbPb 
+µ
+µ
+ 
+→
+ 
+ψ
+Photo-produced J/
+2.0 < y < 4.5
+Data
+No overlap effects
+Overlap effects
+0
+0.05
+0.1
+0.15
+0.2
+]c
+ [GeV/
+T
+p
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3
+−
+10
+×
+]
+-1
+)c
+ [(GeV/
+y
+d
+T
+p
+/d
+ψ
+J/
+Y
+2
+d
+LHCb
+ = 5 TeV
+NN
+s
+PbPb 
+µ
+µ
+ 
+→
+ 
+ψ
+Photo-produced J/
+2.0 < y < 4.5
+2.0 < y < 4.5
+9.2
+±
+> = 19.7
+part
+N
+<
+Data
+No overlap effects
+Overlap effects
+Fig. 5: The results of the differential photo-production yields of J/ψ versus the
+rapidity and the number of participants in collisions. Double-differential yields of
+the coherent J/ψ produced in peripheral PbPb collisions as a function of pT (right).
+4. Conclusion
+We report the results of coherent J/ψ, ψ(2S) production cross sections
+in UPCs, as well as their ratio. It is the first coherent ψ(2S) and ψ(2S) to
+J/ψ ratio measurement in forward rapidity region for UPC at LHC, and
+the differential cross section of coherent J/ψ and ψ(2S) production in PbPb
+UPC is also measured as a function of p∗
+T for the first time. Also, it is the
+most precise measurement of J/ψ production in UPCs. The production of
+photo-produced J/ψ mesons in peripheral PbPb collisions is measured and
+by far it is the most precise coherent photo-produced J/ψ by LHCb.
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+arXiv:2108.02681.
+
diff --git a/qNAyT4oBgHgl3EQfZPf6/content/tmp_files/load_file.txt b/qNAyT4oBgHgl3EQfZPf6/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..16b7aefdee2e8875dfdd9fce24c1008d4d8faac0
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+++ b/qNAyT4oBgHgl3EQfZPf6/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf,len=309
+page_content='Quarkonia production in ultra-peripheral PbPb collisions  at LHCb∗  Xiaolin Wang on behalf of the LHCb collaboration  Guangdong Provincial Key Laboratory of Nuclear Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' Guangdong-Hong Kong  Joint Laboratory of Quantum Matter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' Institute of Quantum Matter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' South China  Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' Guangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' China  Received January 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' 2023  Measurements of coherent charmonium production cross sections to-  gether with their ratio in ultra-peripheral PbPb collisions are studied at  a nucleon-nucleon centre-of-mass energy of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='02 TeV, the differential cross-  sections are measured as a function of rapidity and transverse momentum,  separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' The photo-production of J/ψ mesons at low transverse momen-  tum is studied in peripheral PbPb collisions, which confirms coherent J/ψ  production in hadronic collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' These latest results significantly improve  previous measurements and are compared with some theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' Introduction  The LHCb detector is a single-arm forward spectrometer covering the  pseudorapidity range 2 < η < 5 [1,2], which has great performance with pre-  cise vertex reconstruction, high particle momentum resolution, and excellent  particle identification capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' Measurements of quarkonia production in  (ultra-)peripheral heavy-ion collisions play an important role in studying  the photon-nucleus interaction, the mechanism of vector meson production,  and the partonic structure of nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' Meanwhile, coherent photo-production  would provide an excellent laboratory to study the nuclear shadowing effects  and the initial state of heavy ion collisions at small-x at the LHC [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' Charmonia production in ultra-peripheral PbPb collisions at  LHCb [4]  Ultra-peripheral collisions(UPCs) occur when the impact parameter is  larger than the sum of the radii of two incoming nuclei [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' The two ions  ∗ Presented at ”Diffraction and Low-x 2022”, Corigliano Calabro (Italy), September  24-30, 2022  (1)  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='00222v1  [hep-ex]  31 Dec 2022  2  template  printed on January 3, 2023  interact via their cloud of semi-real photons, and photon-nuclear interactions  dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' In UPCs, J/ψ and ψ(2S) mesons are produced from the colorless  reaction of a photon from one nucleus and a Pomeron from the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' If the  photon interacts with a Pomeron emitted by the whole nucleus, there would  be no nucleus break-up, and this is called coherent production, which is the  process we are going to study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  The charmonia candidates are reconstructed through the J/ψ → µ+µ−  and ψ(2S) → µ+µ− decay channels using a PbPb data sample corresponding  to an integrated luminosity of 228 ± 10 µb−1, which was collected by the  LHCb experiment in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' According to the characteristics of UPCs, only  events with low activity could be kept in the selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' After that, the signal  extraction is done through two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' A fit on the dimuon invariant mass  spectrum is needed to estimate the non-resonant background yields within  the J/ψ and ψ(2S) mass windows, then the fits to J/ψ and ψ(2S) transverse  momentum distributions of selected candidates are performed to determine  the coherent production events from the inclusive charmonia yields, as shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  The measured differential cross sections as a function of y∗ and p∗  T of  coherent J/ψ and ψ(2S) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' 2, where the starred notation  indicates that the observable is defined in the nucleus-nucleus centre-of-  mass frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' The differential cross-section ratio of coherent ψ(2S) to J/ψ  production is calculated as a function of rapidity for the first time and  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' Compared with theoretical predictions, which are grouped  as perturbative-QCD calculations [6,7] and colour-glass-condensate (CGC)  models [8–15], it could be found that the measurements are in agreement  with most of the prediction curves in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' Study of J/ψ photo-production in PbPb peripheral collisions  at √sNN = 5 TeV [16]  In peripheral collisions, the impact parameter is less than the sum of  radii of the two colliding nuclei, so there is not only photo-nuclear interaction  but also hadronic interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' For hadronic productions, the gluon-gluon  interaction produces a pair of charm quarks and lastly goes into the J/ψ  meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' In this case, J/ψ meson will have transverse momentum typically  between 1 and 2 GeV/c, and for coherent photo-production, J/ψ would have  a very small transverse momentum, less than 300 MeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' Thanks to the  high precision of the LHCb experiment, there are two visible clear peaks  in the distribution of dimuon ln(p∗2  T ), which could be used to discriminate  photo-produced and hadronically produced J/ψ, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  The differential photo-production yields of J/ψ versus the rapidity, the  number of participants in the collision, and the double-differential J/ψ photo-  template  printed on January 3, 2023  3  3000  3500  4000  ]  2c  [MeV/ µ  +  µ  m  10  2  10  3  10  4  10  5  10  )  2c  Candidates / (5 MeV/  Data  Fit  ψ  J/  (2S)  ψ  Background  LHCb  = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='02 TeV  NN  s  PbPb  < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='5  y*  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='0 <  15  −  10  −  5  −  0  )]  2c/  2  ) [ln(GeV  2  T  p  ln(  0  200  400  600  800  1000  )] )  2c/  2  Events / ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='125 [ln(GeV  Data  Fit  ψ  J/  Coherent  ψ  J/  Incoherent  X  +  ψ  J/  →  (2S)  ψ  Non-reson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  LHCb  = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='02 TeV  NN  s  PbPb  < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='5  y*  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='0 <  J/ψ  15  −  10  −  5  −  0  )]  2c/  2  ) [ln(GeV  2  T  p  ln(  0  20  40  60  80  100  120  140  160  180  200  )] )  2c/  2  Events / ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='3 [ln(GeV  Data  Fit  (2S)  ψ  Coherent  (2S)  ψ  Incoherent  Non-reson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  LHCb  = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='02 TeV  NN  s  PbPb  < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='5  y*  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='0 <  ψ(2S)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' 1: Fit to the invariant mass distribution of dimuon candidates (up) and the  ln(p∗2  T ) distribution fit of dimuon candidates within the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='0 < y∗ < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='5 range for  J/ψ candidates (left) and ψ(2S) candidates (right), where the starred notation  indicates that the observable is defined in the nucleus-nucleus centre-of-mass frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  production yields versus transverse momentum as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' For the  phenomenological models, one scenario does not consider the destructive  effect due to the overlap between the two nuclei, whereas the other takes it  into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' In general, the trends of J/ψ photo-production measurements  and theoretical predictions are consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' Meanwhile, the two theoretical  curves do not show a significant difference because the collisions are peripheral  and the nuclear overlapping effect is expected to increase in more central  collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' Currently, the measurement is limited to low values of Npart due  to detector limitation, and the measurement of pT is consistent with the  coherent J/ψ photo-production;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' the peak value of transverse momentum is  around 60 MeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  4  template  printed on January 3, 2023  0  1  2  3  4  5  y∗  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
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+page_content='5  dσJ/ψ/dy∗ [mb]  LHCb  PbPb √sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='02 TeV  Coherent J/ψ production  Luminosity unc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' : 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='4%  0  1  2  3  4  5  y∗  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
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+page_content=' : 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='4%  data  stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
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+page_content='  Guzey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  LTA W  LTA S  EPS09  Krelina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  GBW+BT  GBW+POW  KST+BT  GG-hs+BG  M¨antysaari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  No fluct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' +BG  Gon¸calves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  bCGC+BG  bCGC+GLC  IP-SAT+BG  IP-SAT+GLC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='20  p∗  T [GeV/c]  0  10  20  30  40  50  60  d2σJ/ψ/dp∗  Tdy∗ [mb/(GeV/c)]  LHCb  PbPb √sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='02 TeV  Coherent J/ψ production  Luminosity unc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' : 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='4%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='20  p∗  T [GeV/c]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='5  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='0  d2σψ(2S)/dp∗  Tdy∗ [mb/(GeV/c)]  LHCb  PbPb √sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='02 TeV  Coherent ψ(2S) production  2 < y∗ < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='5  Luminosity unc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' : 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='4%  data  stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' unc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' unc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  Guzey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  LTA W  LTA S  EPPS16  M¨antysaari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  Is fluct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' +BG  No fluct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' +BG  Is fluct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' +GLC  No fluct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' +GLC  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' 2: Differential cross-section as a function y∗ (up) and p∗  T (down) for coherent J/ψ  (left) and ψ(2S) (right) production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' Compared to theoretical predictions, which are  grouped as perturbative-QCD calculations (red lines) and colour-glass-condensate  models (blue lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  0  1  2  3  4  5  y∗  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='40  σψ(2S)/σJ/ψ  LHCb  PbPb √sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='02 TeV  Coherent ψ(2S)/J/ψ production  data  stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' unc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' unc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  Guzey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  LTA W  LTA S  EPS09  Krelina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  GBW+BT  GBW+POW  KST+BT  GG-hs+BG  M¨antysaari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  No fluct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' +BG  Gon¸calves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  bCGC+BG  bCGC+GLC  IP-SAT+BG  IP-SAT+GLC  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' 3: Differential cross-section ratio of ψ(2S) to J/ψ as a function of y∗, compared  to theoretical predictions, which are separated into perturbative QCD calculations  (red lines) and colour-glass-condensate models (blue lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  template  printed on January 3, 2023  5  3000  3050  3100  3150  3200  ]  2c  ) [MeV/  −  µ  +  µ  (  m  0  50  100  150  200  250  300  )  2c  Candidates /(5 MeV/  Data  Signal  Background  LHCb  = 5 TeV  NN  s  PbPb  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='9  ±  > = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='6  part  N  <  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='0 < y < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='5  52  ±  = 1764  Total  ψ  J/  N  5  10  15  20  ])  2c/  2  [MeV  /  2  T  p  ln(  0  20  40  60  80  100  120  140  160  180  200  220  Events/(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='40)  LHCb  = 5 TeV  NN  s  PbPb  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='9  ±  > = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='6  part  N  <  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='0 < y < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='5  19  ±  = 277  ψ  J/  N  Data  hadro-produced  ψ  J/  photo-produced  ψ  J/  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' 4: Fit to the invariant mass spectrum of J/ψ candidates (left) and the ln(p2  T)  distribution fit for J/ψ candidates (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  2  3  4  ψ  J/  y  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='6  3  −  10  ×  y  /d  ψ  J/  dY  LHCb  = 5 TeV  NN  s  PbPb  µ  µ  →  ψ  Photo-produced J/  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='2  ±  > = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='7  part  N  <  < 300 MeV/c  T  ψ  J/  p  Data  No overlap effects  Overlap effects  10  20  30  >  part  N  <  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='5  3  −  10  ×  y  /d  ψ  J/  dY  LHCb  = 5 TeV  NN  s  PbPb  µ  µ  →  ψ  Photo-produced J/  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='0 < y < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='5  Data  No overlap effects  Overlap effects  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='2  ]c  [GeV/  T  p  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='5  3  3  −  10  ×  ]  1  )c  [(GeV/  y  d  T  p  /d  ψ  J/  Y  2  d  LHCb  = 5 TeV  NN  s  PbPb  µ  µ  →  ψ  Photo-produced J/  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='0 < y < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='0 < y < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='5  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='2  ±  > = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='7  part  N  <  Data  No overlap effects  Overlap effects  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' 5: The results of the differential photo-production yields of J/ψ versus the  rapidity and the number of participants in collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' Double-differential yields of  the coherent J/ψ produced in peripheral PbPb collisions as a function of pT (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' Conclusion  We report the results of coherent J/ψ, ψ(2S) production cross sections  in UPCs, as well as their ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' It is the first coherent ψ(2S) and ψ(2S) to  J/ψ ratio measurement in forward rapidity region for UPC at LHC, and  the differential cross section of coherent J/ψ and ψ(2S) production in PbPb  UPC is also measured as a function of p∗  T for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' Also, it is the  most precise measurement of J/ψ production in UPCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' The production of  photo-produced J/ψ mesons in peripheral PbPb collisions is measured and  by far it is the most precise coherent photo-produced J/ψ by LHCb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content='  REFERENCES  [1] LHCb Collaboration, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' Alves Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfZPf6/content/2301.00222v1.pdf'}
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+Graphical quantum Clifford-encoder compilers from the ZX calculus
+Andrey Boris Khesin,1, ∗ Jonathan Z. Lu,2, † and Peter W. Shor1, ‡
+1Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA
+2Department of Physics, Harvard University, Cambridge, MA
+(Dated: January 9, 2023)
+We present a quantum compilation algorithm that maps Clifford encoders, an equivalence class of
+quantum circuits that arise universally in quantum error correction, into a representation in the ZX
+calculus. In particular, we develop a canonical form in the ZX calculus and prove canonicity as well
+as efficient reducibility of any Clifford encoder into the canonical form. The diagrams produced by
+our compiler explicitly visualize information propagation and entanglement structure of the encoder,
+revealing properties that may be obscured in the circuit or stabilizer-tableau representation.
+Quantum algorithms have been shown to solve a
+broad range of computational problems ranging from
+search to physical simulation [1–8]. Recent progress
+in the experimental realization of scalable quantum
+computers have further generated excitement about
+the potential of quantum algorithms in practice [9–
+13]. Often, however, a quantum algorithm may not
+be in a representation that is most conducive to how
+one wants to study it. For example, an algorithm
+may be expressed in terms of higher-level unitary
+operations (e.g. a quantum Fourier transform oper-
+ator, continuous rotation gates, etc.), while an actual
+quantum computer can only implement a finite (al-
+beit hopefully universal) set of gates. This is akin to
+the classical motivation for compilers, which map
+high-level programming languages in which algo-
+rithms are expressed into a universal set of boolean-
+circuit gates.
+A similar compiler can be devised
+for the quantum realm, using the famous Solovay-
+Kitaev theorem which has been recently been im-
+proved to rely on fewer assumptions about the finite
+gate set itself [14]. Beyond this fundamental result,
+there has been a recent flurry of results related to
+quantum compilation from high-level operations to
+a finite gate set [15–19]. The choice of finite gate
+set is typically chosen to be Clifford (Hadamard,
+phase, and controlled-NOT gates) set plus the T
+gate, where T = |0⟩ ⟨0| + eiπ/4 |1⟩ ⟨1|, which is uni-
+versal [8, 19].
+Yet the idea of compilation can extend far be-
+yond the realm of high-to-low-level transformations
+of quantum gates; any undesirable property of an al-
+gorithm’s representation can motivate the construc-
+tion of a compiler that maps the algorithm into a
+more useful representation for the task at hand. We
+are specifically motivated by the fact that the circuit
+representation of quantum operations can often blur
+the structure of the output’s entanglement structure
+(in the intuitive sense of understanding how the cir-
+cuit entangles the input qubits just by looking at it).
+The circuit diagram may also obscure the structure
+of information propagation, i.e. which qubits affect
+the values of which other qubits.
+In this letter,
+we take a first step towards
+a compiler that resolves such issues by produc-
+ing a representation that explicitly illustrates in-
+formation propagation and entanglement struc-
+ture.
+Specifically, we consider a restricted set of
+quantum operations—Clifford operations—and map
+them into visual graph diagrams that we design to
+have the above properties. Although Clifford opera-
+tions are not universal, they are used widely in stud-
+ies of fault-tolerant quantum computation [8, 20]
+and quantum coding theory [8, 21–25].
+Building Blocks — Our main tool is the ZX-
+calculus, a graphical language for vectors that has
+become of great interest in quantum information re-
+search [26–30]. The ZX-calculus produces visual di-
+agrams that represent quantum states, circuits, and
+more. As with any formal logical system, the ZX
+calculus has a set of rules which may be iteratively
+applied to transform diagrams into equivalent dia-
+grams. These rules are representations of identities
+in quantum circuits. The ZX-calculus has become
+of more interest than ever in fault-tolerant quan-
+tum computation and quantum compiler theory be-
+cause it can explicitly visualize properties of circuits
+and entanglement in an intuitive manner.
+It has
+recently been applied to a host of quantum compu-
+tation problems, including lattice surgery [31] and
+quantum optimization [32].
+Importantly, the ZX-
+calculus is, for stabilizer tableaus, complete (equal-
+ities of tableaus can be derived from corresponding
+ZX diagrams), sound (vice versa), and universal (ev-
+ery quantum operation can be expressed in the ZX-
+calculus) [26, 33].
+Consequently, there is great potential for the ZX-
+calculus to be not simply a tool for quantum com-
+piler theory, but a quantum representation itself to
+which circuits and, in the stabilizer formalism of
+quantum error correction, stabilizer tableaus, can
+be compiled.
+A requirement for any well-defined
+arXiv:2301.02356v1  [quant-ph]  6 Jan 2023
+
+2
+compiler whose output is a ZX-calculus diagram is
+uniqueness—any two equivalent input representa-
+tions must map to the same ZX-calculus diagram.
+Such a diagram is known as a ZX canonical form
+(ZXCF). Much room for exploration remains for the
+construction of such forms. Recently, Hu and Khesin
+[34] investigated the idea of ZXCFs on states, and
+showed the following.
+Theorem A (Hu and Khesin [34]). There exists a
+ZXCF for stabilizer quantum states, which we call
+HK form.
+Their analysis motivates our construction of a
+ZXCF for quantum circuits (though we restrict our-
+selves to Clifford circuits). More precisely, we com-
+pile Clifford encoders, which are equivalence classes
+of circuits.
+Definition B. Two circuits are equivalent Clifford
+encoders if they have the same image over all possi-
+ble input states. That is, two circuits C1 and C2 are
+equivalent encoders if there exists a unitary U such
+that C1 = C2U or C2 = C1U.
+An
+encoder
+can
+always
+be—and
+is
+most
+naturally—represented by a Clifford circuit (as op-
+posed to a non-Clifford circuit). Encoders moreover
+have the same expressive power as incomplete stabi-
+lizer tableaus—stabilizer matrices of size (n−k)×n
+that do not fully specify a state. Incomplete stabi-
+lizer tableaus are the crux of the Gottesman-Knill
+theorem, which proves the efficient classical simu-
+latability of Clifford circuits [35] and are also used
+to specify encodings (hence the name Clifford en-
+coders) of the logical qubits in quantum error cor-
+rection. In fact, incomplete stabilizer tableaus are
+themselves Clifford encoders, and are used widely in
+error-correcting protocols [24, 25, 36]. As such, we
+will consider these tableaus to be a possible form of
+the compiler input, in addition to encoders repre-
+sented as a Clifford circuit.
+We will first construct a ZXCF that illustrates in-
+formation propagation and entanglement. We will
+then describe the full compilation algorithm and
+show that it is efficient. Lastly, we show that the
+compiler map is well-defined by proving the canon-
+icity of the ZXCF.
+Canonical Form Construction — We follow the
+standard notation of ZX-calculus graph diagram-
+matics, specified in Backens [33]; we refer the reader
+there for the basics of the ZX calculus and trans-
+formation rules. Green nodes are associated with Z
+operators, and red with X. Each node may be as-
+sociated with a local Clifford operator (which may
+be expressed as a phase which is a multiple of π/2),
+and each edge may have a Hadamard gate on it. We
+color an edge blue if it has a Hadamard gate on it
+(some notations draw a box on the edge instead).
+The circuit takes n − k input qubits to n output
+qubits, for k ≥ 0. We define our ZXCF in this paper
+to be in encoder-respecting form if the diagram is
+structured as follows.
+Definition C. An encoder-respecting form D has
+only green nodes, and is structured as a semi-
+bipartite graph (input-associated nodes and output-
+associated nodes). The input cluster I has n − k
+nodes associated with the n − k input qubits of the
+corresponding encoder, and the output cluster O has
+n nodes. Each input node has a free (not connected
+to any other nodes) edge—the input edge. Similarly,
+each output node has a free output edge. The out-
+put nodes are numbered from 1 to n—in order from
+left to right on an incomplete stabilizer tableau or
+top to bottom on Clifford circuit output wires. For
+convenience, we refer to the node v connected to an
+output edge numbered i as the output node num-
+bered i.
+The design of the encoder-respecting form graphi-
+cally illustrates how information propagates from in-
+put to output (which edges connect I to O) as well
+as the entanglement structure (which edges connect
+O to O) of the underlying encoder. Note that the
+structure of D gives rise to a natural binary “partial
+adjacency matrix” MD of size (n − k) × n, which
+describes the edges between I and O like a usual
+graph adjacency matrix representation.
+Given a good structure, all that remains in a com-
+piler is to ensure a canonical form, so that the map
+from encoder to ZX diagrams is well-defined.
+We
+constrain an encoder-respecting form into a ZXCF
+via four additional rules.
+1. Edge Rule: The ZXCF must have exactly one
+Z node per free edge and every internal edge
+in our ZXCF must have a Hadamard gate on
+it.
+2. Hadamard Rule: No output node v may both
+have a Hadamard gate on its output edge and
+have edges connecting v to lower-numbered
+nodes or input nodes.
+3. RREF Rule:
+MD must be in reduced row-
+echelon form (RREF).
+4. Clifford Rule: Let P ⊆ O be the nodes associ-
+ated with the pivot columns of the RREF ma-
+trix MD, so that |P| = n−k. There can be no
+local Clifford operations {I, S, Z, SZ, H, HZ}
+
+3
+FIG. 1.
+Example of an encoder-respecting form and
+some ways it might violate the 4 rules.
+on input or pivot nodes, or their free edges.
+There can be no input-input edges or pivot-
+pivot edges.
+To provide some intuition on these 4 rules, Fig. 1
+depicts a generic example of some possible violations
+to the rules.
+In the graphical representation of the ZXCF, pivot
+nodes correspond to a subset of nodes in O that con-
+nect to exactly one input; the subset is such that P
+is in one-to-one correspondence with I. The Clifford
+rule extends the semi-bipartite constraint on edges
+by disallowing connections between pivots.
+These four rules are sufficient to produce a ZXCF;
+we will show the following.
+Theorem D. Any Clifford ZX encoder has a unique
+equivalent ZXCF satisfying the Edge, Hadamard,
+RREF, and Clifford rules. There is an efficient al-
+gorithm to canonicalize the ZX encoder.
+We will first describe the transform to a ZXCF.
+First, we will apply the following lemma.
+Lemma E. There exists an efficient transformation
+of a ZX encoder diagram with n − k input edges and
+n output edges into a corresponding ZX presentation
+of a stabilizer state in HK form, with n + (n − k) =
+2n − k output edges.
+The proof of this lemma is relegated to the Ap-
+pendix.
+Application of Lemma E results in a HK form,
+which resembles the diagrammatics in Backens [33].
+In particular, there is one Z node per vertex of the
+graph in the HK form, with any internal edge hav-
+ing a Hadamard gate on it. Free edges can only have
+Hadamard gates on them if their phase is a multi-
+ple of π, corresponding to the local Clifford gates H
+and HZ in HK form, and if the associated node is
+not connected to any lower-numbered nodes. Nodes
+whose free edges have no Hadamard gate are free to
+have any multiple of π
+2 as a phase, corresponding to
+local Clifford operations I, S, Z, or SZ, respectively.
+We are now equipped with a HK ZX diagram that
+represents a state.
+This diagram has only output
+edges.
+To return it into an encoder diagram, we
+turn the appropriate n − k output edges back into
+input edges.
+In the circuit representation, this is
+equivalent to turning bra’s into ket’s and vice versa
+to map between an operator and a state. For exam-
+ple, |00⟩ ⟨1|−|11⟩ ⟨0| ↔ |001⟩−|110⟩. This operation
+gives us an equivalent encoder diagram in HK form.
+Note that HK form by definition is in encoder-
+respecting form. It also satisfies the Edge rule and
+an analogous Hadamard rule—there are no input
+edges in HK form so the analogous rule is enforced
+only on lowered-numbered nodes, but if we number
+the nodes from the beginning such that the input
+nodes are lowered-numbered than all output nodes,
+then the transformed encoder-HK will satisfy our
+ZXCF Hadamard rule.
+So, all that remains is to
+simplify the diagram to obey the RREF and Clifford
+rules. The former is given by the following theorem,
+which is proven in the Appendix.
+Theorem F. An encoder diagram in HK form can
+be efficiently transformed to satisfy the RREF rule,
+while continuing to satisfy the Edge and Hadamard
+rules.
+The only remaining task is to enforce the Clif-
+ford rule. First, if there are any edges between in-
+put nodes (those in I), we can simply remove them.
+(They correspond to controlled-Z operations, and
+we can take off any unitary operation on the input
+by the encoder definition.) The same goes for local
+Cliffords on I.
+All that is left to manipulate are the nodes in P.
+For any pivot node with a non-zero phase, we use
+Eq. (9) in Hu and Khesin [34] and apply it to the
+input node vin associated with that pivot node. As
+shown there, this operation identically applies a lo-
+cal complementation about the input node, which
+notably does not change the entries of MD. How-
+ever, this operation also increases the phase of each
+neighbors of vin by π
+2 (due to multiplication by S)
+so we repeat this process until the phase vanishes.
+Lastly, if there are any edges between pivots p1
+and p2, we can remove them by applying Eq. (10)
+from Hu and Khesin [34] to their pair of associated
+input nodes v1 and v2. This swaps the sets of neigh-
+bours of v1 and v2, denoted N1 and N2 (which we
+could always swap back at will). This will also tog-
+gle (i.e.
+if there were an edge, now there is not,
+
+1
+(4)
+X
+X
+3
+X
+n4
+and vice versa) any edge connected to N1 × N2,
+which will include exactly the one pivot-pivot edge
+we wish to flip. (We define an edge between a node
+v∗ ∈ N1 ∩ N2 and itself to be a Z gate, which adds
+a phase of π to v∗.) Note that this operation does
+not violate the RREF rule because once this process
+is complete we can simply swap the neighbors back
+(without any toggling) via a row operation. It also
+does not violate the Hadamard rule as there cannot
+be Hadamard gates on the output edges of nodes
+in either N1 or N2; they were removed earlier via
+Eq. (10) of Hu and Khesin [34]. We repeat until no
+pivot-pivot edges are found.
+With that step, the ZXCF transformation to
+ZXCF is complete. Although this process may seem
+lengthy, all of these steps can be done systematically
+in an efficient manner, without having to go back
+to fix earlier rules. While the algorithm as stated is
+polynomial time, we believe it is possible to add opti-
+mizations that further improve the time complexity
+for practical implementation.
+Compilation Process — With the canonical form
+in hand, we proceed to describe a method for com-
+piling down the three representations we consider
+in in this letter (ZX encoder diagram, circuit, and
+tableau) into a ZXCF.
+In the previous sections we showed how to compile
+a ZX encoder diagram into a ZXCF. Moreover, in the
+Appendix proof of Lemma E, we showed that this
+compilation is actulaly done by first transforming
+into a circuit representation, so that step is covered
+as well. All that remains is the incomplete stabilizer
+tableau.
+The most direct method of compilation is to trans-
+form the tableau into a Clifford circuit representa-
+tion, and then apply the above. We will construct
+our circuit in reverse, going from outputs to inputs.
+If our tableau has n columns and k rows, we begin
+by drawing n output wires. We then apply the fol-
+lowing procedure for each row of the tableau. Apply
+a Clifford operation U that turns the first stabilizer
+into Z1 = Z ⊗ I ⊗ · · · ⊗ I, using the technique of the
+Gottesman-Knill theorem.
+We multiply the remaining rows by the first until
+the entire first column of the tableau has only I, with
+the exception of the first row. We then post-select on
+the +1 result of a computational basis measurement
+by applying ⟨0|.
+When reading the circuit in the
+forward direction, this is equivalent to initializing a
+work qubit in the |0⟩ state. The effect of U and the
+measurement is equivalent to applying I+P
+2 , where
+P is the first row of the stabilizer tableau. This is
+equivalent to post-selecting on the +1 measurement
+result of that stabilizer. Having measured the qubit,
+we remove its corresponding row and column from
+the tableau, and repeat on a tableau of n − 1 qubits
+and k −1 rows, until there are no rows left, at which
+point the remaining n − k wires that have not been
+capped by measurements become our inputs.
+Proof of Canonicity — We next prove that our
+proposed ZXCF is indeed canonical. In other words,
+two equivalent stabilizer tableaus—generators of the
+same subspace of the n-qubit Hilbert space—will
+map to the same ZXCF. We proceed by a counting
+argument. Since any incomplete stabilizer tableau
+can be turned into a ZXCF, it suffices to prove that
+there are an equal number of incomplete stabilizer
+tableaus as there are possible ZXCF diagrams.
+Lemma G. The number of stabilizer tableaus on n
+qubits with k rows is equal to the number of ZXCF
+diagrams with n outputs and n − k inputs.
+First, the number of incomplete stabilizer tableaus
+with k stabilizers on n qubits is
+k�
+i=1
+2 · 4n/2i−1 − 2 · 2i−1
+k�
+i=1
+2k − 2i−1
+=
+k
+�
+i=1
+22n−i+2 − 2i
+2k − 2i−1
+.
+(1)
+For each row i, there are 2 · 4n possible Pauli strings
+(including the sign), but the requirement that they
+commute with previous rows divides the count by
+2i−1. Of these 2 · 4n/2i−1 valid strings, 2i−1 strings
+are linear combinations of previous strings, with a
+factor of 2 for strings that differ by a sign from pre-
+vious strings.
+We have overcounted, however, since any two
+tableaus that generate the same subspace are equiv-
+alent.
+Thus, we need to divide by the number of
+choices of k generators for a given subspace of char-
+acteristic 2,
+k�
+i=1
+2k − 2i−1.
+Next, the number of ZXCF diagrams for the same
+n and k is given by the following function evaluated
+at p = o = 0.
+f(n, k, p, o) =
+�
+1
+if n = k = 0
+An,k,p,o + Bn,k,p,o
+else
+(2)
+where
+An,k,p,o = 2of(n − 1, k, p + 1, o)
+(3)
+if n ̸= k and An,k,p,o = 0 if n = k and where
+Bn,k,p,o = (22p+o+2 + 2)f(n − 1, k − 1, p, o + 1) (4)
+if k ̸= 0 and Bn,k,p,o = 0 if k = 0.
+
+5
+This function is computed with a base case of an
+empty tableau when n = k = 0. To derive f, imag-
+ine that, starting with two empty bins, we must
+assign the n − k output nodes to be pivots (case
+A) or non-pivots (case B), where pivots need to be
+matched with input nodes. The current number of
+pivots is tracked by p and the number of non-pivot
+outputs is tracked by o.
+Suppose we want the next output node to be a
+pivot (in P). Since there is a one-to-one correspon-
+dence between pivots and inputs, we can add pivots
+only if n > k. The matching between pivots and
+input nodes is fully constrained by the RREF rule,
+which sorts the inputs and pivots together. The Clif-
+ford rule says that no pivot nodes may have local
+Clifford operations, so we just need to choose the
+edges connecting them to nodes we have already as-
+signed. Since there are no pivot-pivot edges and the
+pivot connects to only one input, we have exactly
+2o possibilities. Having made an assignment, p in-
+creases by 1, and n decreases by 1 (both input and
+output are decremented since the pivot matches with
+an input).
+Suppose instead we want the next output node
+to be a non-pivot (in O). Then we have to choose
+the local Clifford operation as well as its edges to
+previously assigned vertices. If we choose to connect
+the node to none of the p assigned inputs, p assigned
+pivots, or o assigned vertices, then we are allowed to
+place any of the 6 local Cliffords on the node.
+If
+any of those edges are present, however, we cannot
+apply a Hadamard to the output edge due to the
+Hadamard rule. Hence, 4 choices remain for the local
+Clifford operation. This works out to a total of 4 ·
+(22p+o −1)+6 = 22p+o+2 +2 possibilities. To finish,
+we decrement the number of output qubits without
+changing the number of inputs.
+We can check that the recursion is solved by
+f(n, k, p, o) =
+2o(n−k)
+k
+�
+i=1
+(2n+1 − 2i)(2n−i+1+2p+o + 1)
+2k − 2i−1
+.
+(5)
+One
+can
+verify
+by
+standard
+induction
+that
+f(n, k, 0, 0) gives the same expression as Eq. (1),
+proving that ZXCF is indeed canonical.
+Application to Quantum Codes — Consider, for
+example of a ZXCF, the nine-qubit error-correcting
+code, due to Shor [24], which uses 9 physical qubits
+to encode 1 logical qubit. The Shor code may be
+represented by the stabilizer tableau
+�
+�����������
+Z
+Z
+I
+I
+I
+I
+I
+I
+I
+Z
+I
+Z
+I
+I
+I
+I
+I
+I
+I
+I
+I
+Z
+Z
+I
+I
+I
+I
+I
+I
+I
+Z
+I
+Z
+I
+I
+I
+I
+I
+I
+I
+I
+I
+Z
+Z
+I
+I
+I
+I
+I
+I
+I
+Z
+I
+Z
+X X X X X X
+I
+I
+I
+X X X
+I
+I
+I
+X X X
+�
+�����������
+.
+(6)
+Application of our compilation method yields the
+ZXCF given in Fig. 2(a). There are three identical
+sectors of the outputs, two with Hadamarded out-
+puts and one without. This resembles our expecta-
+tions from examination of the un-normalized qubit
+representation of the Shor code, (|000⟩ ± |111⟩)⊗3.
+We may compare this to a relatively simple but un-
+canonical form in Fig. 2(b) which one might heuristi-
+cally construct. We created Fig. 2(b) by using a ZX
+rule wherein a Z node with two Hadamarded edges
+can be replaced with a single non-Hadamarded edge.
+This diagram has some similar visualization—and is
+in some senses simpler—but has no obvious associa-
+tion of nodes with the qubits, which blurs interpre-
+tations about information propagation or entangle-
+ment structure.
+FIG. 2.
+A ZXCF of the nine-qubit code (a) versus a
+simpler (but not canonical) form (b).
+There is an additional simplification we can make
+to our ZXCF when the encoder in question is for
+error correction. In particular, when one transmits
+
+3
+5
+8
+66
+encoded qubits, the set of errors that can be cor-
+rected does not change if we apply local Clifford op-
+erations to the encoding, since the space of p-qubit
+Pauli errors is preserved. As a result, we can disre-
+gard the Hadamarded output edges and the phases
+on the output nodes off error-correcting encoders.
+Any code thus has an equivalent representation up
+to local Cliffords whose ZXCF consists of nothing
+more than a semi-bipartite graph. We have there-
+fore proven:
+Theorem H. For any stabilizer code, there exists
+at least one equivalent code that has a presentation
+as a semi-bipartite graph with no local Clifford oper-
+ations.
+Further examples are provided in the Appendix.
+Conclusion and Outlook — In this letter, we
+introduced
+a
+quantum
+compiler
+mapping
+Clif-
+ford encoders (or equivalently incomplete stabilizer
+tableaus) into a canonical ZX-calculus form.
+The
+representation of our canonical form as a semi-
+bipartite graph between input and output qubits ex-
+plicitly visualizes information propagation from in-
+put to output, as well as entanglement structure of
+the output.
+We then proved the algorithmic effi-
+ciency and correctness of the compiler.
+Our compilation technique takes a first step to-
+wards the use of ZX-calculi as representations for a
+useful (in terms of information and entanglement)
+visualization of quantum operations. Whether the
+ZXCF technique can be generalized beyond Clifford
+gates—as our construction relies crucially on them—
+is an important next step in the exploration of ZX-
+based quantum compilers. An interesting direction
+of further research is the possibility of applying our
+ZXCF to the decomposition of magic states into lin-
+ear combinations of stabilizer states. Another pos-
+sible direction is the analysis of the correspondence
+between the quality of a code and its properties in
+the ZXCF representation. A better understanding
+of such a correspondence may inform better ways to
+design new quantum codes.
+Acknowledgments — ABK was supported by the
+National Science Foundation (NSF) under Grant No.
+CCF-1729369. PWS was supported by the NSF un-
+der Grant No. CCF-1729369, by the NSF Science
+and Technology Center for Science of Information
+under Grant No. CCF-0939370, by the U.S. Depart-
+ment of Energy, Office of Science, National Quantum
+Information Science Research Centers, Co-design
+Center for Quantum Advantage (C2QA) under con-
+tract number DE-SC0012704., and by NTT Re-
+search Award AGMT DTD 9.24.20.
+∗ khesin@mit.edu
+† lujz@mit.edu
+‡ shor@math.mit.edu
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+number of non-clifford gates in quantum circuits,
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+lizer quantum mechanics, New Journal of Physics
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+tion of stabilizer circuits, Physical Review A 70,
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+tum error correction, Proceedings of the Royal So-
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+five-qubit error correcting code, Physical Review
+Letters 86, 5811 (2001).
+Additional Proofs
+We begin with the proof of Lemma E. Generally,
+an operator can be transformed into a state by writ-
+ing the operator as a linear combination of projectors
+|φ⟩ ⟨ψ|, and then flipping the bra’s into ket’s. This
+process is known as the Choi–Jamio�lkowski isomor-
+phism. We describe how to perform an analogous
+transformation in the ZX calculus, along with the
+canonicalization into HK form.
+Proof. We will turn the encoder diagram into a Clif-
+ford circuit, and then map the output of that circuit
+(with input the all-|+⟩ state) to HK form. Any X
+(red) nodes in the ZX encoder diagram can be trans-
+formed into Z (green) nodes with the same phase
+surrounded by Hadamard gates. Now, we can inter-
+pret the ZX encoder diagram as a state by treating
+the input edges as output edges (the ZX version of
+the Choi–Jamio�lkowski isomorphism).
+Our ZX diagram now computes a state, so we can
+express it entirely in terms of the following elemen-
+tary operations and their circuit analogs. First, we
+could have a Z node with only one output or only
+one output. In a circuit, this is a |+⟩ qubit or a post-
+selected ⟨+| measurement, respectively. We can also
+have a Hadamarded edge or a node with π
+2 phase,
+which are represented as H or S gates in circuits,
+respectively. Lastly, we can have a Z node of degree
+3, with either 2 inputs and 1 output or 1 input and
+2 outputs that acts as a merge or a split. This is
+equivalent to applying a CX operation between two
+qubits where the second qubit is either initialized
+to |0⟩ before the CX gate or post-selected by the
+measurement ⟨0| after the CX gate, respectively.
+Any ZX diagram can be expressed in terms of
+these operations [33]. Furthermore, as we apply each
+of these operations we can keep track of the current
+HK form for our state. (The Hu and Khesin [34]
+method shows us how to keep a diagram in HK form
+as each operation is applied.) All of these steps can
+
+8
+be done efficiently [34], so this gives us an efficient
+procedure for turning a ZX diagram into a corre-
+sponding stabilizer state in HK form.
+We next give the proof of Theorem F.
+Proof. The key step is to show that the operations
+corresponding to row operations on the partial adja-
+cency matrix MD can be done without violating the
+Edge and Hadamard rules. To do so, let us examine
+the action of a stabilizer on the encoder diagram.
+The stabilizer is a Pauli matrix. Suppose we ap-
+ply any row—a Pauli string that stabilizes the out-
+put of the encoder. (The row can be a string of X’s
+and Z’s with a global phase.) That is, this string is
+sent through the output edges of the diagram. As
+the operators pass through the local Clifford opera-
+tions on the output vertices, they turn into different
+operators. For example, consider a X operator (rep-
+resented by a X node with a π phase) that passes
+through a free output edge of an output node v. (It
+may be useful to draw an example to follow along.)
+By the π rule of ZX calculus, this results in a global
+phase, negation of the local phase on v, as well as a X
+node on each internal neighboring edge connected to
+v. As each X node moves along the internal edge, it
+will be conjugated past the internal Hadamard gate,
+arriving as a Z on the other side of the edge.
+In sum, the total action of the entire stabilizer
+will visually look a choreographed dance where the
+nodes act out the following steps.
+1. Pauli matrices commute past the H gates on
+the output edges, with some of them changing
+colour (X ↔ Z).
+2. Z nodes from the stabilizer simply add their
+phase to the nodes in the diagram, while the
+X nodes split into one X node for every inter-
+nal edge connected to the output node v, and
+result in a phase flip on every neighbour of v.
+Once this dance is complete, all the added phases
+necessarily cancel and the added global phase re-
+turns to 1, since the string is a stabilizer.
+But note that any single X (or Y represented as
+iXZ) sent through an output node v (after it has
+already passed through a Hadamard on the output
+edge if one exists) may result in some operators (rep-
+resented as X or Z nodes) that hit an input node via
+an internal edge connected to the input node. But at
+the end of the day, all of the operators on the input
+edges must cancel out due to stabilization. This im-
+plies that each input node must be connected to an
+even number of nodes on which the stabilizers apply
+X or Y gates after passing through the Hadamard
+gates on the free output edges.
+Consequently, for any pair of input nodes, we are
+free to connect all of the neighbors of one to the other
+via (Hadamarded) internal edges without changing
+the parity effect of the stabilizers described above.
+This corresponds to adding two rows in MD.
+Finally, we note that since any pair of internal
+edges have Hadamard gates on them, if we have two
+edges between the same pair of Z nodes, we can re-
+move them both by the rules of ZX calculus.
+In
+other words, we are effectively free to perform arbi-
+trary row operations on MD mod 2.
+This allows us to row-reduce MD to satisfy the
+RREF rule.
+Further Examples
+We provide some additional examples of our ZX-
+CFs, specifically on the most well-known quantum
+error-correcting codes. The ZXCFs are obtained by
+applying the procedure applied in the main text. As
+with our main-text notation, the blue edges have
+Hadamard operations on them. Our first example is
+the seven-qubit code, due to Steane [36]. The code
+is represented by an incomplete stabilizer matrix
+�
+�������
+I
+I
+I
+X X X X
+I
+X X
+I
+I
+X X
+X
+I
+X
+I
+X
+I
+X
+I
+I
+I
+Z
+Z
+Z
+Z
+I
+Z
+Z
+I
+I
+Z
+Z
+Z
+I
+Z
+I
+Z
+I
+Z
+�
+�������
+.
+(7)
+In our ZXCF, this takes the form given in Figure 3.
+The nodes of the code are simply the corners of a
+cube.
+FIG. 3. ZXCF of the seven-qubit code. The 8 nodes in
+the diagram form the vertices of a boolean cube.
+Note that the stabilizers of the 7-qubit code are
+positioned at exactly the 1-indices in the binary rep-
+resentation of a node label. We can see elegant sym-
+metries in Fig. 3 with the positions of the nodes
+
+CO
+5
+2
+6
+49
+and their expressions in binary. The reason for why
+the basis vectors adjacent to 0 are not 1, 2, and 4,
+which are 001, 010, and 100 in binary, is because
+the rules of the ZXCF require that the output edges
+with Hadamard gates not be connected to lower-
+numbered nodes, placing nodes 1, 2, and 4 in the
+same column in the diagram.
+We next consider the five-qubit code, which is the
+smallest code one can achieve for correction of an
+arbitrary single-qubit error and has been studied ex-
+perimentally [25, 37]. Its incomplete stabilizer rep-
+resentation is given by
+�
+���
+X Z
+Z X
+I
+I
+X Z
+Z X
+X
+I
+X Z
+Z
+Z X
+I
+X Z
+�
+��� .
+(8)
+Our procedure produces the ZXCF shown in Fig. 4.
+FIG. 4. ZXCF of the five-qubit code.
+Note that the stabilizer tableau is just 4 cyclic
+permutations of the first row.
+the ZXCF reflects
+this cyclic structure in the output explicitly.
+
+2
+3
+4
+5
\ No newline at end of file
diff --git a/wNE0T4oBgHgl3EQfcABc/content/tmp_files/load_file.txt b/wNE0T4oBgHgl3EQfcABc/content/tmp_files/load_file.txt
new file mode 100644
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--- /dev/null
+++ b/wNE0T4oBgHgl3EQfcABc/content/tmp_files/load_file.txt
@@ -0,0 +1,495 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf,len=494
+page_content='Graphical quantum Clifford-encoder compilers from the ZX calculus  Andrey Boris Khesin,1, ∗ Jonathan Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Lu,2, † and Peter W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Shor1, ‡  1Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA  2Department of Physics, Harvard University, Cambridge, MA  (Dated: January 9, 2023)  We present a quantum compilation algorithm that maps Clifford encoders, an equivalence class of  quantum circuits that arise universally in quantum error correction, into a representation in the ZX  calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' In particular, we develop a canonical form in the ZX calculus and prove canonicity as well  as efficient reducibility of any Clifford encoder into the canonical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' The diagrams produced by  our compiler explicitly visualize information propagation and entanglement structure of the encoder,  revealing properties that may be obscured in the circuit or stabilizer-tableau representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Quantum algorithms have been shown to solve a  broad range of computational problems ranging from  search to physical simulation [1–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Recent progress  in the experimental realization of scalable quantum  computers have further generated excitement about  the potential of quantum algorithms in practice [9–  13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Often, however, a quantum algorithm may not  be in a representation that is most conducive to how  one wants to study it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' For example, an algorithm  may be expressed in terms of higher-level unitary  operations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' a quantum Fourier transform oper-  ator, continuous rotation gates, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' ), while an actual  quantum computer can only implement a finite (al-  beit hopefully universal) set of gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' This is akin to  the classical motivation for compilers, which map  high-level programming languages in which algo-  rithms are expressed into a universal set of boolean-  circuit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  A similar compiler can be devised  for the quantum realm, using the famous Solovay-  Kitaev theorem which has been recently been im-  proved to rely on fewer assumptions about the finite  gate set itself [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Beyond this fundamental result,  there has been a recent flurry of results related to  quantum compilation from high-level operations to  a finite gate set [15–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' The choice of finite gate  set is typically chosen to be Clifford (Hadamard,  phase, and controlled-NOT gates) set plus the T  gate, where T = |0⟩ ⟨0| + eiπ/4 |1⟩ ⟨1|, which is uni-  versal [8, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Yet the idea of compilation can extend far be-  yond the realm of high-to-low-level transformations  of quantum gates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' any undesirable property of an al-  gorithm’s representation can motivate the construc-  tion of a compiler that maps the algorithm into a  more useful representation for the task at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' We  are specifically motivated by the fact that the circuit  representation of quantum operations can often blur  the structure of the output’s entanglement structure  (in the intuitive sense of understanding how the cir-  cuit entangles the input qubits just by looking at it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  The circuit diagram may also obscure the structure  of information propagation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' which qubits affect  the values of which other qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  In this letter,  we take a first step towards  a compiler that resolves such issues by produc-  ing a representation that explicitly illustrates in-  formation propagation and entanglement struc-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Specifically, we consider a restricted set of  quantum operations—Clifford operations—and map  them into visual graph diagrams that we design to  have the above properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Although Clifford opera-  tions are not universal, they are used widely in stud-  ies of fault-tolerant quantum computation [8, 20]  and quantum coding theory [8, 21–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Building Blocks — Our main tool is the ZX-  calculus, a graphical language for vectors that has  become of great interest in quantum information re-  search [26–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' The ZX-calculus produces visual di-  agrams that represent quantum states, circuits, and  more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' As with any formal logical system, the ZX  calculus has a set of rules which may be iteratively  applied to transform diagrams into equivalent dia-  grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' These rules are representations of identities  in quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' The ZX-calculus has become  of more interest than ever in fault-tolerant quan-  tum computation and quantum compiler theory be-  cause it can explicitly visualize properties of circuits  and entanglement in an intuitive manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  It has  recently been applied to a host of quantum compu-  tation problems, including lattice surgery [31] and  quantum optimization [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Importantly, the ZX-  calculus is, for stabilizer tableaus, complete (equal-  ities of tableaus can be derived from corresponding  ZX diagrams), sound (vice versa), and universal (ev-  ery quantum operation can be expressed in the ZX-  calculus) [26, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Consequently, there is great potential for the ZX-  calculus to be not simply a tool for quantum com-  piler theory, but a quantum representation itself to  which circuits and, in the stabilizer formalism of  quantum error correction, stabilizer tableaus, can  be compiled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  A requirement for any well-defined  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='02356v1  [quant-ph]  6 Jan 2023  2  compiler whose output is a ZX-calculus diagram is  uniqueness—any two equivalent input representa-  tions must map to the same ZX-calculus diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Such a diagram is known as a ZX canonical form  (ZXCF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Much room for exploration remains for the  construction of such forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Recently, Hu and Khesin  [34] investigated the idea of ZXCFs on states, and  showed the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Theorem A (Hu and Khesin [34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' There exists a  ZXCF for stabilizer quantum states, which we call  HK form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Their analysis motivates our construction of a  ZXCF for quantum circuits (though we restrict our-  selves to Clifford circuits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' More precisely, we com-  pile Clifford encoders, which are equivalence classes  of circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Two circuits are equivalent Clifford  encoders if they have the same image over all possi-  ble input states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' That is, two circuits C1 and C2 are  equivalent encoders if there exists a unitary U such  that C1 = C2U or C2 = C1U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  An  encoder  can  always  be—and  is  most  naturally—represented by a Clifford circuit (as op-  posed to a non-Clifford circuit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Encoders moreover  have the same expressive power as incomplete stabi-  lizer tableaus—stabilizer matrices of size (n−k)×n  that do not fully specify a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Incomplete stabi-  lizer tableaus are the crux of the Gottesman-Knill  theorem, which proves the efficient classical simu-  latability of Clifford circuits [35] and are also used  to specify encodings (hence the name Clifford en-  coders) of the logical qubits in quantum error cor-  rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' In fact, incomplete stabilizer tableaus are  themselves Clifford encoders, and are used widely in  error-correcting protocols [24, 25, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' As such, we  will consider these tableaus to be a possible form of  the compiler input, in addition to encoders repre-  sented as a Clifford circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  We will first construct a ZXCF that illustrates in-  formation propagation and entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' We will  then describe the full compilation algorithm and  show that it is efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Lastly, we show that the  compiler map is well-defined by proving the canon-  icity of the ZXCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Canonical Form Construction — We follow the  standard notation of ZX-calculus graph diagram-  matics, specified in Backens [33];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' we refer the reader  there for the basics of the ZX calculus and trans-  formation rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Green nodes are associated with Z  operators, and red with X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Each node may be as-  sociated with a local Clifford operator (which may  be expressed as a phase which is a multiple of π/2),  and each edge may have a Hadamard gate on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' We  color an edge blue if it has a Hadamard gate on it  (some notations draw a box on the edge instead).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  The circuit takes n − k input qubits to n output  qubits, for k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' We define our ZXCF in this paper  to be in encoder-respecting form if the diagram is  structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Definition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' An encoder-respecting form D has  only green nodes, and is structured as a semi-  bipartite graph (input-associated nodes and output-  associated nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' The input cluster I has n − k  nodes associated with the n − k input qubits of the  corresponding encoder, and the output cluster O has  n nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Each input node has a free (not connected  to any other nodes) edge—the input edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Similarly,  each output node has a free output edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' The out-  put nodes are numbered from 1 to n—in order from  left to right on an incomplete stabilizer tableau or  top to bottom on Clifford circuit output wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' For  convenience, we refer to the node v connected to an  output edge numbered i as the output node num-  bered i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  The design of the encoder-respecting form graphi-  cally illustrates how information propagates from in-  put to output (which edges connect I to O) as well  as the entanglement structure (which edges connect  O to O) of the underlying encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Note that the  structure of D gives rise to a natural binary “partial  adjacency matrix” MD of size (n − k) × n, which  describes the edges between I and O like a usual  graph adjacency matrix representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Given a good structure, all that remains in a com-  piler is to ensure a canonical form, so that the map  from encoder to ZX diagrams is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  We  constrain an encoder-respecting form into a ZXCF  via four additional rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Edge Rule: The ZXCF must have exactly one  Z node per free edge and every internal edge  in our ZXCF must have a Hadamard gate on  it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Hadamard Rule: No output node v may both  have a Hadamard gate on its output edge and  have edges connecting v to lower-numbered  nodes or input nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' RREF Rule:  MD must be in reduced row-  echelon form (RREF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Clifford Rule: Let P ⊆ O be the nodes associ-  ated with the pivot columns of the RREF ma-  trix MD, so that |P| = n−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' There can be no  local Clifford operations {I, S, Z, SZ, H, HZ}  3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Example of an encoder-respecting form and  some ways it might violate the 4 rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  on input or pivot nodes, or their free edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  There can be no input-input edges or pivot-  pivot edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  To provide some intuition on these 4 rules, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' 1  depicts a generic example of some possible violations  to the rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  In the graphical representation of the ZXCF, pivot  nodes correspond to a subset of nodes in O that con-  nect to exactly one input;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' the subset is such that P  is in one-to-one correspondence with I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' The Clifford  rule extends the semi-bipartite constraint on edges  by disallowing connections between pivots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  These four rules are sufficient to produce a ZXCF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  we will show the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Any Clifford ZX encoder has a unique  equivalent ZXCF satisfying the Edge, Hadamard,  RREF, and Clifford rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' There is an efficient al-  gorithm to canonicalize the ZX encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  We will first describe the transform to a ZXCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  First, we will apply the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' There exists an efficient transformation  of a ZX encoder diagram with n − k input edges and  n output edges into a corresponding ZX presentation  of a stabilizer state in HK form, with n + (n − k) =  2n − k output edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  The proof of this lemma is relegated to the Ap-  pendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Application of Lemma E results in a HK form,  which resembles the diagrammatics in Backens [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  In particular, there is one Z node per vertex of the  graph in the HK form, with any internal edge hav-  ing a Hadamard gate on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Free edges can only have  Hadamard gates on them if their phase is a multi-  ple of π, corresponding to the local Clifford gates H  and HZ in HK form, and if the associated node is  not connected to any lower-numbered nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Nodes  whose free edges have no Hadamard gate are free to  have any multiple of π  2 as a phase, corresponding to  local Clifford operations I, S, Z, or SZ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  We are now equipped with a HK ZX diagram that  represents a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  This diagram has only output  edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  To return it into an encoder diagram, we  turn the appropriate n − k output edges back into  input edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  In the circuit representation, this is  equivalent to turning bra’s into ket’s and vice versa  to map between an operator and a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' For exam-  ple, |00⟩ ⟨1|−|11⟩ ⟨0| ↔ |001⟩−|110⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' This operation  gives us an equivalent encoder diagram in HK form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Note that HK form by definition is in encoder-  respecting form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' It also satisfies the Edge rule and  an analogous Hadamard rule—there are no input  edges in HK form so the analogous rule is enforced  only on lowered-numbered nodes, but if we number  the nodes from the beginning such that the input  nodes are lowered-numbered than all output nodes,  then the transformed encoder-HK will satisfy our  ZXCF Hadamard rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  So, all that remains is to  simplify the diagram to obey the RREF and Clifford  rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' The former is given by the following theorem,  which is proven in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Theorem F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' An encoder diagram in HK form can  be efficiently transformed to satisfy the RREF rule,  while continuing to satisfy the Edge and Hadamard  rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  The only remaining task is to enforce the Clif-  ford rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' First, if there are any edges between in-  put nodes (those in I), we can simply remove them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  (They correspond to controlled-Z operations, and  we can take off any unitary operation on the input  by the encoder definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=') The same goes for local  Cliffords on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  All that is left to manipulate are the nodes in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  For any pivot node with a non-zero phase, we use  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' (9) in Hu and Khesin [34] and apply it to the  input node vin associated with that pivot node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' As  shown there, this operation identically applies a lo-  cal complementation about the input node, which  notably does not change the entries of MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' How-  ever, this operation also increases the phase of each  neighbors of vin by π  2 (due to multiplication by S)  so we repeat this process until the phase vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Lastly, if there are any edges between pivots p1  and p2, we can remove them by applying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' (10)  from Hu and Khesin [34] to their pair of associated  input nodes v1 and v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' This swaps the sets of neigh-  bours of v1 and v2, denoted N1 and N2 (which we  could always swap back at will).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' This will also tog-  gle (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  if there were an edge, now there is not,  1  (4)  X  X  3  X  n4  and vice versa) any edge connected to N1 × N2,  which will include exactly the one pivot-pivot edge  we wish to flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' (We define an edge between a node  v∗ ∈ N1 ∩ N2 and itself to be a Z gate, which adds  a phase of π to v∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=') Note that this operation does  not violate the RREF rule because once this process  is complete we can simply swap the neighbors back  (without any toggling) via a row operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' It also  does not violate the Hadamard rule as there cannot  be Hadamard gates on the output edges of nodes  in either N1 or N2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' they were removed earlier via  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' (10) of Hu and Khesin [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' We repeat until no  pivot-pivot edges are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  With that step, the ZXCF transformation to  ZXCF is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Although this process may seem  lengthy, all of these steps can be done systematically  in an efficient manner, without having to go back  to fix earlier rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' While the algorithm as stated is  polynomial time, we believe it is possible to add opti-  mizations that further improve the time complexity  for practical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Compilation Process — With the canonical form  in hand, we proceed to describe a method for com-  piling down the three representations we consider  in in this letter (ZX encoder diagram, circuit, and  tableau) into a ZXCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  In the previous sections we showed how to compile  a ZX encoder diagram into a ZXCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Moreover, in the  Appendix proof of Lemma E, we showed that this  compilation is actulaly done by first transforming  into a circuit representation, so that step is covered  as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' All that remains is the incomplete stabilizer  tableau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  The most direct method of compilation is to trans-  form the tableau into a Clifford circuit representa-  tion, and then apply the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' We will construct  our circuit in reverse, going from outputs to inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  If our tableau has n columns and k rows, we begin  by drawing n output wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' We then apply the fol-  lowing procedure for each row of the tableau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Apply  a Clifford operation U that turns the first stabilizer  into Z1 = Z ⊗ I ⊗ · · · ⊗ I, using the technique of the  Gottesman-Knill theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  We multiply the remaining rows by the first until  the entire first column of the tableau has only I, with  the exception of the first row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' We then post-select on  the +1 result of a computational basis measurement  by applying ⟨0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  When reading the circuit in the  forward direction, this is equivalent to initializing a  work qubit in the |0⟩ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' The effect of U and the  measurement is equivalent to applying I+P  2 , where  P is the first row of the stabilizer tableau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' This is  equivalent to post-selecting on the +1 measurement  result of that stabilizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Having measured the qubit,  we remove its corresponding row and column from  the tableau, and repeat on a tableau of n − 1 qubits  and k −1 rows, until there are no rows left, at which  point the remaining n − k wires that have not been  capped by measurements become our inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Proof of Canonicity — We next prove that our  proposed ZXCF is indeed canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' In other words,  two equivalent stabilizer tableaus—generators of the  same subspace of the n-qubit Hilbert space—will  map to the same ZXCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' We proceed by a counting  argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Since any incomplete stabilizer tableau  can be turned into a ZXCF, it suffices to prove that  there are an equal number of incomplete stabilizer  tableaus as there are possible ZXCF diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Lemma G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' The number of stabilizer tableaus on n  qubits with k rows is equal to the number of ZXCF  diagrams with n outputs and n − k inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  First, the number of incomplete stabilizer tableaus  with k stabilizers on n qubits is  k�  i=1  2 · 4n/2i−1 − 2 · 2i−1  k�  i=1  2k − 2i−1  =  k  �  i=1  22n−i+2 − 2i  2k − 2i−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  (1)  For each row i, there are 2 · 4n possible Pauli strings  (including the sign), but the requirement that they  commute with previous rows divides the count by  2i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Of these 2 · 4n/2i−1 valid strings, 2i−1 strings  are linear combinations of previous strings, with a  factor of 2 for strings that differ by a sign from pre-  vious strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  We have overcounted, however, since any two  tableaus that generate the same subspace are equiv-  alent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Thus, we need to divide by the number of  choices of k generators for a given subspace of char-  acteristic 2,  k�  i=1  2k − 2i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Next, the number of ZXCF diagrams for the same  n and k is given by the following function evaluated  at p = o = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  f(n, k, p, o) =  �  1  if n = k = 0  An,k,p,o + Bn,k,p,o  else  (2)  where  An,k,p,o = 2of(n − 1, k, p + 1, o)  (3)  if n ̸= k and An,k,p,o = 0 if n = k and where  Bn,k,p,o = (22p+o+2 + 2)f(n − 1, k − 1, p, o + 1) (4)  if k ̸= 0 and Bn,k,p,o = 0 if k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  5  This function is computed with a base case of an  empty tableau when n = k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' To derive f, imag-  ine that, starting with two empty bins, we must  assign the n − k output nodes to be pivots (case  A) or non-pivots (case B), where pivots need to be  matched with input nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' The current number of  pivots is tracked by p and the number of non-pivot  outputs is tracked by o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Suppose we want the next output node to be a  pivot (in P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Since there is a one-to-one correspon-  dence between pivots and inputs, we can add pivots  only if n > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' The matching between pivots and  input nodes is fully constrained by the RREF rule,  which sorts the inputs and pivots together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' The Clif-  ford rule says that no pivot nodes may have local  Clifford operations, so we just need to choose the  edges connecting them to nodes we have already as-  signed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Since there are no pivot-pivot edges and the  pivot connects to only one input, we have exactly  2o possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Having made an assignment, p in-  creases by 1, and n decreases by 1 (both input and  output are decremented since the pivot matches with  an input).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Suppose instead we want the next output node  to be a non-pivot (in O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Then we have to choose  the local Clifford operation as well as its edges to  previously assigned vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' If we choose to connect  the node to none of the p assigned inputs, p assigned  pivots, or o assigned vertices, then we are allowed to  place any of the 6 local Cliffords on the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  If  any of those edges are present, however, we cannot  apply a Hadamard to the output edge due to the  Hadamard rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Hence, 4 choices remain for the local  Clifford operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' This works out to a total of 4 ·  (22p+o −1)+6 = 22p+o+2 +2 possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' To finish,  we decrement the number of output qubits without  changing the number of inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  We can check that the recursion is solved by  f(n, k, p, o) =  2o(n−k)  k  �  i=1  (2n+1 − 2i)(2n−i+1+2p+o + 1)  2k − 2i−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  (5)  One  can  verify  by  standard  induction  that  f(n, k, 0, 0) gives the same expression as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' (1),  proving that ZXCF is indeed canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Application to Quantum Codes — Consider, for  example of a ZXCF, the nine-qubit error-correcting  code, due to Shor [24], which uses 9 physical qubits  to encode 1 logical qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' The Shor code may be  represented by the stabilizer tableau  �  �����������  Z  Z  I  I  I  I  I  I  I  Z  I  Z  I  I  I  I  I  I  I  I  I  Z  Z  I  I  I  I  I  I  I  Z  I  Z  I  I  I  I  I  I  I  I  I  Z  Z  I  I  I  I  I  I  I  Z  I  Z  X X X X X X  I  I  I  X X X  I  I  I  X X X  �  �����������  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  (6)  Application of our compilation method yields the  ZXCF given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' There are three identical  sectors of the outputs, two with Hadamarded out-  puts and one without.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' This resembles our expecta-  tions from examination of the un-normalized qubit  representation of the Shor code, (|000⟩ ± |111⟩)⊗3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  We may compare this to a relatively simple but un-  canonical form in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' 2(b) which one might heuristi-  cally construct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' We created Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' 2(b) by using a ZX  rule wherein a Z node with two Hadamarded edges  can be replaced with a single non-Hadamarded edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  This diagram has some similar visualization—and is  in some senses simpler—but has no obvious associa-  tion of nodes with the qubits, which blurs interpre-  tations about information propagation or entangle-  ment structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  A ZXCF of the nine-qubit code (a) versus a  simpler (but not canonical) form (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  There is an additional simplification we can make  to our ZXCF when the encoder in question is for  error correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' In particular, when one transmits  3  5  8  66  encoded qubits, the set of errors that can be cor-  rected does not change if we apply local Clifford op-  erations to the encoding, since the space of p-qubit  Pauli errors is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' As a result, we can disre-  gard the Hadamarded output edges and the phases  on the output nodes off error-correcting encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Any code thus has an equivalent representation up  to local Cliffords whose ZXCF consists of nothing  more than a semi-bipartite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' We have there-  fore proven:  Theorem H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' For any stabilizer code, there exists  at least one equivalent code that has a presentation  as a semi-bipartite graph with no local Clifford oper-  ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Further examples are provided in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Conclusion and Outlook — In this letter, we  introduced  a  quantum  compiler  mapping  Clif-  ford encoders (or equivalently incomplete stabilizer  tableaus) into a canonical ZX-calculus form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  The  representation of our canonical form as a semi-  bipartite graph between input and output qubits ex-  plicitly visualizes information propagation from in-  put to output, as well as entanglement structure of  the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  We then proved the algorithmic effi-  ciency and correctness of the compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Our compilation technique takes a first step to-  wards the use of ZX-calculi as representations for a  useful (in terms of information and entanglement)  visualization of quantum operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Whether the  ZXCF technique can be generalized beyond Clifford  gates—as our construction relies crucially on them—  is an important next step in the exploration of ZX-  based quantum compilers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' An interesting direction  of further research is the possibility of applying our  ZXCF to the decomposition of magic states into lin-  ear combinations of stabilizer states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Another pos-  sible direction is the analysis of the correspondence  between the quality of a code and its properties in  the ZXCF representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' A better understanding  of such a correspondence may inform better ways to  design new quantum codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Acknowledgments — ABK was supported by the  National Science Foundation (NSF) under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  CCF-1729369.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' PWS was supported by the NSF un-  der Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' CCF-1729369, by the NSF Science  and Technology Center for Science of Information  under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' CCF-0939370, by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Depart-  ment of Energy, Office of Science, National Quantum  Information Science Research Centers, Co-design  Center for Quantum Advantage (C2QA) under con-  tract number DE-SC0012704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
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+page_content=' Series A: Mathematical, Physical  and Engineering Sciences 452, 2551 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  [37] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Knill, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Laflamme, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Martinez, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Ne-  grevergne, Benchmarking quantum computers: The  five-qubit error correcting code, Physical Review  Letters 86, 5811 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Additional Proofs  We begin with the proof of Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Generally,  an operator can be transformed into a state by writ-  ing the operator as a linear combination of projectors  |φ⟩ ⟨ψ|, and then flipping the bra’s into ket’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' This  process is known as the Choi–Jamio�lkowski isomor-  phism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' We describe how to perform an analogous  transformation in the ZX calculus, along with the  canonicalization into HK form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' We will turn the encoder diagram into a Clif-  ford circuit, and then map the output of that circuit  (with input the all-|+⟩ state) to HK form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Any X  (red) nodes in the ZX encoder diagram can be trans-  formed into Z (green) nodes with the same phase  surrounded by Hadamard gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Now, we can inter-  pret the ZX encoder diagram as a state by treating  the input edges as output edges (the ZX version of  the Choi–Jamio�lkowski isomorphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Our ZX diagram now computes a state, so we can  express it entirely in terms of the following elemen-  tary operations and their circuit analogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' First, we  could have a Z node with only one output or only  one output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' In a circuit, this is a |+⟩ qubit or a post-  selected ⟨+| measurement, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' We can also  have a Hadamarded edge or a node with π  2 phase,  which are represented as H or S gates in circuits,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Lastly, we can have a Z node of degree  3, with either 2 inputs and 1 output or 1 input and  2 outputs that acts as a merge or a split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' This is  equivalent to applying a CX operation between two  qubits where the second qubit is either initialized  to |0⟩ before the CX gate or post-selected by the  measurement ⟨0| after the CX gate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Any ZX diagram can be expressed in terms of  these operations [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Furthermore, as we apply each  of these operations we can keep track of the current  HK form for our state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' (The Hu and Khesin [34]  method shows us how to keep a diagram in HK form  as each operation is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=') All of these steps can  8  be done efficiently [34], so this gives us an efficient  procedure for turning a ZX diagram into a corre-  sponding stabilizer state in HK form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  We next give the proof of Theorem F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' The key step is to show that the operations  corresponding to row operations on the partial adja-  cency matrix MD can be done without violating the  Edge and Hadamard rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' To do so, let us examine  the action of a stabilizer on the encoder diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  The stabilizer is a Pauli matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Suppose we ap-  ply any row—a Pauli string that stabilizes the out-  put of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' (The row can be a string of X’s  and Z’s with a global phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=') That is, this string is  sent through the output edges of the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' As  the operators pass through the local Clifford opera-  tions on the output vertices, they turn into different  operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' For example, consider a X operator (rep-  resented by a X node with a π phase) that passes  through a free output edge of an output node v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' (It  may be useful to draw an example to follow along.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=')  By the π rule of ZX calculus, this results in a global  phase, negation of the local phase on v, as well as a X  node on each internal neighboring edge connected to  v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' As each X node moves along the internal edge, it  will be conjugated past the internal Hadamard gate,  arriving as a Z on the other side of the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  In sum, the total action of the entire stabilizer  will visually look a choreographed dance where the  nodes act out the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Pauli matrices commute past the H gates on  the output edges, with some of them changing  colour (X ↔ Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Z nodes from the stabilizer simply add their  phase to the nodes in the diagram, while the  X nodes split into one X node for every inter-  nal edge connected to the output node v, and  result in a phase flip on every neighbour of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Once this dance is complete, all the added phases  necessarily cancel and the added global phase re-  turns to 1, since the string is a stabilizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  But note that any single X (or Y represented as  iXZ) sent through an output node v (after it has  already passed through a Hadamard on the output  edge if one exists) may result in some operators (rep-  resented as X or Z nodes) that hit an input node via  an internal edge connected to the input node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' But at  the end of the day, all of the operators on the input  edges must cancel out due to stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' This im-  plies that each input node must be connected to an  even number of nodes on which the stabilizers apply  X or Y gates after passing through the Hadamard  gates on the free output edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Consequently, for any pair of input nodes, we are  free to connect all of the neighbors of one to the other  via (Hadamarded) internal edges without changing  the parity effect of the stabilizers described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  This corresponds to adding two rows in MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Finally, we note that since any pair of internal  edges have Hadamard gates on them, if we have two  edges between the same pair of Z nodes, we can re-  move them both by the rules of ZX calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  In  other words, we are effectively free to perform arbi-  trary row operations on MD mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  This allows us to row-reduce MD to satisfy the  RREF rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Further Examples  We provide some additional examples of our ZX-  CFs, specifically on the most well-known quantum  error-correcting codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' The ZXCFs are obtained by  applying the procedure applied in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' As  with our main-text notation, the blue edges have  Hadamard operations on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Our first example is  the seven-qubit code, due to Steane [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' The code  is represented by an incomplete stabilizer matrix  �  �������  I  I  I  X X X X  I  X X  I  I  X X  X  I  X  I  X  I  X  I  I  I  Z  Z  Z  Z  I  Z  Z  I  I  Z  Z  Z  I  Z  I  Z  I  Z  �  �������  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  (7)  In our ZXCF, this takes the form given in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  The nodes of the code are simply the corners of a  cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' ZXCF of the seven-qubit code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' The 8 nodes in  the diagram form the vertices of a boolean cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Note that the stabilizers of the 7-qubit code are  positioned at exactly the 1-indices in the binary rep-  resentation of a node label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' We can see elegant sym-  metries in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' 3 with the positions of the nodes  CO  5  2  6  49  and their expressions in binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' The reason for why  the basis vectors adjacent to 0 are not 1, 2, and 4,  which are 001, 010, and 100 in binary, is because  the rules of the ZXCF require that the output edges  with Hadamard gates not be connected to lower-  numbered nodes, placing nodes 1, 2, and 4 in the  same column in the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  We next consider the five-qubit code, which is the  smallest code one can achieve for correction of an  arbitrary single-qubit error and has been studied ex-  perimentally [25, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' Its incomplete stabilizer rep-  resentation is given by  �  ���  X Z  Z X  I  I  X Z  Z X  X  I  X Z  Z  Z X  I  X Z  �  ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  (8)  Our procedure produces the ZXCF shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content=' ZXCF of the five-qubit code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  Note that the stabilizer tableau is just 4 cyclic  permutations of the first row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  the ZXCF reflects  this cyclic structure in the output explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
+page_content='  2  3  4  5' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfcABc/content/2301.02356v1.pdf'}
diff --git a/y9AzT4oBgHgl3EQfCvrc/content/tmp_files/2301.00966v1.pdf.txt b/y9AzT4oBgHgl3EQfCvrc/content/tmp_files/2301.00966v1.pdf.txt
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+Chemotaxis of chiral swimmers
+External chemical gradient leads to efficient swimming of chiral swimmers
+Ruma Maity1 and P.S. Burada2, a)
+1)Department of Physics, Indian Institute Science Education and Research Bhopal,
+Bhopal 462066, India
+2)Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur 721302,
+India
+External gradients can strongly influence the collective behavior of microswimmers. In
+this paper, we study the behavior of two hydrodynamically interacting self-propelled chiral
+swimmers, in the low-Reynolds number regime, under the influence of an external linear
+chemical gradient. We use the generalized squirmer model called the chiral squirmer, a
+spherically shaped body with an asymmetric surface slip velocity, to represent the swim-
+mer. We find that the external gradient favors the attraction between the swimmers and,
+in some situations, leads to a bounded state in which the swimmers move in a highly syn-
+chronous manner. Further, due to this cooperative motion, swimmers efficiently reach the
+chemical target compared to the individual swimmers. This study may help to understand
+the collective behavior of chiral swimmers and to design synthetic microswimmers for
+targeted drug delivery.
+a)Electronic mail: psburada@phy.iitkgp.ac.in
+1
+arXiv:2301.00966v1  [physics.bio-ph]  3 Jan 2023
+
+Chemotaxis of chiral swimmers
+I.
+INTRODUCTION
+Microorganisms are tiny, but their collective behavior can influence the climate and human life
+in various ways. For example, massive plankton blooms in the ocean, harmful red tides along the
+coastline, and bioconvection1. Individual microorganisms exhibit various propulsive mechanisms
+depending on their size, shape, etc.2. When microorganisms, e.g., E-Coli, sperm cells, Parame-
+sium, and Volvox, swim in a fluid, the viscous force from the surrounding medium dominates over
+the inertia of the body3. The corresponding Reynolds number is nearly zero for these microswim-
+mers. The hydrodynamic interaction among the microswimmers may lead to long-range rotational
+order4,5. It is due to the existence of dynamical instability in the system. Note that it is possible to
+stabilize the instability by various means, for example, chemical signaling6. Biofilm formation is
+an apt example of collective sensing of chemical stimulus by a bacteria colony7.
+A popular squirmer model8,9, a sphere with an axisymmetric surface slip, is used to understand
+the hydrodynamic behavior of spherical or nearly spherical-shaped microorganisms. Note that the
+squirmer can exhibit translational motion only. However, in general, many microorganisms exhibit
+translational and rotational motion. Resulting in swimming in helical paths and exhibiting chiral
+flows10,11. Notably, the non-axisymmetric flow feature of a swimmer is common in literature12–14.
+Therefore, the generalized squirmer model called the chiral squirmer11,12,15 is more applicable
+for studying the collective behavior of spherical self-propelled bodies. In the chiral squirmer
+model, the tangential slip velocity on the body’s surface considers both the body’s translational
+and rotational degree of freedom. Also, note that a pair of chiral swimmers exhibit a peculiar
+bounded state, in addition to commonly observed states such as convergence and divergence states,
+due to the non-axisymmetric nature of the body as reported in our earlier works11,15. When the
+separation distance between the swimmers is significant compared to the swimmer’s size, the
+swimmer’s flow field affects the other swimmers’ migration velocity and vice versa. However,
+when the swimmers approach each other, the lubrication forces and torques arise and control the
+hydrodynamic behavior of the swimmers15–17.
+In general, microorganisms sense the external stimulus in their vicinity and respond to it by
+moving towards18 or away from it. This directional movement in the presence of external chemical
+stimulus is known as chemotaxis. It plays a vital role in biological pattern formation19, altering
+the viscosity of the medium20,21, generating turbulent flows22, etc. Chemotaxis of sperm cells, E.
+coli, Dictyostelium, Paramecium, Tetrahymena thermophila, Amoeba proteus, etc. is ubiquitous in
+2
+
+Chemotaxis of chiral swimmers
+nature23–25. Artificial swimmers are also able to exhibit chemotaxis under controlled experimental
+conditions. Unlike the swimming strategies of the living microorganisms, different mechanisms
+like the Marangoni effect26, the active diffusion27, the interfacial tension28, and the extraction of
+work from active medium29 prevail for artificial swimmers to respond to the chemical gradient.
+In this article, we study the response of a pair of chiral swimmers to the external chemical gra-
+dient by considering the near- and far-field hydrodynamic interaction between them. The strength
+of the gradient solely controls the response of a single swimmer to the external gradient. How-
+ever, for a pair of swimmers, the hydrodynamic interaction between them plays a crucial role in
+the chemotaxis of the swimmers. The latter plays a constructive role in the success of chemo-
+taxis. The paper is organized as follows. The general chiral squirmer model is briefly discussed
+in section II. In section IV, we analyze the hydrodynamic behavior of two chiral swimmers in the
+presence of a linear chemical gradient. The main conclusions are provided in section VI.
+II.
+HYDRODYNAMICS OF A CHIRAL SQUIRMER
+The low Reynolds number swimmers3,8 obey the Stokes equation30 given by
+η∇2u = ∇p,
+(1)
+where η is the viscosity of the fluid around the body, u is the velocity field, and p is the correspond-
+ing pressure field which plays the role of a Lagrange multiplier to impose the incompressibility
+constraint ∇·u = 0. In the chiral squirmer model, the effective slip velocity11 on the surface of a
+spherical non-deformable body of radius a is prescribed by
+us =
+∞
+∑
+l=1
+l
+∑
+m=−l
+�
+−βlm ∇s
+�
+Pm
+l (cosθ)eimφ�
++γlm ˆr×∇s
+�
+Pm
+l (cosθ)eimφ��
+,
+(2)
+where ∇s = eθ∂/∂θ + eφ(1/sinθ)(∂/∂φ) is the surface gradient operator, ˆr is the unit vector
+in radial direction, Pm
+l (cosθ)eimφ are non-normalized spherical harmonics, where Pm
+l (cosθ) are
+the associated Legendre polynomials of order m and degree l. The complex coefficients βlm and
+γlm are the surface slip velocity mode amplitudes. We introduce the real and imaginary parts of
+these amplitudes as βlm = β r
+lm + imβ i
+lm and γlm = γr
+lm + imγi
+lm with complex conjugates β ∗
+lm =
+(−1)mβl,−m and γ∗
+lm = (−1)mγl,−m, respectively.
+Interestingly, the swimming velocity V and the rotation rate Ω of the swimmer can be obtained
+directly from the surface slip velocity Eq. 2, using the reciprocal theorem, without explicitly calcu-
+lating the corresponding flow- and stress-field (see Ref.31). They can be obtained in the body-fixed
+3
+
+Chemotaxis of chiral swimmers
+n
+b
+t
+Ω
+V
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+Puller
+Pusher
+<latexit sha1_base64="iB0Tu58sIY84cY0sU0HTPKXDkA=">A
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+2V1bX1js7RV3t7Z3duvHBy2dZwqQlsk5rHqhlhTziRtGWY47SaKYhFy2gnHd7nfeaJKs1g+mklCA4GHkWMYGMl3xfYjJTIavhs2q9U3b
+o7A1omXkGqUKDZr3z5g5ikgkpDONa657mJCTKsDCOcTst+qmCyRgPac9SiQXVQTbLPEWnVhmgKFb2SYNm6u+NDAutJyK0k3lGvejl4n
+9eLzXRTZAxmaSGSjI/FKUcmRjlBaABU5QYPrE8VsVkRGWGFibE1lW4K3+OVl0j6ve1f1y4eLauO2qKMEx3ACNfDgGhpwD01oAYEnuE
+V3pzUeXHenY/56IpT7BzBHzifP806kYs=</latexit>(a)
+<latexit sha1_base64="tEAmV2DH7ZzibWtM6Ftrp9yMZs=">AB83icbVDLSgMxFL3js9ZX1aWbYBHqpsyIr2XRjcsK9gGdoWTST
+BuaZIYkI5Shv+HGhSJu/Rl3/o2ZdhbaeiBwOde7skJE860cd1vZ2V1bX1js7RV3t7Z3duvHBy2dZwqQlsk5rHqhlhTziRtGWY47SaKYhFy2gnHd7nfeaJKs1g+mklCA4GHkWMYGMl3xfYjJTIauHZtF+punV3BrRMvIJUoUCzX/nyBzFJBZWGcKx1z3MTE2RYGUY4nZb9VNMEkz
+Ee0p6lEguqg2yWeYpOrTJAUazskwbN1N8bGRZaT0RoJ/OMetHLxf+8XmqimyBjMkNlWR+KEo5MjHKC0ADpigxfGIJorZrIiMsMLE2JrKtgRv8cvLpH1e967qlw8X1cZtUcJjuEauDBNTgHprQAgIJPMrvDmp8+K8Ox/z0RWn2DmCP3A+fwDOwJGM</latexit>(b)
+FIG. 1.
+Chiral squirmer (a) with a prescribed surface slip velocity in the body-fixed reference frame n,
+b, and t. The corresponding velocity and rotation of the swimmer are V and Ω, respectively. The types of
+swimmers are shown in (b). For the puller-type swimmer β r
+20 > 0 and for the pusher-type swimmer β r
+20 < 0
+(see the text for details). β r
+20 = 0 corresponds to the neutral swimmer (not shown).
+reference frame as V = 2(β r
+11, β i
+11, β r
+10)/3 and Ω = (γ r
+11, γ i
+11, γ r
+10)/a11. Without loss of generality,
+the frame n,b, and t can be chosen such that t points in the direction of velocity. Accordingly, we
+have β r
+11 = β i
+11 = 0 and write β r
+10 = 3v/2 such that v = |V| is the speed of the swimmer. Thus, the
+velocity and rotation rate of the chiral swimmer read11,
+V = vt ,
+(3)
+Ω = γ r
+11
+a n+ γ i
+11
+a b+ γ r
+10
+a t.
+(4)
+Also, for simplicity, we choose that the swimmer has the rotation rate in the n−t plane only. With
+this choice, we have γ i
+11 = 0. In addition, we choose the magnitude of the rotation as |Ω| = v/a
+such that the components of the rotation rate can be expressed as γ r
+11/a = (v/a)sinχ ,γ i
+11/a = 0
+and γ r
+10/a = (v/a)cosχ, where χ is the angle between V and Ω. Fig. 1(a) depicts the chiral
+squirmer with non-axisymmetric surface slip (Eq. 2), its swimming direction, and the rotation
+rate. The corresponding equations of motion of the swimmer can be obtained using the force- and
+torque-balance conditions as
+˙q = V, ˙n = Ω×n, ˙b = Ω×b, ˙t = Ω×t,
+(5)
+where q is the swimmer’s position, and the dot represents the derivative with respect to time. For
+V ∥ Ω, we get χ = 0, and the resulting swimming path is a straight line. In this case, the swimmer
+rotates around the axis of motion. For χ = π/2, the chiral swimmer moves in a circular path in a
+plane. For other values of χ, the path of the chiral swimmer is helix (see Refs.11,15).
+Using the boundary conditions, u = us at the surface of the chiral squirmer and u → 0 as r → ∞,
+i.e., in the laboratory frame of reference, the Stokes equation (Eq. 1) can be solved analytically to
+4
+
+Chemotaxis of chiral swimmers
+obtain the corresponding flow- and pressure-field11,15. They read,
+ulf(r) = 3v
+2
+a3
+r3
+�
+P1(t· ˆr) ˆr− t
+3
+�
++3β r
+20
+�a4
+r4 − a2
+r2
+�
+P2(t· ˆr) ˆr
++β r
+20
+a4
+r4 P′
+2 (t· ˆr)[(t· ˆr)ˆr−t]−γ r
+20
+a3
+r3 P′
+2 (t· ˆr)t× ˆr,
+(6)
+plf(r) = −2η β r
+20
+a2
+r3 P2 (t· ˆr) ,
+(7)
+where t is the swimming direction, r is the distance from the center of the swimmer where the
+flow field is determined, ˆr = r/r is the unit radial vector, P2(x) denotes a second-order Legendre
+polynomial, and P′
+2 = dP2/dx with x = t · ˆr = cosθ. The flow field in the body-fixed frame (bf)
+can be obtained from that in the lab frame (lf) as ubf(r) = ulf(r)−V−Ω×r. Note that in Eq. (6),
+we have only written terms up to l = 2 and do not consider higher-order contributions since they
+decay more rapidly with r. Besides, to have a minimal model, we have ignored l = 2 modes with
+m ̸= 0. However, it is straightforward to include the additional terms in the analysis. The sign of
+β r
+20 decides the nature of the swimming. For example, β r
+20 > 0 corresponds to puller-type swimmer
+and β r
+20 < 0 corresponds to pusher-type swimmer (see Fig. 1(b)). While pullers have an extensile
+force dipole, resulting, e.g., from the front part of the body, pushers have a contractile force dipole
+stemming, e.g., from the rear part of the body2.
+III.
+HYDRODYNAMIC INTERACTION IN THE PRESENCE OF LUBRICATION
+FORCES
+At low Reynolds numbers, microswimmers strongly influence their fluidic environment, as
+do their nearby swimmers. When the swimmers approach each other, the lubrication forces and
+torques arise, which play a vital role in the hydrodynamic behavior of the swimmers15,16,32. It has
+been reported recently that a pair of chiral squirmers portray various behaviors, e.g., divergence,
+convergence, and even a bounded state11 as a result of their mutual hydrodynamic interaction.
+However, considering the lubrication forces and torques (as in the dense suspension of swimmers),
+the swimmers do not exhibit convergence behavior15. In this work, we include the lubrication
+effects. The lubrication force acting on a swimmer can be calculated by knowing the velocity field
+of the nearby swimmer16,32. Due to this lubrication force, the corresponding velocity contribution
+to a chiral swimmer from the other swimmer, and vice versa, is15
+U = −2a2Bε lnε ,
+(8)
+5
+
+Chemotaxis of chiral swimmers
+where ε is half of the separation distance between the swimmers, B = β r(1)
+10 t13 −β r(2)
+10 t23, β r(1)
+10
+and
+β r(2)
+10
+are l = 1 and m = 0 modes (see the text after Eq. 2) of swimmer one and two, respectively,
+t13 = t1·eZ, t23 = t2·eZ, eZ is the unit vector along the Z direction, and t1 and t2 are the correspond-
+ing orientations of the swimmers (see Ref.15 for more details). Note that the lubrication torques
+are of the order O(ε1/2), and which can be neglected in the limit ε → 0.
+The equations of motion of a swimmer in the presence of another swimmer, by considering
+lubrication forces into account, read as
+˙qi = Vi +Ulub
+i
++
+2
+∑
+j=1;i̸= j
+uj(qi j,nj,bj,tj)
+�
+����
+˙ni
+˙bi
+˙ti
+�
+���� =
+�
+Ωi +
+2
+∑
+j=1;i̸=j
+ω j(qi j,nj,bj,tj)
+�
+×
+�
+����
+ni
+bi
+ti
+�
+���� ,
+(9)
+where Ulub
+i
+= U(cosθ ′ti −sinθ ′ni) is the additional velocity contribution arising due to the other
+swimmer in the lubrication region, θ ′ = cos−1(ti ·eZ), and the vorticity field ω = (∇×u)/2. Note
+that Ulub
+i
+= 0 for R > 2(a+ε), and Ui +
+2
+∑
+j=1;i̸= j
+uj =
+2
+∑
+j=1;i̸=j
+ω j = 0 for R ≤ 2(a+ε), where ε ≪ a
+and R = |qi j| is the radial distance between the swimmers.
+IV.
+INFLUENCE OF EXTERNAL LINEAR CHEMICAL GRADIENT
+Consider that the two swimmers are subjected to an external chemical concentration field c(q);
+see the schematic diagram Fig. 2. As the swimmers sense the gradient, they move from a lower-
+to a higher-concentration region. Consequently, the local concentration field that the swimmers
+experience changes with time. The stimulus level (s = c(q(t))) on the swimmer’s body is thus a
+function of time. A swimmer’s response to the stimulus is related to the relative change in the
+chemoattractant concentration (∆c/c) in the vicinity33. The binding of the chemoattractant to the
+body’s receptors, known as activation, and the unbinding of that, known as deactivation, triggers
+the internal chemotactic signaling system of the body. In our model systems, it results in the
+modification of the slip coefficients (Eq. 2) of the swimmer, which leads change in the velocity
+and rotation rate of the swimmer34–37.
+To study the chemotaxis of two swimmers, we use the Barkai–Leibler model34,38,39 for the
+adaptation and relaxation mechanism of the swimmer to the external chemical gradient. Note that
+6
+
+Chemotaxis of chiral swimmers
+x
+y
+z
+~V2
+~⌦2
+~V1
+~⌦1
+<latexit sha1_base64="wb5u7cXp/b4L1vn0rD9RfT60=">AB6HicdVDJSgNBEK2JW4xb1KOXxiB4G
+mbE7Rj04jERs0AyhJ5OTdKmZ6G7RwhDvsCLB0W8+kne/Bt7kgiuD5p+vFdFVT0/EVxpx3m3CguLS8srxdXS2vrG5lZ5e6ep4lQybLBYxLtU4WCR9jQXAtsJxJp6Ats+aPL3G/doVQ8jm70OEvpIOIB5xRbaT6da9cexjJwf
+5TVx7+jsVmKPWK791+zFLQ4w0E1Spjusk2suo1JwJnJS6qcKEshEdYMfQiIaovGy6IQcGKVPgliaF2kyVb92ZDRUahz6pjKkeqh+ern4l9dJdXDuZTxKUo0Rmw0KUkF0TPKrSZ9LZFqMDaFMcrMrYUMqKdMm5IJ4fNS8j9p
+HtnuqX1SP65UL+ZxFGEP9uEQXDiDKlxBDRrAOEeHuHJurUerGfrZVZasOY9u/AN1usHwV2M7A=</latexit>R
+FIG. 2.
+Schematic diagram of two hydrodynamically interacting chiral swimmers in the presence of a
+linear chemical gradient applied only along the x direction. The separation distance between the swimmers
+is R.
+this model is for information processing and memory via the internal biochemical network, and it
+has been used widely for bacterial and sperm chemotaxis34,35. The ability of the system to adapt
+and respond to the external stimulus can be captured by a simple dynamical system,
+σ ˙ab = pb(sb +s)−ab ,
+(10a)
+µ ˙pb = pb(1−ab),
+(10b)
+where σ is the relaxation time, µ is the adaptation time, ab(t) is the dimensionless output variable,
+pb(t) is the dynamic sensitivity related to adaptation, and sb(t) arises from the background activity
+of receptors in the absence of the chemical stimulus. The chemoattractant has a dimension of
+concentration. Note that these equations are valid for weak concentration gradients only. Under a
+constant stimulus s(t) = Sc, the system reaches a steady state for which ab = 1 and pb = 1/(sb +
+Sc). Therefore, the dynamic sensitivity (pb) maintains an inverse relation with sb and s. With
+increasing stimulus level, pb decreases. The system here is totally adaptive as ab(t) is independent
+of Sc.
+In the presence of an external stimulus, the slip coefficients (Eq. 2) of the swimmer are modified
+as,
+X = X(0) +X(1) (ab(t)−1) ,
+(11)
+where X = {βlm,γlm}, the unperturbed slip coefficients are X(0) =
+�
+β (0)
+lm ,γ(0)
+lm
+�
+, and the pertur-
+bation due to the external gradient is X(1) =
+�
+β (1)
+lm ,γ(1)
+lm
+�
+. Here, we consider a linear chemical
+gradient given by34,
+c(q) = c0 +c1 ·q,
+(12)
+7
+
+Chemotaxis of chiral swimmers
+where c1 = ∇c(q) = c1i is the chemical gradient applied in the x direction and q = ix+jy+kz is
+the position vector. The chemotactic stimulus is S(t) = c(q(t))34.
+Using Eq. (9), we numerically calculate the trajectories of a pair of chiral swimmers, which
+are hydrodynamically interacting, in the presence of a linear chemical gradient. Note that the
+velocity and the rotation rate of the chiral squirmer are functions of the slip coefficients, which are
+affected by the chemical gradient; see Eq. 11. As a result, the hydrodynamic interaction between
+the swimmers is affected. In the present work, we consider chiral swimmers having velocities of
+equal magnitudes, i.e., |V1| = |V2| = v. However, the rotation rates of the swimmers are in general
+different and read, Ω1 = v(sinχ1,0,cosχ1)/a for swimmer one and Ω2 = v(sinχ2,0,cosχ2)/a for
+swimmer two. Chemical and hydrodynamic perturbations in the velocity and rotation rate of the
+swimmers change the corresponding torsion and curvature of the swimmers’ helical trajectories.
+Hence their movements are altered.
+As discussed, a swimmer’s flow field influences the neighboring swimmer’s motion. The hy-
+drodynamic flow field of a swimmer (Eq. (6)) comprises l = 1,2 modes as leading order terms.
+The l = 2 mode is important as it plays a vital role in determining the hydrodynamic interaction
+between the swimmers. The slip coefficients corresponding to l = 2 mode are assumed to have the
+following form: 3β r
+20 = 3γ r
+20 = λ1 for squirmer one, and 3β r
+20 = 3γ r
+20 = λ2 for squirmer two. The
+former assumptions are made to minimize the parameters of the problem. Note that for λ1 ̸= λ2,
+the flow fields of the swimmers are different. Thus, variation in χi (angle between Vi and Ωi)
+and ±λi (i = 1,2) (the strength of the flow field), quantitatively measure the nature of the interac-
+tion between the chiral squirmers and gives rise to several interesting swimming characteristics.
+Notably, different initial configurations also influence the interaction between the swimmers. We
+found that, out of various possible initial configurations for the swimmers, swimmers interact with
+each other for an extended period only in the planar configuration11 as is the case in the present
+study. Here, at time t = 0, both the swimmers are on the xy plane, with an initial orientation along
+the z−direction and an initial separation distance R0. Our previous study (see Ref.11) explored the
+interaction between the two chiral swimmers with the former configuration, which exhibited some
+new behavior (bounded state) in addition to the ones (convergence and divergence state) displayed
+by axisymmetric swimmers. We mainly focus on how the external chemical gradient influences
+these observed behaviors in the present work.
+8
+
+Chemotaxis of chiral swimmers
+0
+2
+4
+λ/v
+0
+π
+6
+π
+3
+χ
+(a) Pusher - Pusher
+0
+2
+4
+λ/v
+0
+π
+6
+π
+3
+χ
+(b) Puller - Puller
+0
+2
+4
+λ/v
+0
+π
+6
+π
+3
+χ
+(c) Pusher - Puller
+0
+2
+4
+λ/v
+0
+π
+6
+π
+3
+χ
+(d) Puller - Pusher
+FIG. 3. (a)-(d) Depicting swimming states of hydrodynamically interacting chiral squirmers performing
+chemotaxis, where the strength of the chemical gradient c1 = 0.1. The symbols imply the following states.
+Square: bounded state (BS), closed circles: monotonic divergence state (MD), open circles: divergence
+state (D), plus: locked state, and cross symbols: parallel motion. The two swimmers initially start at
+q1 = (9,9,0)a and q2 = (3,3,0)a. The corresponding initial velocities and rotation rates of the swimmers
+are V1 = V2 = v(0,0,1) and Ω1 = Ω2 = v(cosχ,0,sinχ)/a, respectively.
+A.
+State diagram in the χ - λ plane
+Fig. 3 depicts the hydrodynamic behavior of two chiral swimmers in the presence of an external
+chemical gradient. When λ1 = λ2 = λ and χ1 = χ2 = χ, the swimmers are identical (see the Fig. 3
+caption). The swimmers portray various behaviors for varying λ/v and χ. As reported earlier15,
+in the absence of a chemical gradient, two chiral swimmers exhibit divergence, monotonic di-
+vergence, and bounded states (see Fig. 4) only due to the lubrication effects. These observed
+states arise due to pure hydrodynamic interaction between the swimmers. However, the chemical
+gradient converts some monotonic divergence states into divergence states and some divergence
+states into bounded states (see Fig. 4). Indeed, the chemical stimulus favors the attraction between
+the swimmers. The density of bounded states is more for the odd combination of swimmers like
+pusher-puller or vice-versa (see fig. 3(c) and (d)). For other combinations, swimmers rarely exhibit
+bounded states (see fig. 3(a) and (b)). The occurrence of bounded motion in even combinations of
+swimmers is due to the presence of the chemical gradient. Note that in the case of axisymmetric
+swimmers (without chirality), the hydrodynamic behavior of pusher-puller and puller-pusher-type
+swimmers is the same. However, in the case of chiral swimmers, the hydrodynamic behavior of
+different swimmers exhibit different behaviors11. It is because the flow pattern of a puller-type
+chiral swimmer is different from a pusher-type chiral swimmer11.
+Note that for χ = 0 and λ ̸= 0, swimmers move in a straight line in the z direction, attract each
+9
+
+Chemotaxis of chiral swimmers
+other, and converge to a locked state15. In this state, the swimmers approach a minimum distance
+and stay together. For χ = 0 (straight line motion) and χ = π/2 (circular motion), the neutral
+swimmers, i.e., λ = 0, move parallel to each other without influencing each other11. Not only the
+chemical gradient but the initial separation distance R0 (at t = 0) between the swimmers also plays
+a vital role in the hydrodynamic interaction between the swimmers. Interestingly, as R0 increases,
+the bounded state is more favorable. It is reported in detail in our earlier studies (see Refs.11,15).
+We also study the interaction between non-identical swimmers, i.e., λ1 ̸= λ2 and χ1 ̸= χ2 (see
+Appendix.A). For a fixed χ = π/3 and varying flowfield strengths (λ1,λ2), we observe mainly
+divergence states and a few bounded states for the different combinations of the swimmers (see
+fig. 6(a)). Identical or nearly identical swimmers mostly exhibit bounded motion primarily for
+the puller-pusher combination (see fig. 6(a), top right panel). However, other combinations also
+exhibit bounded motion when λ1 = λ2 = 0 (see fig. 6(a)). The former implies that non-identical
+field strength is unfavorable for the bounded motion. On the other hand, we can fix the flow-
+field strengths at λ = v and vary relative orientations of the swimmers (χ1,χ2) (see fig. 6(b)).
+Non-identical orientations are associated with divergence and monotonic divergence states. See
+Appendix A for a detailed discussion.
+Fig. 4 depicts the trajectories and the radial separation distance between the swimmers in the
+presence and absence of the chemical gradient. As mentioned before, the simultaneous effect of
+the hydrodynamic interaction and the chemical stimulus converts some of the divergence states
+into bounded states (see figs. 4 (b),(e)). Here, the stimulus increases the attraction between the
+swimmers. As a result, for the bounded state, the swimmers tend to align in the direction of the
+chemical gradient. The oscillation in the separation distance (R) between the swimmers decreases
+(see figs. 4(b),(c),(e),(f)) when compared to the stimulus-free situation. Whereas for the monotonic
+divergence state, the swimmers diverge at a slower rate than the stimulus-free case (see figs. 4
+(a),(d)). As reported earlier, in the case of a single chiral swimmer, the body completely aligns its
+helix axis to the direction of the gradient36. In the two-swimmer case, none of the swimmers can
+make their helix axis parallel to the direction of the gradient because of the hydrodynamic flow
+field generated by the neighboring swimmer. In particular, it is due to the dominating term in the
+hydrodynamic flow field o(1/r2) and the weak chemical gradient we apply.
+10
+
+Chemotaxis of chiral swimmers
+ 0
+ 100
+ 200
+ 0
+ 10
+ 20
+ 0
+ 50
+ 100
+ 0
+ 50
+ 100  150
+ 0
+ 10
+ 20
+ 0
+ 100
+ 200
+ 0
+ 200  400  600
+ 0
+ 10
+ 20
+ 0
+ 100
+ 200
+ 300
+x
+a
+y
+a
+z
+a
+(a)
+x
+a
+y
+a
+z
+a
+(b)
+x
+a
+y
+a
+z
+a
+(c)
+ 0
+ 10
+ 20
+ 30
+ 0
+ 200
+ 400
+c1 : 0.00
+ : 0.01
+ : 0.10
+ 0
+ 2.5
+ 5
+ 7.5
+ 0
+ 100
+ 200
+ 300
+ 0
+ 5
+ 10
+ 0
+ 500
+ 1000
+ 1500
+tv/a
+R
+a
+(d)
+tv/a
+R
+a
+(e)
+tv/a
+R
+a
+(f)
+FIG. 4.
+(a)-(c) Show the trajectories of two swimmers in the presence and in the absence of a linear
+chemical gradient for monotonic divergence (MD), divergence (D), and bounded states (BS). Here, the
+strength of the linear gradient c1 = 0.05. The other parameters are, χ = π/6,λ1 = −2, and λ2 = 2 for
+D state, χ = π/3,λ1 = −3,λ2 = 3 for MD state, and χ = π/3,λ1 = 1,λ2 = −1 for BS state. (d)-(f) The
+corresponding radial separation distance R between the swimmers as a function of time. Here, lengths are
+scaled by the radius of the swimmer a and time scaled by τ = v/a.
+V.
+DEPENDENCE OF RELATIVE SUCCESS TIME ON CHEMICAL GRADIENT
+A question may arise whether a pair of swimmers or a single swimmer is efficient in swimming
+in the direction of the external chemical gradient. To answer this, we measure the relative success
+time defined as ∆τ = (t0 −tc)/t0. Here, t0 is the time a pair of swimmers or a single swimmer
+takes to reach a particular reference point on the x axis in the absence of the chemical gradient,
+and tc is the same in the presence of the chemical gradient. Note that ∆τ increases with c1 and
+saturates for higher c1. The latter is due to the saturation of the internal signaling network of the
+bodies39. Henceforth, the network cannot modify its translational and rotational velocities further.
+We find pair swimming is more efficient than individual swimming (see Fig. 5). It is because
+the combined effect of the hydrodynamic interaction between the swimmers and chemical per-
+turbation increases the motility of the swimmers. Interestingly, the bounded swimmers are more
+11
+
+Chemotaxis of chiral swimmers
+0
+0.1
+0.2
+0.3
+0.4
+∆τ
+10−5
+10−4
+10−3
+10−2
+10−1
+c1
+χ = π
+3 , λ1 = 2, λ2 = −2 (BS)
+χ = 7π
+24 , λ1 = −2, λ2 = 2 (BS)
+χ = π
+4 , λ1 = −2, λ2 = 2 (BS)
+χ = π
+3 (Single)
+(a)
+0
+0.1
+0.2
+0.3
+∆τ
+10−5
+10−4
+10−3
+10−2
+10−1
+c1
+χ = π
+4 , λ1 = −2, λ2 = −2 (D)
+χ = π
+6 , λ1 = −2, λ2 = −2 (D)
+χ = π
+8 , λ1 = −2, λ2 = −2 (D)
+χ = π
+3 (Single)
+(b)
+FIG. 5.
+The relative success time ∆τ = (t0 −tc)/t0 as a function of the chemical gradient strength c1 for
+various observed states, (a) for bounded states and (b) for divergence states. The diamond symbols for the
+single swimmer. Here, lines connect the data points.
+efficient than others (see Fig. 5(a)), especially for lower values of χ. However, the influence of χ
+on chemotactic success rate is minimal for diverging motion (see Fig. 5(b)). Note that, the range
+of c1 is from o(10−6) to o(10−1). For weak gradients, divergence motion persists. With increas-
+ing strength of c1, as mentioned earlier, some divergence states get converted into bounded states.
+Note that for values of χ ∼ π/2, swimmers effectively move in circular paths, and correspondingly
+the success rate is low (not shown).
+VI.
+CONCLUSIONS
+In this work, we have studied the combined behavior of two chiral swimmers by considering
+the lubrication effect in the presence of a linear chemical gradient. We have used a generalized
+squirmer model called the chiral squirmer. In general, a pair of chiral swimmers can exhibit
+bounded, divergence, convergence, and monotonic convergence states without a chemical gradient.
+However, only bounded, divergence, and monotonic divergence states are observed in the current
+study. The convergence and monotonic convergence states are absent because of the lubrication
+force between the swimmers. Due to this lubrication force, swimmers repel each other when they
+approach each other.
+Interestingly, a chemical gradient converts some divergence- and monotonic-divergence states
+into bounded states. The chemical gradient favors the attraction between the swimmers. It leads
+to efficient swimming. We have presented detailed state diagrams depicting the hydrodynamic
+12
+
+Chemotaxis of chiral swimmers
+behavior of two chiral swimmers in the presence of a chemical gradient. We have computed
+the relative success time of a pair of swimmers in aligning their movement toward the chemical
+gradient, whose strength is controlled by the parameter c1. We find that pair swimming is more ef-
+ficient than individual swimming as the combined effect of the hydrodynamic interaction between
+the two swimmers and chemical perturbation increases the motility of the swimmers. Interestingly,
+the bounded swimmers are more efficient than others.
+This study is not limited to chemotaxis but can be generalized to various external gradients, e.g.,
+phototaxis and gyrotaxis. For these cases, one needs to identify how the slip coefficients of the
+chiral swimmer can be modified according to the external gradients. Also, the present study helps
+design artificial micro locomotors which can be used for drug delivery and targeted applications.
+VII.
+ACKNOWLEDGEMENTS
+The authors acknowledge the financial support of the Indian Institute of Technology Kharagpur,
+India.
+VIII.
+DATA AVAILABILITY
+The data supporting this study’s findings are available from the corresponding author upon
+reasonable request.
+Appendix A: State diagrams of λ1 - λ2 and χ1 - χ2
+Fig. 6(a) depicts the λ1−λ2 state diagram of two chiral swimmers, in the presence of a chemical
+gradient, for fixed χ1 and χ2. The additive contributions of hydrodynamic interaction and chem-
+ical perturbations drive the swimmers away from each other. Only the swimmers with identical
+or nearly identical field strengths (|λ1| = |λ2|) exhibit bounded motion (see Fig. 6(a)). Moreover,
+identical combinations of swimmers, i.e., puller-puller or pusher-pusher, mostly display a diver-
+gence state with a rare occurrence of bounded states. The probability of observing a bounded
+state is more if the first swimmer is a puller and the second swimmer is a pusher. However, the
+opposite combination does not work while |λ1| ̸= 0,|λ2| ̸= 0. The reason is the asymmetry in the
+swimmer’s flow field about the orientation of the swimmer. At different positions around a chiral
+swimmer, the neighboring swimmer experiences different hydrodynamic forces. As a result, the
+13
+
+Chemotaxis of chiral swimmers
+−4
+−2
+λ2/v
+−4
+−2
+λ1/v
+Pusher - Pusher
+0
+2
+4 λ1/v
+0
+2
+4
+λ2/v
+Puller - Puller
+0
+2
+4
+λ2/v
+−4
+−2
+λ1/v
+Pusher - Puller
+0
+2
+4 λ1/v
+−4
+−2
+0
+λ2/v
+Puller - Pusher
+0
+π
+6
+π
+3
+χ2
+0
+π
+6
+π
+3
+χ1
+Pusher - Pusher
+0
+π
+6
+π
+3
+χ1
+0
+π
+6
+π
+3
+χ2
+Puller - Puller
+0
+π
+6
+π
+3
+χ2
+0
+π
+6
+π
+3
+χ1
+Pusher - Puller
+0
+π
+6
+π
+3
+χ1
+0
+π
+6
+π
+3
+χ2
+Puller - Pusher
+(a)
+(b)
+FIG. 6. State diagrams corresponding to two hydrodynamically interacting chiral swimmers in the presence
+of chemotaxis. (a) For varying field-strengths (λ1,λ2) and (b) for (χ1,χ2). See the main text for details.
+The initial conditions and symbol descriptions are the same as in figure 3. We set χ1 = χ2 = π/3 for the
+left panel and |λ1| = |λ2| = |V| = v for the right panel.
+interaction between the swimmers is different for their different relative positions (see Ref.11 for
+more details).
+Fig. 6(b) shows the χ1−χ2 state diagram of two chiral swimmers, in the presence of a chemical
+gradient, for fixed λ1 and λ2. The swimmers do not exhibit bounded motion unless χ1 = χ2.
+As mentioned before, similar combinations of the swimmers, i.e., puller-puller or pusher-pusher,
+exhibit bounded states rarely. However, the opposite combinations, i.e., puller-pusher or pusher-
+puller, do exhibit more bounded states (see Fig. 6 (b)). In the absence of a chemical gradient, more
+monotonic divergence states are observed (see Ref.11). However, the presence of the chemical field
+increases the attraction between the swimmers and converts some of the monotonic divergence
+(MD) states into divergence (D) states (see Fig. 6(b)).
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+16
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf,len=579
+page_content='Chemotaxis of chiral swimmers  External chemical gradient leads to efficient swimming of chiral swimmers  Ruma Maity1 and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Burada2, a)  1)Department of Physics, Indian Institute Science Education and Research Bhopal,  Bhopal 462066, India  2)Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur 721302,  India  External gradients can strongly influence the collective behavior of microswimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' In  this paper, we study the behavior of two hydrodynamically interacting self-propelled chiral  swimmers, in the low-Reynolds number regime, under the influence of an external linear  chemical gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' We use the generalized squirmer model called the chiral squirmer, a  spherically shaped body with an asymmetric surface slip velocity, to represent the swim-  mer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' We find that the external gradient favors the attraction between the swimmers and,  in some situations, leads to a bounded state in which the swimmers move in a highly syn-  chronous manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Further, due to this cooperative motion, swimmers efficiently reach the  chemical target compared to the individual swimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' This study may help to understand  the collective behavior of chiral swimmers and to design synthetic microswimmers for  targeted drug delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  a)Electronic mail: psburada@phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='iitkgp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='in  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='00966v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='bio-ph]  3 Jan 2023  Chemotaxis of chiral swimmers  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  INTRODUCTION  Microorganisms are tiny, but their collective behavior can influence the climate and human life  in various ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' For example, massive plankton blooms in the ocean, harmful red tides along the  coastline, and bioconvection1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Individual microorganisms exhibit various propulsive mechanisms  depending on their size, shape, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' When microorganisms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=', E-Coli, sperm cells, Parame-  sium, and Volvox, swim in a fluid, the viscous force from the surrounding medium dominates over  the inertia of the body3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The corresponding Reynolds number is nearly zero for these microswim-  mers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The hydrodynamic interaction among the microswimmers may lead to long-range rotational  order4,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' It is due to the existence of dynamical instability in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Note that it is possible to  stabilize the instability by various means, for example, chemical signaling6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Biofilm formation is  an apt example of collective sensing of chemical stimulus by a bacteria colony7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  A popular squirmer model8,9, a sphere with an axisymmetric surface slip, is used to understand  the hydrodynamic behavior of spherical or nearly spherical-shaped microorganisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Note that the  squirmer can exhibit translational motion only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' However, in general, many microorganisms exhibit  translational and rotational motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Resulting in swimming in helical paths and exhibiting chiral  flows10,11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Notably, the non-axisymmetric flow feature of a swimmer is common in literature12–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  Therefore, the generalized squirmer model called the chiral squirmer11,12,15 is more applicable  for studying the collective behavior of spherical self-propelled bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' In the chiral squirmer  model, the tangential slip velocity on the body’s surface considers both the body’s translational  and rotational degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Also, note that a pair of chiral swimmers exhibit a peculiar  bounded state, in addition to commonly observed states such as convergence and divergence states,  due to the non-axisymmetric nature of the body as reported in our earlier works11,15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' When the  separation distance between the swimmers is significant compared to the swimmer’s size, the  swimmer’s flow field affects the other swimmers’ migration velocity and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' However,  when the swimmers approach each other, the lubrication forces and torques arise and control the  hydrodynamic behavior of the swimmers15–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  In general, microorganisms sense the external stimulus in their vicinity and respond to it by  moving towards18 or away from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' This directional movement in the presence of external chemical  stimulus is known as chemotaxis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' It plays a vital role in biological pattern formation19, altering  the viscosity of the medium20,21, generating turbulent flows22, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Chemotaxis of sperm cells, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  coli, Dictyostelium, Paramecium, Tetrahymena thermophila, Amoeba proteus, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' is ubiquitous in  2  Chemotaxis of chiral swimmers  nature23–25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Artificial swimmers are also able to exhibit chemotaxis under controlled experimental  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Unlike the swimming strategies of the living microorganisms, different mechanisms  like the Marangoni effect26, the active diffusion27, the interfacial tension28, and the extraction of  work from active medium29 prevail for artificial swimmers to respond to the chemical gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  In this article, we study the response of a pair of chiral swimmers to the external chemical gra-  dient by considering the near- and far-field hydrodynamic interaction between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The strength  of the gradient solely controls the response of a single swimmer to the external gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' How-  ever, for a pair of swimmers, the hydrodynamic interaction between them plays a crucial role in  the chemotaxis of the swimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The latter plays a constructive role in the success of chemo-  taxis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The general chiral squirmer model is briefly discussed  in section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' In section IV, we analyze the hydrodynamic behavior of two chiral swimmers in the  presence of a linear chemical gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The main conclusions are provided in section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  HYDRODYNAMICS OF A CHIRAL SQUIRMER  The low Reynolds number swimmers3,8 obey the Stokes equation30 given by  η∇2u = ∇p,  (1)  where η is the viscosity of the fluid around the body, u is the velocity field, and p is the correspond-  ing pressure field which plays the role of a Lagrange multiplier to impose the incompressibility  constraint ∇·u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' In the chiral squirmer model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' the effective slip velocity11 on the surface of a  spherical non-deformable body of radius a is prescribed by  us =  ∞  ∑  l=1  l  ∑  m=−l  �  −βlm ∇s  �  Pm  l (cosθ)eimφ�  +γlm ˆr×∇s  �  Pm  l (cosθ)eimφ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  (2)  where ∇s = eθ∂/∂θ + eφ(1/sinθ)(∂/∂φ) is the surface gradient operator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' ˆr is the unit vector  in radial direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Pm  l (cosθ)eimφ are non-normalized spherical harmonics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' where Pm  l (cosθ) are  the associated Legendre polynomials of order m and degree l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The complex coefficients βlm and  γlm are the surface slip velocity mode amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' We introduce the real and imaginary parts of  these amplitudes as βlm = β r  lm + imβ i  lm and γlm = γr  lm + imγi  lm with complex conjugates β ∗  lm =  (−1)mβl,−m and γ∗  lm = (−1)mγl,−m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  Interestingly, the swimming velocity V and the rotation rate Ω of the swimmer can be obtained  directly from the surface slip velocity Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 2, using the reciprocal theorem, without explicitly calcu-  lating the corresponding flow- and stress-field (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' They can be obtained in the body-fixed  3  Chemotaxis of chiral swimmers  n  b  t  Ω  V  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='Puller  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='Pusher  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='<latexit sha1_base64="iB0Tu58sIY84cY0sU0HTPKXDkA=">A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='AB83icbVDLSgMxFL3js9ZX1aWbYBHqpsyIr2XRjcsK9gGdoWTSTBuaZIYkI5Shv+HGhSJu/Rl3/o2ZdhbaeiBwOde7skJE860cd1vZ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='2V1bX1js7RV3t7Z3duvHBy2dZwqQlsk5rHqhlhTziRtGWY47SaKYhFy2gnHd7nfeaJKs1g+mklCA4GHkWMYGMl3xfYjJTIavhs2q9U3b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='o7A1omXkGqUKDZr3z5g5ikgkpDONa657mJCTKsDCOcTst+qmCyRgPac9SiQXVQTbLPEWnVhmgKFb2SYNm6u+NDAutJyK0k3lGvejl4n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='9eLzXRTZAxmaSGSjI/FKUcmRjlBaABU5QYPrE8VsVkRGWGFibE1lW4K3+OVl0j6ve1f1y4eLauO2qKMEx3ACNfDgGhpwD01oAYEnuE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='V3pzUeXHenY/56IpT7BzBHzifP806kYs=</latexit>(a)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='<latexit sha1_base64="tEAmV2DH7ZzibWtM6Ftrp9yMZs=">AB83icbVDLSgMxFL3js9ZX1aWbYBHqpsyIr2XRjcsK9gGdoWTST  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='BuaZIYkI5Shv+HGhSJu/Rl3/o2ZdhbaeiBwOde7skJE860cd1vZ2V1bX1js7RV3t7Z3duvHBy2dZwqQlsk5rHqhlhTziRtGWY47SaKYhFy2gnHd7nfeaJKs1g+mklCA4GHkWMYGMl3xfYjJTIauHZtF+punV3BrRMvIJUoUCzX/nyBzFJBZWGcKx1z3MTE2RYGUY4nZb9VNMEkz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='Ee0p6lEguqg2yWeYpOrTJAUazskwbN1N8bGRZaT0RoJ/OMetHLxf+8XmqimyBjMkNlWR+KEo5MjHKC0ADpigxfGIJorZrIiMsMLE2JrKtgRv8cvLpH1e967qlw8X1cZtUcJjuEauDBNTgHprQAgIJPMrvDmp8+K8Ox/z0RWn2DmCP3A+fwDOwJGM</latexit>(b)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  Chiral squirmer (a) with a prescribed surface slip velocity in the body-fixed reference frame n,  b, and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The corresponding velocity and rotation of the swimmer are V and Ω, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The types of  swimmers are shown in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' For the puller-type swimmer β r  20 > 0 and for the pusher-type swimmer β r  20 < 0  (see the text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' β r  20 = 0 corresponds to the neutral swimmer (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  reference frame as V = 2(β r  11, β i  11, β r  10)/3 and Ω = (γ r  11, γ i  11, γ r  10)/a11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Without loss of generality,  the frame n,b, and t can be chosen such that t points in the direction of velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Accordingly, we  have β r  11 = β i  11 = 0 and write β r  10 = 3v/2 such that v = |V| is the speed of the swimmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Thus, the  velocity and rotation rate of the chiral swimmer read11,  V = vt ,  (3)  Ω = γ r  11  a n+ γ i  11  a b+ γ r  10  a t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  (4)  Also, for simplicity, we choose that the swimmer has the rotation rate in the n−t plane only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' With  this choice, we have γ i  11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' In addition, we choose the magnitude of the rotation as |Ω| = v/a  such that the components of the rotation rate can be expressed as γ r  11/a = (v/a)sinχ ,γ i  11/a = 0  and γ r  10/a = (v/a)cosχ, where χ is the angle between V and Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 1(a) depicts the chiral  squirmer with non-axisymmetric surface slip (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 2), its swimming direction, and the rotation  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The corresponding equations of motion of the swimmer can be obtained using the force- and  torque-balance conditions as  ˙q = V, ˙n = Ω×n, ˙b = Ω×b, ˙t = Ω×t,  (5)  where q is the swimmer’s position, and the dot represents the derivative with respect to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' For  V ∥ Ω, we get χ = 0, and the resulting swimming path is a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' In this case, the swimmer  rotates around the axis of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' For χ = π/2, the chiral swimmer moves in a circular path in a  plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' For other values of χ, the path of the chiral swimmer is helix (see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='11,15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  Using the boundary conditions, u = us at the surface of the chiral squirmer and u → 0 as r → ∞,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=', in the laboratory frame of reference, the Stokes equation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 1) can be solved analytically to  4  Chemotaxis of chiral swimmers  obtain the corresponding flow- and pressure-field11,15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' They read,  ulf(r) = 3v  2  a3  r3  �  P1(t· ˆr) ˆr− t  3  �  +3β r  20  �a4  r4 − a2  r2  �  P2(t· ˆr) ˆr  +β r  20  a4  r4 P′  2 (t· ˆr)[(t· ˆr)ˆr−t]−γ r  20  a3  r3 P′  2 (t· ˆr)t× ˆr,  (6)  plf(r) = −2η β r  20  a2  r3 P2 (t· ˆr) ,  (7)  where t is the swimming direction, r is the distance from the center of the swimmer where the  flow field is determined, ˆr = r/r is the unit radial vector, P2(x) denotes a second-order Legendre  polynomial, and P′  2 = dP2/dx with x = t · ˆr = cosθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The flow field in the body-fixed frame (bf)  can be obtained from that in the lab frame (lf) as ubf(r) = ulf(r)−V−Ω×r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Note that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' (6),  we have only written terms up to l = 2 and do not consider higher-order contributions since they  decay more rapidly with r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Besides, to have a minimal model, we have ignored l = 2 modes with  m ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' However, it is straightforward to include the additional terms in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The sign of  β r  20 decides the nature of the swimming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' For example, β r  20 > 0 corresponds to puller-type swimmer  and β r  20 < 0 corresponds to pusher-type swimmer (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' While pullers have an extensile  force dipole, resulting, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=', from the front part of the body, pushers have a contractile force dipole  stemming, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=', from the rear part of the body2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  HYDRODYNAMIC INTERACTION IN THE PRESENCE OF LUBRICATION  FORCES  At low Reynolds numbers, microswimmers strongly influence their fluidic environment, as  do their nearby swimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' When the swimmers approach each other, the lubrication forces and  torques arise, which play a vital role in the hydrodynamic behavior of the swimmers15,16,32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' It has  been reported recently that a pair of chiral squirmers portray various behaviors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=', divergence,  convergence, and even a bounded state11 as a result of their mutual hydrodynamic interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  However, considering the lubrication forces and torques (as in the dense suspension of swimmers),  the swimmers do not exhibit convergence behavior15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' In this work, we include the lubrication  effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The lubrication force acting on a swimmer can be calculated by knowing the velocity field  of the nearby swimmer16,32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Due to this lubrication force, the corresponding velocity contribution  to a chiral swimmer from the other swimmer, and vice versa, is15  U = −2a2Bε lnε ,  (8)  5  Chemotaxis of chiral swimmers  where ε is half of the separation distance between the swimmers, B = β r(1)  10 t13 −β r(2)  10 t23, β r(1)  10  and  β r(2)  10  are l = 1 and m = 0 modes (see the text after Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 2) of swimmer one and two, respectively,  t13 = t1·eZ, t23 = t2·eZ, eZ is the unit vector along the Z direction, and t1 and t2 are the correspond-  ing orientations of the swimmers (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='15 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Note that the lubrication torques  are of the order O(ε1/2), and which can be neglected in the limit ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  The equations of motion of a swimmer in the presence of another swimmer, by considering  lubrication forces into account, read as  ˙qi = Vi +Ulub  i  +  2  ∑  j=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='i̸= j  uj(qi j,nj,bj,tj)  �  ����  ˙ni  ˙bi  ˙ti  �  ���� =  �  Ωi +  2  ∑  j=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='i̸=j  ω j(qi j,nj,bj,tj)  �  ×  �  ����  ni  bi  ti  �  ���� ,  (9)  where Ulub  i  = U(cosθ ′ti −sinθ ′ni) is the additional velocity contribution arising due to the other  swimmer in the lubrication region, θ ′ = cos−1(ti ·eZ), and the vorticity field ω = (∇×u)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Note  that Ulub  i  = 0 for R > 2(a+ε), and Ui +  2  ∑  j=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='i̸= j  uj =  2  ∑  j=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='i̸=j  ω j = 0 for R ≤ 2(a+ε), where ε ≪ a  and R = |qi j| is the radial distance between the swimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  INFLUENCE OF EXTERNAL LINEAR CHEMICAL GRADIENT  Consider that the two swimmers are subjected to an external chemical concentration field c(q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  see the schematic diagram Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' As the swimmers sense the gradient, they move from a lower-  to a higher-concentration region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Consequently, the local concentration field that the swimmers  experience changes with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The stimulus level (s = c(q(t))) on the swimmer’s body is thus a  function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' A swimmer’s response to the stimulus is related to the relative change in the  chemoattractant concentration (∆c/c) in the vicinity33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The binding of the chemoattractant to the  body’s receptors, known as activation, and the unbinding of that, known as deactivation, triggers  the internal chemotactic signaling system of the body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' In our model systems, it results in the  modification of the slip coefficients (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 2) of the swimmer, which leads change in the velocity  and rotation rate of the swimmer34–37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  To study the chemotaxis of two swimmers, we use the Barkai–Leibler model34,38,39 for the  adaptation and relaxation mechanism of the swimmer to the external chemical gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Note that  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='Chemotaxis of chiral swimmers  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='y  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='~V2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='~⌦2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='~V1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='~⌦1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='<latexit sha1_base64="wb5u7cXp/b4L1vn0rD9RfT60=">AB6HicdVDJSgNBEK2JW4xb1KOXxiB4G  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='mbE7Rj04jERs0AyhJ5OTdKmZ6G7RwhDvsCLB0W8+kne/Bt7kgiuD5p+vFdFVT0/EVxpx3m3CguLS8srxdXS2vrG5lZ5e6ep4lQybLBYxLtU4WCR9jQXAtsJxJp6Ats+aPL3G/doVQ8jm70OEvpIOIB5xRbaT6da9cexjJwf  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='5TVx7+jsVmKPWK791+zFLQ4w0E1Spjusk2suo1JwJnJS6qcKEshEdYMfQiIaovGy6IQcGKVPgliaF2kyVb92ZDRUahz6pjKkeqh+ern4l9dJdXDuZTxKUo0Rmw0KUkF0TPKrSZ9LZFqMDaFMcrMrYUMqKdMm5IJ4fNS8j9p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='HtnuqX1SP65UL+ZxFGEP9uEQXDiDKlxBDRrAOEeHuHJurUerGfrZVZasOY9u/AN1usHwV2M7A=</latexit>R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  Schematic diagram of two hydrodynamically interacting chiral swimmers in the presence of a  linear chemical gradient applied only along the x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The separation distance between the swimmers  is R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  this model is for information processing and memory via the internal biochemical network, and it  has been used widely for bacterial and sperm chemotaxis34,35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The ability of the system to adapt  and respond to the external stimulus can be captured by a simple dynamical system,  σ ˙ab = pb(sb +s)−ab ,  (10a)  µ ˙pb = pb(1−ab),  (10b)  where σ is the relaxation time, µ is the adaptation time, ab(t) is the dimensionless output variable,  pb(t) is the dynamic sensitivity related to adaptation, and sb(t) arises from the background activity  of receptors in the absence of the chemical stimulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The chemoattractant has a dimension of  concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Note that these equations are valid for weak concentration gradients only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Under a  constant stimulus s(t) = Sc, the system reaches a steady state for which ab = 1 and pb = 1/(sb +  Sc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Therefore, the dynamic sensitivity (pb) maintains an inverse relation with sb and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' With  increasing stimulus level, pb decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The system here is totally adaptive as ab(t) is independent  of Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  In the presence of an external stimulus, the slip coefficients (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 2) of the swimmer are modified  as,  X = X(0) +X(1) (ab(t)−1) ,  (11)  where X = {βlm,γlm}, the unperturbed slip coefficients are X(0) =  �  β (0)  lm ,γ(0)  lm  �  , and the pertur-  bation due to the external gradient is X(1) =  �  β (1)  lm ,γ(1)  lm  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Here, we consider a linear chemical  gradient given by34,  c(q) = c0 +c1 ·q,  (12)  7  Chemotaxis of chiral swimmers  where c1 = ∇c(q) = c1i is the chemical gradient applied in the x direction and q = ix+jy+kz is  the position vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The chemotactic stimulus is S(t) = c(q(t))34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' (9), we numerically calculate the trajectories of a pair of chiral swimmers, which  are hydrodynamically interacting, in the presence of a linear chemical gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Note that the  velocity and the rotation rate of the chiral squirmer are functions of the slip coefficients, which are  affected by the chemical gradient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' As a result, the hydrodynamic interaction between  the swimmers is affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' In the present work, we consider chiral swimmers having velocities of  equal magnitudes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=', |V1| = |V2| = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' However, the rotation rates of the swimmers are in general  different and read, Ω1 = v(sinχ1,0,cosχ1)/a for swimmer one and Ω2 = v(sinχ2,0,cosχ2)/a for  swimmer two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Chemical and hydrodynamic perturbations in the velocity and rotation rate of the  swimmers change the corresponding torsion and curvature of the swimmers’ helical trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  Hence their movements are altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  As discussed, a swimmer’s flow field influences the neighboring swimmer’s motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The hy-  drodynamic flow field of a swimmer (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' (6)) comprises l = 1,2 modes as leading order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  The l = 2 mode is important as it plays a vital role in determining the hydrodynamic interaction  between the swimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The slip coefficients corresponding to l = 2 mode are assumed to have the  following form: 3β r  20 = 3γ r  20 = λ1 for squirmer one, and 3β r  20 = 3γ r  20 = λ2 for squirmer two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The  former assumptions are made to minimize the parameters of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Note that for λ1 ̸= λ2,  the flow fields of the swimmers are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Thus, variation in χi (angle between Vi and Ωi)  and ±λi (i = 1,2) (the strength of the flow field), quantitatively measure the nature of the interac-  tion between the chiral squirmers and gives rise to several interesting swimming characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  Notably, different initial configurations also influence the interaction between the swimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' We  found that, out of various possible initial configurations for the swimmers, swimmers interact with  each other for an extended period only in the planar configuration11 as is the case in the present  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Here, at time t = 0, both the swimmers are on the xy plane, with an initial orientation along  the z−direction and an initial separation distance R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Our previous study (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='11) explored the  interaction between the two chiral swimmers with the former configuration, which exhibited some  new behavior (bounded state) in addition to the ones (convergence and divergence state) displayed  by axisymmetric swimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' We mainly focus on how the external chemical gradient influences  these observed behaviors in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  8  Chemotaxis of chiral swimmers  0  2  4  λ/v  0  π  6  π  3  χ  (a) Pusher - Pusher  0  2  4  λ/v  0  π  6  π  3  χ  (b) Puller - Puller  0  2  4  λ/v  0  π  6  π  3  χ  (c) Pusher - Puller  0  2  4  λ/v  0  π  6  π  3  χ  (d) Puller - Pusher  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' (a)-(d) Depicting swimming states of hydrodynamically interacting chiral squirmers performing  chemotaxis, where the strength of the chemical gradient c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The symbols imply the following states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  Square: bounded state (BS), closed circles: monotonic divergence state (MD), open circles: divergence  state (D), plus: locked state, and cross symbols: parallel motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The two swimmers initially start at  q1 = (9,9,0)a and q2 = (3,3,0)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The corresponding initial velocities and rotation rates of the swimmers  are V1 = V2 = v(0,0,1) and Ω1 = Ω2 = v(cosχ,0,sinχ)/a, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  State diagram in the χ - λ plane  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 3 depicts the hydrodynamic behavior of two chiral swimmers in the presence of an external  chemical gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' When λ1 = λ2 = λ and χ1 = χ2 = χ, the swimmers are identical (see the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 3  caption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The swimmers portray various behaviors for varying λ/v and χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' As reported earlier15,  in the absence of a chemical gradient, two chiral swimmers exhibit divergence, monotonic di-  vergence, and bounded states (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 4) only due to the lubrication effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' These observed  states arise due to pure hydrodynamic interaction between the swimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' However, the chemical  gradient converts some monotonic divergence states into divergence states and some divergence  states into bounded states (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Indeed, the chemical stimulus favors the attraction between  the swimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The density of bounded states is more for the odd combination of swimmers like  pusher-puller or vice-versa (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 3(c) and (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' For other combinations, swimmers rarely exhibit  bounded states (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 3(a) and (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The occurrence of bounded motion in even combinations of  swimmers is due to the presence of the chemical gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Note that in the case of axisymmetric  swimmers (without chirality), the hydrodynamic behavior of pusher-puller and puller-pusher-type  swimmers is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' However, in the case of chiral swimmers, the hydrodynamic behavior of  different swimmers exhibit different behaviors11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' It is because the flow pattern of a puller-type  chiral swimmer is different from a pusher-type chiral swimmer11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  Note that for χ = 0 and λ ̸= 0, swimmers move in a straight line in the z direction, attract each  9  Chemotaxis of chiral swimmers  other, and converge to a locked state15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' In this state, the swimmers approach a minimum distance  and stay together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' For χ = 0 (straight line motion) and χ = π/2 (circular motion), the neutral  swimmers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=', λ = 0, move parallel to each other without influencing each other11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Not only the  chemical gradient but the initial separation distance R0 (at t = 0) between the swimmers also plays  a vital role in the hydrodynamic interaction between the swimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Interestingly, as R0 increases,  the bounded state is more favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' It is reported in detail in our earlier studies (see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='11,15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  We also study the interaction between non-identical swimmers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=', λ1 ̸= λ2 and χ1 ̸= χ2 (see  Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' For a fixed χ = π/3 and varying flowfield strengths (λ1,λ2), we observe mainly  divergence states and a few bounded states for the different combinations of the swimmers (see  fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 6(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Identical or nearly identical swimmers mostly exhibit bounded motion primarily for  the puller-pusher combination (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 6(a), top right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' However, other combinations also  exhibit bounded motion when λ1 = λ2 = 0 (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 6(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The former implies that non-identical  field strength is unfavorable for the bounded motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' On the other hand, we can fix the flow-  field strengths at λ = v and vary relative orientations of the swimmers (χ1,χ2) (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 6(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  Non-identical orientations are associated with divergence and monotonic divergence states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' See  Appendix A for a detailed discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 4 depicts the trajectories and the radial separation distance between the swimmers in the  presence and absence of the chemical gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' As mentioned before, the simultaneous effect of  the hydrodynamic interaction and the chemical stimulus converts some of the divergence states  into bounded states (see figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 4 (b),(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Here, the stimulus increases the attraction between the  swimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' As a result, for the bounded state, the swimmers tend to align in the direction of the  chemical gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The oscillation in the separation distance (R) between the swimmers decreases  (see figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 4(b),(c),(e),(f)) when compared to the stimulus-free situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Whereas for the monotonic  divergence state, the swimmers diverge at a slower rate than the stimulus-free case (see figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 4  (a),(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' As reported earlier, in the case of a single chiral swimmer, the body completely aligns its  helix axis to the direction of the gradient36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' In the two-swimmer case, none of the swimmers can  make their helix axis parallel to the direction of the gradient because of the hydrodynamic flow  field generated by the neighboring swimmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' In particular, it is due to the dominating term in the  hydrodynamic flow field o(1/r2) and the weak chemical gradient we apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  10  Chemotaxis of chiral swimmers  0  100  200  0  10  20  0  50  100  0  50  100  150  0  10  20  0  100  200  0  200  400  600  0  10  20  0  100  200  300  x  a  y  a  z  a  (a)  x  a  y  a  z  a  (b)  x  a  y  a  z  a  (c)  0  10  20  30  0  200  400  c1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='00  : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='01  : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='10  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='5  5  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='5  0  100  200  300  0  5  10  0  500  1000  1500  tv/a  R  a  (d)  tv/a  R  a  (e)  tv/a  R  a  (f)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  (a)-(c) Show the trajectories of two swimmers in the presence and in the absence of a linear  chemical gradient for monotonic divergence (MD), divergence (D), and bounded states (BS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Here, the  strength of the linear gradient c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The other parameters are, χ = π/6,λ1 = −2, and λ2 = 2 for  D state, χ = π/3,λ1 = −3,λ2 = 3 for MD state, and χ = π/3,λ1 = 1,λ2 = −1 for BS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' (d)-(f) The  corresponding radial separation distance R between the swimmers as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Here, lengths are  scaled by the radius of the swimmer a and time scaled by τ = v/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  DEPENDENCE OF RELATIVE SUCCESS TIME ON CHEMICAL GRADIENT  A question may arise whether a pair of swimmers or a single swimmer is efficient in swimming  in the direction of the external chemical gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' To answer this, we measure the relative success  time defined as ∆τ = (t0 −tc)/t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Here, t0 is the time a pair of swimmers or a single swimmer  takes to reach a particular reference point on the x axis in the absence of the chemical gradient,  and tc is the same in the presence of the chemical gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Note that ∆τ increases with c1 and  saturates for higher c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The latter is due to the saturation of the internal signaling network of the  bodies39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Henceforth, the network cannot modify its translational and rotational velocities further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  We find pair swimming is more efficient than individual swimming (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' It is because  the combined effect of the hydrodynamic interaction between the swimmers and chemical per-  turbation increases the motility of the swimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Interestingly, the bounded swimmers are more  11  Chemotaxis of chiral swimmers  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='4  ∆τ  10−5  10−4  10−3  10−2  10−1  c1  χ = π  3 , λ1 = 2, λ2 = −2 (BS)  χ = 7π  24 , λ1 = −2, λ2 = 2 (BS)  χ = π  4 , λ1 = −2, λ2 = 2 (BS)  χ = π  3 (Single)  (a)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='3  ∆τ  10−5  10−4  10−3  10−2  10−1  c1  χ = π  4 , λ1 = −2, λ2 = −2 (D)  χ = π  6 , λ1 = −2, λ2 = −2 (D)  χ = π  8 , λ1 = −2, λ2 = −2 (D)  χ = π  3 (Single)  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  The relative success time ∆τ = (t0 −tc)/t0 as a function of the chemical gradient strength c1 for  various observed states, (a) for bounded states and (b) for divergence states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The diamond symbols for the  single swimmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Here, lines connect the data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  efficient than others (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 5(a)), especially for lower values of χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' However, the influence of χ  on chemotactic success rate is minimal for diverging motion (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 5(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Note that, the range  of c1 is from o(10−6) to o(10−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' For weak gradients, divergence motion persists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' With increas-  ing strength of c1, as mentioned earlier, some divergence states get converted into bounded states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  Note that for values of χ ∼ π/2, swimmers effectively move in circular paths, and correspondingly  the success rate is low (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  CONCLUSIONS  In this work, we have studied the combined behavior of two chiral swimmers by considering  the lubrication effect in the presence of a linear chemical gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' We have used a generalized  squirmer model called the chiral squirmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' In general, a pair of chiral swimmers can exhibit  bounded, divergence, convergence, and monotonic convergence states without a chemical gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  However, only bounded, divergence, and monotonic divergence states are observed in the current  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The convergence and monotonic convergence states are absent because of the lubrication  force between the swimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Due to this lubrication force, swimmers repel each other when they  approach each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  Interestingly, a chemical gradient converts some divergence- and monotonic-divergence states  into bounded states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The chemical gradient favors the attraction between the swimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' It leads  to efficient swimming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' We have presented detailed state diagrams depicting the hydrodynamic  12  Chemotaxis of chiral swimmers  behavior of two chiral swimmers in the presence of a chemical gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' We have computed  the relative success time of a pair of swimmers in aligning their movement toward the chemical  gradient, whose strength is controlled by the parameter c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' We find that pair swimming is more ef-  ficient than individual swimming as the combined effect of the hydrodynamic interaction between  the two swimmers and chemical perturbation increases the motility of the swimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Interestingly,  the bounded swimmers are more efficient than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  This study is not limited to chemotaxis but can be generalized to various external gradients, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=',  phototaxis and gyrotaxis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' For these cases, one needs to identify how the slip coefficients of the  chiral swimmer can be modified according to the external gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Also, the present study helps  design artificial micro locomotors which can be used for drug delivery and targeted applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  The authors acknowledge the financial support of the Indian Institute of Technology Kharagpur,  India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  DATA AVAILABILITY  The data supporting this study’s findings are available from the corresponding author upon  reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  Appendix A: State diagrams of λ1 - λ2 and χ1 - χ2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 6(a) depicts the λ1−λ2 state diagram of two chiral swimmers, in the presence of a chemical  gradient, for fixed χ1 and χ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The additive contributions of hydrodynamic interaction and chem-  ical perturbations drive the swimmers away from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Only the swimmers with identical  or nearly identical field strengths (|λ1| = |λ2|) exhibit bounded motion (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 6(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Moreover,  identical combinations of swimmers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=', puller-puller or pusher-pusher, mostly display a diver-  gence state with a rare occurrence of bounded states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The probability of observing a bounded  state is more if the first swimmer is a puller and the second swimmer is a pusher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' However, the  opposite combination does not work while |λ1| ̸= 0,|λ2| ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The reason is the asymmetry in the  swimmer’s flow field about the orientation of the swimmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' At different positions around a chiral  swimmer, the neighboring swimmer experiences different hydrodynamic forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' As a result, the  13  Chemotaxis of chiral swimmers  −4  −2  λ2/v  −4  −2  λ1/v  Pusher - Pusher  0  2  4 λ1/v  0  2  4  λ2/v  Puller - Puller  0  2  4  λ2/v  −4  −2  λ1/v  Pusher - Puller  0  2  4 λ1/v  −4  −2  0  λ2/v  Puller - Pusher  0  π  6  π  3  χ2  0  π  6  π  3  χ1  Pusher - Pusher  0  π  6  π  3  χ1  0  π  6  π  3  χ2  Puller - Puller  0  π  6  π  3  χ2  0  π  6  π  3  χ1  Pusher - Puller  0  π  6  π  3  χ1  0  π  6  π  3  χ2  Puller - Pusher  (a)  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' State diagrams corresponding to two hydrodynamically interacting chiral swimmers in the presence  of chemotaxis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' (a) For varying field-strengths (λ1,λ2) and (b) for (χ1,χ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' See the main text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  The initial conditions and symbol descriptions are the same as in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' We set χ1 = χ2 = π/3 for the  left panel and |λ1| = |λ2| = |V| = v for the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  interaction between the swimmers is different for their different relative positions (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='11 for  more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 6(b) shows the χ1−χ2 state diagram of two chiral swimmers, in the presence of a chemical  gradient, for fixed λ1 and λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' The swimmers do not exhibit bounded motion unless χ1 = χ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  As mentioned before, similar combinations of the swimmers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=', puller-puller or pusher-pusher,  exhibit bounded states rarely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' However, the opposite combinations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=', puller-pusher or pusher-  puller, do exhibit more bounded states (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 6 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' In the absence of a chemical gradient, more  monotonic divergence states are observed (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' However, the presence of the chemical field  increases the attraction between the swimmers and converts some of the monotonic divergence  (MD) states into divergence (D) states (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' 6(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
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+page_content=' Maity and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Burada, Physical Review E 106, 054613 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  16S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Wang and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Ardekani, Physical Review E 87, 063010 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  17R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Di Leonardo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Dell’Arciprete, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Angelani, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Iebba, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
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+page_content='  18D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Kirchman, Microbial Ecology 28, 255 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  19D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Saintillan and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Shelley, Physical Review Letters 100, 178103 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  20A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Sokolov and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Aranson, Physical review letters 103, 148101 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  21B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Haines, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Sokolov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Aranson, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Berlyand, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Karpeev, Physical Review E  80, 041922 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content='  22J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
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+page_content=' Heidenreich, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Drescher, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Wensink, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
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+page_content='  23S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Larsen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Macnab, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Koshland, Nature 249, 74 (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
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+page_content=' Alvarez, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Friedrich, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Wilson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Pascal, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Colin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Pichlo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Rennhack,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Brenker, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Kaupp, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
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+page_content='  25A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Sarvestani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Amir, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' T, Cell Biochem Biophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
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+page_content='  26C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Jin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Kr¨uger, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
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+page_content=' Maass, Proceedings of the National Academy of Sciences 114,  5089 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Kopp, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Sen, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Velegol, Physical review letters 99,  178103 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
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+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Paxton, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
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+page_content=' Olmeda, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
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+page_content=' Cao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
+page_content=' Crespi, Journal of the American Chemical Society 126, 13424 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
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+page_content=' Savel’ev, Physical Review E 94,  012613 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9AzT4oBgHgl3EQfCvrc/content/2301.00966v1.pdf'}
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+arXiv:2301.11754v1  [cs.IT]  27 Jan 2023
+1
+Information-Theoretic Privacy-Preserving
+Schemes Based On Perfect Privacy
+Borzoo Rassouli1 and Deniz G¨und¨uz2
+1 Nokia Bell Labs, Munich, Germany
+2 Department of Electrical and Electronic Engineering, Imperial College London, London, UK
+borzoo.rassouli@nokia-bell-labs.com, d.gunduz@imperial.ac.uk
+Abstract
+Consider a pair of random variables (X, Y ) distributed according to a given joint distribution pXY . A curator
+wishes to maximally disclose information about Y , while limiting the information leakage incurred on X. Adopting
+mutual information to measure both utility and privacy of this information disclosure, the problem is to maximize
+I(Y ; U), subject to I(X; U) ≤ ǫ, where U denotes the released random variable and ǫ is a given privacy threshold.
+Two settings are considered, where in the first one, the curator has access to (X, Y ), and hence, the optimization
+is over pU|XY , while in the second one, the curator can only observe Y and the optimization is over pU|Y . In
+both settings, the utility-privacy trade-off is investigated from theoretical and practical perspective. More specifically,
+several privacy-preserving schemes are proposed in these settings based on generalizing the notion of statistical
+independence. Moreover, closed-form solutions are provided in certain scenarios. Finally, convexity arguments are
+provided for the utility-privacy trade-off as functionals of the joint distribution pXY .
+I. INTRODUCTION
+Consider a situation in which Alice wants to release some useful information about herself to Bob, represented by
+random variable Y , and she receives some utility from this disclosure of information. At the same time, she wishes
+to conceal from Bob some private information which depends on Y , represented by X. To this end, instead of
+letting Bob have a direct access to Y , a privacy-preserving mapping/data release mechanism1 is applied, whereby
+a distorted version of Y , denoted by U, is revealed to Bob. In this context, privacy and utility are competing goals:
+The more Y is distorted by the privacy-preserving mapping, the less information can Bob infer about X, but also
+the less the utility that can be obtained. This trade-off is the very result of the dependencies between X and Y .
+Stated formally in a general context, consider a triplet of random variables (X, Y, W), distributed according to
+the given/known joint probability mass function (pmf) pXY W . Let X denote the private/sensitive data that the
+user/curator wants to conceal, Y denote the public/useful data the user wishes to disclose, and W denote the
+observable data that the curator observes, which can be regarded as a noisy version of (X, Y ). Assume that the
+privacy-preserving mapping takes W as input, and maps it to the released data, denoted by U. In this scenario,
+1The terms ”mapping”, ”mechanism”, ”scheme” and ”algorithm” are used interchangeably in this paper.
+
+2
+(X, Y )−W −U form a Markov chain, and the privacy-preserving mapping is captured by the conditional distribution
+pU|W . We assume that all the alphabets/supports X, Y, W are finite sets.
+As stated, for the triplet (X, Y, W), a privacy-preserving mapping can be constructed by obtaining pmfs pU|W (·|w)
+for each w ∈ W such that pU|W meets certain conditions corresponding to the utility/privacy requirements.
+Equivalently, this can be carried out by obtaining U, pW|U(·|u), ∀u ∈ U, such that pW is preserved in the
+joint distribution pWU and those conditions are met. We call the former approach forward construction/model and
+the latter one backward construction/model.
+In this paper, we are solely interested in the special cases of full data observation and public data observation
+which refer to the settings in which the privacy-preserving mapping has direct access to both the private and public
+data (i.e., W = (X, Y )) and only to the public data (i.e., W = Y ), respectively.
+By adopting mutual information as the measure of both utility and privacy (i.e., I(Y ; U), and I(X; U), respec-
+tively), the optimal utility-privacy (U-P) trade-off in the public data observation model is defined as
+gǫ(X, Y ) ≜
+max
+pU|Y :
+X−Y −U
+I(X;U)≤ǫ
+I(Y ; U),
+(1)
+and in the full data observation model, the optimal U-P trade-off can be formulated as
+Gǫ(X, Y ) ≜
+max
+pU|XY :
+I(X;U)≤ǫ
+I(Y ; U),
+(2)
+where the effective range of ǫ is [0, I(X; Y )] by noting that both (1) and (2) have the upper bound of H(Y ) which
+is attained by setting U ≜ Y , which in turn results in I(X; U) = I(X; Y ).2
+Perfect privacy [1] refers to the stringent constraint of X ⊥⊥ U, i.e., ǫ = 0 in the U-P trade-off. Assume that
+we have an algorithm which satisfies this constraint. In other words, once applied to W (with W = (X, Y ) or Y
+depending on the model involved), this algorithm releases U such that I(X; U) = 0. Select an arbitrary conditional
+pmf pZ|XY , and construct a private-public pair as (Z, Y ) ∼ �
+x pXY (x, ·)pZ|XY (·|x, ·). Applying the algorithm
+in this new context (with W = (Z, Y ) or Y ), we get U ′ such that I(Z; U ′) = 0. The utility obtained is a lower
+bound on the original optimal U-P trade-off at ǫ = I(X; U ′), and by changing pZ|XY and repeating this process,
+we can sweep the whole range of ǫ, i.e., [0, I(X; Y )]. This simple observation is the basis of the privacy-preserving
+schemes proposed in this paper.
+The information-theoretic view of privacy has gained increasing attention recently, with an incomplete list of
+related literature being [1]–[13]. In [2], a general statistical inference framework is proposed to capture the loss
+of privacy in legitimate transactions of data. In [3], the privacy-utility trade-off under the log-loss cost function is
+considered, called as the privacy funnel, which is closely related to the information bottleneck introduced in [14]
+(see also [4]). In [15], sharp bounds on the optimal privacy-utility trade-off for the privacy funnel are derived, and an
+alternative characterization of the perfect privacy condition (also studied in [16] in a different context) is proposed.
+2That the maximums in (1) and (2) exist follows from standard arguments in real analysis (compactness, continuity) as in [1]. Furthermore,
+when (X, Y ) ∼ pXY , the quantities gǫ(X, Y ) and Gǫ(X, Y ) are written interchangeably as gǫ(pXY ) and Gǫ(pXY ), respectively.
+
+3
+Measuring both the privacy and the utility in terms of mutual information, perfect privacy is fully characterized in
+[17] for the binary case.
+The current paper contributes to this context as follows.
+• Upper and lower bounds on G0(X, Y ) are proposed and their tightness is investigated.
+• Based on the aforementioned bounds, and the simplex method [18], a closed-from solution for G0(X, Y ) is
+derived when X is binary, or (|X|, |Y|) = (3, 2). In this context, it is shown that for fixed conditional pmf
+pY |X, the optimal privacy-preserving mapping does not depend on pX, which is of practical interest when the
+curator is unaware of the distribution of the private data.
+• It is shown that for fixed pX, Gǫ(X, Y ) is concave in pY |X, and for fixed pY |X, both G0(X, Y ) and g0(X, Y )
+are convex in pX.
+• Based on the lower bound on G0(X, Y ), a privacy-preserving scheme is presented as a lower bound on the
+optimal U-P trade-off in the case of full data observation.
+• When X is binary, an algorithmic lower bound on g0(X, Y ) is presented, which is optimal when |Y| = 3.
+• Based on this algorithm, a privacy-preserving scheme is presented as a lower bound on the optimal U-P
+trade-off in the case of public data observation.
+Notations. Random variables are denoted by capital letters (X, Y , etc.), their realizations by lower case letters
+(x, y, etc.), and their alphabets by capital letters in calligraphic font (X, Y, etc.). Matrices, and vectors are denoted
+by bold capital and bold lower case letters, respectively. The rank of matrix A is denoted by rank(A). For integers
+m, n, we have the discrete interval [m : n] ≜ {m, m + 1, . . . , n} if m ≤ n, and ∅ (the empty set), otherwise.
+The set [1 : n] is written in short as [n]. Given two positive integers a, b, a modulo b is abbreviated as a mod b,
+and S mod b denotes {x mod b| ∀x ∈ S}. Given two pmfs p, q, the Kullback–Leibler divergence from q to p is
+defined as3 D(p||q) ≜ �
+x p(x) log2
+p(x)
+q(x). All the logarithms in this paper are to the base of 2. For 0 ≤ t ≤ 1,
+¯t ≜ 1 − t and Hb(t) ≜ −t log t − ¯t log ¯t denotes the binary entropy function (with the convention 0 log 0 ≜ 0). For
+an event E, the indicator 1{E} is one when E occurs, and zero, otherwise. The domain of function f is denoted by
+dom(f), and throughout the paper, if there are more than one candidate for arg minx∈dom(f) f(x), one is selected
+arbitrarily. For a real number x, we define (x)+ ≜ max{0, x}. Define the support of a given pair (X, Y ) ∼ pXY
+as supp(X, Y ) ≜ {(x, y) ∈ X × Y|pXY (x, y) > 0}. Finally, throughout this paper, we encounter summations of
+the form �
+u∈U(·), i.e., summation over the elements of U. If U happens to be the empty set, this summation is
+defined as zero.
+II. PRELIMINARIES
+Throughout the paper, the U-P plane refers to the 2-dimensional plane, in which the horizontal and vertical axes
+denote the privacy-leakage I(X; U) and utility I(Y ; U), respectively. Furthermore, we say that a point P = (Px, Py)
+is achievable on this plane if there exists a joint distribution pXY · pU|XY (in the case of full data observation), or
+pXY · pU|Y (in the case of public data observation), such that I(X; U) = Px, and I(Y ; U) = Py.
+3We assume that p is absolutely continuous with respect to q, i.e., q(x) = 0 implies p(x) = 0, otherwise, D(p||q) ≜ ∞.
+
+4
+Remark 1. When ǫ ∈ [0, I(X, Y )], both gǫ(X, Y ) and Gǫ(X, Y ) are concave and strictly increasing functions of ǫ
+that lie above the lines connecting (0, g0(X, Y )), and (0, G0(X, Y )) to (I(X; Y ), H(Y )), respectively. Furthermore,
+they both lie under the line connecting (0, H(Y |X)) to (I(X; Y ), H(Y )). Also, for an optimal mapping in (1) or
+(2), we have I(X; U) = ǫ. Finally, both g0(X, Y ) and G0(X, Y ) can be obtained via a linear program (LP).
+Proof. The concavity of gǫ(X, Y ) in ǫ is shown in [5, lemma 2]. That of Gǫ(X, Y ) follows similarly4. We also
+have
+gǫ(X, Y ) ≤ Gǫ(X, Y )
+(3)
+≤
+max
+pU|X,Y :
+I(X;U)≤ǫ
+I(X, Y ; U)
+=
+max
+pU|X,Y :
+I(X;U)≤ǫ
+I(X; U) + I(Y ; U|X)
+≤ ǫ + H(Y |X), ǫ ∈ [0, I(X; Y )],
+(4)
+where (3) is by definition. This means that both gǫ(X, Y ) and Gǫ(X, Y ) lie under the line connecting (0, H(Y |X))
+to (I(X; Y ), H(Y )). Unless in the degenerate case of X ⊥⊥ Y (which results in ǫ ∈ [0, I(X; Y )] = {0}), we have
+H(Y |X) < H(Y ), and hence, both gǫ(X, Y ) and Gǫ(X, Y ), being concave functions, must be strictly increasing in
+ǫ. As a result, they lie above the lines connecting (0, g0(X, Y )), and (0, G0(X, Y )) to (I(X; Y ), H(Y )), respectively.
+By noting that in maximizing a convex functional, the maximum occurs at an extreme point, we conclude that
+for any maximizer in (1) or (2), we have I(X; U) = ǫ, when ǫ ∈ [0, I(X; Y )].
+As shown in [1], g0(X, Y ) is obtained via an LP. More specifically, in the Markov chain X − Y − U, if we
+consider the backward model, the conditional pmf pY |U(·|u) must belong to a convex polytope determined by
+the condition X ⊥⊥ U, i.e., pX(·) = �
+y pX|Y (·|y)pY |U(y|u). Since H(Y |U = u) is a concave functional of
+pY |U(·|u), and its minimum occurs at an extreme point, we first obtain the extreme points of the aforementioned
+convex polytope, and what remains is to allocate proper weights, i.e., pU(·), such that H(Y |U) is minimized subject
+to pY (·) = �
+u pY |U(·|u)pU(u), which is an LP. Similarly, by considering the Markov chain X − (X, Y ) − U,
+G0(X, Y ) can be obtained through an LP.
+Definition 1. The upper concave envelope of a set of points P ≜ {(xi, yi)}n
+i=1 in R2 is defined as5
+uce[P](·) ≜ inf{f(x)| dom(f) = [min
+i
+xi, max
+i
+xi], f is concave , f(xi) ≥ yi, ∀i ∈ [n]}.
+Figure 1 provides an example of the upper concave envelope (solid line) of a set of points (filled circles).
+Remark 2. From Remark 1 and Definition 1, it is obvious that if the points in P are achievable on the U-P plane,
+{(x, uce[P](x)) | ∀x ∈ [mini xi, maxi xi]} is also achievable, and hence, uce[P](·) serves as a lower bound on the
+optimal utility-privacy trade-off6.
+4It can be shown via example that the claim of strict concavity is too strong for these two curves.
+5The lower convex envelope of P is the negative of the upper concave envelope of P− ≜ {(xi, yi)|(xi, −yi) ∈ P, ∀i ∈ [n]}.
+6Since (I(X; Y ), H(Y )) is an achievable point, we can include it in P. Therefore, the resulting uce[P](·) will be non-decreasing.
+
+5
+Fig. 1: The upper concave envelope of a set of points.
+The privacy-preserving algorithms in this paper find achievable points based on a generalization of statistical
+independence. The sole purpose of the following definition is to simplify the explanation of this generalization.
+Definition 2. For a pair of random variables (X, U) ∼ pXU, and k ∈ [|X| − 1], if
+����
+�
+x ∈ X
+���� pX|U(x|u) = pX(x), ∀u ∈ U
+����� ≥ k,
+(5)
+or equivalently,
+����
+�
+x ∈ X
+���� pU|X(·|x) = pU(·)
+����� ≥ k,
+(6)
+we say that X is at least k-independent of U, which is denoted by X
+k
+⊥⊥ U.
+Note that i) X
+k
+⊥⊥ U =⇒ X
+k−1
+⊥⊥ U, k ∈ [2 : |X| − 1], ii) X
+|X |−1
+⊥⊥
+U =⇒ X ⊥⊥ U, and iii) this is an asymmetric
+notion, i.e., X
+k
+⊥⊥ U ̸=⇒ U
+k
+⊥⊥ X. Furthermore, if X − U − Z form a Markov chain, from (6), we conclude that
+X
+k
+⊥⊥ U results in X
+k
+⊥⊥ Z.
+III. FULL DATA OBSERVATION MODEL
+In this Section, we assume that the curator has access to both X and Y , i.e., W = (X, Y ), and propose an
+achievable scheme, i.e., a lower bound on Gǫ(X, Y ), which is defined in (2). To this end, we find achievable points
+
+6
+x1y1
+x1y2
+x2y1
+x2y2
+x3y1
+x3y2
+u1
+u2
+u3
+u4
+u5
+u6
+u7
+u8
+Fig. 2: An illustrative representation of Lemma 1 for (X, Y ) ∈ {x1, x2, x3} × {y1, y2}. If uk is connected to pair
+(xi, yj), we have p(xi, yj|uk) = p(xi), (i, j, k) ∈ [3] × [2] × [8].
+on the U-P trade-off and propose their upper concave envelope as the lower bound.
+The following Lemma is central to the analysis in this Section.
+Lemma 1. For an optimal pU∗|XY in the evaluation of G0(X, Y ), we must have
+|{y ∈ Y|p(x, y|u∗) > 0}| = 1, ∀(x, u∗) ∈ X × U∗,
+(7)
+which results in
+H(Y |X, U ∗) = 0,
+(8)
+where U ∗ ∈ U∗ is induced by the optimal mapping pU∗|XY . In other words, for any (x, u∗) ∈ X × U∗, there must
+exist yx,u∗ ∈ Y such that
+p(x, yx,u∗|u∗) = p(x), p(x, y|u∗) = 0, ∀y ∈ Y\{yx,u∗},
+(9)
+which results in a lower bound on the cardinality of |U∗| as
+|U∗| ≥ max
+x∈X |{y ∈ Y|p(y|x) > 0}|.
+Proof. The proof is provided in Appendix A.7
+Lemma 1 is exemplified in Figure 2 in which (X, Y ) ∈ {x1, x2, x3} × {y1, y2}. Let pi ≜ p(xi), i ∈ [3]. As
+(9) requires, for each realization of U, there is exactly one link to subgroup i of nodes, i.e., {(xi, yj)}2
+j=1 with
+transition probability pi.
+7This Lemma is also given in [7, Lemma 5]. Since it was also independently provided in [19, Lemma 3] by the authors of the current
+manuscript, it is mentioned here.
+
+7
+Remark 3. If Y is not a function of X, we have G0(X, Y ) > 0 by [1, Theorem 4] which is resulted by a U such
+that X ⊥⊥ U and H(Y |X, U) = 0 according to Lemma 1. Obviously, if Y is a function of X, we can select U as
+an arbitrary singleton. This observation provides an alternative proof for the functional representation lemma [20,
+p. 626].
+In order to propose a lower bound on Gǫ(X, Y ), we start with ǫ = 0, which is equivalent to X ⊥⊥ U. It is
+already known that G0(X, Y ) can be obtained via an LP whose dimension is the total number of extreme points of
+the convex polytope stated earlier in Remark 1. However, according to Lemma 1, these extreme points are already
+known. They are all the conditional pmfs pXY |U that satisfy the property in (9). As a result, the dimension of
+the LP involved in evaluating G0(X, Y ) is �
+x∈X |{y ∈ Y|p(y|x) > 0}| ≤ |Y||X |. Note that even assuming a
+polynomial time complexity for the algorithm used for solving the LP, unless |X| and |Y| are small or the matrix
+of joint distribution PXY is sparse, the problem becomes computationally intractable in terms of time and space.
+Therefore, a tractable method is desirable.
+In what follows, an algorithm (Algorithm 1) is proposed in Theorem 1 that provides a lower bound on G0(X, Y ).
+This algorithm is proved to be optimal in Theorem 2 when X is binary or (|X|, |Y|) = (3, 2). Finally, building
+upon Algorithm 1, Proposition 1 presents Algorithm 2 which produces a privacy-preserving mapping as a lower
+bound on Gǫ(X, Y ), ǫ ∈ [0, I(X; Y )].
+Theorem 1. For a given pair (X, Y ) ∼ pXY , we have
+G0(X, Y ) ≥
+�
+H(Y ) −
+�
+1 −
+�
+y
+min
+x p(y|x)
+�
+min{H(X), log |Y|}
+�+
+.
+(10)
+Proof. Define the index set
+I ≜ {y ∈ Y|p(y|x) > 0, ∀x ∈ X},
+(11)
+and relabel the elements of Y = {y1, y2, . . . , y|Y|} such that the first |I| elements belong to I. The algorithm
+proceeds as follows. First, |I| mass points for U are created, which are denoted by uj(j ∈ [|I|]), each having
+pU(uj) = minx pY |X(yj|x), respectively, such that for all x ∈ X, we have p(x, yk|uj) = p(x), if j = k, and 0,
+otherwise (j, k ∈ [|I|]). It is evident that thus far, the posterior pX(·|uj) remains the same as the prior pX(·), which
+is in line with the condition of X ⊥⊥ U. Moreover, these mass points are such that pY |U(yj|uj) = 1, resulting in
+H(Y |U = uj) = 0, j ∈ [|Y|]. Afterwards, the iterations begin. In each iteration i, a mass point ui+|I| is created
+such that pX(·|ui+|I|) = pX(·), and the conditional pmf of the pair (X, Y ) conditioned on {U = ui+|I|} has
+the same mass probabilities as in pX(·) resulting in H(X, Y |U = ui+|I|) = H(X). Hence, H(Y |U = ui+|I|) ≤
+H(X), i ≥ 1 (note that we also have the trivial upper bound H(Y |U = ui+|I|) ≤ log |Y|). The procedure is
+provided in Algorithm 1, and the algorithm terminates at some iteration N, where N ≤ |supp(X, Y )| − |I|, which
+is further tightened in Remark 5.
+The rationale behind this algorithm becomes clear by considering the backward construction as follows. Let each
+realization (x, y) of (X, Y ) be denoted by a node. Arrange these nodes in a long column vector as in Figure 2. In this
+configuration, we divide the nodes into |X| subgroups of nodes: The first subgroup of nodes is {(x1, yi)|i ∈ [|Y|]},
+
+8
+Algorithm 1 A lower bound on G0(X, Y ).
+1: function ALGORITHM1(pY |X)
+2:
+p(uj|x, y) = minx∈X p(y|x)
+p(y|x)
+· 1{y=yj}, ∀j ∈ [|I|], (x, y) ∈ supp(X, Y )
+3:
+a1(x, y) = p(y|x) − minx p(y|x), ∀(x, y) ∈ supp(X, Y )
+4:
+i = 1
+5:
+while maxx,y ai(x, y) ̸= 0 do
+6:
+a∗
+i = minx,y{ai(x, y)|ai(x, y) > 0}
+7:
+(x∗, y∗) = arg minx,y{ai(x, y)|ai(x, y) > 0}
+8:
+fi(x) = arg miny{ai(x, y)|ai(x, y) > 0}, ∀x ∈ X
+9:
+p(ui+|I||x, y) =
+a∗
+i
+p(y|x) ·
+�
+1{ai(x,y∗)>0,y=y∗} + 1{ai(x,y∗)=0,y=fi(x)}
+�
+, ∀(x, y) ∈ supp(X, Y )
+10:
+ai+1(x, y) = ai(x, y) − a∗
+i ·
+�
+1{ai(x,y∗)>0,y=y∗} + 1{ai(x,y∗)=0,y=fi(x)}
+�
+, ∀(x, y) ∈ supp(X, Y )
+11:
+i = i + 1
+12:
+end while
+13:
+return pU|X,Y
+14: end function
+the second subgroup is {(x2, yi)|i ∈ [|Y|]}, and so on. Obviously, the sum of the mass probabilities of the nodes
+in the i-th subgroup is pX(xi), ∀i ∈ [|X|]. Therefore, if in this construction, from each mass point (or node) u,
+there is one connection/link to only one of the nodes in the first subgroup with transition probability pX(x1), one
+link to only one of the nodes in the second subgroup with transition probability pX(x2), and so on, which is what
+Lemma 1 implies, we have pX|U(·|u) = pX(·), and H(Y |U = u) ≤ H(X, Y |U = u) = H(X). However, if in
+the first |I| realizations of U, the links that connect each u ∈ {u1, . . . , u|I|} to |X| subgroups arrive at nodes that
+have the same second coordinate, i.e., y, we get H(Y |U = u) = 0 for these |I| realizations of U. Also, if a node
+u is to be connected to |X| nodes sharing the same second coordinate, e.g., {(xi, y)}|X |
+i=1 for some y ∈ Y, we must
+have p(u) ≤ p(y|xi), ∀i ∈ [|X|]. This is needed to guarantee that the requirement p(xi, y|u) = p(xi), i ∈ [|X|]
+does not violate the preservation of pXY in pXY U.8 Therefore, we set p(u) equal to its maximum allowable
+value, i.e., minx p(y|x). The aforementioned procedure is captured in step 2 of the algorithm by making the
+convention 0
+0 · 0 ≜ 0. Subsequently, the event containing the first |I| realizations of U occurs with probability of
+�
+y∈I minx p(y|x) = �
+y∈Y minx p(y|x), which results in H(Y |U) ≤ (1−�
+y minx p(y|x)) min{H(X), log |Y|}.
+The concern in this backward construction is to preserve the original distribution pXY in the resulting joint
+pmf pXY U. Since in the construction of U, it is known from our impositions that if p(x, y|u) ̸= 0, for some
+(x, y) ∈ supp(X, Y ), then we must have p(x, y|u) = p(x), we observe that the preservation of pXY boils down to
+that of the conditional pmf pY |X. In other words, denoting the set of all realizations u that have a link to (x, y) by
+8Since otherwise, we have p(u) > p(y|xj), for some j ∈ [|X|]. This results in pXY U(xj, y, u) = p(u)p(xj, y|u) = p(u)p(xj) > p(xj, y),
+which results in p(xj, y) induced by pXY U being greater than the original pXY (xj, y).
+
+9
+Ux,y, i.e., Ux,y ≜ {u ∈ U|p(x, y|u) ̸= 0}, ∀(x, y) ∈ supp(X, Y ), the preservation of pXY is equivalent to
+p(x, y) =
+�
+u∈U
+p(x, y, u)
+=
+�
+u∈Ux,y
+p(x, y|u)p(u)
+= p(x)
+�
+u∈Ux,y
+p(u), ∀(x, y) ∈ supp(X, Y ),
+which is in turn equivalent to
+p(y|x) =
+�
+u∈Ux,y
+p(u), ∀(x, y) ∈ supp(X, Y ).
+Therefore, we only need to make sure that the mass probabilities of all the nodes u that are connected to (x, y) sum
+up to p(y|x). To this end, we harness a waterfilling-like procedure, in which the water levels denote the remaining
+probabilities which need to be ”filled”. In step 3, the water levels are set as a1(x, y) by taking into account the
+assignment in step 2. In other words, for each node (x, y), the amount of minx p(y|x) has already been filled by
+the links from uj, ∀j ∈ [|I|] in step 2. At each iteration i, node ui+|I| is created to fill the minimum water level
+denoted by a∗
+i in step 6. A/the minimizer is denoted by (x∗, y∗) in step 7. Note that in this step and step 8, if there
+are multiple minimizers, one is selected arbitrarily9. In step 8, fi(x) denotes a/the minimum non-zero water level
+in subgroup x at iteration i. We create ui+|I|, and set p(ui+|I|) ≜ a∗
+i , and connect this node to |X| nodes, each
+belonging to one subgroup, with the transition probability of p(x1) for the link to subgroup 1, p(x2) for the link
+to subgroup 2, and so on. In doing so, we take this intuition into account that points with common y-coordinates
+are desirable, as this allocation is in line with lowering H(Y |U). Hence, in each subgroup x (x ∈ X), if the water
+level corresponding to (x, y∗), i.e., ai(x, y∗), is non-zero, this point is selected, otherwise, the point corresponding
+to a/the minimum water level of this subgroup is selected, i.e., (x, fi(x)). This is given in step 9 of the algorithm,
+and in step 10, the water levels are updated.
+Since in step 3 (prior to the iterations), the water levels of at least |I| nodes are filled, and at each iteration,
+at least one water level gets filled, i.e., ai+1(x∗, y∗) becomes zero (which occurs in step 10), the algorithm
+terminates after at most |supp(X, Y )| − |I| iterations. With this pU|XY , we get X ⊥⊥ U, and H(Y |U) ≤ (1 −
+�
+y minx p(y|x)) min{H(X), log |Y|}, which proves the lower bound in (10).
+The following example clarifies the steps in Algorithm 1.
+Example 1. Let (X, Y ) ∈ {x1, x2, x3} × {y1, y2, y3} be distributed according to the joint pmf PXY = PY |XpX
+as
+PX,Y =
+
+
+0.2
+0.4
+0.6
+0.5
+0.2
+0.3
+0.3
+0.4
+0.1
+
+
+
+
+p1
+p2
+p3
+
+ ,
+9Although at the expense of making the algorithm more complicated, one can propose a better selection (in terms of lowering H(Y |U)), we
+do not discuss it here.
+
+10
+where pi ≜ pX(xi), and column i of PY |X represents pY |X(·|xi), ∀i ∈ [3]. The reason for representing the mass
+probabilities of X as parameters, i.e., pi’s, rather than numerical values is this interesting property the design of
+a privacy-preserving mapping via Algorithm 1 does not depend on pX, which is elaborated further in Remark 4.
+Figure 3a illustrates step 2 of the algorithm. On the left hand side of this figure, the elements of X × Y are
+arranged into 3 (= |X|) subgroups in a column, and their corresponding probabilities are shown on their left side.
+In this example, we have I = Y, hence, we create 3 (= |I|) realizations of U, denoted by ui, i ∈ [3], with the
+corresponding probabilities of minx p(yi|x), which are shown on the right side of these points. Afterwards, each
+ui is connected to x1yi, x2yi, x3yi, with transition probabilities of pi, respectively. This is equivalent to step 2 of
+the algorithm.
+In Figure 3b, we have the same set of mass points xiyj’s whose mass probabilities have been updated by
+taking into account Figure 3a. In other words, each mass point xiyj has the remaining probability of p(xi, yj) −
+p(xi, yj|uj)p(uj) (= pip(yj|xi) − pip(uj) = pia1(xi, yj), where a1(·, ·) is defined in step 3) to be filled with other
+realizations of U. These remaining probabilities are shown on the left side of xiyj’s. Iteration 1 starts, and u4 is
+created, whose aim is to fill the minimum (non-zero) remaining probability , which is that of x∗y∗( = x3y2 in this
+example). This u4 is connected to x3y2, and x1y2 (whose y-coordinate is in common with x3y2), and x2y1, which
+has the minimum (non-zero) water level in the subgroup of x2 (since the water level of x2y2 is zero). These links
+are created bearing in mind that any connection to subgroup i has the transition probability of pi, i ∈ [3].
+Taking into account the connections in Figure 3b, the remaining probabilities are again updated in Figure 3c,
+shown on the left side of xiyj’s. Iteration 2 starts, and realization u5 is created in a similar way.
+Again, taking into account the connections in Figure 3c, the remaining probabilities are updated in Figure 3d,
+shown on the left side of xiyj’s. Iteration 3 starts, and realization u6 is created.
+The remaining probabilities are updated in Figure 3e where we are left with only one non-zero probability in
+each subgroup, i.e., 0.2. In iteration 4, which is the last one, u7 is created to fill all the remaining water levels,
+and the algorithm terminates after 4 iterations.
+Finally, the output of this algorithm is shown in Figure 4, where the transition probabilities in Figure 4a represent
+pXY |U, and those in Figure 4b represent pU|XY . From Figure 4a, it is obvious that pX|U(·|u) = pX(·), ∀u ∈ U,
+and hence, X ⊥⊥ U. Also, H(Y |U = u) = 0, ∀u ∈ {u1, u2, u3}, and H(Y |U = u) ≤ H(X), ∀u ∈ {u4, u5, u6, u7}.
+Therefore, G0(X, Y ) ≥ I(Y ; U) ≥
+�
+H(Y ) − �u7
+u4 p(u)H(X)
+�+ = (H(Y ) − 0.5H(X))+ .10
+Remark 4. (pX-invariance) An advantage of the achievable scheme in Algorithm 1 is that it does not depend on
+the distribution of the private data, i.e., pX(·). In other words, the privacy-preserving mapping pU|XY obtained
+via Algorithm 1 can be derived regardless of the knowledge about pX(·), as long as pY |X is given. This can be
+verified by the fact that none of the 14 steps of Algorithm 1 rely on the knowledge of pX.11 This is the reason
+that in Example 1, the mass probabilities of X are given only as parameters p1, p2, p3, and as it can be verified
+10Note that in this example, H(X) ≤ log |Y| = log 3.
+11Note that in the explanation of Algorithm 1, we indeed made use of pX, but this should not be confusing, since that explanation is about
+the backward construction pXY |U.
+
+11
+p1 × 0.2 : x1y1
+p1 × 0.5 : x1y2
+p1 × 0.3 : x1y3
+p2 × 0.4 : x2y1
+p2 × 0.2 : x2y2
+p2 × 0.4 : x2y3
+p3 × 0.6 : x3y1
+p3 × 0.3 : x3y2
+p3 × 0.1 : x3y3
+u1 : 0.2
+u2 : 0.2
+u3 : 0.1
+p1
+p1
+p1
+p2
+p2
+p2
+p3
+p3
+p3
+(a)
+p1 × 0 : x1y1
+p1 × 0.3 : x1y2
+p1 × 0.2 : x1y3
+p2 × 0.2 : x2y1
+p2 × 0 : x2y2
+p2 × 0.3 : x2y3
+p3 × 0.4 : x3y1
+p3 × 0.1 : x3y2
+p3 × 0 : x3y3
+u4 : 0.1
+p1
+p2
+p3
+(b)
+p1 × 0 : x1y1
+p1 × 0.2 : x1y2
+p1 × 0.2 : x1y3
+p2 × 0.1 : x2y1
+p2 × 0 : x2y2
+p2 × 0.3 : x2y3
+p3 × 0.4 : x3y1
+p3 × 0 : x3y2
+p3 × 0 : x3y3
+u5 : 0.1
+p1
+p2
+p3
+(c)
+p1 × 0 : x1y1
+p1 × 0.2 : x1y2
+p1 × 0.1 : x1y3
+p2 × 0 : x2y1
+p2 × 0 : x2y2
+p2 × 0.3 : x2y3
+p3 × 0.3 : x3y1
+p3 × 0 : x3y2
+p3 × 0 : x3y3
+u6 : 0.1
+p1
+p2
+p3
+(d)
+Fig. 3: An illustrative representation of Algorithm 1.
+
+12
+p1 × 0 : x1y1
+p1 × 0.2 : x1y2
+p1 × 0 : x1y3
+p2 × 0 : x2y1
+p2 × 0 : x2y2
+p2 × 0.2 : x2y3
+p3 × 0.2 : x3y1
+p3 × 0 : x3y2
+p3 × 0 : x3y3
+u7 : 0.2
+p1
+p2
+p3
+(e)
+Fig. 3: An illustrative representation of Algorithm 1 (cont.).
+in Figure 4b, the mapping pU|XY does not depend on a specific choice of them. This feature of Algorithm 1 is not
+only of practical interest (e.g., when the distribution of the private data is unknown or difficult to estimate), but
+also helpful in theory, as used in Corollary 3.1. Finally, it is important to emphasize that for a fixed pY |X, it is the
+proposed privacy-preserving mapping pU|XY that is pX-invariant, not the resulting utility, i.e., I(Y ; U).
+Remark 5. (Number of iterations) In the explanation of Algorithm 1, it is stated that since at each iteration of
+the algorithm, at least one water level is filled, and the algorithm terminates after all these levels are filled, the
+number of iterations is upper bounded by the number of non-zero remaining probabilities prior to the start of the
+iterations, which is at most |supp(X, Y )| − |I|. While this is correct, we observe that, as in Figure 3d, in the very
+last iteration we have exactly |X| non-zero and equal water levels, one in each subgroup, that are filled together
+in one iteration. This is a direct consequence of the fact that at each step of producing a new realization for U
+in the algorithm, i) each subgroup of nodes has the same amount of total water levels, and ii) each subgroup of
+nodes undergoes the same amount of decrement in water levels. As a result, in the last iteration of the algorithm,
+we are left with |X| equal (non-zero) remaining water levels to be filled at once with the last realization of U.
+Therefore, the algorithm terminates after N iterations with N ≤ |supp(X, Y )| − |I| − |X| + 1. Moreover, since at
+each iteration, we get a realization for U, and we already have |I| realizations before the iterations start, we have
+|U| ≤ N + |I| = |supp(X, Y )| − |X| + 1.
+The following Lemma is needed to obtain an upper bound on G0(X, Y ) in the sequel.
+
+13
+p1 × 0.2 : x1y1
+p1 × 0.5 : x1y2
+p1 × 0.3 : x1y3
+p2 × 0.4 : x2y1
+p2 × 0.2 : x2y2
+p2 × 0.4 : x2y3
+p3 × 0.6 : x3y1
+p3 × 0.3 : x3y2
+p3 × 0.1 : x3y3
+u1 : 0.2
+u2 : 0.2
+u3 : 0.1
+u4 : 0.1
+u5 : 0.1
+u6 : 0.1
+u7 : 0.2
+p1
+p1
+p1
+p2
+p2
+p2
+p3
+p3
+p3
+p1
+p2
+p3
+p1
+p2
+p3
+p1
+p2
+p3
+p1
+p2
+p3
+(a) Backward construction: pX,Y,U = pX,Y |U · pU.
+p1 × 0.2 : x1y1
+p1 × 0.5 : x1y2
+p1 × 0.3 : x1y3
+p2 × 0.4 : x2y1
+p2 × 0.2 : x2y2
+p2 × 0.4 : x2y3
+p3 × 0.6 : x3y1
+p3 × 0.3 : x3y2
+p3 × 0.1 : x3y3
+u1 : 0.2
+u2 : 0.2
+u3 : 0.1
+u4 : 0.1
+u5 : 0.1
+u6 : 0.1
+u7 : 0.2
+1
+2
+5
+1
+3
+1
+2
+1
+1
+4
+1
+3
+2
+3
+1
+1
+3
+1
+4
+1
+3
+1
+3
+1
+4
+1
+6
+1
+5
+1
+4
+1
+6
+2
+5
+1
+2
+1
+3
+(b) Forward construction: pX,Y,U = pX,Y · pU|X,Y
+Fig. 4: The output of Algorithm 1.
+Lemma 2. Let f(X) be a function of X ∼ p such that it has at least two realizations. We have
+H(f(X)) ≥ Hb(min
+x p(x)).
+(12)
+Proof. The proof is provided in Appendix B.
+Theorem 2. For a given pair (X, Y ) ∼ pXY , we have
+G0(X, Y ) ≤ H(Y ) −
+�
+1 −
+�
+y
+min
+x p(y|x)
+�
+Hb
+�
+min
+x pX(x)
+�
+.
+(13)
+Proof. If Y is a singleton, i.e., |Y| = 1, we have G0(X, Y ) = 0 and (13) follows, since minx p(y|x) = 1. Therefore,
+in what follows, we assume that |Y| ≥ 2.
+From Lemma 1, in an optimal mapping pU|XY , for any (x, u) ∈ X × U, there exists exactly one yx,u ∈ Y such
+that p(x, yx,u, u) > 0, and we have p(x, yx,u|u) = p(x). For any y ∈ Y, let Uy ≜ {u ∈ U|p(x, y, u) > 0, ∀x ∈ X}
+be the set of realizations of U which are connected to pairs (x1, y), (x2, y), . . . , (x|X |, y). Define ˜U ≜ ∪y∈YUy.
+Since Uy′ ∩ Uy′′ = ∅ when y′ ̸= y′′, it is immediate that H(Y |U = u) = 0, ∀u ∈ ˜U. Since Y conditioned on
+{U = u}, ∀u ̸∈ ˜U is a function of X which has at least two realizations, from Lemma 2, we get H(Y |U = u) ≥
+Hb (minx pX(x)) , ∀u ̸∈ ˜U.
+
+14
+For an arbitrary y ∈ Y, we have
+p(x)p(y|x) =
+�
+u∈U
+p(x, y, u)
+=
+�
+u∈Uy
+p(x, y, u) +
+�
+u̸∈Uy
+p(x, y, u)
+≥
+�
+u∈Uy
+p(u)p(x, y|u)
+= p(x)Pr{U ∈ Uy}, ∀x ∈ X,
+(14)
+where (14) follows from having p(x, y|u) = p(x), ∀u ∈ Uy. Hence, we have Pr{U ∈ Uy} ≤ minx p(y|x), ∀y ∈ Y.
+Noting that ˜U is the union of disjoint sets Uy, ∀y ∈ Y, we get
+Pr{U ∈ ˜U} =
+�
+y
+Pr{U ∈ Uy}
+≤
+�
+y
+min
+x p(y|x).
+(15)
+We can write
+H(Y |U) =
+�
+u∈U
+p(u)H(Y |U = u)
+=
+�
+u∈ ˜U
+p(u)H(Y |U = u) +
+�
+u̸∈ ˜U
+p(u)H(Y |U = u)
+≥
+�
+u∈ ˜U
+p(u) · 0 +
+�
+u̸∈ ˜U
+p(u)Hb(min
+x pX(x))
+=
+�
+1 − Pr{U ∈ ˜U}
+�
+Hb(min
+x pX(x))
+≥
+�
+1 −
+�
+y
+min
+x p(y|x)
+�
+Hb(min
+x pX(x)),
+(16)
+where (16) follows from (15). This proves (13).
+Example 2. Consider (X, Y ) ∼ pXY , in which X is uniformly distributed over [0 : K − 1], where K > 2 is an
+arbitrary integer. Let Y conditioned on {X = x} be uniformly distributed on [x : x + K − 2] mod K. Hence, Y
+is also uniformly distributed over [0 : K − 1]. In this setting, the upper bounds in (4) and (13) are
+H(Y |X) = log(K − 1)
+H(Y ) −
+�
+1 −
+�
+y
+min
+x p(y|x)
+�
+Hb(min
+x p(x)) = K − 1
+K
+log(K − 1).
+Obviously, the bound in (13) is tighter in this example, and it can be readily verified that Algorithm 1 achieves it.
+The following lemma is needed in its following Theorem.
+
+15
+Lemma 3. For the mass probabilities p1, p2, p3 (i.e., pi ≥ 0, i ∈ [3], �
+i pi = 1), we have
+Hb(pi) ≤ Hb(pj) + Hb(pk), i, j, k ∈ [3], j ̸= k,
+(17)
+where Hb(·) denotes the binary entropy function.
+Proof. The proof is provided in Appendix C.
+Theorem 3. When X is binary, Algorithm 1 is optimal, and we have
+G0(X, Y ) = H(Y ) −
+�
+1 −
+�
+y
+min
+x p(y|x)
+�
+H(X).
+(18)
+Also, when (|X|, |Y|) = (3, 2), Algorithm 1 is optimal, and letting Y ≜ {y1, y2}, if we label the elements of X
+according to x1 ≜ minx∈X p(y1|x), x2 ≜ minx∈X p(y2|x), we have
+G0(X, Y ) = H(Y ) − (p(y1|x3) − p(y1|x1)) Hb(p1) − (p(y2|x3) − p(y2|x2)) Hb(p2),
+(19)
+where pi ≜ p(xi), i ∈ [3].
+Proof. When X is binary, we have H(X) = Hb(minx p(x)), and from (13) and Theorem 1, (18) is obtained, and
+Algorithm 1 achieves it.
+When (|X|, |Y|) = (3, 2), we prove the optimality by the simplex method [18]. As already stated, G0(X, Y ) can
+be obtained via an LP. The problem is to find values for pU(·) in Figure 2 such that H(Y |U) is minimized and
+pXY (·, ·) = �
+u pXY |U(·, ·|u)pU(u). For i, j, k ∈ [2], let Pijk denote the probability of that u which is connected
+to (x1, yi), (x2, yj), (x3, yk) with transition probabilities p1, p2 and p3, respectively. For example, in Figure 2, we
+have P111 = p(u1), P112 = p(u2), P121 = p(u3), and so on. As a result, the LP minimizes H(Y |U), which is
+P111 · 0 + P112Hb(p3) + P121Hb(p2) + P122Hb(p1) + P211Hb(p1) + P212Hb(p2) + P221Hb(p3) + P222 · 0 (20)
+over the non-negative values of Pijk, i, j, k ∈ [2] such that
+P111 + P112 + P121 + P122 = p(y1|x1)
+P121 + P122 + P221 + P222 = p(y2|x2)
+P111 + P121 + P211 + P221 = p(y1|x3)
+P111 + P112 + P121 + P122 + P211 + P212 + P221 + P222 = 1.
+(21)
+Changing the order of the variables, the simplex tableau for this LP is provided in Table I. By performing Gaussian
+elimination (i.e., subtracting row 1 from row 3, and then subtracting the sum of row 1, row 2, and the resulting
+row 3 from row 4), we obtain a canonical tableau as in Table II. Note that all the elements of the rightmost column
+are non-negative due to the initial convention x1 ≜ minx∈X p(y1|x), x2 ≜ minx∈X p(y2|x).
+The first four columns of the tableau in Table II form a basis, and
+[P111, P222, P211, P212, P112, P121,P122, P221]T =
+[p(y1|x1), p(y2|x2), p(y1|x3) − p(y1|x1), p(y2|x3) − p(y2|x2), 0, 0, 0, 0]T
+(22)
+
+16
+P111
+P222
+P211
+P212
+P112
+P121
+P122
+P221
+1
+0
+0
+0
+1
+1
+1
+0
+p(y1|x1)
+0
+1
+0
+0
+0
+1
+1
+1
+p(y2|x2)
+1
+0
+1
+0
+0
+1
+0
+1
+p(y1|x3)
+1
+1
+1
+1
+1
+1
+1
+1
+1
+TABLE I: Simplex tableau of order 4 and dimension 8.
+P111
+P222
+P211
+P212
+P112
+P121
+P122
+P221
+1
+0
+0
+0
+1
+1
+1
+0
+p(y1|x1)
+0
+1
+0
+0
+0
+1
+1
+1
+p(y2|x2)
+0
+0
+1
+0
+-1
+0
+-1
+1
+p(y1|x3) − p(y1|x1)
+0
+0
+0
+1
+1
+-1
+0
+-1
+p(y2|x3) − p(y2|x2)
+TABLE II: Canonical form.
+is a basic feasible solution, which results in
+H(Y |U) = (p(y1|x3) − p(y1|x1)) Hb(p1) + (p(y2|x3) − p(y2|x2)) Hb(p2).
+(23)
+In order to show that no other feasible solution outperforms (22), i.e., resulting in a smaller H(Y |U) than (23),
+we proceed as follows. For any feasible solution ˜Pijk, i, j, k ∈ [2], with some calculations, we get
+H(Y |U) = (p(y1|x3) − p(y1|x1)) Hb(p1) + (p(y2|x3) − p(y2|x2)) Hb(p2)
+(24)
++ (Hb(p3) + Hb(p1) − Hb(p2)) ˜P112 + 2Hb(p2) ˜P121
+(25)
++ 2Hb(p1) ˜P122 + (Hb(p3) + Hb(p2) − Hb(p1)) ˜P221.
+(26)
+The RHS in (24) is equal to (23). From Lemma 3 and non-negativity of entropy, all the remaining terms in (25)
+and (26) are non-negative. As a result, no other feasible solution can produce a smaller H(Y |U) than (23), which
+proves (19).
+Corollary 3.1. When |X| = 2, or (|X|, |Y|) = (3, 2), for fixed pY |X, the optimal p∗
+U|X,Y is pX-invariant 12, and
+G0(X, Y ) is convex in pX.
+Proof. The first part of this claim is proved by combining the optimality of Algorithm 1 in Theorem 3, and Remark
+4.
+In what follows, we provide two methods to prove the second part of the claim. The first method makes use of
+pX-invariance, while the second one relies on directly inspecting G0(X, Y ) in (18) and (19).
+12This, however, does not hold in general.
+
+17
+A. First method
+Fix λ ∈ (0, 1). Let pW|XY , pV |XY and pU|XY be optimal solutions in the evaluation of G0(p′
+X ·pY |X), G0(p′′
+X ·
+pY |X) and G0
+�
+(λp′
+X + ¯λp′′
+X) · pY |X
+�
+, respectively. From Corollary 3.1, we can set W = V = U, and write
+pW|XY = pV |XY = pU|XY .
+(27)
+In the sequel, by pXY W , pXY V and pXY U, we are referring to p′
+X · pY |X · pW|XY , p′′
+X · pY |X · pV |XY and
+(λp′
+X + ¯λp′′
+X) · pY |X · pU|XY , respectively, which share the same support, denoted by S, and induce
+pY U = λpY W + ¯λpY V .
+(28)
+For any tuple (x, y, w) ∈ S, we have
+pW (w) = pXY W (x, y, w)
+pXY |W (x, y|w)
+= p′
+X(x)pY |X(y|x)pW|XY (w|x, y)
+pX|W (x|w)pY |XW (y|x, w)
+= p′
+X(x)pY |X(y|x)pW|XY (w|x, y)
+p′
+X(x)
+(29)
+= pY |X(y|x)pW|XY (w|x, y),
+(30)
+where (29) follows from i) having X ⊥⊥ W in pXY W , and ii) having pY |XW (y|x, w) = 1, since (x, y, w) ∈ S.
+From (27) and (30), we get that pU(·) = pW (·) = pV (·), which in conjunction with (28) and convexity of
+I(A; B) in pA|B for fixed pB, results in
+I(Y ; U) ≤ λI(Y ; W) + ¯λI(Y ; V ).
+This is equivalent to
+G0
+�
+(λp′
+X + ¯λp′′
+X) · pY |X
+�
+≤ λG0(p′
+X · pY |X) + ¯λG0(p′′
+X · pY |X),
+which completes the proof.
+B. Second method
+According to [21], for fixed pY |X, the minimum value of λ, for which H(Y ) − λH(X) is a convex functional
+of pX is suppX s∗(X; Y ), where s∗(X; Y ) ≜ supZ:Z−X−Y,Z̸⊥⊥X
+I(Z;Y )
+I(Z;X) is the strong data processing coefficient.
+Since we have 1 − �
+y minx p(y|x) ≥ suppX s∗(X; Y ) (see [6, Corollary 6]), we conclude that G0(X, Y ) in (18)
+is convex in pX for fixed pY |X.
+For fixed pY |X, denoting αi ≜ p(y1|xi), i ∈ [3] for simplicity, (19) becomes
+f(p1, p2) ≜ G0(X, Y ) = Hb(py) − (α3 − α1)Hb(p1) − (α2 − α3)Hb(p2),
+(31)
+where py ≜ (α1 − α3)p1 + (α2 − α3)p2 + α3.
+
+18
+To prove the convexity of G0 in pX, we show that f is convex in (p1, p2). By some calculations, the Hessian
+matrix of f is obtained as
+∇2f =
+
+a + b
+√ac
+√ac
+c + d
+
+ ,
+(32)
+with a ≜ − (α1−α3)2
+py(1−py) , b ≜
+α3−α1
+p1(1−p1), c ≜ − (α2−α3)2
+py(1−py) and d ≜
+α2−α3
+p2(1−p2).
+The characteristic polynomial of ∇2f is λ2−(a+b+c+d)λ+ad+b(c+d) whose roots determine the eigenvalues.
+From the initial convention, we have α1 ≤ α3 ≤ α2. This, in conjunction with the inequality αi ¯αj + αj ¯αi ≥
+|αi − αj|, i, j ∈ [3] results in a + b + c + d ≥ 0 and ad + b(c + d) ≥ 0, which in turn means that the eigenvalues
+of ∇2f are non-negative. Therefore, ∇2f is positive semi-definite and f is convex in (p1, p2).
+The convexity result in Corollary 3.1 is not specific to |X| = 2 or (|X|, |Y|) = (3, 2) as the second part of the
+following Theorem indicates.
+Theorem 4. For given ǫ ≥ 0 and pX, Gǫ(X, Y ) is concave in pY |X. Furthermore, for given pY |X, G0(X, Y ) and
+g0(X, Y ) are convex in pX.
+Proof. The first part of the claim is proved as follows. Fix ǫ ≥ 0. Given two conditional pmfs p′
+Y |X and p′′
+Y |X, let
+pW|XY and pV |XY be maximizers of Gǫ(pX · p′
+Y |X) and Gǫ(pX · p′′
+Y |X) in (2), respectively. In other words, when
+(X, Y ) is distributed according to pX · p′
+Y |X (or pX · p′′
+Y |X), an optimal privacy-preserving mapping is pW|XY (or
+pV |XY ). In the sequel, pXY W and pXY V refer to pX · p′
+Y |X · pW|XY and pX · p′′
+Y |X · pV |XY , respectively. Without
+loss of optimality, select the alphabets W and V, such that W ∩ V = ∅. Fix λ ∈ (0, 1), and let U ≜ W ∪ V. Let
+(X, Y ) ∼ pX · (λp′
+Y |X + ¯λp′′
+Y |X), and define the following conditional pmf
+pU|XY (u|x, y) ≜ λp′(y|x)pW|XY (u|x, y) · 1{u∈W} + ¯λp′′(y|x)pV |XY (u|x, y) · 1{u∈V}
+λp′(y|x) + ¯λp′′(y|x)
+,
+(33)
+for all (u, x, y) ∈ U × supp(X, Y ).
+In what follows, we show that the joint pmf pXY U induced by (33) results in I(X; U) ≤ ǫ and I(Y ; U) ≥
+λI(Y ; W) + ¯λI(Y ; V ), which completes the proof.
+The construction in (33) results in
+pU(u) = λpW (u) · 1{u∈W} + ¯λpV (u) · 1{u∈V}
+(34)
+pXY |U(x, y|u) = pXY |W (x, y|u) · 1{u∈W} + pXY |V (x, y|u) · 1{u∈V}, ∀(u, x, y) ∈ U × supp(X, Y ).
+(35)
+Let E ≜ 1{U∈W} be a binary r.v. defined as a function of U. From (34), we have pE(1) = λ. Also,
+pE|X(1|x) =
+�
+y
+pEY |X(1, y|x)
+=
+�
+y
+pY |X(y|x)pE|XY (1|x, y)
+=
+�
+y
+(λp′(y|x) + ¯λp′′(y|x))pE|XY (1|x, y)
+=
+�
+y
+(λp′(y|x) + ¯λp′′(y|x))
+�
+u∈W
+pU|XY (u|x, y)
+
+19
+=
+�
+y
+(λp′(y|x) + ¯λp′′(y|x))
+�
+u∈W
+λp′(y|x)pW|XY (u|x, y)
+λp′(y|x) + ¯λp′′(y|x)
+(36)
+=
+�
+y
+(λp′(y|x) + ¯λp′′(y|x))
+λp′(y|x)
+λp′(y|x) + ¯λp′′(y|x)
+= λ, ∀x ∈ X
+(37)
+where (36) results from (33), and (37) results in E ⊥⊥ X with pE(1) = λ.
+We can write
+pXY U|E(x, y, u|e) = pU(u)pXY |U(x, y|u)pE|U(e|u)
+pE(e)
+= pU(u)
+�
+pXY |W (x, y|u) · 1{e=1} + pXY |V (x, y|u) · 1{e=0}
+�
+λ · 1{e=1} + ¯λ · 1{e=0}
+(38)
+= pXY W (x, y, u) · 1{e=1} + pXY V (x, y, u) · 1{e=0}, e ∈ {0, 1},
+(39)
+where (38) and (39) follow from (35) and (34). Therefore,
+I(X; U) = I(X; U, E)
+(40)
+= I(X; U|E)
+(41)
+= λI(X; U|E = 1) + ¯λI(X; U|E = 0)
+= λI(X; W) + ¯λI(X; V )
+(42)
+≤ ǫ,
+(43)
+where in (40), E is a deterministic function of U, and (41) results from X ⊥⊥ E. We obtain (42) from (39).
+From (43), we are allowed to write
+Gǫ
+�
+pX · (λp′
+Y |X + ¯λp′′
+Y |X)
+�
+≥ I(Y ; U)
+= I(Y ; U, E)
+≥ I(Y ; U|E)
+= λI(Y ; U|E = 1) + ¯λI(Y ; U|E = 0)
+= λI(Y ; W) + ¯λI(Y ; V )
+(44)
+= λGǫ(pX · p′
+Y |X) + ¯λGǫ(pX · p′′
+Y |X),
+(45)
+where (44) follows from (39). Finally, by noting that ǫ and λ were chosen arbitrarily, (45) proves that for fixed pX,
+Gǫ(pX · pY |X) is concave in pY |X.
+To prove the second part of the claim, we proceed as follows. Fix pY |X and λ ∈ (0, 1). Given two pmfs p′
+X and
+p′′
+X, let pX = λp′
+X + ¯λp′′
+X. Let p∗
+U|XY be an optimal mapping in the evaluation of G0(pX · pY |X), which induces
+U ∗ ∼ pU∗. Let Iq denote I(Y ; U) when (X, Y, U) ∼ q · pY |X · p∗
+U|XY . Obviously, G0(pX · pY |X) = IpX.
+
+20
+When (X, Y, U) ∼ q · pY |X · p∗
+U|XY , where q is an arbitrary pmf on X, we have
+pU|X(·|x) =
+�
+y
+pY |X(y|x)p∗
+U|XY (·|x, y)
+= pU∗|X(·|x)
+= pU∗(·),
+(46)
+and hence, X ⊥⊥ U. Therefore, we have by definition
+Iq ≤ G0(q · pY |X).
+(47)
+When (X, Y, U) ∼ q · pY |X · p∗
+U|XY , we have
+pY |U(·|·) =
+�
+x
+q(x)pY |XU
+=
+�
+x
+q(x)
+pY |X(·|x)p∗
+U|XY (·|x, ·)
+pU∗(·)
+,
+(48)
+which means that the conditional pmf pY |U derived from (λp′
+X + ¯λp′′
+X) · pY |X · p∗
+U|XY is λ times the conditional
+pmf pY |U derived from p′
+X · pY |X · p∗
+U|XY plus ¯λ times the conditional pmf pY |U derived from p′′
+X · pY |X · p∗
+U|XY .
+Hence, we can write
+G0((λp′
+X + ¯λp′′
+X) · pY |X) = Iλp′
+X+¯λp′′
+X
+≤ λIp′
+X + ¯λIp′′
+X
+(49)
+≤ λG0(p′
+X · pY |X) + ¯λG0(p′′
+X · pY |X),
+(50)
+where (49) results from the convexity of I(Y ; U) in pY |U for fixed pU, and (50) follows from (47).
+Finally, the above analysis for proving the convexity of G0(X, Y ) remains valid if X − Y − U form a Markov
+chain, and we replace p∗
+U|XY with p∗
+U|Y , which is an optimal mapping in the evaluation of g0(pX ·pY |X). Therefore,
+for fixed pY |X, g0(X, Y ) is also convex in pX.
+Example 3. Let (X, Y ) ∈ {x1, x2} × {y1, y2}, in which p ≜ p(x1), and the transition from X to Y follows
+a general binary asymmetric channel (BAC) with cross over probabilities α, β ∈ [0, 1], i.e., α ≜ p(y2|x1) and
+β ≜ p(y1|x2). Therefore, we have q ≜ p(y1) = p¯α + ¯pβ. From Theorem 2, we have that
+G0(X, Y ) = Hb(q) −
+�
+1 − min{α, ¯β} − min{β, ¯α}
+�
+Hb(p),
+= Hb(q) − |α − ¯β|Hb(p).
+(51)
+It is already known that for a given p, G0(X, Y ) is concave in (α, β), and the maximizer is (α∗, β∗) = ( 1
+2, 1
+2),
+which results in X ⊥⊥ Y , and H(Y ) = 1.
+For a given (α, β), G0(X, Y ) is convex in p. Therefore, by setting
+d
+dpG0(X, Y ) = 0, we solve for p∗ as as
+p∗ =
+β
+α + β 1{α< ¯β} +
+¯β
+¯α + ¯β 1{α> ¯β},
+
+21
+and when α = ¯β, p∗ is arbitrary, since we get G0(X, Y ) = Hb(β) = Hb(α) irrespective of the value of p due to
+the independence of X and Y .
+In the special case of α = β, i.e., if the transition from X to Y is a binary symmetric channel (BSC(α)), we
+have that p∗ = 1
+2, if α ̸= 1
+2, and any number in [0, 1] if α = 1
+2. Therefore, we get G0(X, Y ) ≥ 2 min{α, ¯α}. Taking
+Corollary 3.1 into account, this means that if for a pair (X, Y ) whose pY |X is BSC(α), the curator is unaware of
+pX, he can still obtain the optimal mapping and make sure that the utility of this release is at least 2 min{α, ¯α}.
+As another case, let (X, Y ) ∈ {x1, x2} × {y0, y1, y2}, in which p(x1) ≜ p, and the transition from X to Y is a
+binary eraser channel with erasure probability of e, i.e., BEC(e), in which p(yi|xj) = e1{i=0} + ¯e1{i=j}, (i, j) ∈
+[0 : 2] × [2]. Denote the entropy of a ternary random variable with mass probabilities p1, p2, p3 by H(p1, p2, p3).
+From Theorem 2, we have
+G0(X, Y ) = H(Y ) − (1 −
+�
+y
+min
+x p(y|x))H(X)
+= H(p¯e, e, ¯p¯e) − ¯eHb(p)
+= Hb(e),
+and U ≜ 1{Y =y0} attains it. Moreover, since this mapping is only from Y to U, we conclude that g0(X, Y ) = Hb(e).
+By changing the role of X and Y , and applying the second part of Theorem 3, we get
+G0(Y, X) = Hb(p) − pHb(¯p¯e) − ¯pHb(p¯e),
+where U ≜ X · 1{Y =y0} + ˜X · 1{Y ̸=y0}, in which ˜X ∈ X is a Bernoulli random variable independent of (X, Y )
+with p ˜
+X(x1) = p, achieves it. We also have g0(Y, X) = 0 from [1, Corollary 2] by noting that the nullity of PY |X
+is zero.13
+Thus far, we have presented an achievable scheme in Algorithm 1 as a lower bound on G0(X, Y ). Based on
+this scheme, we proceed to present a privacy-preserving algorithm as a lower bound on Gǫ(X, Y ). On the U-P
+plane, the privacy restriction becomes stricter as we move from right to left. The rightmost point (I(X; Y ), H(Y ))
+is achieved when there is no constraint on privacy, while the leftmost point (0, G0(X, Y )) relates to the strictest
+privacy restriction, which is statistical independence between the private and the released data. Therefore, it makes
+sense to obtain achievable points on the U-P plane starting from no privacy constraint and increasing the restrictions
+incrementally until we reach the requirement X ⊥⊥ U. One approach is as follows. Let R ≜ (I(X; Y ), H(Y )) denote
+the rightmost point on the U-P trade-off curve. In order to obtain an achievable point P1, we impose the requirement
+that X must be at least 1-independent of U, i.e., X
+1
+⊥⊥ U. In other words, we require that X must have at least
+one realization, call it x1, whose posterior probability p(x1|u) is the same as the prior p(x1) for any realization u
+13With some back of the envelope calculations, it can be verified that for an M-ary erasure channel (M ≥ 2), in which (X, Y ) ∈
+{x1, . . . , xM } × {y0, y1, . . . , yM} and p(yi|xj) = e1{i=0} + ¯e1{i=j}, (i, j) ∈ [0 : M] × [M], we have g0(X, Y ) = G0(X, Y ) =
+H(Y |X) = Hb(e) attained by U ≜ 1{Y =y0}. Furthermore, denoting the probability vector associated with pX by p, we have g0(Y, X) = 0
+and G0(Y, X) = H(X) − �M
+i=1 piH(¯ep + e1i) attained by U ≜ X · 1{Y =y0} + ˜
+X · 1{Y ̸=y0}, where 1i is the i-th standard unit vector,
+and ˜
+X ⊥⊥ (X, Y ) is distributed according to pX.
+
+22
+of U. Letting 0 not be an element of X, define an auxiliary random variable Z1 ≜ X · 1{X=x1}. The constraint
+of having X at least 1-independent of the released data can be satisfied by designing a privacy-preserving scheme
+via Algorithm 1 for the new pair (Z1, Y ), i.e., pU1|Z1Y , which guarantees Z1 ⊥⊥ U1, or equivalently X
+1
+⊥⊥ U1. The
+mapping pU1|XY , which is induced by pU1|Z1Y , results in the achievable point P1 ≜ (I(X; U1), I(Y ; U1)). To get
+P2, we require that X must be at least 2-independent of the released data. To this end select arbitrary x1, x2 ∈ X,
+and define Z2 ≜ X · 1{X∈{x1,x2}}, and obtain a mapping pU2|Z2Y via Algorithm 1, which satisfies X
+2
+⊥⊥ U2. The
+corresponding pU2|XY produces P2 ≜ (I(X; U2), I(Y ; U2)). This procedure continues providing achievable points
+until we reach the constraint X ⊥⊥ U, which is taken care of by applying Algorithm 1 to the pair (X, Y ). Finally,
+the upper concave envelope of the set of these achievable points results in a lower bound on Gǫ(X, Y ), which is
+formally presented in the following Proposition..
+Proposition 1. (Privacy-preserving mapping - a lower bound on Gǫ(X, Y )) Let (X, Y ) ∼ pXY be given. Let
+k ∈ [|X| − 2] and X ′ be an arbitrary subset of X with size k. Without loss of generality, assume that 0 ̸∈ X, and
+let Z be a function of X defined as
+Z ≜ X · 1{X∈X ′}.
+(52)
+Applying the achievable scheme in Theorem 1, i.e., Algorithm 1, to the pair (Z, Y ) results in a mapping pU|ZY ,
+such that Z ⊥⊥ U, or equivalently, X
+k
+⊥⊥ U. Calculate pU|XY from pU|ZY , and set PX ′ ≜ (I(X; U), I(Y ; U)).
+Apply Algorithm 1 to (X, Y ) and obtain an achievable point denoted by L. Let P ≜ ∪X ′⊂X {PX ′}∪{L, R} denote
+the set of all the achievable points obtained so far. Finally, we have the U-P trade-off uce[P](ǫ) as a lower bound
+on Gǫ(X, Y ), ∀ǫ ∈ [0, I(X; Y )].
+For the tuple (Z, X, Y, U) in Proposition 1, it can be readily verified that
+pZ(z) = pX(z) · 1{z̸=0} + (
+�
+x∈X \X ′
+pX(x)) · 1{z=0}
+pY |Z(y|z) = pY |X(y|z) · 1{z̸=0} +
+�
+x∈X \X ′ pXY (x, y)
+�
+x∈X \X ′ pX(x)
+· 1{z=0},
+(53)
+pU|XY (u|x, y) = pU|ZY (u|x, y) · 1{x∈X ′} + pU|ZY (u|0, y) · 1{x̸∈X ′}.
+The quantity I(Y ; U) is obtained after pU|ZY is obtained in the algorithm (unless |X ′| = 1, for which Theorem 2
+gives a closed-form solution), while I(X; U) can be obtained prior to the algorithm as
+I(X; U) = I(X, Z; U)
+(54)
+= I(Z; U) + I(X; U|Z)
+= I(X; U|Z)
+(55)
+= I(X; U, Y |Z)
+(56)
+
+23
+= I(X; Y |Z) + I(X; U|Z, Y )
+= I(X; Y |Z)
+(57)
+= I(X; Y ) − I(Z; Y ),
+(58)
+where (54) follows from defining Z as a function of X in (52), and (55) results from Z ⊥⊥ U in Algorithm 1. The
+equality in (56) follows from satisfying the conditions of Lemma 1 in Algorithm 1. In other words, conditioned
+on the event {Z = z}, Y is a function of U, or equivalently H(Y |Z, U) = 0. The equality in (57) results from
+the Markov chain X − (Z, Y ) − U, since U is generated by applying Algorithm 1 to the pair (Z, Y ). Finally, (58)
+follows from the fact that Z − X − Y form a Markov chain.
+The procedure in Proposition 1 is computationally complex when |X| is large, since there are 2|X | − |X| − 2
+nonempty subsets of X with size at most |X| − 2. Therefore, for large |X|, we can restrict the analysis to a fixed
+collection of subsets denoted by {Xk} for k ∈ [|X| − 2], in which |Xk| = k and Xj ⊂ Xk if j ≤ k. Let (Zk, Uk)
+be the same as (Z, U) in Proposition 1 when X ′ = Xk, k ∈ [|X| − 2]. On the U-P plane, the slope of the line
+connecting R (= (I(X; Y ), H(Y ))) to (I(X; Uk), I(Y ; Uk)) is
+mk ≜
+H(Y ) − I(Y ; Uk)
+I(X; Y ) − I(X; Uk)
+= H(Y ) − I(Y ; Uk)
+I(Zk; Y )
+(59)
+≤
+min
+�
+H(Y ),
+�
+1 − �
+y minz pY |Zk(y|z)
+�
+min{H(Zk), log |Y|}
+�
+I(Zk; Y )
+,
+(60)
+where (59) and (60) follow from (58) and (10), respectively. We denote the upper bound in (60) by f(pXY , Xk).
+Knowing that Gǫ(X, Y ) is a concave and non-decreasing curve, a heuristic approach is to select X1 such that m1
+is minimized, and an even simpler approach would be to minimize the upper bound, i.e., f(pXY , X1). Therefore,
+we set
+xk ≜
+arg min
+x∈X \Xk−1:
+Xk=Xk−1∪{x}
+f(pXY , Xk),
+∀k ∈ [|X| − 2], X0 ≜ ∅.
+(61)
+The procedures of this achievable scheme are provided in Algorithm 2.
+Remark 6. (Non-algorithmic U-P trade-off) The achievable points in Proposition 1 are obtained after applying
+Algorithm 1 to each constructed pair (Z, Y ). More specifically, it is the y coordinate of these points that are
+obtained after the application of the algorithm, since the x coordinates are already known prior to the algorithm
+as in (53). If we replace these y coordinates with their corresponding lower bounds according to (10), we obtain
+a new set of achievable points. Obviously, these points lie below the initial set of points, but they are obtained
+without the need for the algorithm. Therefore, preserving (52) and its preceding assumptions in Proposition 1, we
+
+24
+Algorithm 2 Privacy-preserving mapping (simplified version of Proposition 1)
+1: function ALGORITHM2(pX,Y )
+2:
+Select the collection {Xk}, k ∈ [|X| − 2], according to (61).
+3:
+pU|XY ≜ Algorithm1(pY |X)
+4:
+P ≜ {(0, I(Y ; U)), (I(X; Y ), H(Y ))}
+5:
+Z0 ≜ 0
+6:
+k = 1
+7:
+while I(Zk−1; Y ) ̸= I(X; Y ) do
+8:
+Zk ≜ X · 1{X∈Xk}
+9:
+pUk|ZkY ≜ Algorithm1(pY |Zk)
+10:
+P ≜ P ∪ {(I(X; Uk), I(Y ; Uk))}
+11:
+k = k + 1
+12:
+end while
+13:
+return uce[P](·)
+14: end function
+set
+˜PX ′ ≜
+�
+I(X; Y ) − I(X; Z) ,
+�
+H(Y ) −
+�
+1 −
+�
+y
+min
+z
+pY |Z(y|z)
+�
+min{H(Z), log |Y|}
+�+ �
+(62)
+˜L ≜
+�
+0 ,
+�
+H(Y ) −
+�
+1 −
+�
+y
+min
+x pY |X(y|x)
+�
+min{H(X), log |Y|}
+�+ �
+,
+(63)
+and ˜P ≜ ∪X ′⊂X { ˜PX ′}∪{˜L, R}. The U-P trade-off uce[ ˜
+P](ǫ) is a non-algorithmic lower bound on Gǫ(X, Y ), ∀ǫ ∈
+[0, I(X; Y )]. Needless to say that this can also be applied to the simplified scheme (for large |X|) discussed in
+Algorithm 2.
+IV. PUBLIC DATA OBSERVATION
+In this section, we assume that the curator has access to only Y , and propose an achievable scheme, i.e., a lower
+bound on gǫ(X, Y ), defined in (1). We start with ǫ = 0, i.e., X ⊥⊥ U. An algorithm is proposed (Algorithm 3) that
+provides a lower bound on g0(X, Y ). Afterwards, this algorithm is used to generate a privacy-preserving mapping,
+which results in a lower bound on gǫ(X, Y ), ǫ ∈ [0 : I(X; Y )].
+Like the previous section, we start with a simple theoretical result.
+Lemma 4. ( [1, Theorem 1]) For an optimal mapping pU∗|Y in the evaluation of g0(X, Y ), we have
+|{y ∈ Y|p(y|u∗) > 0}| ≤ rank(PX|Y ), ∀u∗ ∈ U∗,
+(64)
+where PX|Y is an |X| × |Y| matrix with (i, j)-th entry equal to pX|Y (xi|yj).
+
+25
+Lemma 4 implies that in the evaluation of g0(X, Y ), if X is binary, for any u∗ ∈ U∗ (corresponding to an/the
+optimal solution), there exist at most two realizations of Y , denoted by y1, y2 ∈ Y, such that p(y1|u∗), p(y2|u∗) > 0.
+It is also obvious that if there exists only one y0 ∈ Y, such that p(y0|u∗) > 0 (and hence, p(y0|u∗) = 1), the
+condition X ⊥⊥ U indicates that this y0 must satisfy pX|Y (·|y0) = pX(·). In other words, if there exists no such y0
+satisfying pX|Y (·|y0) = pX(·), we must have |{y ∈ Y|p(y|u∗) > 0}| = 2, ∀u∗ ∈ U∗ for binary X. Therefore, in the
+achievable scheme, it makes sense to build a mapping pU|Y , such that its corresponding pY |U is in line with this
+observation. To this end, we start with the backward model, i.e., pY |U, by imposing that i) for all the realizations
+u of U, the condition in lemma 4 must be satisfied, ii) the pmf pY must be preserved in pY,U. The results are
+provided in the following Proposition. Throughout this section, we exclude the trivial case of X ⊥⊥ Y .
+Proposition 2. (A lower bound on g0(X, Y ) for binary X.) Let X ≜ {x0, x1}. First, if there exists a mass point
+ˆy ∈ Y, for which p(x0|ˆy) = p(x0), we create a corresponding u, such that p(u|y) = 1{y=ˆy}, ∀y ∈ Y. The set of
+all such ˆy’s is denoted by B ≜ {y1, . . . , y|B|}.14 Therefore, |B| realizations of U are created according to p(ui|y) ≜
+1{y=yi}, ∀i ∈ [|B|], ∀y ∈ Y. Furthermore, we have H(Y |U = u) = 0 and p(x0|u) = p(x0), ∀u ∈ {u1, . . . , u|B|}.
+Next, Y\B is considered. Note that there is no element y of this set for which p(x0|y) = p(x0). Hence, in line with
+lemma 4, we create realizations of U, each of which connected to exactly two elements of this set. Having in mind
+that we require to have p(x0|u) = p(x0) for any u ∈ U, we conclude that each of these newly created u’s must be
+connected to two elements y0, y′
+0 ∈ Y\B such that p(x0) can be written as a convex combination of p(x0|y0) and
+p(x0|y′
+0). In other words, we must have either p(x0|y0) < p(x0) < p(x0|y′
+0) or p(x0|y0) > p(x0) > p(x0|y′
+0). In this
+light, the set Y\B is divided into disjoint sets Y0 ≜ {y ∈ Y|p(x0|y) < p(x0)}, and Y′
+0 ≜ {y ∈ Y|p(x0|y) > p(x0)}.
+The purpose of this division is to make sure that p(x0) can be written as a convex combination of an arbitrary
+element of Y0 and an arbitrary element of Y′
+0. Therefore, if we create a mass point (or node) u, and connect it to
+one node in Y0, and another node in Y′
+0, with proper weights, the posterior p(x0|u) remains the same as the prior
+p(x0), which is in accordance with the condition X ⊥⊥ U. This is carried out in an iterative way, where at each
+iteration i, a mass point ui+|B| is created that is connected only to two mass points of Y, i.e., y0 from Y0, and y′
+0
+from Y′
+0, with proper weights such that p(x0|ui+|B|) = p(x0), i.e., p(y0|ui+|B|) = f(y0, y′
+0) ≜
+p(x0|y′
+0)−p(x0)
+p(x0|y′
+0)−p(x0|y0),
+and p(y′
+0|ui+|B|) = ¯f(y0, y′
+0) ≜ 1 − f(y0, y′
+0). This results in H(Y |U = ui+|B|) = Hb(f(y0, y′
+0)). Note that the
+selection of a pair (y0, y′
+0) ∈ Y0 ×Y′
+0 can be done arbitrarily; however, in order to minimize H(Y |U) heuristically,
+an asymmetric selection is carried out, i.e., y0 is the point whose corresponding p(x0|y0) is the farthest from p(x0)
+among the points in Y0, whereas, y′
+0 is the point whose corresponding p(x0|y′
+0) is the closest to p(x0) among the
+points in Y′
+0.
+In order to preserve the marginal pmf of Y in the resulting pair (Y, U), a water filling approach is utilized,
+whereby at each iteration i, the water levels of y0, y′
+0 are updated. More specifically, once y0 ∈ Y0, and y′
+0 ∈ Y′
+0
+are selected, the algorithm fills the water level of at least one of them. In the first iteration, the water levels are
+the mass probabilities p(y0) and p(y′
+0). Knowing that p(y0|u1+|B|) = f(y0, y′
+0), and p(y′
+0|u1+|B|) = ¯f(y0, y′
+0),
+14Needless to say that if p(x0|y) ̸= p(x0), ∀y ∈ Y, we have B = ∅, and |B| = 0. Also, the elements of Y have been relabeled in
+accordance with the definition of B.
+
+26
+we need to assign a mass probability to u1+|B| such that at least one of the conditions i) p(u1+|B|)f(y0, y′
+0) =
+p(y0), p(u1+|B|) ¯f(y0, y′
+0) ≤ p(y′
+0) or ii) p(u1+|B|) ¯f(y0, y′
+0) = p(y′
+0), p(u1+|B|)f(y0, y′
+0) ≤ p(y0) is valid, which is
+equivalent to having at least one water level filled and the other one not exceeded. This results in the assignment
+p(u1+|B|) ≜ min{
+p(y0)
+f(y0,y′
+0),
+p(y′
+0)
+¯f(y0,y′
+0)}. Afterwards, the water levels of y0 and y′
+0 are modified, and the algorithm
+moves on to the next iteration.
+Since at each iteration, at least one water level corresponding to an element of Y\B is filled, and at the very
+last iteration, the remaining two water levels are filled at once15, the algorithm terminates after N iterations for
+some N ≤ |Y| − |B| − 1, which results in |U| ≤ |Y| − 1. Let fmax ≜ max(y0,y′
+0)∈Y0×Y′
+0 f(y0, y′
+0). As mentioned
+before, H(Y |U = u) = 0, ∀u ∈ {u1, . . . , u|B|}. Moreover, since the conditional pmf of Y given any realization
+u ∈ U\{u1, . . . , u|B|} has two mass probabilities, i.e., f(y0, y′
+0), ¯f(y0, y′
+0) for some (y0, y′
+0) ∈ Y0 × Y′
+0, we have
+H(Y |U = u) ≤ Hb(fmax), ∀u ∈ U\{u1, . . . , u|B|}. As a result, we get H(Y |U) ≤ (1 − �
+y∈B p(y))Hb(fmax), and
+I(Y ; U) ≥
+�
+H(Y ) − (1 − �
+y∈B p(y))Hb(fmax)
+�+
+≥ (H(Y ) − 1)+. The aforementioned procedures are provided
+in Algorithm 3.
+Algorithm 3 A lower bound on g0(X, Y ) for binary X.
+1: function ALGORITHM3(pXY )
+2:
+B ≜ {y ∈ Y|p(x0|y) = p(x0)} = {y1, y2, . . . , y|B|}
+3:
+p(ui|y) = 1{y=yi}, ∀i ∈ [|B|], ∀y ∈ Y
+4:
+Y0 ≜ {y ∈ Y|p(x0|y) < p(x0)}, Y′
+0 ≜ {y ∈ Y|p(x0|y) > p(x0)}
+5:
+f(y, y′) ≜
+p(x0|y′)−p(x0)
+p(x0|y′)−p(x0|y), ¯f(y, y′) ≜ 1 − f(y, y′), ∀(y, y′) ∈ Y0 × Y′
+0
+6:
+a1(y) = p(y), ∀y ∈ Y\B
+7:
+i = 1
+8:
+while maxy ai(y) ̸= 0 do
+9:
+y0 = arg miny∈Y0{p(x0|y)|ai(y) > 0}, y′
+0 = arg miny∈Y′
+0{p(x0|y)|ai(y) > 0}
+10:
+p(ui+|B||y0) = f(y0,y′
+0)
+p(y0)
+min{ ai(y0)
+f(y0,y′
+0),
+ai(y′
+0)
+¯f(y0,y′
+0)}
+11:
+p(ui+|B||y′
+0) =
+¯
+f(y0,y′
+0)
+p(y′
+0)
+min{ ai(y0)
+f(y0,y′
+0),
+ai(y′
+0)
+¯f(y0,y′
+0)}
+12:
+p(ui+|B||y) = 0, ∀y ∈ Y\{y0, y′
+0}
+13:
+ai+1(y) = ai(y) − p(ui+|B|)
+�
+f(y0, y′
+0)1{y=y0} + ¯f(y0, y′
+0)1{y=y′
+0}
+�
+, ∀y ∈ Y\B
+14:
+i = i + 1
+15:
+end while
+16:
+return pU|Y
+17: end function
+15since otherwise, after one more iteration, we are left with a mass point y′ ∈ Y\B, such that p(x0|y′) = p(x0). This is a contradiction,
+since all such mass points are already contained in B.
+
+27
+Example 4. Consider the pair (X, Y ) ∈ {x0, x1} × {y1, y2, y3, y4}, with
+pY =
+�
+1
+2
+1
+4
+1
+8
+1
+8
+�T
+,
+PX|Y =
+
+0.3
+0.8
+0.5
+0.4
+0.7
+0.2
+0.5
+0.6
+
+ .
+(65)
+We have B = ∅, since there is no y ∈ Y for which p(x0|y) = p(x0) (= 0.4625). We have Y0 = {y1, y4}, and
+Y′
+0 = {y2, y3}. In step 6 of the algorithm, we have the first water levels as a1 = [0.5, 0.25, 0.125, 0.125]T, which
+is the same as the mass probabilities of Y . Figure 5 provides an illustrative explanation of the iterations in the
+algorithm, where the probabilities for the pair (X, Y ) are according to (65). In the first three subfigures, the
+transition probabilities are from U to Y , while in the last subfigure, it is from Y to U.
+In the first iteration, we have y0 = y1, y′
+0 = y3 according to step 9. Hence, u1 is created which connects
+to y1, y3 with transition probabilities f(y1, y3) = 0.1875, and ¯f(y1, y3) = 1 − f(y1, y3), respectively. We set
+p(u1) ≜ min{ a1(y1)
+f(y1,y3),
+a1(y3)
+¯
+f(y1,y3)} = 0.154 to fill the water level a1(y3). The water levels are updated, and we get
+a2 = [0.4712, 0.25, 0, 0.125]T shown on the RHS of yi’s in Figure 5b. In iteration 2, considered in the same figure, we
+get y0 = y1, y′
+0 = y2. Hence, u2 is created which connects to y1, y2 with transition probabilities f(y1, y2) = 0.675,
+and ¯f(y1, y2) = 1 − f(y1, y2), respectively. We set p(u2) ≜ min{ a2(y1)
+f(y1,y2),
+a2(y2)
+¯
+f(y1,y2)} = 0.698 to fill the water level
+a2(y1). Hence, we get the update a3 = [0, 0.0231, 0, 0.125]T, which is shown in Figure 5c. In the last iteration, we
+have y0 = y4, y′
+0 = y2. Hence, u3 is created which connects to y4, y2 with transition probabilities f(y4, y2) = 0.845,
+and ¯f(y4, y2) = 1 − f(y4, y2), respectively. We set p(u3) ≜ min{ a3(y4)
+f(y4,y2),
+a3(y2)
+¯
+f(y4,y2)} = 0.148 to fill the water levels
+a3(y2) and a3(y4). Finally, we get a4 = [0, 0, 0, 0]T, and the algorithm terminates after 3 iterations. The output of
+the algorithm, i.e., pU|Y , is shown in Figure 5d, which results in a utility of I(Y ; U) = 0.9063 bits. It is interesting
+to observe that this pU|Y actually coincides with the optimal solution obtained in [19, Example 1] via linear
+programming. Therefore, for the (X, Y ) distributed according to (65), we have g0(X, Y ) = 0.9063.
+Theorem 5. If |Y| = 3, Algorithm 3 is optimal, i.e., it achieves g0(X, Y ).
+Proof. Let X ≜ {x0, x1} and B ≜ {y ∈ Y|p(x0|y) = p(x0)}. We have either |B| = 0 or |B| = 1, since otherwise,
+X ⊥⊥ Y , and U ∗ = Y .
+First, assume |B| = 1. Therefore, with a proper relabling of the elements in Y, we have p(x0|y1) = p(x0),
+p(x0|y2) > p(x0), and P(x0|y3) < p(x0). 16 For a mapping pU|Y which results in X ⊥⊥ U, define Ui ≜ {u ∈
+U|p(yi|u) > 0}, hence, U = ∪3
+i=1Ui. For an optimal mapping, we must have p(y1|u) = 1, ∀u ∈ U1, since otherwise,
+we have either |{y ̸= y1|p(y|u) > 0}| = 1 or 2, where the former results in p(x0|u) ̸= p(x0), which violates the
+condition X ⊥⊥ U, and the latter violates Lemma 4, which states that each u must be connected to at most two
+realizations of Y . As a result U1 ∩ (U2 ∪ U3) = ∅ and Pr{U ∈ U1} = p(y1). Furthermore, we have U2 = U3, since
+otherwise, we get X ̸⊥⊥ U. For any realization u ∈ U2, we must have p(y3|u) = 1 − p(y2|u) = f(y3, y2), with
+16That both p(x0|y2) and p(x0|y3) cannot be lower or greater than p(x0) is obvious, since otherwise, we get p(x0) < p(x0), which is
+absurd.
+
+28
+y1 : 0.5
+y2 : 0.25
+y3 : 0.125
+y4 : 0.125
+u1 : 0.154
+0.3
+0.7
+0.8
+0.2
+0.5
+0.5
+0.4
+0.6
+0.1875
+0.8125
+(a) Iteration 1.
+y1 : 0.4712
+y2 : 0.25
+y3 : 0
+y4 : 0.125
+u2 : 0.698
+0.3
+0.7
+0.8
+0.2
+0.5
+0.5
+0.4
+0.6
+0.675
+0.325
+(b) Iteration 2.
+y1 : 0
+y2 : 0.0231
+y3 : 0
+y4 : 0.125
+u3 : 0.148
+0.3
+0.7
+0.8
+0.2
+0.5
+0.5
+0.4
+0.6
+0.155
+0.845
+(c) Iteration 3
+y1 : 0.5
+y2 : 0.25
+y3 : 0.125
+y4 : 0.125
+u1 : 0.154
+u2 : 0.698
+u2 : 0.148
+0.3
+0.7
+0.8
+0.2
+0.5
+0.5
+0.4
+0.6
+0.0577
+1
+0.9423
+0.9075
+0.0925
+1
+(d) The output pU|Y .
+Fig. 5: Illustration of Example 4.
+f(·, ·) defined in Proposition 2, since otherwise, the condition X ⊥⊥ U is violated. Therefore, we have
+H(Y |U) =
+�
+u∈U1
+H(Y |U = u) +
+�
+u∈U2
+H(Y |U = u)
+= 0 +
+�
+u∈U2
+Hb(f(y3, y2))
+= (1 − p(y1))Hb(f(y3, y2)),
+which is obtained via Algorithm 3, and we get g0(X, Y ) = Hb(p(y1)).
+Next, assume |B| = 0. With a proper relabling of the elements in Y, we have either p(x0|y1) > p(x0) >
+p(x0|yi), i ∈ {2, 3}, or p(x0|y1) < p(x0) < p(x0|yi), i ∈ {2, 3}. We only consider the former, as the proof for the
+latter follows similarly. Let Ui, i ∈ [3], be defined as before. We have U = U1, since otherwise, for any u ∈ U\U1,
+p(x0|u) < p(x0), and hence, X ̸⊥⊥ U. Furthermore, U2 ∩ U3 = ∅, since other wise, in conjunction with U = U1,
+there exists u ∈ U such that p(yi|u) > 0, ∀i ∈ [3], which violates the condition in Lemma 4. For any i ∈ {2, 3}
+and any realization u ∈ Ui, we must have p(yi|u) = 1 − p(y1|u) = f(yi, y1), since otherwise, X ̸⊥⊥ U. Moreover,
+
+29
+since p(yi) = �
+u∈Ui p(u)p(yi|u), we get Pr{U ∈ Ui} =
+p(yi)
+f(yi,y1), i ∈ {2, 3}. Finally, we have
+H(Y |U) =
+3
+�
+i=2
+�
+u∈Ui
+H(Y |U = u)
+=
+3
+�
+i=2
+p(yi)
+f(yi, y1)Hb(f(yi, y1)),
+which is attained by Algorithm 3.
+Based on the achievable scheme in Proposition 2, we can now propose a privacy-preserving as a lower bound
+on gǫ(X, Y ).
+Proposition 3. (Privacy-preserving mapping - a lower bound on gǫ(X, Y ).) Let (X, Y ) ∼ pXY be given, and let
+S = (x1, x2, . . . , x|X |−1) be an arbitrary ordered (|X|− 1)-tuple of the elements in X. Set U0 ≜ Y . The algorithm
+starts off from this point by decreasing the privacy-leakage step by step as follows. In the first step, define the binary
+random variable ˆX1 ∈ {0, 1} as ˆX1 ≜ 1{X=x1}. Since ˆX1 is a function of X, ˆX1 − X − Y − U0 form a Markov
+chain. Since ˆX1 is binary, by applying Algorithm 3 in Proposition 2 to the pair ( ˆX1, U0), pU1|U0 is generated such
+that X − Y − U0 − U1 form a Markov chain, and ˆX1 ⊥⊥ U1, or equivalently, pX|U1(x1|u) = pX(x1), ∀u ∈ U1.
+Hence, X is at least 1-independent of U1. Set P1 ≜ (I(X; U1), I(Y ; U1)). The algorithm proceeds in an iterative
+way as follows. After building the Markov chain ˆXi−X−Y −Ui−1, i ∈ [2 : |X|−1], in which ˆXi ≜ 1{X=xi}, apply
+Algorithm 3 to ( ˆ
+Xi, Ui−1) to generate pUi|Ui−1, such that X − Y − Ui−1 − Ui form a Markov chain and ˆXi ⊥⊥ Ui,
+or equivalently, X is at least i-independent of Ui. Set Pi ≜ (I(X; Ui), I(Y ; Ui)), and let PS ≜ {Pi}|X |−1
+i=1
+denote
+the set of achievable points for the ordered tuple S introduced earlier. Finally, let P ≜ ∪SPS ∪ {I(X; Y ), H(Y )}.
+A lower bound on gǫ(X, Y ) is provided by uce[P](·).
+For a fixed tuple S in Proposition 3, we have
+I(Y ; Ui) = H(Y ) − H(Y, Ui−1|Ui) + H(Ui−1|Y, Ui)
+≥ I(Y ; Ui−1) − H(Ui−1|Ui)
+(66)
+≥ I(Y ; Ui−1) − 1, ∀i ∈ [|X| − 1],
+(67)
+where (66) follows from the Markov chain Y − Ui−1 − Ui and non-negativity of entropy, and (67) results from the
+fact that according to Algorithm 3, Ui−1 conditioned on any realization ui of Ui has at most two non-zero mass
+probabilities, and hence, H(Ui−1|Ui) ≤ 1.
+We also have
+I(X; Ui) = I( ˆXi, X; Ui)
+(68)
+= I(X; Ui| ˆXi)
+(69)
+≤ I(X; Ui−1| ˆXi)
+(70)
+= I(X; Ui−1) − I( ˆXi; Ui−1), ∀i ∈ [|X| − 1],
+(71)
+
+30
+where (68) follows from having defined ˆXi as a function of X, (69) results from ˆXi ⊥⊥ Ui, and finally, (70) is
+from the application of data processing inequality in ˆXi − X − Ui−1 − Ui.
+The procedure in Proposition 3 can be computationally complex when |X| is large, as the total number of ordered
+(|X| − 1)-tuples is |X|!. Therefore, this calls for a simpler scheme when |X| is large. Let S = (x1, . . . , x|X |−1)
+be a given ordered tuple. In the first iteration of the algorithm, a release random variable U1 is generated, which
+results in a utility greater than or equal to (H(Y ) − 1)+ from (67), since U0 = Y , and a privacy-leakage lower
+than or equal to I(X; Y ) − I( ˆX1; Y ) from (71). Since gǫ(X, Y ) is concave and non-decreasing in ǫ, if a utility
+within 1 bit of H(Y ) is to be achieved in iteration 1, a heuristic approach is to choose an x1 ∈ X which is
+likely to result in the maximum drop in the privacy-leakage, i.e., I( ˆX1; Y ). In other words, on the U-P plane,
+the algorithm tries to depart from the rightmost point (I(X; Y ), H(Y )) with the lowest slope. As a result, we set
+x1 ≜ arg maxx∈X I(1{X=x}; U0), and the algorithm proceeds to provide U1 in X − Y − U1. Following the same
+rationale, we set x2 ≜ arg maxx∈X \{x1} I(1{X=x}; U1), and so on.
+The procedure of this simplified scheme is presented in Algorithm 4.
+Algorithm 4 Privacy-preserving mapping - a lower bound on gǫ(X, Y ).
+1: function ALGORITHM4(pXY )
+2:
+U0 ≜ Y
+3:
+x1 ≜ arg maxx∈X I(1{X=x}; U0)
+4:
+P ≜ {(I(X; U0), I(Y ; U0))}
+5:
+i = 1
+6:
+while I(X; Ui−1) ̸= 0 do
+7:
+ˆXi ≜ 1{X=xi}
+8:
+pUi|Ui−1 ≜ Algorithm3(p ˆ
+XiUi−1)
+9:
+P ≜ P ∪ {(I(X; Ui), I(Y ; Ui))}
+10:
+i = i + 1
+11:
+xi ≜ arg maxx∈X \{x1,...,xi−1} I(1{X=x}; Ui−1)
+12:
+end while
+13:
+return uce[P](·)
+14: end function
+V. NUMERICAL RESULTS
+In this section, the performance of the proposed privacy-preserving schemes are evaluated. Prior to this evaluation,
+we note that
+g0(X, Y ) ≥ gL
+0 ≜
+�
+H(Y ) − log rank(PX|Y )
+�+
+(72)
+G0(X, Y ) ≥ GL
+0 ≜
+�
+H(Y ) − min{H(X), log rank(PX|Y )}
+�+ ,
+(73)
+where PX|Y is the matrix form of the conditional pmf pX|Y .
+
+31
+The lower bound in (72) is from [1, Corollary 2] and (73) results from i) G0(X, Y ) ≥ g0(X, Y ) along with (72)
+and ii) the fact that according to Lemma 1, for an optimal mapping p∗
+U|XY , we have H(Y |X, U ∗) = 0, and hence
+G0(X, Y ) = I(Y ; U ∗)
+= I(X, Y ; U ∗) − I(X; U ∗|Y )
+= I(Y ; U ∗|X) − I(X; U ∗|Y )
+≥ H(Y |X) − H(Y |X, U ∗) − H(X|Y )
+= H(Y |X) − H(X|Y )
+= H(Y ) − H(X).
+A. Full data observation
+Consider the following joint pmf on X × Y = [8]2 generated randomly17.
+pX =
+
+
+0.175
+0.089
+0.146
+0.026
+0.077
+0.167
+0.145
+0.175
+
+
+,
+PY |X =
+
+
+0.130
+0.233
+0.159
+0.045
+0.185
+0.158
+0.039
+0.051
+0.007
+0.061
+0.072
+0.117
+0.046
+0.054
+0.067
+0.065
+0.168
+0.251
+0.217
+0.106
+0.034
+0.107
+0.219
+0.160
+0.185
+0.011
+0.008
+0.154
+0.141
+0.147
+0.066
+0.123
+0.134
+0.099
+0.100
+0.169
+0.271
+0.188
+0.212
+0.091
+0.150
+0.016
+0.087
+0.180
+0.096
+0.202
+0.063
+0.216
+0.147
+0.035
+0.175
+0.066
+0.165
+0.115
+0.242
+0.152
+0.078
+0.293
+0.182
+0.162
+0.063
+0.029
+0.091
+0.143
+
+
+.
+(74)
+Figure 6 illustrates the lower bounds on Gǫ(X, Y ) in Proposition 1 and Remark 6. The filled black circles are
+the achievable points in Proposition 1 whose upper concave envelope is drawn in solid black line. The empty red
+circles correspond to the non-algorithmic achievable points in Remark 6 whose upper concave envelope is plotted in
+solid red line. The top dashed blue line is the line connecting (0, H(Y |X)) to (I(X; Y ), H(Y )), while the bottom
+dashed blue line corresponds to the line connecting (0, GL
+0 ), where GL
+0 is given in (73), to (I(X; Y ), H(Y )).
+Figure 7 illustrates the lower bounds on Gǫ(X, Y ) in the simplified version of Proposition 1, i.e., Algorithm 2. The
+filled black circles are the achievable points in Algorithm 2 whose upper concave envelope is drawn in solid black
+line. The empty red circles correspond to the non-algorithmic achievable points in Remark 6 when applied to the
+procedure in Algorithm 2 whose upper concave envelope is plotted in solid red line. The dashed blue lines are as
+mentioned earlier.
+B. Public data observation
+Figure 8 illustrates the lower bounds on gǫ(X, Y ) in Proposition 3 and its simplified version, i.e., Algorithm 4.
+The filled black circles are the achievable points in Proposition 3 whose upper concave envelope is drawn in solid
+17Each probability vector is obtain by normalizing an 8-dimensional vector whose elements have been i.i.d. generated according to uniform
+distribution over the interval [0, 1].
+
+32
+0
+0.05
+0.1
+0.15
+0.2
+0
+0.5
+1
+1.5
+2
+2.5
+3
+Fig. 6: Lower bounds on Gǫ(X, Y ) in Proposition 1 and Remark 6.
+black line. The empty red circles correspond to the achievable points in Algorithm 4 whose upper concave envelope
+is plotted in solid red line. The top dashed blue line is as mentioned earlier in the full data observation, while the
+bottom dashed blue line is the line connecting (0, gL
+0 ), where gL
+0 is given in (72), to (I(X; Y ), H(Y )).
+VI. CONCLUSIONS
+Information-theoretic privacy is considered in this paper, in which a curator, aware of the joint distribution pXY ,
+wishes to maximize I(Y ; U) subject to I(X; U) ≤ ǫ. The optimization is over pU|XY (or pU|Y ) when curator has
+access to (X, Y ) (or only Y ), and trade-off is captured by Gǫ(X, Y ) (or gǫ(X, Y )). The problem is investigated
+from theoretical and practical point of view.
+APPENDIX A
+Fix an arbitrary x0 ∈ X. Assume that for some u0 ∈ U∗, we have p(x0, y′|u0) > 0, and p(x0, y′′|u0) > 0 with
+y′ ̸= y′′. It is shown that this cannot be optimal by construction. In other words, a mapping p ˆU|XY is constructed
+such that I(X; ˆU) = I(X; U ∗) and I(Y ; ˆU) > I(Y ; U ∗), which disproves the optimality of pU∗|XY .
+
+33
+0
+0.05
+0.1
+0.15
+0.2
+0
+0.5
+1
+1.5
+2
+2.5
+3
+Fig. 7: Lower bounds on Gǫ(X, Y ) in Algorithm 2.
+Assume the random variable ˆU ∈ (U∗\{u0}) ∪ {u′
+0, u′′
+0} with u′
+0, u′′
+0 ̸∈ U∗ such that
+pX,Y | ˆU(x0, y′|u′
+0), pX,Y | ˆU(x0, y′′|u′′
+0) ≜ pX,Y |U(x0, y′|u0) + pX,Y |U(x0, y′′|u0)
+pX,Y | ˆU(x0, y′′|u′
+0), pX,Y | ˆU(x0, y′|u′′
+0) ≜ 0,
+p ˆU(u′
+0) ≜ pU(u0)
+pX,Y |U(x0, y′|u0)
+pX,Y |U(x0, y′|u0) + pX,Y |U(x0, y′′|u0)
+p ˆU(u′′
+0) ≜ pU(u0)
+pX,Y |U(x0, y′′|u0)
+pX,Y |U(x0, y′|u0) + pX,Y |U(x0, y′′|u0)
+pX,Y, ˆU(x, y, u) ≜ pX,Y,U(x, y, u), ∀(x, y, u) ∈ X × Y × ( ˆU\{u′
+0, u′′
+0})
+pX,Y, ˆU(x, y|u) ≜ pX,Y,U(x, y|u0), ∀u ∈ {u′
+0, u′′
+0}, ∀(x, y) ̸= (x0, y′), (x0, y′′).
+It can be verified that in this construction, i) the marginal pX,Y is preserved in (X, Y, ˆU), ii) H(X|U ∗) = H(X| ˆU)
+(which results from having pX, ˆU(·, u) = pX,U∗(·, u), ∀u ∈ U∗\{u0, u′
+0, u′′
+0}, and pX|U∗(·|u0) = pX| ˆU(·|u′
+0) =
+pX| ˆU(·|u′′
+0), with p(u0) = p(u′
+0)+p(u′′
+0)), and iii) H(Y | ˆU) < H(Y |U) due to strict concavity of entropy. Therefore,
+we have constructed p ˆU|X,Y , such that I(X; ˆU) = I(X; U ∗), and I(Y ; U ∗) < I(Y ; ˆU) which contradicts the
+attainability of G0(X, Y ) by pU∗|X,Y . Hence, by noting that x0 was chosen arbitrarily, we obtain
+|{y ∈ Y|p(x, y|u∗) > 0}| ≤ 1, ∀(x, u∗) ∈ X × U∗,
+
+34
+Fig. 8: Lower bounds on gǫ(X, Y ) in Proposition 3 and Algorithm 4.
+which is equivalent to (7) and (8) when X ⊥⊥ U.
+APPENDIX B
+Let x∗ ≜ arg minx p(x) and define X0 ≜ {x ∈ X|f(x) = f(x∗)}. Hence, we have
+p(x∗) ≤ min {Pr{X ∈ X0}, 1 − Pr{X ∈ X0}} .
+(75)
+Define Z ≜ 1{f(X)=f(x∗)} as a function of f(X). Since f(X) has at least two realizations, Z is a Bernoulli random
+variable with parameter Pr{X ∈ X0}. Therefore, we can write
+H(f(X)) ≥ H(Z)
+(76)
+= Hb (Pr{X ∈ X0})
+≥ Hb(p(x∗)),
+(77)
+where (76) follows from defining Z as a function of f(X), and (77) results from (75).
+APPENDIX C
+If i = j or i = k, the proof is complete by the non-negativity of entropy. Also, if any of pi, pj, pk is 0 or 1, the
+proof is complete, since it results in one of the following trivial possibilities i) 0 ≤ 0, ii) Hb(pi) ≤ Hb(1 − pi),
+
+2.5
+2
+;U)
+1
+1.5
+PointsinPinProposition3
+uceipinProposition3
+PointsinPinAlgorithm4
+ucepin Algorithm 4
+0.5
+0.05
+0.1
+0.15
+0.2
+I(X;U)35
+or iii) the non-negativity of entropy. Therefore, we assume that the indices i, j, k are all distinct and non of the
+mass probabilities is 0 or 1. Since the RHS of (17) is symmetric with respect to pj, pk, i.e., it doesn’t change if
+we exchange pj and pk, without loss of generality, assume that pj ≤ pk. Therefore, we have
+Hb(pi) = Hb(pj + pk)
+< Hb(pk) + H′
+b(pk)pj
+(78)
+≤ Hb(pk) + H′
+b(pj)pj
+(79)
+< Hb(pk) + H(pj),
+(80)
+where (78) results from Taylor expansion of Hb(·) and its strict concavity, i.e., H′′
+b (·) < 0. The latter also results
+in (79) and (80), i.e., H′
+b(pk) ≤ H′
+b(pj) since pj ≤ pk, and H′
+b(pj)pj < Hb(pj).
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf,len=1442
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='11754v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='IT]  27 Jan 2023  1  Information-Theoretic Privacy-Preserving  Schemes Based On Perfect Privacy  Borzoo Rassouli1 and Deniz G¨und¨uz2  1 Nokia Bell Labs, Munich, Germany  2 Department of Electrical and Electronic Engineering, Imperial College London, London, UK  borzoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='rassouli@nokia-bell-labs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='com, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='gunduz@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='uk  Abstract  Consider a pair of random variables (X, Y ) distributed according to a given joint distribution pXY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' A curator  wishes to maximally disclose information about Y , while limiting the information leakage incurred on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Adopting  mutual information to measure both utility and privacy of this information disclosure, the problem is to maximize  I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U), subject to I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) ≤ ǫ, where U denotes the released random variable and ǫ is a given privacy threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Two settings are considered, where in the first one, the curator has access to (X, Y ), and hence, the optimization  is over pU|XY , while in the second one, the curator can only observe Y and the optimization is over pU|Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In  both settings, the utility-privacy trade-off is investigated from theoretical and practical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' More specifically,  several privacy-preserving schemes are proposed in these settings based on generalizing the notion of statistical  independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Moreover, closed-form solutions are provided in certain scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Finally, convexity arguments are  provided for the utility-privacy trade-off as functionals of the joint distribution pXY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' INTRODUCTION  Consider a situation in which Alice wants to release some useful information about herself to Bob, represented by  random variable Y , and she receives some utility from this disclosure of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' At the same time, she wishes  to conceal from Bob some private information which depends on Y , represented by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' To this end, instead of  letting Bob have a direct access to Y , a privacy-preserving mapping/data release mechanism1 is applied, whereby  a distorted version of Y , denoted by U, is revealed to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In this context, privacy and utility are competing goals:  The more Y is distorted by the privacy-preserving mapping, the less information can Bob infer about X, but also  the less the utility that can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' This trade-off is the very result of the dependencies between X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Stated formally in a general context, consider a triplet of random variables (X, Y, W), distributed according to  the given/known joint probability mass function (pmf) pXY W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Let X denote the private/sensitive data that the  user/curator wants to conceal, Y denote the public/useful data the user wishes to disclose, and W denote the  observable data that the curator observes, which can be regarded as a noisy version of (X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Assume that the  privacy-preserving mapping takes W as input, and maps it to the released data, denoted by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In this scenario,  1The terms ”mapping”, ”mechanism”, ”scheme” and ”algorithm” are used interchangeably in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  2  (X, Y )−W −U form a Markov chain, and the privacy-preserving mapping is captured by the conditional distribution  pU|W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' We assume that all the alphabets/supports X, Y, W are finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  As stated, for the triplet (X, Y, W), a privacy-preserving mapping can be constructed by obtaining pmfs pU|W (·|w)  for each w ∈ W such that pU|W meets certain conditions corresponding to the utility/privacy requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Equivalently, this can be carried out by obtaining U, pW|U(·|u), ∀u ∈ U, such that pW is preserved in the  joint distribution pWU and those conditions are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' We call the former approach forward construction/model and  the latter one backward construction/model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  In this paper, we are solely interested in the special cases of full data observation and public data observation  which refer to the settings in which the privacy-preserving mapping has direct access to both the private and public  data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', W = (X, Y )) and only to the public data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', W = Y ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  By adopting mutual information as the measure of both utility and privacy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U), and I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U), respec-  tively), the optimal utility-privacy (U-P) trade-off in the public data observation model is defined as  gǫ(X, Y ) ≜  max  pU|Y :  X−Y −U  I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='U)≤ǫ  I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U),  (1)  and in the full data observation model, the optimal U-P trade-off can be formulated as  Gǫ(X, Y ) ≜  max  pU|XY :  I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='U)≤ǫ  I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U),  (2)  where the effective range of ǫ is [0, I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y )] by noting that both (1) and (2) have the upper bound of H(Y ) which  is attained by setting U ≜ Y , which in turn results in I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) = I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2  Perfect privacy [1] refers to the stringent constraint of X ⊥⊥ U, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', ǫ = 0 in the U-P trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Assume that  we have an algorithm which satisfies this constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In other words, once applied to W (with W = (X, Y ) or Y  depending on the model involved), this algorithm releases U such that I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Select an arbitrary conditional  pmf pZ|XY , and construct a private-public pair as (Z, Y ) ∼ �  x pXY (x, ·)pZ|XY (·|x, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Applying the algorithm  in this new context (with W = (Z, Y ) or Y ), we get U ′ such that I(Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U ′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The utility obtained is a lower  bound on the original optimal U-P trade-off at ǫ = I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U ′), and by changing pZ|XY and repeating this process,  we can sweep the whole range of ǫ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', [0, I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' This simple observation is the basis of the privacy-preserving  schemes proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  The information-theoretic view of privacy has gained increasing attention recently, with an incomplete list of  related literature being [1]–[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In [2], a general statistical inference framework is proposed to capture the loss  of privacy in legitimate transactions of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In [3], the privacy-utility trade-off under the log-loss cost function is  considered, called as the privacy funnel, which is closely related to the information bottleneck introduced in [14]  (see also [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In [15], sharp bounds on the optimal privacy-utility trade-off for the privacy funnel are derived, and an  alternative characterization of the perfect privacy condition (also studied in [16] in a different context) is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  2That the maximums in (1) and (2) exist follows from standard arguments in real analysis (compactness, continuity) as in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Furthermore,  when (X, Y ) ∼ pXY , the quantities gǫ(X, Y ) and Gǫ(X, Y ) are written interchangeably as gǫ(pXY ) and Gǫ(pXY ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  3  Measuring both the privacy and the utility in terms of mutual information, perfect privacy is fully characterized in  [17] for the binary case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  The current paper contributes to this context as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Upper and lower bounds on G0(X, Y ) are proposed and their tightness is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Based on the aforementioned bounds, and the simplex method [18], a closed-from solution for G0(X, Y ) is  derived when X is binary, or (|X|, |Y|) = (3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In this context, it is shown that for fixed conditional pmf  pY |X, the optimal privacy-preserving mapping does not depend on pX, which is of practical interest when the  curator is unaware of the distribution of the private data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  It is shown that for fixed pX, Gǫ(X, Y ) is concave in pY |X, and for fixed pY |X, both G0(X, Y ) and g0(X, Y )  are convex in pX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Based on the lower bound on G0(X, Y ), a privacy-preserving scheme is presented as a lower bound on the  optimal U-P trade-off in the case of full data observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  When X is binary, an algorithmic lower bound on g0(X, Y ) is presented, which is optimal when |Y| = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Based on this algorithm, a privacy-preserving scheme is presented as a lower bound on the optimal U-P  trade-off in the case of public data observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Random variables are denoted by capital letters (X, Y , etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ), their realizations by lower case letters  (x, y, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ), and their alphabets by capital letters in calligraphic font (X, Y, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Matrices, and vectors are denoted  by bold capital and bold lower case letters, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The rank of matrix A is denoted by rank(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' For integers  m, n, we have the discrete interval [m : n] ≜ {m, m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' , n} if m ≤ n, and ∅ (the empty set), otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  The set [1 : n] is written in short as [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Given two positive integers a, b, a modulo b is abbreviated as a mod b,  and S mod b denotes {x mod b| ∀x ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Given two pmfs p, q, the Kullback–Leibler divergence from q to p is  defined as3 D(p||q) ≜ �  x p(x) log2  p(x)  q(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' All the logarithms in this paper are to the base of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' For 0 ≤ t ≤ 1,  ¯t ≜ 1 − t and Hb(t) ≜ −t log t − ¯t log ¯t denotes the binary entropy function (with the convention 0 log 0 ≜ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' For  an event E, the indicator 1{E} is one when E occurs, and zero, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The domain of function f is denoted by  dom(f), and throughout the paper, if there are more than one candidate for arg minx∈dom(f) f(x), one is selected  arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' For a real number x, we define (x)+ ≜ max{0, x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Define the support of a given pair (X, Y ) ∼ pXY  as supp(X, Y ) ≜ {(x, y) ∈ X × Y|pXY (x, y) > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Finally, throughout this paper, we encounter summations of  the form �  u∈U(·), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', summation over the elements of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' If U happens to be the empty set, this summation is  defined as zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' PRELIMINARIES  Throughout the paper, the U-P plane refers to the 2-dimensional plane, in which the horizontal and vertical axes  denote the privacy-leakage I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) and utility I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Furthermore, we say that a point P = (Px, Py)  is achievable on this plane if there exists a joint distribution pXY · pU|XY (in the case of full data observation), or  pXY · pU|Y (in the case of public data observation), such that I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) = Px, and I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) = Py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  3We assume that p is absolutely continuous with respect to q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', q(x) = 0 implies p(x) = 0, otherwise, D(p||q) ≜ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  4  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' When ǫ ∈ [0, I(X, Y )], both gǫ(X, Y ) and Gǫ(X, Y ) are concave and strictly increasing functions of ǫ  that lie above the lines connecting (0, g0(X, Y )), and (0, G0(X, Y )) to (I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ), H(Y )), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Furthermore,  they both lie under the line connecting (0, H(Y |X)) to (I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ), H(Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Also, for an optimal mapping in (1) or  (2), we have I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) = ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Finally, both g0(X, Y ) and G0(X, Y ) can be obtained via a linear program (LP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The concavity of gǫ(X, Y ) in ǫ is shown in [5, lemma 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' That of Gǫ(X, Y ) follows similarly4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' We also  have  gǫ(X, Y ) ≤ Gǫ(X, Y )  (3)  ≤  max  pU|X,Y :  I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='U)≤ǫ  I(X, Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U)  =  max  pU|X,Y :  I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='U)≤ǫ  I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) + I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U|X)  ≤ ǫ + H(Y |X), ǫ ∈ [0, I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y )],  (4)  where (3) is by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' This means that both gǫ(X, Y ) and Gǫ(X, Y ) lie under the line connecting (0, H(Y |X))  to (I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ), H(Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Unless in the degenerate case of X ⊥⊥ Y (which results in ǫ ∈ [0, I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y )] = {0}), we have  H(Y |X) < H(Y ), and hence, both gǫ(X, Y ) and Gǫ(X, Y ), being concave functions, must be strictly increasing in  ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' As a result, they lie above the lines connecting (0, g0(X, Y )), and (0, G0(X, Y )) to (I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ), H(Y )), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  By noting that in maximizing a convex functional, the maximum occurs at an extreme point, we conclude that  for any maximizer in (1) or (2), we have I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) = ǫ, when ǫ ∈ [0, I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  As shown in [1], g0(X, Y ) is obtained via an LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' More specifically, in the Markov chain X − Y − U, if we  consider the backward model, the conditional pmf pY |U(·|u) must belong to a convex polytope determined by  the condition X ⊥⊥ U, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', pX(·) = �  y pX|Y (·|y)pY |U(y|u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Since H(Y |U = u) is a concave functional of  pY |U(·|u), and its minimum occurs at an extreme point, we first obtain the extreme points of the aforementioned  convex polytope, and what remains is to allocate proper weights, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', pU(·), such that H(Y |U) is minimized subject  to pY (·) = �  u pY |U(·|u)pU(u), which is an LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Similarly, by considering the Markov chain X − (X, Y ) − U,  G0(X, Y ) can be obtained through an LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The upper concave envelope of a set of points P ≜ {(xi, yi)}n  i=1 in R2 is defined as5  uce[P](·) ≜ inf{f(x)| dom(f) = [min  i  xi, max  i  xi], f is concave , f(xi) ≥ yi, ∀i ∈ [n]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Figure 1 provides an example of the upper concave envelope (solid line) of a set of points (filled circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' From Remark 1 and Definition 1, it is obvious that if the points in P are achievable on the U-P plane,  {(x, uce[P](x)) | ∀x ∈ [mini xi, maxi xi]} is also achievable, and hence, uce[P](·) serves as a lower bound on the  optimal utility-privacy trade-off6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  4It can be shown via example that the claim of strict concavity is too strong for these two curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  5The lower convex envelope of P is the negative of the upper concave envelope of P− ≜ {(xi, yi)|(xi, −yi) ∈ P, ∀i ∈ [n]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  6Since (I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ), H(Y )) is an achievable point, we can include it in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Therefore, the resulting uce[P](·) will be non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' 1: The upper concave envelope of a set of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  The privacy-preserving algorithms in this paper find achievable points based on a generalization of statistical  independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The sole purpose of the following definition is to simplify the explanation of this generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' For a pair of random variables (X, U) ∼ pXU, and k ∈ [|X| − 1], if  ����  �  x ∈ X  ���� pX|U(x|u) = pX(x), ∀u ∈ U  ����� ≥ k,  (5)  or equivalently,  ����  �  x ∈ X  ���� pU|X(·|x) = pU(·)  ����� ≥ k,  (6)  we say that X is at least k-independent of U, which is denoted by X  k  ⊥⊥ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Note that i) X  k  ⊥⊥ U =⇒ X  k−1  ⊥⊥ U, k ∈ [2 : |X| − 1], ii) X  |X |−1  ⊥⊥  U =⇒ X ⊥⊥ U, and iii) this is an asymmetric  notion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', X  k  ⊥⊥ U ̸=⇒ U  k  ⊥⊥ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Furthermore, if X − U − Z form a Markov chain, from (6), we conclude that  X  k  ⊥⊥ U results in X  k  ⊥⊥ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' FULL DATA OBSERVATION MODEL  In this Section, we assume that the curator has access to both X and Y , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', W = (X, Y ), and propose an  achievable scheme, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', a lower bound on Gǫ(X, Y ), which is defined in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' To this end, we find achievable points  6  x1y1  x1y2  x2y1  x2y2  x3y1  x3y2  u1  u2  u3  u4  u5  u6  u7  u8  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' 2: An illustrative representation of Lemma 1 for (X, Y ) ∈ {x1, x2, x3} × {y1, y2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' If uk is connected to pair  (xi, yj), we have p(xi, yj|uk) = p(xi), (i, j, k) ∈ [3] × [2] × [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  on the U-P trade-off and propose their upper concave envelope as the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  The following Lemma is central to the analysis in this Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' For an optimal pU∗|XY in the evaluation of G0(X, Y ), we must have  |{y ∈ Y|p(x, y|u∗) > 0}| = 1, ∀(x, u∗) ∈ X × U∗,  (7)  which results in  H(Y |X, U ∗) = 0,  (8)  where U ∗ ∈ U∗ is induced by the optimal mapping pU∗|XY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In other words, for any (x, u∗) ∈ X × U∗, there must  exist yx,u∗ ∈ Y such that  p(x, yx,u∗|u∗) = p(x), p(x, y|u∗) = 0, ∀y ∈ Y\\{yx,u∗},  (9)  which results in a lower bound on the cardinality of |U∗| as  |U∗| ≥ max  x∈X |{y ∈ Y|p(y|x) > 0}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The proof is provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='7  Lemma 1 is exemplified in Figure 2 in which (X, Y ) ∈ {x1, x2, x3} × {y1, y2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Let pi ≜ p(xi), i ∈ [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' As  (9) requires, for each realization of U, there is exactly one link to subgroup i of nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', {(xi, yj)}2  j=1 with  transition probability pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  7This Lemma is also given in [7, Lemma 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Since it was also independently provided in [19, Lemma 3] by the authors of the current  manuscript, it is mentioned here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  7  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' If Y is not a function of X, we have G0(X, Y ) > 0 by [1, Theorem 4] which is resulted by a U such  that X ⊥⊥ U and H(Y |X, U) = 0 according to Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Obviously, if Y is a function of X, we can select U as  an arbitrary singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' This observation provides an alternative proof for the functional representation lemma [20,  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' 626].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  In order to propose a lower bound on Gǫ(X, Y ), we start with ǫ = 0, which is equivalent to X ⊥⊥ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' It is  already known that G0(X, Y ) can be obtained via an LP whose dimension is the total number of extreme points of  the convex polytope stated earlier in Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' However, according to Lemma 1, these extreme points are already  known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' They are all the conditional pmfs pXY |U that satisfy the property in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' As a result, the dimension of  the LP involved in evaluating G0(X, Y ) is �  x∈X |{y ∈ Y|p(y|x) > 0}| ≤ |Y||X |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Note that even assuming a  polynomial time complexity for the algorithm used for solving the LP, unless |X| and |Y| are small or the matrix  of joint distribution PXY is sparse, the problem becomes computationally intractable in terms of time and space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Therefore, a tractable method is desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  In what follows, an algorithm (Algorithm 1) is proposed in Theorem 1 that provides a lower bound on G0(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  This algorithm is proved to be optimal in Theorem 2 when X is binary or (|X|, |Y|) = (3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Finally, building  upon Algorithm 1, Proposition 1 presents Algorithm 2 which produces a privacy-preserving mapping as a lower  bound on Gǫ(X, Y ), ǫ ∈ [0, I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' For a given pair (X, Y ) ∼ pXY , we have  G0(X, Y ) ≥  �  H(Y ) −  �  1 −  �  y  min  x p(y|x)  �  min{H(X), log |Y|}  �+  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  (10)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Define the index set  I ≜ {y ∈ Y|p(y|x) > 0, ∀x ∈ X},  (11)  and relabel the elements of Y = {y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' , y|Y|} such that the first |I| elements belong to I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The algorithm  proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' First, |I| mass points for U are created, which are denoted by uj(j ∈ [|I|]), each having  pU(uj) = minx pY |X(yj|x), respectively, such that for all x ∈ X, we have p(x, yk|uj) = p(x), if j = k, and 0,  otherwise (j, k ∈ [|I|]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' It is evident that thus far, the posterior pX(·|uj) remains the same as the prior pX(·), which  is in line with the condition of X ⊥⊥ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Moreover, these mass points are such that pY |U(yj|uj) = 1, resulting in  H(Y |U = uj) = 0, j ∈ [|Y|].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Afterwards, the iterations begin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In each iteration i, a mass point ui+|I| is created  such that pX(·|ui+|I|) = pX(·), and the conditional pmf of the pair (X, Y ) conditioned on {U = ui+|I|} has  the same mass probabilities as in pX(·) resulting in H(X, Y |U = ui+|I|) = H(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Hence, H(Y |U = ui+|I|) ≤  H(X), i ≥ 1 (note that we also have the trivial upper bound H(Y |U = ui+|I|) ≤ log |Y|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The procedure is  provided in Algorithm 1, and the algorithm terminates at some iteration N, where N ≤ |supp(X, Y )| − |I|, which  is further tightened in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  The rationale behind this algorithm becomes clear by considering the backward construction as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Let each  realization (x, y) of (X, Y ) be denoted by a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Arrange these nodes in a long column vector as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In this  configuration, we divide the nodes into |X| subgroups of nodes: The first subgroup of nodes is {(x1, yi)|i ∈ [|Y|]},  8  Algorithm 1 A lower bound on G0(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  1: function ALGORITHM1(pY |X)  2:  p(uj|x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y) = minx∈X p(y|x)  p(y|x)  1{y=yj},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ∀j ∈ [|I|],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y) ∈ supp(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y )  3:  a1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y) = p(y|x) − minx p(y|x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ∀(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y) ∈ supp(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y )  4:  i = 1  5:  while maxx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='y ai(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y) ̸= 0 do  6:  a∗  i = minx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='y{ai(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y)|ai(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y) > 0}  7:  (x∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y∗) = arg minx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='y{ai(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y)|ai(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y) > 0}  8:  fi(x) = arg miny{ai(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y)|ai(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y) > 0},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ∀x ∈ X  9:  p(ui+|I||x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y) =  a∗  i  p(y|x) ·  �  1{ai(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='y∗)>0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='y=y∗} + 1{ai(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='y∗)=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='y=fi(x)}  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ∀(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y) ∈ supp(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y )  10:  ai+1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y) = ai(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y) − a∗  i ·  �  1{ai(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='y∗)>0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='y=y∗} + 1{ai(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='y∗)=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='y=fi(x)}  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ∀(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y) ∈ supp(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y )  11:  i = i + 1  12:  end while  13:  return pU|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='Y  14: end function  the second subgroup is {(x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' yi)|i ∈ [|Y|]},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Obviously, the sum of the mass probabilities of the nodes  in the i-th subgroup is pX(xi), ∀i ∈ [|X|].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Therefore, if in this construction, from each mass point (or node) u,  there is one connection/link to only one of the nodes in the first subgroup with transition probability pX(x1), one  link to only one of the nodes in the second subgroup with transition probability pX(x2), and so on, which is what  Lemma 1 implies, we have pX|U(·|u) = pX(·), and H(Y |U = u) ≤ H(X, Y |U = u) = H(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' However, if in  the first |I| realizations of U, the links that connect each u ∈ {u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' , u|I|} to |X| subgroups arrive at nodes that  have the same second coordinate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', y, we get H(Y |U = u) = 0 for these |I| realizations of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Also, if a node  u is to be connected to |X| nodes sharing the same second coordinate, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', {(xi, y)}|X |  i=1 for some y ∈ Y, we must  have p(u) ≤ p(y|xi), ∀i ∈ [|X|].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' This is needed to guarantee that the requirement p(xi, y|u) = p(xi), i ∈ [|X|]  does not violate the preservation of pXY in pXY U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='8 Therefore, we set p(u) equal to its maximum allowable  value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', minx p(y|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The aforementioned procedure is captured in step 2 of the algorithm by making the  convention 0  0 · 0 ≜ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Subsequently, the event containing the first |I| realizations of U occurs with probability of  �  y∈I minx p(y|x) = �  y∈Y minx p(y|x), which results in H(Y |U) ≤ (1−�  y minx p(y|x)) min{H(X), log |Y|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  The concern in this backward construction is to preserve the original distribution pXY in the resulting joint  pmf pXY U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Since in the construction of U, it is known from our impositions that if p(x, y|u) ̸= 0, for some  (x, y) ∈ supp(X, Y ), then we must have p(x, y|u) = p(x), we observe that the preservation of pXY boils down to  that of the conditional pmf pY |X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In other words, denoting the set of all realizations u that have a link to (x, y) by  8Since otherwise, we have p(u) > p(y|xj), for some j ∈ [|X|].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' This results in pXY U(xj, y, u) = p(u)p(xj, y|u) = p(u)p(xj) > p(xj, y),  which results in p(xj, y) induced by pXY U being greater than the original pXY (xj, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  9  Ux,y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', Ux,y ≜ {u ∈ U|p(x, y|u) ̸= 0}, ∀(x, y) ∈ supp(X, Y ), the preservation of pXY is equivalent to  p(x, y) =  �  u∈U  p(x, y, u)  =  �  u∈Ux,y  p(x, y|u)p(u)  = p(x)  �  u∈Ux,y  p(u), ∀(x, y) ∈ supp(X, Y ),  which is in turn equivalent to  p(y|x) =  �  u∈Ux,y  p(u), ∀(x, y) ∈ supp(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Therefore, we only need to make sure that the mass probabilities of all the nodes u that are connected to (x, y) sum  up to p(y|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' To this end, we harness a waterfilling-like procedure, in which the water levels denote the remaining  probabilities which need to be ”filled”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In step 3, the water levels are set as a1(x, y) by taking into account the  assignment in step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In other words, for each node (x, y), the amount of minx p(y|x) has already been filled by  the links from uj, ∀j ∈ [|I|] in step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' At each iteration i, node ui+|I| is created to fill the minimum water level  denoted by a∗  i in step 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' A/the minimizer is denoted by (x∗, y∗) in step 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Note that in this step and step 8, if there  are multiple minimizers, one is selected arbitrarily9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In step 8, fi(x) denotes a/the minimum non-zero water level  in subgroup x at iteration i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' We create ui+|I|, and set p(ui+|I|) ≜ a∗  i , and connect this node to |X| nodes, each  belonging to one subgroup, with the transition probability of p(x1) for the link to subgroup 1, p(x2) for the link  to subgroup 2, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In doing so, we take this intuition into account that points with common y-coordinates  are desirable, as this allocation is in line with lowering H(Y |U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Hence, in each subgroup x (x ∈ X), if the water  level corresponding to (x, y∗), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', ai(x, y∗), is non-zero, this point is selected, otherwise, the point corresponding  to a/the minimum water level of this subgroup is selected, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', (x, fi(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' This is given in step 9 of the algorithm,  and in step 10, the water levels are updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Since in step 3 (prior to the iterations), the water levels of at least |I| nodes are filled, and at each iteration,  at least one water level gets filled, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', ai+1(x∗, y∗) becomes zero (which occurs in step 10), the algorithm  terminates after at most |supp(X, Y )| − |I| iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' With this pU|XY , we get X ⊥⊥ U, and H(Y |U) ≤ (1 −  �  y minx p(y|x)) min{H(X), log |Y|}, which proves the lower bound in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  The following example clarifies the steps in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Let (X, Y ) ∈ {x1, x2, x3} × {y1, y2, y3} be distributed according to the joint pmf PXY = PY |XpX  as  PX,Y =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  p1  p2  p3  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb ,  9Although at the expense of making the algorithm more complicated, one can propose a better selection (in terms of lowering H(Y |U)), we  do not discuss it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  10  where pi ≜ pX(xi), and column i of PY |X represents pY |X(·|xi), ∀i ∈ [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The reason for representing the mass  probabilities of X as parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', pi’s, rather than numerical values is this interesting property the design of  a privacy-preserving mapping via Algorithm 1 does not depend on pX, which is elaborated further in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Figure 3a illustrates step 2 of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' On the left hand side of this figure, the elements of X × Y are  arranged into 3 (= |X|) subgroups in a column, and their corresponding probabilities are shown on their left side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  In this example, we have I = Y, hence, we create 3 (= |I|) realizations of U, denoted by ui, i ∈ [3], with the  corresponding probabilities of minx p(yi|x), which are shown on the right side of these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Afterwards, each  ui is connected to x1yi, x2yi, x3yi, with transition probabilities of pi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' This is equivalent to step 2 of  the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  In Figure 3b, we have the same set of mass points xiyj’s whose mass probabilities have been updated by  taking into account Figure 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In other words, each mass point xiyj has the remaining probability of p(xi, yj) −  p(xi, yj|uj)p(uj) (= pip(yj|xi) − pip(uj) = pia1(xi, yj), where a1(·, ·) is defined in step 3) to be filled with other  realizations of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' These remaining probabilities are shown on the left side of xiyj’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Iteration 1 starts, and u4 is  created, whose aim is to fill the minimum (non-zero) remaining probability , which is that of x∗y∗( = x3y2 in this  example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' This u4 is connected to x3y2, and x1y2 (whose y-coordinate is in common with x3y2), and x2y1, which  has the minimum (non-zero) water level in the subgroup of x2 (since the water level of x2y2 is zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' These links  are created bearing in mind that any connection to subgroup i has the transition probability of pi, i ∈ [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Taking into account the connections in Figure 3b, the remaining probabilities are again updated in Figure 3c,  shown on the left side of xiyj’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Iteration 2 starts, and realization u5 is created in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Again, taking into account the connections in Figure 3c, the remaining probabilities are updated in Figure 3d,  shown on the left side of xiyj’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Iteration 3 starts, and realization u6 is created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  The remaining probabilities are updated in Figure 3e where we are left with only one non-zero probability in  each subgroup, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In iteration 4, which is the last one, u7 is created to fill all the remaining water levels,  and the algorithm terminates after 4 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Finally, the output of this algorithm is shown in Figure 4, where the transition probabilities in Figure 4a represent  pXY |U, and those in Figure 4b represent pU|XY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' From Figure 4a, it is obvious that pX|U(·|u) = pX(·), ∀u ∈ U,  and hence, X ⊥⊥ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Also, H(Y |U = u) = 0, ∀u ∈ {u1, u2, u3}, and H(Y |U = u) ≤ H(X), ∀u ∈ {u4, u5, u6, u7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Therefore, G0(X, Y ) ≥ I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) ≥  �  H(Y ) − �u7  u4 p(u)H(X)  �+ = (H(Y ) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='5H(X))+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='10  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' (pX-invariance) An advantage of the achievable scheme in Algorithm 1 is that it does not depend on  the distribution of the private data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', pX(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In other words, the privacy-preserving mapping pU|XY obtained  via Algorithm 1 can be derived regardless of the knowledge about pX(·), as long as pY |X is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' This can be  verified by the fact that none of the 14 steps of Algorithm 1 rely on the knowledge of pX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='11 This is the reason  that in Example 1, the mass probabilities of X are given only as parameters p1, p2, p3, and as it can be verified  10Note that in this example, H(X) ≤ log |Y| = log 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  11Note that in the explanation of Algorithm 1, we indeed made use of pX, but this should not be confusing, since that explanation is about  the backward construction pXY |U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  11  p1 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2 : x1y1  p1 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='5 : x1y2  p1 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='3 : x1y3  p2 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='4 : x2y1  p2 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2 : x2y2  p2 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='4 : x2y3  p3 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='6 : x3y1  p3 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='3 : x3y2  p3 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1 : x3y3  u1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2  u2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2  u3 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1  p1  p1  p1  p2  p2  p2  p3  p3  p3  (a)  p1 × 0 : x1y1  p1 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='3 : x1y2  p1 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2 : x1y3  p2 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2 : x2y1  p2 × 0 : x2y2  p2 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='3 : x2y3  p3 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='4 : x3y1  p3 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1 : x3y2  p3 × 0 : x3y3  u4 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1  p1  p2  p3  (b)  p1 × 0 : x1y1  p1 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2 : x1y2  p1 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2 : x1y3  p2 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1 : x2y1  p2 × 0 : x2y2  p2 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='3 : x2y3  p3 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='4 : x3y1  p3 × 0 : x3y2  p3 × 0 : x3y3  u5 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1  p1  p2  p3  (c)  p1 × 0 : x1y1  p1 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2 : x1y2  p1 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1 : x1y3  p2 × 0 : x2y1  p2 × 0 : x2y2  p2 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='3 : x2y3  p3 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='3 : x3y1  p3 × 0 : x3y2  p3 × 0 : x3y3  u6 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1  p1  p2  p3  (d)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' 3: An illustrative representation of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  12  p1 × 0 : x1y1  p1 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2 : x1y2  p1 × 0 : x1y3  p2 × 0 : x2y1  p2 × 0 : x2y2  p2 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2 : x2y3  p3 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2 : x3y1  p3 × 0 : x3y2  p3 × 0 : x3y3  u7 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2  p1  p2  p3  (e)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' 3: An illustrative representation of Algorithm 1 (cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  in Figure 4b, the mapping pU|XY does not depend on a specific choice of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' This feature of Algorithm 1 is not  only of practical interest (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', when the distribution of the private data is unknown or difficult to estimate), but  also helpful in theory, as used in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Finally, it is important to emphasize that for a fixed pY |X, it is the  proposed privacy-preserving mapping pU|XY that is pX-invariant, not the resulting utility, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' (Number of iterations) In the explanation of Algorithm 1, it is stated that since at each iteration of  the algorithm, at least one water level is filled, and the algorithm terminates after all these levels are filled, the  number of iterations is upper bounded by the number of non-zero remaining probabilities prior to the start of the  iterations, which is at most |supp(X, Y )| − |I|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' While this is correct, we observe that, as in Figure 3d, in the very  last iteration we have exactly |X| non-zero and equal water levels, one in each subgroup, that are filled together  in one iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' This is a direct consequence of the fact that at each step of producing a new realization for U  in the algorithm, i) each subgroup of nodes has the same amount of total water levels, and ii) each subgroup of  nodes undergoes the same amount of decrement in water levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' As a result, in the last iteration of the algorithm,  we are left with |X| equal (non-zero) remaining water levels to be filled at once with the last realization of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Therefore, the algorithm terminates after N iterations with N ≤ |supp(X, Y )| − |I| − |X| + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Moreover, since at  each iteration, we get a realization for U, and we already have |I| realizations before the iterations start, we have  |U| ≤ N + |I| = |supp(X, Y )| − |X| + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  The following Lemma is needed to obtain an upper bound on G0(X, Y ) in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  13  p1 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2 : x1y1  p1 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='5 : x1y2  p1 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='3 : x1y3  p2 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='4 : x2y1  p2 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2 : x2y2  p2 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='4 : x2y3  p3 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='6 : x3y1  p3 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='3 : x3y2  p3 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1 : x3y3  u1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2  u2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2  u3 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1  u4 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1  u5 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1  u6 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1  u7 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2  p1  p1  p1  p2  p2  p2  p3  p3  p3  p1  p2  p3  p1  p2  p3  p1  p2  p3  p1  p2  p3  (a) Backward construction: pX,Y,U = pX,Y |U · pU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  p1 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2 : x1y1  p1 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='5 : x1y2  p1 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='3 : x1y3  p2 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='4 : x2y1  p2 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2 : x2y2  p2 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='4 : x2y3  p3 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='6 : x3y1  p3 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='3 : x3y2  p3 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1 : x3y3  u1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2  u2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2  u3 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1  u4 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1  u5 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1  u6 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1  u7 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2  1  2  5  1  3  1  2  1  1  4  1  3  2  3  1  1  3  1  4  1  3  1  3  1  4  1  6  1  5  1  4  1  6  2  5  1  2  1  3  (b) Forward construction: pX,Y,U = pX,Y · pU|X,Y  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' 4: The output of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Let f(X) be a function of X ∼ p such that it has at least two realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' We have  H(f(X)) ≥ Hb(min  x p(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  (12)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The proof is provided in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' For a given pair (X, Y ) ∼ pXY , we have  G0(X, Y ) ≤ H(Y ) −  �  1 −  �  y  min  x p(y|x)  �  Hb  �  min  x pX(x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  (13)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' If Y is a singleton, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', |Y| = 1, we have G0(X, Y ) = 0 and (13) follows, since minx p(y|x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Therefore,  in what follows, we assume that |Y| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  From Lemma 1, in an optimal mapping pU|XY , for any (x, u) ∈ X × U, there exists exactly one yx,u ∈ Y such  that p(x, yx,u, u) > 0, and we have p(x, yx,u|u) = p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' For any y ∈ Y, let Uy ≜ {u ∈ U|p(x, y, u) > 0, ∀x ∈ X}  be the set of realizations of U which are connected to pairs (x1, y), (x2, y), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' , (x|X |, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Define ˜U ≜ ∪y∈YUy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Since Uy′ ∩ Uy′′ = ∅ when y′ ̸= y′′, it is immediate that H(Y |U = u) = 0, ∀u ∈ ˜U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Since Y conditioned on  {U = u}, ∀u ̸∈ ˜U is a function of X which has at least two realizations, from Lemma 2, we get H(Y |U = u) ≥  Hb (minx pX(x)) , ∀u ̸∈ ˜U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  14  For an arbitrary y ∈ Y, we have  p(x)p(y|x) =  �  u∈U  p(x, y, u)  =  �  u∈Uy  p(x, y, u) +  �  u̸∈Uy  p(x, y, u)  ≥  �  u∈Uy  p(u)p(x, y|u)  = p(x)Pr{U ∈ Uy}, ∀x ∈ X,  (14)  where (14) follows from having p(x, y|u) = p(x), ∀u ∈ Uy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Hence, we have Pr{U ∈ Uy} ≤ minx p(y|x), ∀y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Noting that ˜U is the union of disjoint sets Uy, ∀y ∈ Y, we get  Pr{U ∈ ˜U} =  �  y  Pr{U ∈ Uy}  ≤  �  y  min  x p(y|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  (15)  We can write  H(Y |U) =  �  u∈U  p(u)H(Y |U = u)  =  �  u∈ ˜U  p(u)H(Y |U = u) +  �  u̸∈ ˜U  p(u)H(Y |U = u)  ≥  �  u∈ ˜U  p(u) · 0 +  �  u̸∈ ˜U  p(u)Hb(min  x pX(x))  =  �  1 − Pr{U ∈ ˜U}  �  Hb(min  x pX(x))  ≥  �  1 −  �  y  min  x p(y|x)  �  Hb(min  x pX(x)),  (16)  where (16) follows from (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' This proves (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Consider (X, Y ) ∼ pXY , in which X is uniformly distributed over [0 : K − 1], where K > 2 is an  arbitrary integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Let Y conditioned on {X = x} be uniformly distributed on [x : x + K − 2] mod K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Hence, Y  is also uniformly distributed over [0 : K − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In this setting, the upper bounds in (4) and (13) are  H(Y |X) = log(K − 1)  H(Y ) −  �  1 −  �  y  min  x p(y|x)  �  Hb(min  x p(x)) = K − 1  K  log(K − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Obviously, the bound in (13) is tighter in this example, and it can be readily verified that Algorithm 1 achieves it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  The following lemma is needed in its following Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  15  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' For the mass probabilities p1, p2, p3 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', pi ≥ 0, i ∈ [3], �  i pi = 1), we have  Hb(pi) ≤ Hb(pj) + Hb(pk), i, j, k ∈ [3], j ̸= k,  (17)  where Hb(·) denotes the binary entropy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The proof is provided in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' When X is binary, Algorithm 1 is optimal, and we have  G0(X, Y ) = H(Y ) −  �  1 −  �  y  min  x p(y|x)  �  H(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  (18)  Also, when (|X|, |Y|) = (3, 2), Algorithm 1 is optimal, and letting Y ≜ {y1, y2}, if we label the elements of X  according to x1 ≜ minx∈X p(y1|x), x2 ≜ minx∈X p(y2|x), we have  G0(X, Y ) = H(Y ) − (p(y1|x3) − p(y1|x1)) Hb(p1) − (p(y2|x3) − p(y2|x2)) Hb(p2),  (19)  where pi ≜ p(xi), i ∈ [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' When X is binary, we have H(X) = Hb(minx p(x)), and from (13) and Theorem 1, (18) is obtained, and  Algorithm 1 achieves it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  When (|X|, |Y|) = (3, 2), we prove the optimality by the simplex method [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' As already stated, G0(X, Y ) can  be obtained via an LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The problem is to find values for pU(·) in Figure 2 such that H(Y |U) is minimized and  pXY (·, ·) = �  u pXY |U(·, ·|u)pU(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' For i, j, k ∈ [2], let Pijk denote the probability of that u which is connected  to (x1, yi), (x2, yj), (x3, yk) with transition probabilities p1, p2 and p3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' For example, in Figure 2, we  have P111 = p(u1), P112 = p(u2), P121 = p(u3), and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' As a result, the LP minimizes H(Y |U), which is  P111 · 0 + P112Hb(p3) + P121Hb(p2) + P122Hb(p1) + P211Hb(p1) + P212Hb(p2) + P221Hb(p3) + P222 · 0 (20)  over the non-negative values of Pijk, i, j, k ∈ [2] such that  P111 + P112 + P121 + P122 = p(y1|x1)  P121 + P122 + P221 + P222 = p(y2|x2)  P111 + P121 + P211 + P221 = p(y1|x3)  P111 + P112 + P121 + P122 + P211 + P212 + P221 + P222 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  (21)  Changing the order of the variables, the simplex tableau for this LP is provided in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' By performing Gaussian  elimination (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', subtracting row 1 from row 3, and then subtracting the sum of row 1, row 2, and the resulting  row 3 from row 4), we obtain a canonical tableau as in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Note that all the elements of the rightmost column  are non-negative due to the initial convention x1 ≜ minx∈X p(y1|x), x2 ≜ minx∈X p(y2|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  The first four columns of the tableau in Table II form a basis, and  [P111, P222, P211, P212, P112, P121,P122, P221]T =  [p(y1|x1), p(y2|x2), p(y1|x3) − p(y1|x1), p(y2|x3) − p(y2|x2), 0, 0, 0, 0]T  (22)  16  P111  P222  P211  P212  P112  P121  P122  P221  1  0  0  0  1  1  1  0  p(y1|x1)  0  1  0  0  0  1  1  1  p(y2|x2)  1  0  1  0  0  1  0  1  p(y1|x3)  1  1  1  1  1  1  1  1  1  TABLE I: Simplex tableau of order 4 and dimension 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  P111  P222  P211  P212  P112  P121  P122  P221  1  0  0  0  1  1  1  0  p(y1|x1)  0  1  0  0  0  1  1  1  p(y2|x2)  0  0  1  0  1  0  1  1  p(y1|x3) − p(y1|x1)  0  0  0  1  1  1  0  1  p(y2|x3) − p(y2|x2)  TABLE II: Canonical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  is a basic feasible solution, which results in  H(Y |U) = (p(y1|x3) − p(y1|x1)) Hb(p1) + (p(y2|x3) − p(y2|x2)) Hb(p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  (23)  In order to show that no other feasible solution outperforms (22), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', resulting in a smaller H(Y |U) than (23),  we proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' For any feasible solution ˜Pijk, i, j, k ∈ [2], with some calculations, we get  H(Y |U) = (p(y1|x3) − p(y1|x1)) Hb(p1) + (p(y2|x3) − p(y2|x2)) Hb(p2)  (24)  + (Hb(p3) + Hb(p1) − Hb(p2)) ˜P112 + 2Hb(p2) ˜P121  (25)  + 2Hb(p1) ˜P122 + (Hb(p3) + Hb(p2) − Hb(p1)) ˜P221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  (26)  The RHS in (24) is equal to (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' From Lemma 3 and non-negativity of entropy, all the remaining terms in (25)  and (26) are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' As a result, no other feasible solution can produce a smaller H(Y |U) than (23), which  proves (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' When |X| = 2, or (|X|, |Y|) = (3, 2), for fixed pY |X, the optimal p∗  U|X,Y is pX-invariant 12, and  G0(X, Y ) is convex in pX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The first part of this claim is proved by combining the optimality of Algorithm 1 in Theorem 3, and Remark  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  In what follows, we provide two methods to prove the second part of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The first method makes use of  pX-invariance, while the second one relies on directly inspecting G0(X, Y ) in (18) and (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  12This, however, does not hold in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  17  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' First method  Fix λ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Let pW|XY , pV |XY and pU|XY be optimal solutions in the evaluation of G0(p′  X ·pY |X), G0(p′′  X ·  pY |X) and G0  �  (λp′  X + ¯λp′′  X) · pY |X  �  , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' From Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1, we can set W = V = U, and write  pW|XY = pV |XY = pU|XY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  (27)  In the sequel, by pXY W , pXY V and pXY U, we are referring to p′  X · pY |X · pW|XY , p′′  X · pY |X · pV |XY and  (λp′  X + ¯λp′′  X) · pY |X · pU|XY , respectively, which share the same support, denoted by S, and induce  pY U = λpY W + ¯λpY V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  (28)  For any tuple (x, y, w) ∈ S, we have  pW (w) = pXY W (x, y, w)  pXY |W (x, y|w)  = p′  X(x)pY |X(y|x)pW|XY (w|x, y)  pX|W (x|w)pY |XW (y|x, w)  = p′  X(x)pY |X(y|x)pW|XY (w|x, y)  p′  X(x)  (29)  = pY |X(y|x)pW|XY (w|x, y),  (30)  where (29) follows from i) having X ⊥⊥ W in pXY W , and ii) having pY |XW (y|x, w) = 1, since (x, y, w) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  From (27) and (30), we get that pU(·) = pW (·) = pV (·), which in conjunction with (28) and convexity of  I(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' B) in pA|B for fixed pB, results in  I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) ≤ λI(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' W) + ¯λI(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  This is equivalent to  G0  �  (λp′  X + ¯λp′′  X) · pY |X  �  ≤ λG0(p′  X · pY |X) + ¯λG0(p′′  X · pY |X),  which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Second method  According to [21], for fixed pY |X, the minimum value of λ, for which H(Y ) − λH(X) is a convex functional  of pX is suppX s∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ), where s∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ) ≜ supZ:Z−X−Y,Z̸⊥⊥X  I(Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='Y )  I(Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='X) is the strong data processing coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Since we have 1 − �  y minx p(y|x) ≥ suppX s∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ) (see [6, Corollary 6]), we conclude that G0(X, Y ) in (18)  is convex in pX for fixed pY |X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  For fixed pY |X, denoting αi ≜ p(y1|xi), i ∈ [3] for simplicity, (19) becomes  f(p1, p2) ≜ G0(X, Y ) = Hb(py) − (α3 − α1)Hb(p1) − (α2 − α3)Hb(p2),  (31)  where py ≜ (α1 − α3)p1 + (α2 − α3)p2 + α3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  18  To prove the convexity of G0 in pX, we show that f is convex in (p1, p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' By some calculations, the Hessian  matrix of f is obtained as  ∇2f =  \uf8ee  \uf8f0a + b  √ac  √ac  c + d  \uf8f9  \uf8fb ,  (32)  with a ≜ − (α1−α3)2  py(1−py) , b ≜  α3−α1  p1(1−p1), c ≜ − (α2−α3)2  py(1−py) and d ≜  α2−α3  p2(1−p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  The characteristic polynomial of ∇2f is λ2−(a+b+c+d)λ+ad+b(c+d) whose roots determine the eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  From the initial convention, we have α1 ≤ α3 ≤ α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' This, in conjunction with the inequality αi ¯αj + αj ¯αi ≥  |αi − αj|, i, j ∈ [3] results in a + b + c + d ≥ 0 and ad + b(c + d) ≥ 0, which in turn means that the eigenvalues  of ∇2f are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Therefore, ∇2f is positive semi-definite and f is convex in (p1, p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  The convexity result in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1 is not specific to |X| = 2 or (|X|, |Y|) = (3, 2) as the second part of the  following Theorem indicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' For given ǫ ≥ 0 and pX, Gǫ(X, Y ) is concave in pY |X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Furthermore, for given pY |X, G0(X, Y ) and  g0(X, Y ) are convex in pX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The first part of the claim is proved as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Fix ǫ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Given two conditional pmfs p′  Y |X and p′′  Y |X, let  pW|XY and pV |XY be maximizers of Gǫ(pX · p′  Y |X) and Gǫ(pX · p′′  Y |X) in (2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In other words, when  (X, Y ) is distributed according to pX · p′  Y |X (or pX · p′′  Y |X), an optimal privacy-preserving mapping is pW|XY (or  pV |XY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In the sequel, pXY W and pXY V refer to pX · p′  Y |X · pW|XY and pX · p′′  Y |X · pV |XY , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Without  loss of optimality, select the alphabets W and V, such that W ∩ V = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Fix λ ∈ (0, 1), and let U ≜ W ∪ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Let  (X, Y ) ∼ pX · (λp′  Y |X + ¯λp′′  Y |X), and define the following conditional pmf  pU|XY (u|x, y) ≜ λp′(y|x)pW|XY (u|x, y) · 1{u∈W} + ¯λp′′(y|x)pV |XY (u|x, y) · 1{u∈V}  λp′(y|x) + ¯λp′′(y|x)  ,  (33)  for all (u, x, y) ∈ U × supp(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  In what follows, we show that the joint pmf pXY U induced by (33) results in I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) ≤ ǫ and I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) ≥  λI(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' W) + ¯λI(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' V ), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  The construction in (33) results in  pU(u) = λpW (u) · 1{u∈W} + ¯λpV (u) · 1{u∈V}  (34)  pXY |U(x, y|u) = pXY |W (x, y|u) · 1{u∈W} + pXY |V (x, y|u) · 1{u∈V}, ∀(u, x, y) ∈ U × supp(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  (35)  Let E ≜ 1{U∈W} be a binary r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' defined as a function of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' From (34), we have pE(1) = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Also,  pE|X(1|x) =  �  y  pEY |X(1, y|x)  =  �  y  pY |X(y|x)pE|XY (1|x, y)  =  �  y  (λp′(y|x) + ¯λp′′(y|x))pE|XY (1|x, y)  =  �  y  (λp′(y|x) + ¯λp′′(y|x))  �  u∈W  pU|XY (u|x, y)  19  =  �  y  (λp′(y|x) + ¯λp′′(y|x))  �  u∈W  λp′(y|x)pW|XY (u|x, y)  λp′(y|x) + ¯λp′′(y|x)  (36)  =  �  y  (λp′(y|x) + ¯λp′′(y|x))  λp′(y|x)  λp′(y|x) + ¯λp′′(y|x)  = λ, ∀x ∈ X  (37)  where (36) results from (33), and (37) results in E ⊥⊥ X with pE(1) = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  We can write  pXY U|E(x, y, u|e) = pU(u)pXY |U(x, y|u)pE|U(e|u)  pE(e)  = pU(u)  �  pXY |W (x, y|u) · 1{e=1} + pXY |V (x, y|u) · 1{e=0}  �  λ · 1{e=1} + ¯λ · 1{e=0}  (38)  = pXY W (x, y, u) · 1{e=1} + pXY V (x, y, u) · 1{e=0}, e ∈ {0, 1},  (39)  where (38) and (39) follow from (35) and (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Therefore,  I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) = I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U, E)  (40)  = I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U|E)  (41)  = λI(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U|E = 1) + ¯λI(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U|E = 0)  = λI(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' W) + ¯λI(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' V )  (42)  ≤ ǫ,  (43)  where in (40), E is a deterministic function of U, and (41) results from X ⊥⊥ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' We obtain (42) from (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  From (43), we are allowed to write  Gǫ  �  pX · (λp′  Y |X + ¯λp′′  Y |X)  �  ≥ I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U)  = I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U, E)  ≥ I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U|E)  = λI(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U|E = 1) + ¯λI(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U|E = 0)  = λI(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' W) + ¯λI(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' V )  (44)  = λGǫ(pX · p′  Y |X) + ¯λGǫ(pX · p′′  Y |X),  (45)  where (44) follows from (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Finally, by noting that ǫ and λ were chosen arbitrarily, (45) proves that for fixed pX,  Gǫ(pX · pY |X) is concave in pY |X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  To prove the second part of the claim, we proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Fix pY |X and λ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Given two pmfs p′  X and  p′′  X, let pX = λp′  X + ¯λp′′  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Let p∗  U|XY be an optimal mapping in the evaluation of G0(pX · pY |X), which induces  U ∗ ∼ pU∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Let Iq denote I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) when (X, Y, U) ∼ q · pY |X · p∗  U|XY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Obviously, G0(pX · pY |X) = IpX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  20  When (X, Y, U) ∼ q · pY |X · p∗  U|XY , where q is an arbitrary pmf on X, we have  pU|X(·|x) =  �  y  pY |X(y|x)p∗  U|XY (·|x, y)  = pU∗|X(·|x)  = pU∗(·),  (46)  and hence, X ⊥⊥ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Therefore, we have by definition  Iq ≤ G0(q · pY |X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  (47)  When (X, Y, U) ∼ q · pY |X · p∗  U|XY , we have  pY |U(·|·) =  �  x  q(x)pY |XU  =  �  x  q(x)  pY |X(·|x)p∗  U|XY (·|x, ·)  pU∗(·)  ,  (48)  which means that the conditional pmf pY |U derived from (λp′  X + ¯λp′′  X) · pY |X · p∗  U|XY is λ times the conditional  pmf pY |U derived from p′  X · pY |X · p∗  U|XY plus ¯λ times the conditional pmf pY |U derived from p′′  X · pY |X · p∗  U|XY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Hence, we can write  G0((λp′  X + ¯λp′′  X) · pY |X) = Iλp′  X+¯λp′′  X  ≤ λIp′  X + ¯λIp′′  X  (49)  ≤ λG0(p′  X · pY |X) + ¯λG0(p′′  X · pY |X),  (50)  where (49) results from the convexity of I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) in pY |U for fixed pU, and (50) follows from (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Finally, the above analysis for proving the convexity of G0(X, Y ) remains valid if X − Y − U form a Markov  chain, and we replace p∗  U|XY with p∗  U|Y , which is an optimal mapping in the evaluation of g0(pX ·pY |X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Therefore,  for fixed pY |X, g0(X, Y ) is also convex in pX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Let (X, Y ) ∈ {x1, x2} × {y1, y2}, in which p ≜ p(x1), and the transition from X to Y follows  a general binary asymmetric channel (BAC) with cross over probabilities α, β ∈ [0, 1], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', α ≜ p(y2|x1) and  β ≜ p(y1|x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Therefore, we have q ≜ p(y1) = p¯α + ¯pβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' From Theorem 2, we have that  G0(X, Y ) = Hb(q) −  �  1 − min{α, ¯β} − min{β, ¯α}  �  Hb(p),  = Hb(q) − |α − ¯β|Hb(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  (51)  It is already known that for a given p, G0(X, Y ) is concave in (α, β), and the maximizer is (α∗, β∗) = ( 1  2, 1  2),  which results in X ⊥⊥ Y , and H(Y ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  For a given (α, β), G0(X, Y ) is convex in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Therefore, by setting  d  dpG0(X, Y ) = 0, we solve for p∗ as as  p∗ =  β  α + β 1{α< ¯β} +  ¯β  ¯α + ¯β 1{α> ¯β},  21  and when α = ¯β, p∗ is arbitrary, since we get G0(X, Y ) = Hb(β) = Hb(α) irrespective of the value of p due to  the independence of X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  In the special case of α = β, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', if the transition from X to Y is a binary symmetric channel (BSC(α)), we  have that p∗ = 1  2, if α ̸= 1  2, and any number in [0, 1] if α = 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Therefore, we get G0(X, Y ) ≥ 2 min{α, ¯α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Taking  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1 into account, this means that if for a pair (X, Y ) whose pY |X is BSC(α), the curator is unaware of  pX, he can still obtain the optimal mapping and make sure that the utility of this release is at least 2 min{α, ¯α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  As another case, let (X, Y ) ∈ {x1, x2} × {y0, y1, y2}, in which p(x1) ≜ p, and the transition from X to Y is a  binary eraser channel with erasure probability of e, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', BEC(e), in which p(yi|xj) = e1{i=0} + ¯e1{i=j}, (i, j) ∈  [0 : 2] × [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Denote the entropy of a ternary random variable with mass probabilities p1, p2, p3 by H(p1, p2, p3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  From Theorem 2, we have  G0(X, Y ) = H(Y ) − (1 −  �  y  min  x p(y|x))H(X)  = H(p¯e, e, ¯p¯e) − ¯eHb(p)  = Hb(e),  and U ≜ 1{Y =y0} attains it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Moreover, since this mapping is only from Y to U, we conclude that g0(X, Y ) = Hb(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  By changing the role of X and Y , and applying the second part of Theorem 3, we get  G0(Y, X) = Hb(p) − pHb(¯p¯e) − ¯pHb(p¯e),  where U ≜ X · 1{Y =y0} + ˜X · 1{Y ̸=y0}, in which ˜X ∈ X is a Bernoulli random variable independent of (X, Y )  with p ˜  X(x1) = p, achieves it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' We also have g0(Y, X) = 0 from [1, Corollary 2] by noting that the nullity of PY |X  is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='13  Thus far, we have presented an achievable scheme in Algorithm 1 as a lower bound on G0(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Based on  this scheme, we proceed to present a privacy-preserving algorithm as a lower bound on Gǫ(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' On the U-P  plane, the privacy restriction becomes stricter as we move from right to left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The rightmost point (I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ), H(Y ))  is achieved when there is no constraint on privacy, while the leftmost point (0, G0(X, Y )) relates to the strictest  privacy restriction, which is statistical independence between the private and the released data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Therefore, it makes  sense to obtain achievable points on the U-P plane starting from no privacy constraint and increasing the restrictions  incrementally until we reach the requirement X ⊥⊥ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' One approach is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Let R ≜ (I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ), H(Y )) denote  the rightmost point on the U-P trade-off curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In order to obtain an achievable point P1, we impose the requirement  that X must be at least 1-independent of U, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', X  1  ⊥⊥ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In other words, we require that X must have at least  one realization, call it x1, whose posterior probability p(x1|u) is the same as the prior p(x1) for any realization u  13With some back of the envelope calculations, it can be verified that for an M-ary erasure channel (M ≥ 2), in which (X, Y ) ∈  {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' , xM } × {y0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' , yM} and p(yi|xj) = e1{i=0} + ¯e1{i=j}, (i, j) ∈ [0 : M] × [M], we have g0(X, Y ) = G0(X, Y ) =  H(Y |X) = Hb(e) attained by U ≜ 1{Y =y0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Furthermore, denoting the probability vector associated with pX by p, we have g0(Y, X) = 0  and G0(Y, X) = H(X) − �M  i=1 piH(¯ep + e1i) attained by U ≜ X · 1{Y =y0} + ˜  X · 1{Y ̸=y0}, where 1i is the i-th standard unit vector,  and ˜  X ⊥⊥ (X, Y ) is distributed according to pX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  22  of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Letting 0 not be an element of X, define an auxiliary random variable Z1 ≜ X · 1{X=x1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The constraint  of having X at least 1-independent of the released data can be satisfied by designing a privacy-preserving scheme  via Algorithm 1 for the new pair (Z1, Y ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', pU1|Z1Y , which guarantees Z1 ⊥⊥ U1, or equivalently X  1  ⊥⊥ U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The  mapping pU1|XY , which is induced by pU1|Z1Y , results in the achievable point P1 ≜ (I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U1), I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' To get  P2, we require that X must be at least 2-independent of the released data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' To this end select arbitrary x1, x2 ∈ X,  and define Z2 ≜ X · 1{X∈{x1,x2}}, and obtain a mapping pU2|Z2Y via Algorithm 1, which satisfies X  2  ⊥⊥ U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The  corresponding pU2|XY produces P2 ≜ (I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U2), I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' This procedure continues providing achievable points  until we reach the constraint X ⊥⊥ U, which is taken care of by applying Algorithm 1 to the pair (X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Finally,  the upper concave envelope of the set of these achievable points results in a lower bound on Gǫ(X, Y ), which is  formally presented in the following Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='.  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' (Privacy-preserving mapping - a lower bound on Gǫ(X, Y )) Let (X, Y ) ∼ pXY be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Let  k ∈ [|X| − 2] and X ′ be an arbitrary subset of X with size k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Without loss of generality, assume that 0 ̸∈ X, and  let Z be a function of X defined as  Z ≜ X · 1{X∈X ′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  (52)  Applying the achievable scheme in Theorem 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', Algorithm 1, to the pair (Z, Y ) results in a mapping pU|ZY ,  such that Z ⊥⊥ U, or equivalently, X  k  ⊥⊥ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Calculate pU|XY from pU|ZY , and set PX ′ ≜ (I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U), I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Apply Algorithm 1 to (X, Y ) and obtain an achievable point denoted by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Let P ≜ ∪X ′⊂X {PX ′}∪{L, R} denote  the set of all the achievable points obtained so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Finally, we have the U-P trade-off uce[P](ǫ) as a lower bound  on Gǫ(X, Y ), ∀ǫ ∈ [0, I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  For the tuple (Z, X, Y, U) in Proposition 1, it can be readily verified that  pZ(z) = pX(z) · 1{z̸=0} + (  �  x∈X \\X ′  pX(x)) · 1{z=0}  pY |Z(y|z) = pY |X(y|z) · 1{z̸=0} +  �  x∈X \\X ′ pXY (x, y)  �  x∈X \\X ′ pX(x)  1{z=0},  (53)  pU|XY (u|x, y) = pU|ZY (u|x, y) · 1{x∈X ′} + pU|ZY (u|0, y) · 1{x̸∈X ′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  The quantity I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) is obtained after pU|ZY is obtained in the algorithm (unless |X ′| = 1, for which Theorem 2  gives a closed-form solution), while I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) can be obtained prior to the algorithm as  I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) = I(X, Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U)  (54)  = I(Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) + I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U|Z)  = I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U|Z)  (55)  = I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U, Y |Z)  (56)  23  = I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y |Z) + I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U|Z, Y )  = I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y |Z)  (57)  = I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ) − I(Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ),  (58)  where (54) follows from defining Z as a function of X in (52), and (55) results from Z ⊥⊥ U in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The  equality in (56) follows from satisfying the conditions of Lemma 1 in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In other words, conditioned  on the event {Z = z}, Y is a function of U, or equivalently H(Y |Z, U) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The equality in (57) results from  the Markov chain X − (Z, Y ) − U, since U is generated by applying Algorithm 1 to the pair (Z, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Finally, (58)  follows from the fact that Z − X − Y form a Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  The procedure in Proposition 1 is computationally complex when |X| is large, since there are 2|X | − |X| − 2  nonempty subsets of X with size at most |X| − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Therefore, for large |X|, we can restrict the analysis to a fixed  collection of subsets denoted by {Xk} for k ∈ [|X| − 2], in which |Xk| = k and Xj ⊂ Xk if j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Let (Zk, Uk)  be the same as (Z, U) in Proposition 1 when X ′ = Xk, k ∈ [|X| − 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' On the U-P plane, the slope of the line  connecting R (= (I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ), H(Y ))) to (I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Uk), I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Uk)) is  mk ≜  H(Y ) − I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Uk)  I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ) − I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Uk)  = H(Y ) − I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Uk)  I(Zk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y )  (59)  ≤  min  �  H(Y ),  �  1 − �  y minz pY |Zk(y|z)  �  min{H(Zk), log |Y|}  �  I(Zk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y )  ,  (60)  where (59) and (60) follow from (58) and (10), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' We denote the upper bound in (60) by f(pXY , Xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Knowing that Gǫ(X, Y ) is a concave and non-decreasing curve, a heuristic approach is to select X1 such that m1  is minimized, and an even simpler approach would be to minimize the upper bound, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', f(pXY , X1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Therefore,  we set  xk ≜  arg min  x∈X \\Xk−1:  Xk=Xk−1∪{x}  f(pXY , Xk),  ∀k ∈ [|X| − 2], X0 ≜ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  (61)  The procedures of this achievable scheme are provided in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' (Non-algorithmic U-P trade-off) The achievable points in Proposition 1 are obtained after applying  Algorithm 1 to each constructed pair (Z, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' More specifically, it is the y coordinate of these points that are  obtained after the application of the algorithm, since the x coordinates are already known prior to the algorithm  as in (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' If we replace these y coordinates with their corresponding lower bounds according to (10), we obtain  a new set of achievable points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Obviously, these points lie below the initial set of points, but they are obtained  without the need for the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Therefore, preserving (52) and its preceding assumptions in Proposition 1, we  24  Algorithm 2 Privacy-preserving mapping (simplified version of Proposition 1)  1: function ALGORITHM2(pX,Y )  2:  Select the collection {Xk}, k ∈ [|X| − 2], according to (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  3:  pU|XY ≜ Algorithm1(pY |X)  4:  P ≜ {(0, I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U)), (I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ), H(Y ))}  5:  Z0 ≜ 0  6:  k = 1  7:  while I(Zk−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ) ̸= I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ) do  8:  Zk ≜ X · 1{X∈Xk}  9:  pUk|ZkY ≜ Algorithm1(pY |Zk)  10:  P ≜ P ∪ {(I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Uk), I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Uk))}  11:  k = k + 1  12:  end while  13:  return uce[P](·)  14: end function  set  ˜PX ′ ≜  �  I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ) − I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Z) ,  �  H(Y ) −  �  1 −  �  y  min  z  pY |Z(y|z)  �  min{H(Z), log |Y|}  �+ �  (62)  ˜L ≜  �  0 ,  �  H(Y ) −  �  1 −  �  y  min  x pY |X(y|x)  �  min{H(X), log |Y|}  �+ �  ,  (63)  and ˜P ≜ ∪X ′⊂X { ˜PX ′}∪{˜L, R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The U-P trade-off uce[ ˜  P](ǫ) is a non-algorithmic lower bound on Gǫ(X, Y ), ∀ǫ ∈  [0, I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Needless to say that this can also be applied to the simplified scheme (for large |X|) discussed in  Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' PUBLIC DATA OBSERVATION  In this section, we assume that the curator has access to only Y , and propose an achievable scheme, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', a lower  bound on gǫ(X, Y ), defined in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' We start with ǫ = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', X ⊥⊥ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' An algorithm is proposed (Algorithm 3) that  provides a lower bound on g0(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Afterwards, this algorithm is used to generate a privacy-preserving mapping,  which results in a lower bound on gǫ(X, Y ), ǫ ∈ [0 : I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Like the previous section, we start with a simple theoretical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ( [1, Theorem 1]) For an optimal mapping pU∗|Y in the evaluation of g0(X, Y ), we have  |{y ∈ Y|p(y|u∗) > 0}| ≤ rank(PX|Y ), ∀u∗ ∈ U∗,  (64)  where PX|Y is an |X| × |Y| matrix with (i, j)-th entry equal to pX|Y (xi|yj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  25  Lemma 4 implies that in the evaluation of g0(X, Y ), if X is binary, for any u∗ ∈ U∗ (corresponding to an/the  optimal solution), there exist at most two realizations of Y , denoted by y1, y2 ∈ Y, such that p(y1|u∗), p(y2|u∗) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  It is also obvious that if there exists only one y0 ∈ Y, such that p(y0|u∗) > 0 (and hence, p(y0|u∗) = 1), the  condition X ⊥⊥ U indicates that this y0 must satisfy pX|Y (·|y0) = pX(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In other words, if there exists no such y0  satisfying pX|Y (·|y0) = pX(·), we must have |{y ∈ Y|p(y|u∗) > 0}| = 2, ∀u∗ ∈ U∗ for binary X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Therefore, in the  achievable scheme, it makes sense to build a mapping pU|Y , such that its corresponding pY |U is in line with this  observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' To this end, we start with the backward model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', pY |U, by imposing that i) for all the realizations  u of U, the condition in lemma 4 must be satisfied, ii) the pmf pY must be preserved in pY,U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The results are  provided in the following Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Throughout this section, we exclude the trivial case of X ⊥⊥ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' (A lower bound on g0(X, Y ) for binary X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=') Let X ≜ {x0, x1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' First, if there exists a mass point  ˆy ∈ Y, for which p(x0|ˆy) = p(x0), we create a corresponding u, such that p(u|y) = 1{y=ˆy}, ∀y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The set of  all such ˆy’s is denoted by B ≜ {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' , y|B|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='14 Therefore, |B| realizations of U are created according to p(ui|y) ≜  1{y=yi}, ∀i ∈ [|B|], ∀y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Furthermore, we have H(Y |U = u) = 0 and p(x0|u) = p(x0), ∀u ∈ {u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' , u|B|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Next, Y\\B is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Note that there is no element y of this set for which p(x0|y) = p(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Hence, in line with  lemma 4, we create realizations of U, each of which connected to exactly two elements of this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Having in mind  that we require to have p(x0|u) = p(x0) for any u ∈ U, we conclude that each of these newly created u’s must be  connected to two elements y0, y′  0 ∈ Y\\B such that p(x0) can be written as a convex combination of p(x0|y0) and  p(x0|y′  0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In other words, we must have either p(x0|y0) < p(x0) < p(x0|y′  0) or p(x0|y0) > p(x0) > p(x0|y′  0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In this  light, the set Y\\B is divided into disjoint sets Y0 ≜ {y ∈ Y|p(x0|y) < p(x0)}, and Y′  0 ≜ {y ∈ Y|p(x0|y) > p(x0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  The purpose of this division is to make sure that p(x0) can be written as a convex combination of an arbitrary  element of Y0 and an arbitrary element of Y′  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Therefore, if we create a mass point (or node) u, and connect it to  one node in Y0, and another node in Y′  0, with proper weights, the posterior p(x0|u) remains the same as the prior  p(x0), which is in accordance with the condition X ⊥⊥ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' This is carried out in an iterative way, where at each  iteration i, a mass point ui+|B| is created that is connected only to two mass points of Y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', y0 from Y0, and y′  0  from Y′  0, with proper weights such that p(x0|ui+|B|) = p(x0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', p(y0|ui+|B|) = f(y0, y′  0) ≜  p(x0|y′  0)−p(x0)  p(x0|y′  0)−p(x0|y0),  and p(y′  0|ui+|B|) = ¯f(y0, y′  0) ≜ 1 − f(y0, y′  0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' This results in H(Y |U = ui+|B|) = Hb(f(y0, y′  0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Note that the  selection of a pair (y0, y′  0) ∈ Y0 ×Y′  0 can be done arbitrarily;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' however, in order to minimize H(Y |U) heuristically,  an asymmetric selection is carried out, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', y0 is the point whose corresponding p(x0|y0) is the farthest from p(x0)  among the points in Y0, whereas, y′  0 is the point whose corresponding p(x0|y′  0) is the closest to p(x0) among the  points in Y′  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  In order to preserve the marginal pmf of Y in the resulting pair (Y, U), a water filling approach is utilized,  whereby at each iteration i, the water levels of y0, y′  0 are updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' More specifically, once y0 ∈ Y0, and y′  0 ∈ Y′  0  are selected, the algorithm fills the water level of at least one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In the first iteration, the water levels are  the mass probabilities p(y0) and p(y′  0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Knowing that p(y0|u1+|B|) = f(y0, y′  0), and p(y′  0|u1+|B|) = ¯f(y0, y′  0),  14Needless to say that if p(x0|y) ̸= p(x0), ∀y ∈ Y, we have B = ∅, and |B| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Also, the elements of Y have been relabeled in  accordance with the definition of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  26  we need to assign a mass probability to u1+|B| such that at least one of the conditions i) p(u1+|B|)f(y0, y′  0) =  p(y0), p(u1+|B|) ¯f(y0, y′  0) ≤ p(y′  0) or ii) p(u1+|B|) ¯f(y0, y′  0) = p(y′  0), p(u1+|B|)f(y0, y′  0) ≤ p(y0) is valid, which is  equivalent to having at least one water level filled and the other one not exceeded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' This results in the assignment  p(u1+|B|) ≜ min{  p(y0)  f(y0,y′  0),  p(y′  0)  ¯f(y0,y′  0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Afterwards, the water levels of y0 and y′  0 are modified, and the algorithm  moves on to the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Since at each iteration, at least one water level corresponding to an element of Y\\B is filled, and at the very  last iteration, the remaining two water levels are filled at once15, the algorithm terminates after N iterations for  some N ≤ |Y| − |B| − 1, which results in |U| ≤ |Y| − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Let fmax ≜ max(y0,y′  0)∈Y0×Y′  0 f(y0, y′  0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' As mentioned  before, H(Y |U = u) = 0, ∀u ∈ {u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' , u|B|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Moreover, since the conditional pmf of Y given any realization  u ∈ U\\{u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' , u|B|} has two mass probabilities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', f(y0, y′  0), ¯f(y0, y′  0) for some (y0, y′  0) ∈ Y0 × Y′  0, we have  H(Y |U = u) ≤ Hb(fmax), ∀u ∈ U\\{u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' , u|B|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' As a result, we get H(Y |U) ≤ (1 − �  y∈B p(y))Hb(fmax), and  I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) ≥  �  H(Y ) − (1 − �  y∈B p(y))Hb(fmax)  �+  ≥ (H(Y ) − 1)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The aforementioned procedures are provided  in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Algorithm 3 A lower bound on g0(X, Y ) for binary X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  1: function ALGORITHM3(pXY )  2:  B ≜ {y ∈ Y|p(x0|y) = p(x0)} = {y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y|B|}  3:  p(ui|y) = 1{y=yi},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ∀i ∈ [|B|],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ∀y ∈ Y  4:  Y0 ≜ {y ∈ Y|p(x0|y) < p(x0)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y′  0 ≜ {y ∈ Y|p(x0|y) > p(x0)}  5:  f(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y′) ≜  p(x0|y′)−p(x0)  p(x0|y′)−p(x0|y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ¯f(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y′) ≜ 1 − f(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ∀(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y′) ∈ Y0 × Y′  0  6:  a1(y) = p(y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ∀y ∈ Y\\B  7:  i = 1  8:  while maxy ai(y) ̸= 0 do  9:  y0 = arg miny∈Y0{p(x0|y)|ai(y) > 0},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y′  0 = arg miny∈Y′  0{p(x0|y)|ai(y) > 0}  10:  p(ui+|B||y0) = f(y0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='y′  0)  p(y0)  min{ ai(y0)  f(y0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='y′  0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  ai(y′  0)  ¯f(y0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='y′  0)}  11:  p(ui+|B||y′  0) =  ¯  f(y0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='y′  0)  p(y′  0)  min{ ai(y0)  f(y0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='y′  0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  ai(y′  0)  ¯f(y0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='y′  0)}  12:  p(ui+|B||y) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ∀y ∈ Y\\{y0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y′  0}  13:  ai+1(y) = ai(y) − p(ui+|B|)  �  f(y0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y′  0)1{y=y0} + ¯f(y0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y′  0)1{y=y′  0}  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ∀y ∈ Y\\B  14:  i = i + 1  15:  end while  16:  return pU|Y  17: end function  15since otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' after one more iteration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' we are left with a mass point y′ ∈ Y\\B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' such that p(x0|y′) = p(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' This is a contradiction,  since all such mass points are already contained in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  27  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Consider the pair (X, Y ) ∈ {x0, x1} × {y1, y2, y3, y4}, with  pY =  �  1  2  1  4  1  8  1  8  �T  ,  PX|Y =  \uf8ee  \uf8f00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='6  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  (65)  We have B = ∅, since there is no y ∈ Y for which p(x0|y) = p(x0) (= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='4625).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' We have Y0 = {y1, y4}, and  Y′  0 = {y2, y3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In step 6 of the algorithm, we have the first water levels as a1 = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='125, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='125]T, which  is the same as the mass probabilities of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Figure 5 provides an illustrative explanation of the iterations in the  algorithm, where the probabilities for the pair (X, Y ) are according to (65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In the first three subfigures, the  transition probabilities are from U to Y , while in the last subfigure, it is from Y to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  In the first iteration, we have y0 = y1, y′  0 = y3 according to step 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Hence, u1 is created which connects  to y1, y3 with transition probabilities f(y1, y3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1875, and ¯f(y1, y3) = 1 − f(y1, y3), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' We set  p(u1) ≜ min{ a1(y1)  f(y1,y3),  a1(y3)  ¯  f(y1,y3)} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='154 to fill the water level a1(y3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The water levels are updated, and we get  a2 = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='4712, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='25, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='125]T shown on the RHS of yi’s in Figure 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In iteration 2, considered in the same figure, we  get y0 = y1, y′  0 = y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Hence, u2 is created which connects to y1, y2 with transition probabilities f(y1, y2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='675,  and ¯f(y1, y2) = 1 − f(y1, y2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' We set p(u2) ≜ min{ a2(y1)  f(y1,y2),  a2(y2)  ¯  f(y1,y2)} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='698 to fill the water level  a2(y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Hence, we get the update a3 = [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='0231, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='125]T, which is shown in Figure 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In the last iteration, we  have y0 = y4, y′  0 = y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Hence, u3 is created which connects to y4, y2 with transition probabilities f(y4, y2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='845,  and ¯f(y4, y2) = 1 − f(y4, y2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' We set p(u3) ≜ min{ a3(y4)  f(y4,y2),  a3(y2)  ¯  f(y4,y2)} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='148 to fill the water levels  a3(y2) and a3(y4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Finally, we get a4 = [0, 0, 0, 0]T, and the algorithm terminates after 3 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The output of  the algorithm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', pU|Y , is shown in Figure 5d, which results in a utility of I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='9063 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' It is interesting  to observe that this pU|Y actually coincides with the optimal solution obtained in [19, Example 1] via linear  programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Therefore, for the (X, Y ) distributed according to (65), we have g0(X, Y ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='9063.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' If |Y| = 3, Algorithm 3 is optimal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', it achieves g0(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Let X ≜ {x0, x1} and B ≜ {y ∈ Y|p(x0|y) = p(x0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' We have either |B| = 0 or |B| = 1, since otherwise,  X ⊥⊥ Y , and U ∗ = Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  First, assume |B| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Therefore, with a proper relabling of the elements in Y, we have p(x0|y1) = p(x0),  p(x0|y2) > p(x0), and P(x0|y3) < p(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' 16 For a mapping pU|Y which results in X ⊥⊥ U, define Ui ≜ {u ∈  U|p(yi|u) > 0}, hence, U = ∪3  i=1Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' For an optimal mapping, we must have p(y1|u) = 1, ∀u ∈ U1, since otherwise,  we have either |{y ̸= y1|p(y|u) > 0}| = 1 or 2, where the former results in p(x0|u) ̸= p(x0), which violates the  condition X ⊥⊥ U, and the latter violates Lemma 4, which states that each u must be connected to at most two  realizations of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' As a result U1 ∩ (U2 ∪ U3) = ∅ and Pr{U ∈ U1} = p(y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Furthermore, we have U2 = U3, since  otherwise, we get X ̸⊥⊥ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' For any realization u ∈ U2, we must have p(y3|u) = 1 − p(y2|u) = f(y3, y2), with  16That both p(x0|y2) and p(x0|y3) cannot be lower or greater than p(x0) is obvious, since otherwise, we get p(x0) < p(x0), which is  absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  28  y1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='5  y2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='25  y3 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='125  y4 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='125  u1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='154  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
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+page_content='1875  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='8125  (a) Iteration 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  y1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='4712  y2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='25  y3 : 0  y4 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='125  u2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='698  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
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+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
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+page_content='675  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='325  (b) Iteration 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  y1 : 0  y2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='0231  y3 : 0  y4 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='125  u3 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='148  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
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+page_content='155  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='845  (c) Iteration 3  y1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='5  y2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='25  y3 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='125  y4 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='125  u1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='154  u2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='698  u2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='148  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
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+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='0577  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='9423  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='9075  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='0925  1  (d) The output pU|Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' 5: Illustration of Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  f(·, ·) defined in Proposition 2, since otherwise, the condition X ⊥⊥ U is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Therefore, we have  H(Y |U) =  �  u∈U1  H(Y |U = u) +  �  u∈U2  H(Y |U = u)  = 0 +  �  u∈U2  Hb(f(y3, y2))  = (1 − p(y1))Hb(f(y3, y2)),  which is obtained via Algorithm 3, and we get g0(X, Y ) = Hb(p(y1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Next, assume |B| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' With a proper relabling of the elements in Y, we have either p(x0|y1) > p(x0) >  p(x0|yi), i ∈ {2, 3}, or p(x0|y1) < p(x0) < p(x0|yi), i ∈ {2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' We only consider the former, as the proof for the  latter follows similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Let Ui, i ∈ [3], be defined as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' We have U = U1, since otherwise, for any u ∈ U\\U1,  p(x0|u) < p(x0), and hence, X ̸⊥⊥ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Furthermore, U2 ∩ U3 = ∅, since other wise, in conjunction with U = U1,  there exists u ∈ U such that p(yi|u) > 0, ∀i ∈ [3], which violates the condition in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' For any i ∈ {2, 3}  and any realization u ∈ Ui, we must have p(yi|u) = 1 − p(y1|u) = f(yi, y1), since otherwise, X ̸⊥⊥ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Moreover,  29  since p(yi) = �  u∈Ui p(u)p(yi|u), we get Pr{U ∈ Ui} =  p(yi)  f(yi,y1), i ∈ {2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Finally, we have  H(Y |U) =  3  �  i=2  �  u∈Ui  H(Y |U = u)  =  3  �  i=2  p(yi)  f(yi, y1)Hb(f(yi, y1)),  which is attained by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Based on the achievable scheme in Proposition 2, we can now propose a privacy-preserving as a lower bound  on gǫ(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' (Privacy-preserving mapping - a lower bound on gǫ(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=') Let (X, Y ) ∼ pXY be given, and let  S = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' , x|X |−1) be an arbitrary ordered (|X|− 1)-tuple of the elements in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Set U0 ≜ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The algorithm  starts off from this point by decreasing the privacy-leakage step by step as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In the first step, define the binary  random variable ˆX1 ∈ {0, 1} as ˆX1 ≜ 1{X=x1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Since ˆX1 is a function of X, ˆX1 − X − Y − U0 form a Markov  chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Since ˆX1 is binary, by applying Algorithm 3 in Proposition 2 to the pair ( ˆX1, U0), pU1|U0 is generated such  that X − Y − U0 − U1 form a Markov chain, and ˆX1 ⊥⊥ U1, or equivalently, pX|U1(x1|u) = pX(x1), ∀u ∈ U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Hence, X is at least 1-independent of U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Set P1 ≜ (I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U1), I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The algorithm proceeds in an iterative  way as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' After building the Markov chain ˆXi−X−Y −Ui−1, i ∈ [2 : |X|−1], in which ˆXi ≜ 1{X=xi}, apply  Algorithm 3 to ( ˆ  Xi, Ui−1) to generate pUi|Ui−1, such that X − Y − Ui−1 − Ui form a Markov chain and ˆXi ⊥⊥ Ui,  or equivalently, X is at least i-independent of Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Set Pi ≜ (I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Ui), I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Ui)), and let PS ≜ {Pi}|X |−1  i=1  denote  the set of achievable points for the ordered tuple S introduced earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Finally, let P ≜ ∪SPS ∪ {I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ), H(Y )}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  A lower bound on gǫ(X, Y ) is provided by uce[P](·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  For a fixed tuple S in Proposition 3, we have  I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Ui) = H(Y ) − H(Y, Ui−1|Ui) + H(Ui−1|Y, Ui)  ≥ I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Ui−1) − H(Ui−1|Ui)  (66)  ≥ I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Ui−1) − 1, ∀i ∈ [|X| − 1],  (67)  where (66) follows from the Markov chain Y − Ui−1 − Ui and non-negativity of entropy, and (67) results from the  fact that according to Algorithm 3, Ui−1 conditioned on any realization ui of Ui has at most two non-zero mass  probabilities, and hence, H(Ui−1|Ui) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  We also have  I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Ui) = I( ˆXi, X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Ui)  (68)  = I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Ui| ˆXi)  (69)  ≤ I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Ui−1| ˆXi)  (70)  = I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Ui−1) − I( ˆXi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Ui−1), ∀i ∈ [|X| − 1],  (71)  30  where (68) follows from having defined ˆXi as a function of X, (69) results from ˆXi ⊥⊥ Ui, and finally, (70) is  from the application of data processing inequality in ˆXi − X − Ui−1 − Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  The procedure in Proposition 3 can be computationally complex when |X| is large, as the total number of ordered  (|X| − 1)-tuples is |X|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='. Therefore, this calls for a simpler scheme when |X| is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Let S = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' , x|X |−1)  be a given ordered tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In the first iteration of the algorithm, a release random variable U1 is generated, which  results in a utility greater than or equal to (H(Y ) − 1)+ from (67), since U0 = Y , and a privacy-leakage lower  than or equal to I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ) − I( ˆX1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ) from (71).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Since gǫ(X, Y ) is concave and non-decreasing in ǫ, if a utility  within 1 bit of H(Y ) is to be achieved in iteration 1, a heuristic approach is to choose an x1 ∈ X which is  likely to result in the maximum drop in the privacy-leakage, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', I( ˆX1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In other words, on the U-P plane,  the algorithm tries to depart from the rightmost point (I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ), H(Y )) with the lowest slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' As a result, we set  x1 ≜ arg maxx∈X I(1{X=x};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U0), and the algorithm proceeds to provide U1 in X − Y − U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Following the same  rationale, we set x2 ≜ arg maxx∈X \\{x1} I(1{X=x};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U1), and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  The procedure of this simplified scheme is presented in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Algorithm 4 Privacy-preserving mapping - a lower bound on gǫ(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  1: function ALGORITHM4(pXY )  2:  U0 ≜ Y  3:  x1 ≜ arg maxx∈X I(1{X=x};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U0)  4:  P ≜ {(I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U0), I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U0))}  5:  i = 1  6:  while I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Ui−1) ̸= 0 do  7:  ˆXi ≜ 1{X=xi}  8:  pUi|Ui−1 ≜ Algorithm3(p ˆ  XiUi−1)  9:  P ≜ P ∪ {(I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Ui), I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Ui))}  10:  i = i + 1  11:  xi ≜ arg maxx∈X \\{x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=',xi−1} I(1{X=x};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Ui−1)  12:  end while  13:  return uce[P](·)  14: end function  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' NUMERICAL RESULTS  In this section, the performance of the proposed privacy-preserving schemes are evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Prior to this evaluation,  we note that  g0(X, Y ) ≥ gL  0 ≜  �  H(Y ) − log rank(PX|Y )  �+  (72)  G0(X, Y ) ≥ GL  0 ≜  �  H(Y ) − min{H(X), log rank(PX|Y )}  �+ ,  (73)  where PX|Y is the matrix form of the conditional pmf pX|Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  31  The lower bound in (72) is from [1, Corollary 2] and (73) results from i) G0(X, Y ) ≥ g0(X, Y ) along with (72)  and ii) the fact that according to Lemma 1, for an optimal mapping p∗  U|XY , we have H(Y |X, U ∗) = 0, and hence  G0(X, Y ) = I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U ∗)  = I(X, Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U ∗) − I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U ∗|Y )  = I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U ∗|X) − I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U ∗|Y )  ≥ H(Y |X) − H(Y |X, U ∗) − H(X|Y )  = H(Y |X) − H(X|Y )  = H(Y ) − H(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Full data observation  Consider the following joint pmf on X × Y = [8]2 generated randomly17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  pX =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='175  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='089  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='146  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='026  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='077  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='167  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='145  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='175  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  ,  PY |X =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='130  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='233  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='159  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
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+page_content='143  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  (74)  Figure 6 illustrates the lower bounds on Gǫ(X, Y ) in Proposition 1 and Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The filled black circles are  the achievable points in Proposition 1 whose upper concave envelope is drawn in solid black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The empty red  circles correspond to the non-algorithmic achievable points in Remark 6 whose upper concave envelope is plotted in  solid red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The top dashed blue line is the line connecting (0, H(Y |X)) to (I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ), H(Y )), while the bottom  dashed blue line corresponds to the line connecting (0, GL  0 ), where GL  0 is given in (73), to (I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ), H(Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Figure 7 illustrates the lower bounds on Gǫ(X, Y ) in the simplified version of Proposition 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The  filled black circles are the achievable points in Algorithm 2 whose upper concave envelope is drawn in solid black  line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The empty red circles correspond to the non-algorithmic achievable points in Remark 6 when applied to the  procedure in Algorithm 2 whose upper concave envelope is plotted in solid red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The dashed blue lines are as  mentioned earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Public data observation  Figure 8 illustrates the lower bounds on gǫ(X, Y ) in Proposition 3 and its simplified version, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  The filled black circles are the achievable points in Proposition 3 whose upper concave envelope is drawn in solid  17Each probability vector is obtain by normalizing an 8-dimensional vector whose elements have been i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' generated according to uniform  distribution over the interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  32  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='5  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' 6: Lower bounds on Gǫ(X, Y ) in Proposition 1 and Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The empty red circles correspond to the achievable points in Algorithm 4 whose upper concave envelope  is plotted in solid red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The top dashed blue line is as mentioned earlier in the full data observation, while the  bottom dashed blue line is the line connecting (0, gL  0 ), where gL  0 is given in (72), to (I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Y ), H(Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' CONCLUSIONS  Information-theoretic privacy is considered in this paper, in which a curator, aware of the joint distribution pXY ,  wishes to maximize I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) subject to I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U) ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The optimization is over pU|XY (or pU|Y ) when curator has  access to (X, Y ) (or only Y ), and trade-off is captured by Gǫ(X, Y ) (or gǫ(X, Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The problem is investigated  from theoretical and practical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  APPENDIX A  Fix an arbitrary x0 ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Assume that for some u0 ∈ U∗, we have p(x0, y′|u0) > 0, and p(x0, y′′|u0) > 0 with  y′ ̸= y′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' It is shown that this cannot be optimal by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' In other words, a mapping p ˆU|XY is constructed  such that I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ˆU) = I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U ∗) and I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ˆU) > I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U ∗), which disproves the optimality of pU∗|XY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  33  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
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+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='5  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' 7: Lower bounds on Gǫ(X, Y ) in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  Assume the random variable ˆU ∈ (U∗\\{u0}) ∪ {u′  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' u′′  0} with u′  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' u′′  0 ̸∈ U∗ such that  pX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='Y | ˆU(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y′|u′  0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' pX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='Y | ˆU(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y′′|u′′  0) ≜ pX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='Y |U(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y′|u0) + pX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='Y |U(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y′′|u0)  pX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='Y | ˆU(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y′′|u′  0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' pX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='Y | ˆU(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y′|u′′  0) ≜ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  p ˆU(u′  0) ≜ pU(u0)  pX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='Y |U(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y′|u0)  pX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='Y |U(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y′|u0) + pX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='Y |U(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y′′|u0)  p ˆU(u′′  0) ≜ pU(u0)  pX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='Y |U(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y′′|u0)  pX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='Y |U(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y′|u0) + pX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='Y |U(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y′′|u0)  pX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ˆU(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' u) ≜ pX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='U(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' u),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ∀(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' u) ∈ X × Y × ( ˆU\\{u′  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' u′′  0})  pX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ˆU(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y|u) ≜ pX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='U(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y|u0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ∀u ∈ {u′  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' u′′  0},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ∀(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y) ̸= (x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' (x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' y′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  It can be verified that in this construction, i) the marginal pX,Y is preserved in (X, Y, ˆU), ii) H(X|U ∗) = H(X| ˆU)  (which results from having pX, ˆU(·, u) = pX,U∗(·, u), ∀u ∈ U∗\\{u0, u′  0, u′′  0}, and pX|U∗(·|u0) = pX| ˆU(·|u′  0) =  pX| ˆU(·|u′′  0), with p(u0) = p(u′  0)+p(u′′  0)), and iii) H(Y | ˆU) < H(Y |U) due to strict concavity of entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Therefore,  we have constructed p ˆU|X,Y , such that I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ˆU) = I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U ∗), and I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' U ∗) < I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' ˆU) which contradicts the  attainability of G0(X, Y ) by pU∗|X,Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Hence, by noting that x0 was chosen arbitrarily, we obtain  |{y ∈ Y|p(x, y|u∗) > 0}| ≤ 1, ∀(x, u∗) ∈ X × U∗,  34  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' 8: Lower bounds on gǫ(X, Y ) in Proposition 3 and Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  which is equivalent to (7) and (8) when X ⊥⊥ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  APPENDIX B  Let x∗ ≜ arg minx p(x) and define X0 ≜ {x ∈ X|f(x) = f(x∗)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Hence, we have  p(x∗) ≤ min {Pr{X ∈ X0}, 1 − Pr{X ∈ X0}} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  (75)  Define Z ≜ 1{f(X)=f(x∗)} as a function of f(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Since f(X) has at least two realizations, Z is a Bernoulli random  variable with parameter Pr{X ∈ X0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Therefore, we can write  H(f(X)) ≥ H(Z)  (76)  = Hb (Pr{X ∈ X0})  ≥ Hb(p(x∗)),  (77)  where (76) follows from defining Z as a function of f(X), and (77) results from (75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  APPENDIX C  If i = j or i = k, the proof is complete by the non-negativity of entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Also, if any of pi, pj, pk is 0 or 1, the  proof is complete, since it results in one of the following trivial possibilities i) 0 ≤ 0, ii) Hb(pi) ≤ Hb(1 − pi),  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='5  2  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='U)  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='5  PointsinPinProposition3  uceipinProposition3  PointsinPinAlgorithm4  ucepin Algorithm 4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='2  I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='U)35  or iii) the non-negativity of entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Therefore, we assume that the indices i, j, k are all distinct and non of the  mass probabilities is 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Since the RHS of (17) is symmetric with respect to pj, pk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', it doesn’t change if  we exchange pj and pk, without loss of generality, assume that pj ≤ pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Therefore, we have  Hb(pi) = Hb(pj + pk)  < Hb(pk) + H′  b(pk)pj  (78)  ≤ Hb(pk) + H′  b(pj)pj  (79)  < Hb(pk) + H(pj),  (80)  where (78) results from Taylor expansion of Hb(·) and its strict concavity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', H′′  b (·) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' The latter also results  in (79) and (80), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=', H′  b(pk) ≤ H′  b(pj) since pj ≤ pk, and H′  b(pj)pj < Hb(pj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  REFERENCES  [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
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+page_content=' G¨und¨uz, “On perfect privacy,” IEEE Journal on Selected Areas in Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
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+page_content=' Calmon and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
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+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
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+page_content=' Salamatian, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Fawaz, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' M´edard, “From the information bottleneck to the privacy funnel,” in IEEE Information  Theory Workshop (ITW), 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
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+page_content='  [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Sreekumar and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' G¨und¨uz, “Optimal privacy-utility trade-off under a rate constraint,” in 2019 IEEE International Symposium on  Information Theory (ISIT), 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' 2159–2163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Asoodeh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Diaz, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Alajaji, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Linder, “Information extraction under privacy constraints,” Information, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' 1, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content='  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Available: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
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+page_content=' Kamath, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
+page_content=' Nair, “On maximal correlation, hypercontractivity, and the data processing inequality studied  by Erkip and Cover,” https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FKT4oBgHgl3EQfNS2X/content/2301.11754v1.pdf'}
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